R/Kumaraswamy.R
In Amalan-ConStat/R-fitODBOD: Modeling Over Dispersed Binomial Outcome Data Using BMD and ABD
#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' dKUM(p,a,b)
#'
#' @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.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{dKUM} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Kumaraswamy distribution.
#'
#' \code{var} variance of the Kumaraswamy distribution.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#'
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
dKUM<-function(p,a,b)
{
#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))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
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)
{
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]<-a*b*(p[i]^(a-1))*((1-p[i]^a)^(b-1))
}
}
}
}
mean<-b*beta(1+(1/a),b) #according to theory the mean value
variance<-b*beta(1+(2/a),b)-mean^2 #according to theory the variance value
# generating an output in list format consisting pdf,mean and variance
output<-list("pdf"=ans,"mean"=mean,"var"=variance)
return(output)
}
#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' pKUM(p,a,b)
#'
#' @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.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{pKUM} gives the cumulative density values in vector form.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#'
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
pKUM<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#aif so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero and if not providing an error message
#and stopping the function progress
if(a <= 0 | b <= 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]<-1-(1-p[i]^a)^b
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' mazKUM(r,a,b)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param r vector of moments.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{mazKUM} gives the moments about zero in vector form.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
mazKUM<-function(r,a,b)
{
#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))) | any(is.infinite(c(r,a,b))) | any(is.nan(c(r,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, and if not providing an error
#message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
ans<-NULL
#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]<-b*beta(1+(r[i]/a),b)
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' Kumaraswamy Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Kumaraswamy Binomial Distribution.
#'
#' @usage
#' dKumBin(x,n,a,b,it=25000)
#'
#' @param x vector of binomial random variables
#' @param n single value for no of binomial trial
#' @param a single value for shape parameter alpha representing a
#' @param b single value for shape parameter beta representing b
#' @param it number of iterations to converge as a proper
#' probability function replacing infinity
#'
#' @details
#' Mixing Kumaraswamy distribution with Binomial distribution will create the
#' Kumaraswamy Binomial distribution. The probability function and cumulative
#' probability function can be constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability
#' function values.
#'
#' \deqn{P_{KumBin}(x)= ab{n \choose x} \sum_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1) }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{it > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{KumBin}[x]= nbB(1+\frac{1}{a},b) }
#' \deqn{Var_{KumBin}[x]= n^2 b(B(1+\frac{2}{a},b)-bB(1+\frac{1}{a},b)^2)+
#' nb(B(1+\frac{1}{a},b)-B(1+\frac{2}{a},b)) }
#' \deqn{over dispersion= \frac{(bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2)}
#' {(bB(1+\frac{1}{a},b)-(bB(1+\frac{1}{a},b))^2)} }
#'
#' Defined as \eqn{B(a,b)} is the beta 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{dKumBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of the Kumaraswamy Binomial Distribution.
#'
#' \code{var} variance of the Kumaraswamy Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of the Kumaraswamy Distribution.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @examples
#' \dontrun{
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' plot(0,0,main="Kumaraswamy 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,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dKumBin(0:10,10,4,2)$pdf #extracting the pdf values
#' dKumBin(0:10,10,4,2)$mean #extracting the mean
#' dKumBin(0:10,10,4,2)$var #extracting the variance
#' dKumBin(0:10,10,4,2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' 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 1:5)
#' {
#' lines(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pKumBin(0:10,10,4,2) #acquiring the cumulative probability values
#' }
#'
#' @export
dKumBin<-function(x,n,a,b,it=25000)
{
#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,it))) | any(is.infinite(c(x,n,a,b,it))) |any(is.nan(c(x,n,a,b,it))))
{
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)
{
stop("Shape parameters cannot be less than or equal to zero")
}
#checking if number of iterations are less than one
#if so providing an error message and stopping the function progress
else if(it < 1)
{
stop("Number of iterations cannot be less than one")
}
else
{
check<-NULL
ans<-NULL
value<-NULL
ans1<-NULL
value1<-NULL
#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
{
#constructing the probability values for all random variables
y<-0:n
for(i in 1:length(y))
{
j<-0:it
value1[i]<-sum(((-1)^j)*choose(b-1,j)*beta(y[i]+a+a*j,n-y[i]+1))
ans1[i]<-a*b*choose(n,y[i])*value1[i]
}
check<-sum(ans1)
#checking if the sum of all probability values leads upto 1
#if not providing an error message and stopping the function progress
if(check < 0.9999 |check > 1.0001 | any(ans1>1) | any(ans1<0))
{
stop("Shape parameters and number of iterations combination does not create a proper probability mass function")
}
else
{
#for each random variable in the input vector below calculations occur
for(i in 1:length(x))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(x[i]+a+a*j,n-x[i]+1))
ans[i]<-a*b*choose(n,x[i])*value[i]
}
}
}
}
}
mean<-n*b*beta(1+(1/a),b) #according to theory the mean
variance<-(n^2)*b*(beta(1+(2/a),b)-b*(beta(1+(1/a),b))^2)+n*b*(beta(1+(1/a),b)-beta(1+(2/a),b)) #according to theory variance
ove.dis.par<-((b*beta(1+(2/a),b))-(b*beta(1+(1/a),b))^2)/
((b*beta(1+(1/a),b))-(b*beta(1+(1/a),b))^2) #according to theory overdispersion value
# generating an output in list format consisting pdf,mean,variance and overdispersion value
output<-list("pdf"=ans,"mean"=mean,"var"=variance,
"over.dis.para"=ove.dis.par)
return(output)
}
#' Kumaraswamy Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Kumaraswamy Binomial Distribution.
