Nothing
R/Gbeta1.R
In fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
Defines functions
summary.fitMB
print.fitMB
fitMcGBB.default
fitMcGBB
.EstMLEMcGBB
EstMLEMcGBB
NegLLMcGBB
pMcGBB
dMcGBB
mazGBeta1
pGBeta1
dGBeta1
Documented in
dGBeta1
dMcGBB
EstMLEMcGBB
fitMcGBB
mazGBeta1
NegLLMcGBB
pGBeta1
pMcGBB
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' dGBeta1(p,a,b,c)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{dGBeta1} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Generalized Beta Type-1 Distribution.
#'
#' \code{var} variance of the Generalized Beta Type-1 Distribution.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24} .
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
dGBeta1<-function(p,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b,c))) | any(is.infinite(c(p,a,b,c))) | any(is.nan(c(p,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
ans[i]<-(c/beta(a,b))*(p[i]^(a*c-1))*((1-p[i]^c)^(b-1))
}
}
}
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=ans,"mean"=beta(a+b,(1/c))/beta(a,(1/c)),
"var"=(beta(a+b,(2/c))/beta(a,(2/c)))-(beta(a+b,(1/c))/beta(a,(1/c)))^2 ))
}
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' pGBeta1(p,a,b,c)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output \code{pGBeta1} gives the cumulative density values in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
pGBeta1<-function(p,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b,c))) | any(is.infinite(c(p,a,b,c))) | any(is.nan(c(p,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
term<-NULL
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
term[i]<-Re(hypergeo::hypergeo_powerseries(a,1-b,(p[i])^c,a+1))
#checking if the hypergeometric function value is NA(not assigned)values,
#infinite values or NAN(not a number)values if so providing an error message and
#stopping the function progress
if(is.nan(term[i]) | term[i] <= 0 | is.infinite(term[i]) |is.na(term[i]) )
{
stop("Given input values generate error values for Gaussian Hypergeometric Function")
}
else
{
ans[i]<-(((p[i])^(a*c))*term[i])/(a*beta(a,b))
}
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' mazGBeta1(r,a,b,c)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#' @param r vector of moments
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output \code{mazGBeta1} gives the moments about zero in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
mazGBeta1<-function(r,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(r,a,b,c))) | any(is.infinite(c(r,a,b,c))) | any(is.nan(c(r,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 |c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
#for each input values in the vector necessary calculations and conditions are applied
for (i in 1:length(r))
{
#checking if moment values are less than or equal to zero and creating
# an error message as well as stopping the function progress
if(r[i]<=0)
{
stop("Moments cannot be less than or equal to zero")
}
else
{
ans[i]<-beta(a+b,r[i]/c)/beta(a,r[i]/c)
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' McDonald Generalized Beta Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the McDonald Generalized Beta
#' Binomial Distribution.
#'
#' @usage
#' dMcGBB(x,n,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' Mixing Generalized Beta Type-1 Distribution with Binomial distribution
#' the probability function value and cumulative probability function can be constructed
#' and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (\sum_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) ) }
#' \deqn{a,b,c > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{Var_{McGBB}[x]= n^2(\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2) +n(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}) }
#' \deqn{over dispersion= \frac{\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}{\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#'
#' @return
#' The output of \code{dMcGBB} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of McDonald Generalized Beta Binomial Distribution.