#'
#' @usage
#' pKumBin(x,n,a,b,it=25000)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trial.
#' @param a single value for shape parameter alpha representing a.
#' @param b single value for shape parameter beta representing b.
#' @param it number of iterations to converge as a proper
#' probability function replacing infinity.
#'
#' @details
#' Mixing Kumaraswamy distribution with Binomial distribution will create the
#' Kumaraswamy Binomial distribution. The probability function and cumulative
#' probability function can be constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability
#' function values.
#'
#' \deqn{P_{KumBin}(x)= ab{n \choose x} \sum_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1) }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{it > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{KumBin}[x]= nbB(1+\frac{1}{a},b) }
#' \deqn{Var_{KumBin}[x]= (n^2)b(B(1+\frac{2}{a},b)-bB(1+\frac{1}{a},b)^2)+
#' nb(B(1+\frac{1}{a},b)-B(1+\frac{2}{a},b)) }
#' \deqn{over dispersion= \frac{(bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2)}
#' {(bB(1+\frac{1}{a},b)-(bB(1+\frac{1}{a},b))^2)} }
#'
#' Defined as \eqn{B(a,b)} is the beta 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{pKumBin} gives cumulative probability values in vector form.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @examples
#' \dontrun{
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' plot(0,0,main="Kumaraswamy 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,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dKumBin(0:10,10,4,2)$pdf #extracting the pdf values
#' dKumBin(0:10,10,4,2)$mean #extracting the mean
#' dKumBin(0:10,10,4,2)$var #extracting the variance
#' dKumBin(0:10,10,4,2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' 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 1:5)
#' {
#' lines(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pKumBin(0:10,10,4,2) #acquiring the cumulative probability values
#' }
#'
#' @export
pKumBin<-function(x,n,a,b,it=25000)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative probability function
#values are calculated
for(i in 1:length(x))
{
j<-0:x[i]
ans[i]<-sum(dKumBin(j,n,a,b,it)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Kumaraswamy 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 and b
#' and iterations it.
#'
#' @usage
#' NegLLKumBin(x,freq,a,b,it=25000)
#'
#' @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 it number of iterations to converge as a proper probability function
#' replacing infinity.
#'
#' @details
#' \deqn{0 < a,b }
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0 }
#' \deqn{it > 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{NegLLKumBin} will produce a single numeric value.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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{
#' NegLLKumBin(No.D.D,Obs.fre.1,1.3,4.4) #acquiring the negative log likelihood value
#' }
#'
#' @export
NegLLKumBin<-function(x,freq,a,b,it=25000)
{
#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,it))) | any(is.infinite(c(x,freq,a,b,it)))
|any(is.nan(c(x,freq,a,b,it))) )
{
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)
{
stop("Shape parameters cannot be less than or equal to zero")
}
#checking if number of iterations are less than one
#if so providing an error message and stopping the function from progressing further
else if(it < 1)
{
stop("Number of iterations cannot be less than one")
}
else
{
ans1<-NULL
n<-max(x)
value1<-NULL
y<-0:n
#constructing the probability values for all random variables
for(i in 1:length(y))
{
j<-0:it
value1[i]<-sum(((-1)^j)*choose(b-1,j)*beta(y[i]+a+a*j,n-y[i]+1))
ans1[i]<-a*b*choose(n,y[i])*value1[i]
}
check<-sum(ans1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check < 0.9999 |check > 1.0001 | any(ans1>1) | any(ans1<0))
{
stop("Shape parameters and number of iterations combination does not create a proper probability mass function")
}
else
{
#constructing the data set using the random variables vector and frequency vector
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
value<-NULL
for (i in 1:sum(freq))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(data[i]+a+a*j,n-data[i]+1))
}
}
term2<-sum(log(value))
KumBinLL<-sum(freq)*log(a*b)+term1+term2
#calculating the negative log likelihood value and representing as a single output value
return(-KumBinLL)
}
}
}
#' Estimating the shape parameters a and b and iterations for Kumaraswamy Binomial Distribution
#'
#' The function will estimate the shape parameters using the maximum log likelihood method for
#' the Kumaraswamy Binomial distribution when the binomial random variables and
#' corresponding frequencies are given
#'
#' @usage
#' EstMLEKumBin(x,freq,a,b,it,...)
#'
#'
#' @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 it number of iterations to converge as a proper probability function
#' replacing infinity.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#' \deqn{it > 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' \code{EstMLEKumBin} here is used as a wrapper for the \code{mle2} function of
#' \pkg{bbmle} package therefore output is of class of mle2.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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
#' parameters1 <- EstMLEKumBin(x=No.D.D,freq=Obs.fre.1,a=10.1,b=1.1,it=10000)
#'
#' bbmle::coef(parameters1) #extracting the parameters
#' }
#' @export
EstMLEKumBin<-function(x,freq,a,b,it,...)
{
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(.EstMLEKumBin,data=list(x=x,freq=freq),
start = list(a=a,b=b,it=it),...),"NaN")
return(output)
}
.EstMLEKumBin<-function(x,freq,a,b,it)
{
#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
#Kumaraswamy binomial distribution
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
value<-NULL
for (i in 1:sum(freq))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(data[i]+a+a*j,n-data[i]+1))
}
term2<-sum(log(value))
KumBinLL<-sum(freq)*log(a*b)+term1+term2
return(-KumBinLL)
}
#' Fitting the Kumaraswamy Binomial Distribution when binomial random variable, frequency and shape
#' parameters a and b, iterations parameter it are given
#'
#' The function will fit the Kumaraswamy 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 fitKumBin(x,obs.freq,a,b,it)
#'
#' @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 it number of iterations to converge as a proper probability
#' function replacing infinity.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...n}
#' \deqn{obs.freq \ge 0}
#' \deqn{it > 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{fitKumBin} gives the class format \code{fitKB} 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{fitKB} fitted values of \code{dKumBin}.
#'
#' \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{it} estimated it value for iterations.
#'
#' \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{fiited} can be used to
#' extract specific outputs.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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 <- EstMLEKumBin(x=No.D.D,freq=Obs.fre.1,a=10.1,b=1.1,it=10000)
#'
#' bbmle::coef(parameters) #extracting the parameters
#' aKumBin <- bbmle::coef(parameters)[1] #assigning the estimated a
#' bKumBin <- bbmle::coef(parameters)[2] #assigning the estimated b
#' itKumBin <- bbmle::coef(parameters)[3] #assigning the estimated iterations
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' results <- fitKumBin(No.D.D,Obs.fre.1,aKumBin,bKumBin,itKumBin*100)
#' results
#'
#' #extracting the expected frequencies
#' fitted(results)
#'
#' #extracting the residuals
#' residuals(results)
#' }
#'
#' @export
fitKumBin<-function(x,obs.freq,a,b,it)
{
#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,it))) | any(is.infinite(c(x,obs.freq,a,b,it))) |
any(is.nan(c(x,obs.freq,a,b,it))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dKumBin(x,max(x),a,b,it)
#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)-3
#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")
}
#calculating Negative Loglikelihood value and AIC
NegLL<-NegLLKumBin(x,obs.freq,a,b,it)
AICvalue<-2*2+NegLL
#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),
"fitBB"=est,"NegLL"=NegLL,"a"=a,"b"=b,"it"=it,"AIC"=AICvalue,
"over.dis.para"=est$over.dis.para,"call"=match.call())
class(final)<-c("fitKB","fit")
return(final)
}
}
#' @method fitKumBin default
#' @export
fitKumBin.default<-function(x,obs.freq,a,b,it)
{
est<-fitKumBin(x,obs.freq,a,b,it)
return(est)
}
#' @method print fitKB
#' @export
print.fitKB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Kumaraswamy 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 it value :",x$it,"\n\t
X-squared :",x$statistic," ,df :",x$df," ,p-value :",x$p.value,"\n\t
over dispersion :",x$over.dis.para,"\n")
}
#' @method summary fitKB
#' @export
summary.fitKB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Kumaraswamy 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 it value :",object$it,"\n\t
X-squared :",object$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
#' @importFrom stats pchisq
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#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' dKUM(p,a,b)
#'
#' @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.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{dKUM} gives a list format consisting
#'
#' \code{pdf} probability density values in vector form.