#'
#' \code{var} variance of McDonald Generalized Beta Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of McDonald Generalized Beta Binomial Distribution.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.6)
#' plot(0,0,main="Mcdonald generalized beta-binomial probability function graph",
#' xlab="Binomial random variable",ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMcGBB(0:10,10,4,2,1)$pdf #extracting the pdf values
#' dMcGBB(0:10,10,4,2,1)$mean #extracting the mean
#' dMcGBB(0:10,10,4,2,1)$var #extracting the variance
#' dMcGBB(0:10,10,4,2,1)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' points(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' }
#'
#' pMcGBB(0:10,10,4,2,1) #acquiring the cumulative probability values
#'
#' @export
dMcGBB<-function(x,n,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,a,b,c))) | any(is.infinite(c(x,n,a,b,c))) | any(is.nan(c(x,n,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are less than or equal zero ,
#if so providing an error message and stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function progress
if(max(x)>n)
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
else
{
final<-NULL
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
value<-NULL
value<-sum(((-1)^(0:(n-x[i])))*choose(n-x[i],(0:(n-x[i])))*beta((x[i]/c+a+(0:(n-x[i]))/c),b))
final[i]<-choose(n,x[i])*(1/beta(a,b))*value
}
}
}
}
# generating an output in list format consisting pdf,mean,variance and overdispersion value
return(list("pdf"=final,"mean"=n*beta(a+b,1/c)/beta(a,1/c),
"var"=n^2*((beta(a+b,2/c)/beta(a,2/c))-(beta(a+b,1/c)/beta(a,1/c))^2)+
n*((beta(a+b,1/c)/beta(a,1/c))-(beta(a+b,2/c)/beta(a,2/c))),
"over.dis.para"=(((beta(a+b,2/c))/(beta(a,2/c)))-((beta(a+b,1/c))/(beta(a,1/c)))^2)/
(((beta(a+b,1/c))/(beta(a,1/c)))-((beta(a+b,1/c))/(beta(a,1/c)))^2)))
}
#' McDonald Generalized Beta Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the McDonald Generalized Beta
#' Binomial Distribution.
#'
#' @usage
#' pMcGBB(x,n,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' Mixing Generalized Beta Type-1 Distribution with Binomial distribution
#' the probability function value and cumulative probability function can be constructed
#' and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (\sum_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) ) }
#' \deqn{a,b,c > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{Var_{McGBB}[x]= n^2(\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2) +n(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}) }
#' \deqn{over dispersion= \frac{\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}{\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#'
#' @return
#' The output of \code{pMcGBB} gives cumulative probability function values in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.6)
#' plot(0,0,main="Mcdonald generalized beta-binomial probability function graph",
#' xlab="Binomial random variable",ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMcGBB(0:10,10,4,2,1)$pdf #extracting the pdf values
#' dMcGBB(0:10,10,4,2,1)$mean #extracting the mean
#' dMcGBB(0:10,10,4,2,1)$var #extracting the variance
#' dMcGBB(0:10,10,4,2,1)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' points(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' }
#'
#' pMcGBB(0:10,10,4,2,1) #acquiring the cumulative probability values
#'
#' @export
pMcGBB<-function(x,n,a,b,c)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dMcGBB(0:x[i],n,a,b,c)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of McDonald Generalized Beta Binomial Distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the shape parameters a,b and c.
#'
#' @usage
#' NegLLMcGBB(x,freq,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' \deqn{0 < a,b,c }
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,...}
#'
#' @return
#' The output of \code{NegLLMcGBB} will produce a single numeric value.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLMcGBB(No.D.D,Obs.fre.1,.2,.3,1) #acquiring the negative log likelihood value
#'
#' @export
NegLLMcGBB<-function(x,freq,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,a,b,c))) | any(is.infinite(c(x,freq,a,b,c)))
|any(is.nan(c(x,freq,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating a error message as well as stopping the function progress
if(any(c(x,freq) < 0))
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking if shape parameters are less than or equal to zero
#if so creating an error message as well as stopping the function progress
else if( a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
value<-NULL
for (i in 1:sum(freq))
{
value[i]<-sum(((-1)^(0:n-data[i]))*choose(n-data[i],(0:n-data[i]))*
beta((data[i]/c)+a+((0:n-data[i])/c),b))
}
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(n,data[1:sum(freq)])))+sum(log(value))+
sum(freq)*log(1/beta(a,b))))
}
}
}
#' Estimating the shape parameters a,b and c for McDonald Generalized Beta Binomial
#' distribution
#'
#' The function will estimate the shape parameters using the maximum log likelihood method for
#' the McDonald Generalized Beta Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEMcGBB(x,freq,a,b,c,...)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{0 < a,b,c}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' \code{EstMLEMcGBB} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' \dontrun{