#'
#' \code{mean} mean of the Kumaraswamy distribution.
#'
#' \code{var} variance of the Kumaraswamy distribution.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#'
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
dKUM<-function(p,a,b)
{
#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))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
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)
{
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]<-a*b*(p[i]^(a-1))*((1-p[i]^a)^(b-1))
}
}
}
}
mean<-b*beta(1+(1/a),b) #according to theory the mean value
variance<-b*beta(1+(2/a),b)-mean^2 #according to theory the variance value
# generating an output in list format consisting pdf,mean and variance
output<-list("pdf"=ans,"mean"=mean,"var"=variance)
return(output)
}
#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' pKUM(p,a,b)
#'
#' @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.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{pKUM} gives the cumulative density values in vector form.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#'
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
pKUM<-function(p,a,b)
{
#checking if inputs consist NA(not assigned)values, infinite values or NAN(not a number)values
#aif so creating an error message as well as stopping the function progress.
if(any(is.na(c(p,a,b))) | any(is.infinite(c(p,a,b))) | any(is.nan(c(p,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero and if not providing an error message
#and stopping the function progress
if(a <= 0 | b <= 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]<-1-(1-p[i]^a)^b
}
}
#generating an ouput vector of cumulative probability values
return(ans)
}
}
}
#' Kumaraswamy Distribution
#'
#' These functions provide the ability for generating probability density values,
#' cumulative probability density values and moment about zero values for the
#' Kumaraswamy Distribution bounded between [0,1].
#'
#' @usage
#' mazKUM(r,a,b)
#'
#' @param a single value for shape parameter alpha representing as a.
#' @param b single value for shape parameter beta representing as b.
#' @param r vector of moments.
#'
#' @details
#' The probability density function and cumulative density function of a unit
#' bounded Kumaraswamy Distribution with random variable P are given by
#'
#' \deqn{g_{P}(p)= abp^{a-1}(1-p^a)^{b-1} } ; \eqn{0 \le p \le 1}
#' \deqn{G_{P}(p)= 1-(1-p^a)^b} ; \eqn{0 \le p \le 1}
#' \deqn{a,b > 0}
#'
#' The mean and the variance are denoted by
#' \deqn{E[P]= bB(1+\frac{1}{a},b)}
#' \deqn{var[P]= bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2}
#'
#' The moments about zero is denoted as
#' \deqn{E[P^r]= bB(1+\frac{r}{a},b)}
#' \eqn{r = 1,2,3,...}
#'
#' Defined as \eqn{B(a,b)} is the beta 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{mazKUM} gives the moments about zero in vector form.
#'
#' @references
#' Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes.
#' Journal of Hydrology, 46(1), 79-88.
#'
#' Available at : \url{http://dx.doi.org/10.1016/0022-1694(80)90036-0}.
#'
#' Jones, M. C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages.
#' Statistical Methodology, 6(1), 70-81.
#'
#' Available at : \url{http://dx.doi.org/10.1016/j.stamet.2008.04.001}.