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEMcGBB(x=No.D.D,freq=Obs.fre.1,a=0.1,b=0.1,c=0.2)
#'
#' bbmle::coef(parameters) #extracting the parameters
#' }
#' @export
EstMLEMcGBB<-function(x,freq,a,b,c,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
output<-suppressWarnings2(bbmle::mle2(.EstMLEMcGBB,data=list(x=x,freq=freq),
start = list(a=a,b=b,c=c),...),"NaN")
return(output)
}
.EstMLEMcGBB<-function(x,freq,a,b,c)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#Gaussian Hypergeometric Generalized Beta binomial distribution
n<-max(x)
data<-rep(x,freq)
value<-NULL
for (i in 1:sum(freq))
{
value[i]<-sum(((-1)^(0:n-data[i]))*choose(n-data[i],(0:n-data[i]))*
beta((data[i]/c)+a+((0:n-data[i])/c),b))
}
return(-(sum(log(choose(n,data[1:sum(freq)])))+sum(log(value))+
sum(freq)*log(1/beta(a,b))))
}
#' Fitting the McDonald Generalized Beta Binomial distribution when binomial
#' random variable, frequency and shape parameters are given
#'
#' The function will fit the McDonald Generalized Beta Binomial Distribution
#' when random variables, corresponding frequencies and shape parameters are given. It will provide
#' the expected frequencies, chi-squared test statistics value, p value, degree of freedom
#' and over dispersion value so that it can be seen if this distribution fits the data.
#'
#' @usage fitMcGBB(x,obs.freq,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param a single value for shape parameter alpha representing a.
#' @param b single value for shape parameter beta representing b.
#' @param c single value for shape parameter gamma representing c.
#'
#' @details
#' \deqn{0 < a,b,c}
#' \deqn{x = 0,1,2,...}
#' \deqn{obs.freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitMcGBB} gives the class format \code{fitMB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{fitMB} fitted values of \code{dMcGBB}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{a} estimated value for alpha parameter as a.
#'
#' \code{b} estimated value for beta parameter as b.
#'
#' \code{c} estimated value for gamma parameter as c.
#'
#' \code{AIC} AIC value.
#'
#' \code{over.dis.para} over dispersion value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted} can be used to
#' extract specific outputs.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' \dontrun{
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEMcGBB(x=No.D.D,freq=Obs.fre.1,a=0.1,b=0.1,c=3.2)
#'
#' aMcGBB <- bbmle::coef(parameters)[1] #assigning the estimated a
#' bMcGBB <- bbmle::coef(parameters)[2] #assigning the estimated b
#' cMcGBB <- bbmle::coef(parameters)[3] #assigning the estimated c
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' results <- fitMcGBB(No.D.D,Obs.fre.1,aMcGBB,bMcGBB,cMcGBB)
#' results
#'
#' #extracting the expected frequencies
#' fitted(results)
#'
#' #extracting the residuals
#' residuals(results)
#' }
#'
#' @export
fitMcGBB<-function(x,obs.freq,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,a,b,c))) | any(is.infinite(c(x,obs.freq,a,b,c))) |
any(is.nan(c(x,obs.freq,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dMcGBB(x,max(x),a,b,c)
odp<-est$over.dis.para; names(odp)<-NULL
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-4
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#all the above information is mentioned as a message below
#and if the user wishes they can print or not to
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
NegLL<-NegLLMcGBB(x,obs.freq,a,b,c)
names(NegLL)<-NULL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,
"statistic"=round(statistic,4),"df"=df,"p.value"=round(p.value,4),
"fitMB"=est,"NegLL"=NegLL,"a"=a,"b"=b,"c"=c,"AIC"=2*3+2*NegLL,
"over.dis.para"=odp,"call"=match.call())
class(final)<-c("fitMB","fit")
return(final)
}
}
#' @method fitMcGBB default
#' @export
fitMcGBB.default<-function(x,obs.freq,a,b,c)
{
return(fitMcGBB(x,obs.freq,a,b,c))
}
#' @method print fitMB
#' @export
print.fitMB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Mc-Donald Generalized Beta-Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated a parameter :",x$a, " ,estimated b parameter :",x$b,",\n\t
estimated c parameter :",x$c,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n\t
over dispersion :",x$over.dis.para,"\n")
}
#' @method summary fitMB
#' @export
summary.fitMB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Mc-Donald Generalized Beta-Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated a parameter :",object$a," ,estimated b parameter :",object$b,",\n\t
estimated c parameter :",object$c,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
over dispersion :",object$over.dis.para,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
#' @importFrom bbmle mle2
#' @import hypergeo
#' @importFrom stats pchisq
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Defines functions summary.fitMB print.fitMB fitMcGBB.default fitMcGBB .EstMLEMcGBB EstMLEMcGBB NegLLMcGBB pMcGBB dMcGBB mazGBeta1 pGBeta1 dGBeta1
Documented in dGBeta1 dMcGBB EstMLEMcGBB fitMcGBB mazGBeta1 NegLLMcGBB pGBeta1 pMcGBB
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' dGBeta1(p,a,b,c)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output of \code{dGBeta1} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Generalized Beta Type-1 Distribution.
#'
#' \code{var} variance of the Generalized Beta Type-1 Distribution.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24} .
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
dGBeta1<-function(p,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b,c))) | any(is.infinite(c(p,a,b,c))) | any(is.nan(c(p,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
ans[i]<-(c/beta(a,b))*(p[i]^(a*c-1))*((1-p[i]^c)^(b-1))
}
}
}
}
# generating an output in list format consisting pdf,mean and variance
return(list("pdf"=ans,"mean"=beta(a+b,(1/c))/beta(a,(1/c)),
"var"=(beta(a+b,(2/c))/beta(a,(2/c)))-(beta(a+b,(1/c))/beta(a,(1/c)))^2 ))
}
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' pGBeta1(p,a,b,c)
#'
#' @param p vector of probabilities.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output \code{pGBeta1} gives the cumulative density values in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
pGBeta1<-function(p,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b,c))) | any(is.infinite(c(p,a,b,c))) | any(is.nan(c(p,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
term<-NULL
ans<-NULL
#for each input values in the vector necessary calculations and conditions are applied
for(i in 1:length(p))
{
if(p[i]<0 | p[i]>1)
{
stop("Invalid values in the input")
}
else
{
term[i]<-Re(hypergeo::hypergeo_powerseries(a,1-b,(p[i])^c,a+1))
#checking if the hypergeometric function value is NA(not assigned)values,
#infinite values or NAN(not a number)values if so providing an error message and
#stopping the function progress
if(is.nan(term[i]) | term[i] <= 0 | is.infinite(term[i]) |is.na(term[i]) )
{
stop("Given input values generate error values for Gaussian Hypergeometric Function")
}
else
{
ans[i]<-(((p[i])^(a*c))*term[i])/(a*beta(a,b))
}
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Generalized Beta Type-1 Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Generalized Beta Type-1 Distribution bounded between [0,1].
#'
#' @usage
#' mazGBeta1(r,a,b,c)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#' @param r vector of moments
#'
#' @details
#' The probability density function and cumulative density function of a unit bounded
#' Generalized Beta Type-1 Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= \frac{c}{B(a,b)} p^{ac-1} (1-p^c)^{b-1} }; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= \frac{p^{ac}}{aB(a,b)} 2F1(a,1-b;p^c;a+1) } \eqn{0 \le p \le 1}
#' \deqn{a,b,c > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= \frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{var[P]= \frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2 }
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= \frac{B(a+b,\frac{r}{c})}{B(a,\frac{r}{c})} }
#' \eqn{r = 1,2,3,....}
#'
#' Defined as \eqn{B(a,b)} is Beta function.
#' Defined as \eqn{2F1(a,b;c;d)} is Gaussian Hypergeometric function.
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions
#' necessary error messages will be provided to go further.