#'
#' @seealso
#' \code{\link[extraDistr]{Kumaraswamy}}
#'
#' @examples
#' #plotting the random variables and probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Probability density graph",xlab="Random variable",ylab="Probability density values",
#' xlim = c(0,1),ylim = c(0,6))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),dKUM(seq(0,1,by=0.01),a[i],a[i])$pdf,col = col[i])
#' }
#'
#' dKUM(seq(0,1,by=0.01),2,3)$pdf #extracting the probability values
#' dKUM(seq(0,1,by=0.01),2,3)$mean #extracting the mean
#' dKUM(seq(0,1,by=0.01),2,3)$var #extracting the variance
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(4)
#' a <- c(1,2,5,10)
#' plot(0,0,main="Cumulative density graph",xlab="Random variable",ylab="Cumulative density values",
#' xlim = c(0,1),ylim = c(0,1))
#' for (i in 1:4)
#' {
#' lines(seq(0,1,by=0.01),pKUM(seq(0,1,by=0.01),a[i],a[i]),col = col[i])
#' }
#'
#' pKUM(seq(0,1,by=0.01),2,3) #acquiring the cumulative probability values
#' mazKUM(1.4,3,2) #acquiring the moment about zero values
#' mazKUM(2,2,3)-mazKUM(1,2,3)^2 #acquiring the variance for a=2,b=3
#'
#' #only the integer value of moments is taken here because moments cannot be decimal
#' mazKUM(1.9,5.5,6)
#'
#' @export
mazKUM<-function(r,a,b)
{
#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))) | any(is.infinite(c(r,a,b))) | any(is.nan(c(r,a,b))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
#checking if shape parameters are greater than zero, and if not providing an error
#message and stopping the function progress
if(a <= 0 | b <= 0)
{
stop("Shape parameters cannot be less than or equal to zero")
}
else
{
#the moments cannot be a decimal value therefore converting it into an integer
r<-as.integer(r)
ans<-NULL
#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]<-b*beta(1+(r[i]/a),b)
}
}
#generating an ouput vector of moment about zero values
return(ans)
}
}
}
#' Kumaraswamy Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Kumaraswamy Binomial Distribution.
#'
#' @usage
#' dKumBin(x,n,a,b,it=25000)
#'
#' @param x vector of binomial random variables
#' @param n single value for no of binomial trial
#' @param a single value for shape parameter alpha representing a
#' @param b single value for shape parameter beta representing b
#' @param it number of iterations to converge as a proper
#' probability function replacing infinity
#'
#' @details
#' Mixing Kumaraswamy distribution with Binomial distribution will create the
#' Kumaraswamy Binomial distribution. The probability function and cumulative
#' probability function can be constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability
#' function values.
#'
#' \deqn{P_{KumBin}(x)= ab{n \choose x} \sum_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1) }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{it > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{KumBin}[x]= nbB(1+\frac{1}{a},b) }
#' \deqn{Var_{KumBin}[x]= n^2 b(B(1+\frac{2}{a},b)-bB(1+\frac{1}{a},b)^2)+
#' nb(B(1+\frac{1}{a},b)-B(1+\frac{2}{a},b)) }
#' \deqn{over dispersion= \frac{(bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2)}
#' {(bB(1+\frac{1}{a},b)-(bB(1+\frac{1}{a},b))^2)} }
#'
#' Defined as \eqn{B(a,b)} is the beta 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{dKumBin} gives a list format consisting
#'
#' \code{pdf} probability function values in vector form.
#'
#' \code{mean} mean of the Kumaraswamy Binomial Distribution.
#'
#' \code{var} variance of the Kumaraswamy Binomial Distribution.
#'
#' \code{over.dis.para} over dispersion value of the Kumaraswamy Distribution.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @examples
#' \dontrun{
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' plot(0,0,main="Kumaraswamy 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,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dKumBin(0:10,10,4,2)$pdf #extracting the pdf values
#' dKumBin(0:10,10,4,2)$mean #extracting the mean
#' dKumBin(0:10,10,4,2)$var #extracting the variance
#' dKumBin(0:10,10,4,2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' 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 1:5)
#' {
#' lines(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pKumBin(0:10,10,4,2) #acquiring the cumulative probability values
#' }
#'
#' @export
dKumBin<-function(x,n,a,b,it=25000)
{
#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,it))) | any(is.infinite(c(x,n,a,b,it))) |any(is.nan(c(x,n,a,b,it))))
{
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)
{
stop("Shape parameters cannot be less than or equal to zero")
}
#checking if number of iterations are less than one
#if so providing an error message and stopping the function progress
else if(it < 1)
{
stop("Number of iterations cannot be less than one")
}
else
{
check<-NULL
ans<-NULL
value<-NULL
ans1<-NULL
value1<-NULL
#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
{
#constructing the probability values for all random variables
y<-0:n
for(i in 1:length(y))
{
j<-0:it
value1[i]<-sum(((-1)^j)*choose(b-1,j)*beta(y[i]+a+a*j,n-y[i]+1))
ans1[i]<-a*b*choose(n,y[i])*value1[i]
}
check<-sum(ans1)
#checking if the sum of all probability values leads upto 1
#if not providing an error message and stopping the function progress
if(check < 0.9999 |check > 1.0001 | any(ans1>1) | any(ans1<0))
{
stop("Shape parameters and number of iterations combination does not create a proper probability mass function")
}
else
{
#for each random variable in the input vector below calculations occur
for(i in 1:length(x))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(x[i]+a+a*j,n-x[i]+1))
ans[i]<-a*b*choose(n,x[i])*value[i]
}
}
}
}
}
mean<-n*b*beta(1+(1/a),b) #according to theory the mean
variance<-(n^2)*b*(beta(1+(2/a),b)-b*(beta(1+(1/a),b))^2)+n*b*(beta(1+(1/a),b)-beta(1+(2/a),b)) #according to theory variance
ove.dis.par<-((b*beta(1+(2/a),b))-(b*beta(1+(1/a),b))^2)/
((b*beta(1+(1/a),b))-(b*beta(1+(1/a),b))^2) #according to theory overdispersion value
# generating an output in list format consisting pdf,mean,variance and overdispersion value
output<-list("pdf"=ans,"mean"=mean,"var"=variance,
"over.dis.para"=ove.dis.par)
return(output)
}
#' Kumaraswamy Binomial Distribution
#'
#' These functions provide the ability for generating probability function values and
#' cumulative probability function values for the Kumaraswamy Binomial Distribution.