#'
#' @return
#' The output \code{mazGBeta1} gives the moments about zero in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(.1,.2,.3,1.5,2.15)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,10))
#' for (i in 1:5)
#' {
#' lines(seq(0,1,by=0.001),dGBeta1(seq(0,1,by=0.001),a[i],1,2*a[i])$pdf,col = col[i])
#' }
#'
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$pdf #extracting the pdf values
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$mean #extracting the mean
#' dGBeta1(seq(0,1,by=0.01),2,3,1)$var #extracting the variance
#'
#' pGBeta1(0.04,2,3,4) #acquiring the cdf values for a=2,b=3,c=4
#' mazGBeta1(1.4,3,2,2) #acquiring the moment about zero values
#' mazGBeta1(2,3,2,2)-mazGBeta1(1,3,2,2)^2 #acquiring the variance for a=3,b=2,c=2
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazGBeta1(3.2,3,2,2)
#'
#' @export
mazGBeta1<-function(r,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(r,a,b,c))) | any(is.infinite(c(r,a,b,c))) | any(is.nan(c(r,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, if not providing an error message and
#stopping the function progress
if(a <= 0 | b <= 0 |c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
ans<-NULL
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
#for each input values in the vector necessary calculations and conditions are applied
for (i in 1:length(r))
{
#checking if moment values are less than or equal to zero and creating
# an error message as well as stopping the function progress
if(r[i]<=0)
{
stop("Moments cannot be less than or equal to zero")
}
else
{
ans[i]<-beta(a+b,r[i]/c)/beta(a,r[i]/c)
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' McDonald Generalized Beta Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the McDonald Generalized Beta
#' Binomial Distribution.
#'
#' @usage
#' dMcGBB(x,n,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' Mixing Generalized Beta Type-1 Distribution with Binomial distribution
#' the probability function value and cumulative probability function can be constructed
#' and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (\sum_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) ) }
#' \deqn{a,b,c > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{Var_{McGBB}[x]= n^2(\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2) +n(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}) }
#' \deqn{over dispersion= \frac{\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}{\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#'
#' @return
#' The output of \code{dMcGBB} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of McDonald Generalized Beta Binomial Distribution.
#'
#' \code{var} variance of McDonald Generalized Beta Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of McDonald Generalized Beta Binomial Distribution.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.6)
#' plot(0,0,main="Mcdonald generalized beta-binomial probability function graph",
#' xlab="Binomial random variable",ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMcGBB(0:10,10,4,2,1)$pdf #extracting the pdf values
#' dMcGBB(0:10,10,4,2,1)$mean #extracting the mean
#' dMcGBB(0:10,10,4,2,1)$var #extracting the variance
#' dMcGBB(0:10,10,4,2,1)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' points(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' }
#'
#' pMcGBB(0:10,10,4,2,1) #acquiring the cumulative probability values
#'
#' @export
dMcGBB<-function(x,n,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,n,a,b,c))) | any(is.infinite(c(x,n,a,b,c))) | any(is.nan(c(x,n,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are less than or equal zero ,
#if so providing an error message and stopping the function progress
if(a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#checking if at any chance the binomial random variable is greater than binomial trial value
#if so providing an error message and stopping the function progress
if(max(x)>n)
{
stop("Binomial random variable cannot be greater than binomial trial value")
}
#checking if any random variable or trial value is negative if so providig an error message
#and stopping the function progress
else if(any(x<0) | n<0)
{
stop("Binomial random variable or binomial trial value cannot be negative")
}
else
{
final<-NULL
#for each random variable in the input vector below calculations occur
for (i in 1:length(x))
{
value<-NULL
value<-sum(((-1)^(0:(n-x[i])))*choose(n-x[i],(0:(n-x[i])))*beta((x[i]/c+a+(0:(n-x[i]))/c),b))
final[i]<-choose(n,x[i])*(1/beta(a,b))*value
}
}
}
}
# generating an output in list format consisting pdf,mean,variance and overdispersion value
return(list("pdf"=final,"mean"=n*beta(a+b,1/c)/beta(a,1/c),
"var"=n^2*((beta(a+b,2/c)/beta(a,2/c))-(beta(a+b,1/c)/beta(a,1/c))^2)+
n*((beta(a+b,1/c)/beta(a,1/c))-(beta(a+b,2/c)/beta(a,2/c))),
"over.dis.para"=(((beta(a+b,2/c))/(beta(a,2/c)))-((beta(a+b,1/c))/(beta(a,1/c)))^2)/
(((beta(a+b,1/c))/(beta(a,1/c)))-((beta(a+b,1/c))/(beta(a,1/c)))^2)))
}
#' McDonald Generalized Beta Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the McDonald Generalized Beta
#' Binomial Distribution.
#'
#' @usage
#' pMcGBB(x,n,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trials.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' Mixing Generalized Beta Type-1 Distribution with Binomial distribution
#' the probability function value and cumulative probability function can be constructed
#' and are denoted below.
#'
#' The cumulative probability function is the summation of probability function values.
#'
#' \deqn{P_{McGBB}(x)= {n \choose x} \frac{1}{B(a,b)} (\sum_{j=0}^{n-x} (-1)^j {n-x \choose j} B(\frac{x}{c}+a+\frac{j}{c},b) ) }
#' \deqn{a,b,c > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{McGBB}[x]= n\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})} }
#' \deqn{Var_{McGBB}[x]= n^2(\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2) +n(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}) }
#' \deqn{over dispersion= \frac{\frac{B(a+b,\frac{2}{c})}{B(a,\frac{2}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}{\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})}-(\frac{B(a+b,\frac{1}{c})}{B(a,\frac{1}{c})})^2}}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#'
#' @return
#' The output of \code{pMcGBB} gives cumulative probability function values in vector form.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,0.6)
#' plot(0,0,main="Mcdonald generalized beta-binomial probability function graph",
#' xlab="Binomial random variable",ylab="Probability function values",xlim = c(0,10),ylim = c(0,0.5))
#' for (i in 1:5)
#' {
#' lines(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dMcGBB(0:10,10,a[i],2.5,a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dMcGBB(0:10,10,4,2,1)$pdf #extracting the pdf values
#' dMcGBB(0:10,10,4,2,1)$mean #extracting the mean
#' dMcGBB(0:10,10,4,2,1)$var #extracting the variance
#' dMcGBB(0:10,10,4,2,1)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' points(0:10,pMcGBB(0:10,10,a[i],a[i],2),col = col[i])
#' }
#'
#' pMcGBB(0:10,10,4,2,1) #acquiring the cumulative probability values
#'
#' @export
pMcGBB<-function(x,n,a,b,c)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative proability function
#values are calculated
for(i in 1:length(x))
{
ans[i]<-sum(dMcGBB(0:x[i],n,a,b,c)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of McDonald Generalized Beta Binomial Distribution
#'
#' This function will calculate the negative log likelihood value when the vector of binomial random
#' variables and vector of corresponding frequencies are given with the shape parameters a,b and c.
#'
#' @usage
#' NegLLMcGBB(x,freq,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#'
#' @details
#' \deqn{0 < a,b,c }
#' \deqn{freq \ge 0}
#' \deqn{x = 0,1,2,...}
#'
#' @return
#' The output of \code{NegLLMcGBB} will produce a single numeric value.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' NegLLMcGBB(No.D.D,Obs.fre.1,.2,.3,1) #acquiring the negative log likelihood value
#'
#' @export
NegLLMcGBB<-function(x,freq,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,freq,a,b,c))) | any(is.infinite(c(x,freq,a,b,c)))
|any(is.nan(c(x,freq,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if any of the random variables of frequencies are less than zero if so
#creating a error message as well as stopping the function progress
if(any(c(x,freq) < 0))
{
stop("Binomial random variable or frequency values cannot be negative")
}
#checking if shape parameters are less than or equal to zero
#if so creating an error message as well as stopping the function progress
else if( a <= 0 | b <= 0 | c <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#constructing the data set using the random variables vector and frequency vector
n<-max(x)
data<-rep(x,freq)
value<-NULL
for (i in 1:sum(freq))
{
value[i]<-sum(((-1)^(0:n-data[i]))*choose(n-data[i],(0:n-data[i]))*
beta((data[i]/c)+a+((0:n-data[i])/c),b))
}
#calculating the negative log likelihood value and representing as a single output value
return(-(sum(log(choose(n,data[1:sum(freq)])))+sum(log(value))+
sum(freq)*log(1/beta(a,b))))
}
}
}
#' Estimating the shape parameters a,b and c for McDonald Generalized Beta Binomial
#' distribution
#'
#' The function will estimate the shape parameters using the maximum log likelihood method for
#' the McDonald Generalized Beta Binomial distribution when the binomial random
#' variables and corresponding frequencies are given.