#'
#' @usage
#' pKumBin(x,n,a,b,it=25000)
#'
#' @param x vector of binomial random variables.
#' @param n single value for no of binomial trial.
#' @param a single value for shape parameter alpha representing a.
#' @param b single value for shape parameter beta representing b.
#' @param it number of iterations to converge as a proper
#' probability function replacing infinity.
#'
#' @details
#' Mixing Kumaraswamy distribution with Binomial distribution will create the
#' Kumaraswamy Binomial distribution. The probability function and cumulative
#' probability function can be constructed and are denoted below.
#'
#' The cumulative probability function is the summation of probability
#' function values.
#'
#' \deqn{P_{KumBin}(x)= ab{n \choose x} \sum_{j=0}^{it} (-1)^j{b-1 \choose j}B(x+a+aj,n-x+1) }
#' \deqn{a,b > 0}
#' \deqn{x = 0,1,2,...n}
#' \deqn{n = 1,2,3,...}
#' \deqn{it > 0}
#'
#' The mean, variance and over dispersion are denoted as
#' \deqn{E_{KumBin}[x]= nbB(1+\frac{1}{a},b) }
#' \deqn{Var_{KumBin}[x]= (n^2)b(B(1+\frac{2}{a},b)-bB(1+\frac{1}{a},b)^2)+
#' nb(B(1+\frac{1}{a},b)-B(1+\frac{2}{a},b)) }
#' \deqn{over dispersion= \frac{(bB(1+\frac{2}{a},b)-(bB(1+\frac{1}{a},b))^2)}
#' {(bB(1+\frac{1}{a},b)-(bB(1+\frac{1}{a},b))^2)} }
#'
#' Defined as \eqn{B(a,b)} is the beta 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{pKumBin} gives cumulative probability values in vector form.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @examples
#' \dontrun{
#' #plotting the random variables and probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' plot(0,0,main="Kumaraswamy 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,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],lwd=2.85)
#' points(0:10,dKumBin(0:10,10,a[i],a[i])$pdf,col = col[i],pch=16)
#' }
#'
#' dKumBin(0:10,10,4,2)$pdf #extracting the pdf values
#' dKumBin(0:10,10,4,2)$mean #extracting the mean
#' dKumBin(0:10,10,4,2)$var #extracting the variance
#' dKumBin(0:10,10,4,2)$over.dis.para #extracting the over dispersion value
#'
#' #plotting the random variables and cumulative probability values
#' col <- rainbow(5)
#' a <- c(1,2,5,10,.85)
#' plot(0,0,main="Cumulative probability function graph",xlab="Binomial random variable",
#' ylab="Cumulative probability function values",xlim = c(0,10),ylim = c(0,1))
#' for (i in 1:5)
#' {
#' lines(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' points(0:10,pKumBin(0:10,10,a[i],a[i]),col = col[i])
#' }
#'
#' pKumBin(0:10,10,4,2) #acquiring the cumulative probability values
#' }
#'
#' @export
pKumBin<-function(x,n,a,b,it=25000)
{
ans<-NULL
#for each binomial random variable in the input vector the cumulative probability function
#values are calculated
for(i in 1:length(x))
{
j<-0:x[i]
ans[i]<-sum(dKumBin(j,n,a,b,it)$pdf)
}
#generating an ouput vector cumulative probability function values
return(ans)
}
#' Negative Log Likelihood value of Kumaraswamy 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 and b
#' and iterations it.
#'
#' @usage
#' NegLLKumBin(x,freq,a,b,it=25000)
#'
#' @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 it number of iterations to converge as a proper probability function
#' replacing infinity.
#'
#' @details
#' \deqn{0 < a,b }
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0 }
#' \deqn{it > 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{NegLLKumBin} will produce a single numeric value.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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{
#' NegLLKumBin(No.D.D,Obs.fre.1,1.3,4.4) #acquiring the negative log likelihood value
#' }
#'
#' @export
NegLLKumBin<-function(x,freq,a,b,it=25000)
{
#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,it))) | any(is.infinite(c(x,freq,a,b,it)))
|any(is.nan(c(x,freq,a,b,it))) )
{
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)
{
stop("Shape parameters cannot be less than or equal to zero")
}
#checking if number of iterations are less than one
#if so providing an error message and stopping the function from progressing further
else if(it < 1)
{
stop("Number of iterations cannot be less than one")
}
else
{
ans1<-NULL
n<-max(x)
value1<-NULL
y<-0:n
#constructing the probability values for all random variables
for(i in 1:length(y))
{
j<-0:it
value1[i]<-sum(((-1)^j)*choose(b-1,j)*beta(y[i]+a+a*j,n-y[i]+1))
ans1[i]<-a*b*choose(n,y[i])*value1[i]
}
check<-sum(ans1)
#checking if the sum of all probability values leads upto one
#if not providing an error message and stopping the function progress
if(check < 0.9999 |check > 1.0001 | any(ans1>1) | any(ans1<0))
{
stop("Shape parameters and number of iterations combination does not create a proper probability mass function")
}
else
{
#constructing the data set using the random variables vector and frequency vector
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
value<-NULL
for (i in 1:sum(freq))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(data[i]+a+a*j,n-data[i]+1))
}
}
term2<-sum(log(value))
KumBinLL<-sum(freq)*log(a*b)+term1+term2
#calculating the negative log likelihood value and representing as a single output value
return(-KumBinLL)
}
}
}
#' Estimating the shape parameters a and b and iterations for Kumaraswamy Binomial Distribution
#'
#' The function will estimate the shape parameters using the maximum log likelihood method for
#' the Kumaraswamy Binomial distribution when the binomial random variables and
#' corresponding frequencies are given
#'
#' @usage
#' EstMLEKumBin(x,freq,a,b,it,...)
#'
#'
#' @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 it number of iterations to converge as a proper probability function
#' replacing infinity.
#' @param ... mle2 function inputs except data and estimating parameter.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...}
#' \deqn{freq \ge 0}
#' \deqn{it > 0}
#'
#' \strong{NOTE} : If input parameters are not in given domain conditions necessary
#' error messages will be provided to go further.
#'
#' @return
#' \code{EstMLEKumBin} here is used as a wrapper for the \code{mle2} function of
#' \pkg{bbmle} package therefore output is of class of mle2.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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
#' parameters1 <- EstMLEKumBin(x=No.D.D,freq=Obs.fre.1,a=10.1,b=1.1,it=10000)
#'
#' bbmle::coef(parameters1) #extracting the parameters
#' }
#' @export
EstMLEKumBin<-function(x,freq,a,b,it,...)
{
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(.EstMLEKumBin,data=list(x=x,freq=freq),
start = list(a=a,b=b,it=it),...),"NaN")
return(output)
}
.EstMLEKumBin<-function(x,freq,a,b,it)
{
#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
#Kumaraswamy binomial distribution
n<-max(x)
data<-rep(x,freq)
i<-1:sum(freq)
term1<-sum(log(choose(n,data[i])))
value<-NULL
for (i in 1:sum(freq))
{
j<-0:it
value[i]<-sum(((-1)^j)*choose(b-1,j)*beta(data[i]+a+a*j,n-data[i]+1))
}
term2<-sum(log(value))
KumBinLL<-sum(freq)*log(a*b)+term1+term2
return(-KumBinLL)
}
#' Fitting the Kumaraswamy Binomial Distribution when binomial random variable, frequency and shape
#' parameters a and b, iterations parameter it are given
#'
#' The function will fit the Kumaraswamy 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 fitKumBin(x,obs.freq,a,b,it)
#'
#' @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 it number of iterations to converge as a proper probability
#' function replacing infinity.
#'
#' @details
#' \deqn{0 < a,b}
#' \deqn{x = 0,1,2,...n}
#' \deqn{obs.freq \ge 0}
#' \deqn{it > 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{fitKumBin} gives the class format \code{fitKB} 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{fitKB} fitted values of \code{dKumBin}.