#'
#' @usage
#' EstMLEMcGBB(x,freq,a,b,c,...)
#'
#' @param x vector of binomial random variables.
#' @param freq vector of frequencies.
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param c single value for shape parameter gamma representing as c.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{0 < a,b,c}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' \code{EstMLEMcGBB} here is used as a wrapper for the \code{mle2} function of \pkg{bbmle} package
#' therefore output is of class of mle2.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' \dontrun{
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEMcGBB(x=No.D.D,freq=Obs.fre.1,a=0.1,b=0.1,c=0.2)
#'
#' bbmle::coef(parameters) #extracting the parameters
#' }
#' @export
EstMLEMcGBB<-function(x,freq,a,b,c,...)
{
suppressWarnings2 <-function(expr, regex=character())
{
withCallingHandlers(expr, warning=function(w)
{
if (length(regex) == 1 && length(grep(regex, conditionMessage(w))))
{
invokeRestart("muffleWarning")
}
} )
}
output<-suppressWarnings2(bbmle::mle2(.EstMLEMcGBB,data=list(x=x,freq=freq),
start = list(a=a,b=b,c=c),...),"NaN")
return(output)
}
.EstMLEMcGBB<-function(x,freq,a,b,c)
{
#with respective to using bbmle package function mle2 there is no need impose any restrictions
#therefor the output is directly a single numeric value for the negative log likelihood value of
#Gaussian Hypergeometric Generalized Beta binomial distribution
n<-max(x)
data<-rep(x,freq)
value<-NULL
for (i in 1:sum(freq))
{
value[i]<-sum(((-1)^(0:n-data[i]))*choose(n-data[i],(0:n-data[i]))*
beta((data[i]/c)+a+((0:n-data[i])/c),b))
}
return(-(sum(log(choose(n,data[1:sum(freq)])))+sum(log(value))+
sum(freq)*log(1/beta(a,b))))
}
#' Fitting the McDonald Generalized Beta Binomial distribution when binomial
#' random variable, frequency and shape parameters are given
#'
#' The function will fit the McDonald Generalized Beta Binomial Distribution
#' when random variables, corresponding frequencies and shape parameters are given. It will provide
#' the expected frequencies, chi-squared test statistics value, p value, degree of freedom
#' and over dispersion value so that it can be seen if this distribution fits the data.
#'
#' @usage fitMcGBB(x,obs.freq,a,b,c)
#'
#' @param x vector of binomial random variables.
#' @param obs.freq vector of frequencies.
#' @param a single value for shape parameter alpha representing a.
#' @param b single value for shape parameter beta representing b.
#' @param c single value for shape parameter gamma representing c.
#'
#' @details
#' \deqn{0 < a,b,c}
#' \deqn{x = 0,1,2,...}
#' \deqn{obs.freq \ge 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' The output of \code{fitMcGBB} gives the class format \code{fitMB} and \code{fit} consisting a list
#'
#' \code{bin.ran.var} binomial random variables.
#'
#' \code{obs.freq} corresponding observed frequencies.
#'
#' \code{exp.freq} corresponding expected frequencies.
#'
#' \code{statistic} chi-squared test statistics.
#'
#' \code{df} degree of freedom.
#'
#' \code{p.value} probability value by chi-squared test statistic.
#'
#' \code{fitMB} fitted values of \code{dMcGBB}.
#'
#' \code{NegLL} Negative Log Likelihood value.
#'
#' \code{a} estimated value for alpha parameter as a.
#'
#' \code{b} estimated value for beta parameter as b.
#'
#' \code{c} estimated value for gamma parameter as c.
#'
#' \code{AIC} AIC value.
#'
#' \code{over.dis.para} over dispersion value.
#'
#' \code{call} the inputs of the function.
#'
#' Methods \code{summary}, \code{print}, \code{AIC}, \code{residuals} and \code{fitted} can be used to
#' extract specific outputs.