#'
#' \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{it} estimated it value for iterations.
#'
#' \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{fiited} can be used to
#' extract specific outputs.
#'
#' @references
#' Li, X. H., Huang, Y. Y., & Zhao, X. Y. (2011). The Kumaraswamy Binomial Distribution. Chinese Journal
#' of Applied Probability and Statistics, 27(5), 511-521.
#'
#' @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 <- EstMLEKumBin(x=No.D.D,freq=Obs.fre.1,a=10.1,b=1.1,it=10000)
#'
#' bbmle::coef(parameters) #extracting the parameters
#' aKumBin <- bbmle::coef(parameters)[1] #assigning the estimated a
#' bKumBin <- bbmle::coef(parameters)[2] #assigning the estimated b
#' itKumBin <- bbmle::coef(parameters)[3] #assigning the estimated iterations
#'
#' #fitting when the random variable,frequencies,shape parameter values are given.
#' results <- fitKumBin(No.D.D,Obs.fre.1,aKumBin,bKumBin,itKumBin*100)
#' results
#'
#' #extracting the expected frequencies
#' fitted(results)
#'
#' #extracting the residuals
#' residuals(results)
#' }
#'
#' @export
fitKumBin<-function(x,obs.freq,a,b,it)
{
#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,it))) | any(is.infinite(c(x,obs.freq,a,b,it))) |
any(is.nan(c(x,obs.freq,a,b,it))) )
{
stop("NA or Infinite or NAN values in the Input")
}
else
{
est<-dKumBin(x,max(x),a,b,it)
#for given random variables and parameters calculating the estimated probability values
est.prob<-est$pdf
#using the estimated probability values the expected frequencies are calculated
exp.freq<-round((sum(obs.freq)*est.prob),2)
#chi-squared test statistics is calculated with observed frequency and expected frequency
statistic<-sum(((obs.freq-exp.freq)^2)/exp.freq)
#degree of freedom is calculated
df<-length(x)-3
#p value of chi-squared test statistic is calculated
p.value<-1-stats::pchisq(statistic,df)
#all the above information is mentioned as a message below
#and if the user wishes they can print or not to
#checking if df is less than or equal to zero
if(df<0 | df==0)
{
stop("Degrees of freedom cannot be less than or equal to zero")
}
#checking if any of the expected frequencies are less than five and greater than zero, if so
#a warning message is provided in interpreting the results
if(min(exp.freq)<5 && min(exp.freq) > 0)
{
message("Chi-squared approximation may be doubtful because expected frequency is less than 5")
}
#checking if expected frequency is zero, if so providing a warning message in interpreting
#the results
if(min(exp.freq)==0)
{
message("Chi-squared approximation is not suitable because expected frequency approximates to zero")
}
#calculating Negative Loglikelihood value and AIC
NegLL<-NegLLKumBin(x,obs.freq,a,b,it)
AICvalue<-2*2+NegLL
#the final output is in a list format containing the calculated values
final<-list("bin.ran.var"=x,"obs.freq"=obs.freq,"exp.freq"=exp.freq,
"statistic"=round(statistic,4),"df"=df,"p.value"=round(p.value,4),
"fitBB"=est,"NegLL"=NegLL,"a"=a,"b"=b,"it"=it,"AIC"=AICvalue,
"over.dis.para"=est$over.dis.para,"call"=match.call())
class(final)<-c("fitKB","fit")
return(final)
}
}
#' @method fitKumBin default
#' @export
fitKumBin.default<-function(x,obs.freq,a,b,it)
{
est<-fitKumBin(x,obs.freq,a,b,it)
return(est)
}
#' @method print fitKB
#' @export
print.fitKB<-function(x,...)
{
cat("Call: \n")
print(x$call)
cat("\nChi-squared test for Kumaraswamy Binomial Distribution \n\t
Observed Frequency : ",x$obs.freq,"\n\t
expected Frequency : ",x$exp.freq,"\n\t
estimated a parameter :",x$a, " ,estimated b parameter :",x$b,",\n\t
estimated it value :",x$it,"\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 fitKB
#' @export
summary.fitKB<-function(object,...)
{
cat("Call: \n")
print(object$call)
cat("\nChi-squared test for Kumaraswamy Binomial Distribution \n\t
Observed Frequency : ",object$obs.freq,"\n\t
expected Frequency : ",object$exp.freq,"\n\t
estimated a parameter :",object$a," ,estimated b parameter :",object$b,",\n\t
estimated it value :",object$it,"\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
#' @importFrom stats pchisq
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