#'
#' @references
#' Manoj, C., Wijekoon, P. & Yapa, R.D., 2013. The McDonald Generalized Beta-Binomial Distribution: A New
#' Binomial Mixture Distribution and Simulation Based Comparison with Its Nested Distributions in Handling
#' Overdispersion. International Journal of Statistics and Probability, 2(2), pp.24-41.
#'
#' Available at: \doi{10.5539/ijsp.v2n2p24}.
#'
#' Janiffer, N.M., Islam, A. & Luke, O., 2014. Estimating Equations for Estimation of Mcdonald Generalized
#' Beta - Binomial Parameters. , (October), pp.702-709.
#'
#' Roozegar, R., Tahmasebi, S. & Jafari, A.A., 2015. The McDonald Gompertz Distribution: Properties and Applications.
#' Communications in Statistics - Simulation and Computation, (May), pp.0-0.
#'
#' Available at: \doi{10.1080/03610918.2015.1088024}.
#'
#' @seealso
#' \code{\link[bbmle]{mle2}}
#'
#' @examples
#' No.D.D <- 0:7 #assigning the random variables
#' Obs.fre.1 <- c(47,54,43,40,40,41,39,95) #assigning the corresponding frequencies
#'
#' \dontrun{
#' #estimating the parameters using maximum log likelihood value and assigning it
#' parameters <- EstMLEMcGBB(x=No.D.D,freq=Obs.fre.1,a=0.1,b=0.1,c=3.2)
#'
#' aMcGBB <- bbmle::coef(parameters)[1] #assigning the estimated a
#' bMcGBB <- bbmle::coef(parameters)[2] #assigning the estimated b
#' cMcGBB <- bbmle::coef(parameters)[3] #assigning the estimated c
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' results <- fitMcGBB(No.D.D,Obs.fre.1,aMcGBB,bMcGBB,cMcGBB)
#' results
#'
#' #extracting the expected frequencies
#' fitted(results)
#'
#' #extracting the residuals
#' residuals(results)
#' }
#'
#' @export
fitMcGBB<-function(x,obs.freq,a,b,c)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#if so creating an error message as well as stopping the function progress.
if(any(is.na(c(x,obs.freq,a,b,c))) | any(is.infinite(c(x,obs.freq,a,b,c))) |
any(is.nan(c(x,obs.freq,a,b,c))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dMcGBB(x,max(x),a,b,c)
odp<-est$over.dis.para; names(odp)<-NULL
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-4
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#all the above information is mentioned as a message below
#and if the user wishes they can print or not to
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
NegLL<-NegLLMcGBB(x,obs.freq,a,b,c)
names(NegLL)<-NULL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,
"statistic"=round(statistic,4),"df"=df,"p.value"=round(p.value,4),
"fitMB"=est,"NegLL"=NegLL,"a"=a,"b"=b,"c"=c,"AIC"=2*3+2*NegLL,
"over.dis.para"=odp,"call"=match.call())
class(final)<-c("fitMB","fit")
return(final)
}
}
#' @method fitMcGBB default
#' @export
fitMcGBB.default<-function(x,obs.freq,a,b,c)
{
return(fitMcGBB(x,obs.freq,a,b,c))
}
#' @method print fitMB
#' @export
print.fitMB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Mc-Donald Generalized Beta-Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated a parameter :",x$a, " ,estimated b parameter :",x$b,",\n\t
estimated c parameter :",x$c,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n\t
over dispersion :",x$over.dis.para,"\n")
}
#' @method summary fitMB
#' @export
summary.fitMB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Mc-Donald Generalized Beta-Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated a parameter :",object$a," ,estimated b parameter :",object$b,",\n\t
estimated c parameter :",object$c,"\n\t
X-squared :",object$statistic," ,df :",object$df," ,p-value :",object$p.value,"\n\t
over dispersion :",object$over.dis.para,"\n\t
Negative Loglikehood value :",object$NegLL,"\n\t
AIC value :",object$AIC,"\n")
}
#' @importFrom bbmle mle2
#' @import hypergeo
#' @importFrom stats pchisq
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