freund61: Freund's (1961) Bivariate Extension of the Exponential...
In VGAM: Vector Generalized Linear and Additive Models
View source: R/family.bivariate.R
freund61 R Documentation
Freund's (1961) Bivariate Extension of the
Exponential Distribution
Description
Estimate the four parameters of the Freund (1961)
bivariate extension
of the exponential distribution
by maximum likelihood estimation.
Usage
freund61(la = "loglink", lap = "loglink", lb = "loglink",
lbp = "loglink", ia = NULL, iap = NULL, ib = NULL,
ibp = NULL, independent = FALSE, zero = NULL)
Arguments
la, lap, lb, lbp
Link functions applied to the (positive)
parameters \alpha, \alpha',
\beta and \beta', respectively
(the “p” stands for “prime”).
See Links for more choices.
ia, iap, ib, ibp
Initial value for the four parameters respectively.
The default is to estimate them all internally.
independent
Logical.
If TRUE then the parameters are constrained
to satisfy
\alpha=\alpha' and
\beta=\beta',
which implies that y_1 and
y_2 are independent
and each have an ordinary exponential distribution.
zero
A vector specifying which
linear/additive predictors are modelled as intercepts only.
The values can be from the set {1,2,3,4}.
The default is none of them.
See CommonVGAMffArguments for more information.
Details
This model represents one type of bivariate extension
of the exponential
distribution that is applicable to certain problems,
in particular,
to two-component systems which can function if one of
the components
has failed. For example, engine failures in
two-engine planes, paired
organs such as peoples' eyes, ears and kidneys.
Suppose y_1 and y_2 are random variables
representing the lifetimes of
two components A and B
in a two component system.
The dependence between y_1 and y_2
is essentially such that the failure of the B component
changes the parameter of the exponential life distribution
of the A component from \alpha to
\alpha', while the failure of
the A component
changes the parameter of the exponential life distribution
of the B component from \beta to
\beta'.
The joint probability density function is given by
f(y_1,y_2) = \alpha \beta' \exp(-\beta' y_2 -
(\alpha+\beta-\beta')y_1)
for 0 < y_1 < y_2, and
f(y_1,y_2) = \beta \alpha' \exp(-\alpha' y_1 -
(\alpha+\beta-\alpha')y_2)
for 0 < y_2 < y_1.
Here, all four parameters are positive, as well
as the responses
y_1 and y_2.
Under this model, the probability that component A
is the first to fail is
\alpha/(\alpha+\beta).
The time to the first failure is distributed as an
exponential distribution with rate
\alpha+\beta. Furthermore, the
distribution of the time from first failure to failure
of the other component is a mixture of
Exponential(\alpha') and
Exponential(\beta') with proportions
\beta/(\alpha+\beta)
and \alpha/(\alpha+\beta)
respectively.
The marginal distributions are, in general, not exponential.
By default, the linear/additive predictors are
\eta_1=\log(\alpha),
\eta_2=\log(\alpha'),
\eta_3=\log(\beta),
\eta_4=\log(\beta').
A special case is when \alpha=\alpha'
and \beta=\beta', which means that
y_1 and y_2 are independent, and
both have an ordinary exponential distribution with means
1 / \alpha and 1 / \beta
respectively.
Fisher scoring is used,
and the initial values correspond to the MLEs of
an intercept model.
Consequently, convergence may take only one iteration.
Value
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm
and vgam.
Note
To estimate all four parameters, it is necessary to have some
data where y_1<y_2 and y_2<y_1.
The response must be a two-column matrix, with columns
y_1 and y_2.
Currently, the fitted value is a matrix with two columns; the
first column has values
(\alpha'+\beta)/(\alpha' (\alpha+\beta))
for the mean of y_1,
while the second column has values
(\beta'+\alpha)/(\beta' (\alpha+\beta))
for the mean of y_2.
The variance of y_1 is
\frac{(\alpha')^2 + 2 \alpha \beta + \beta^2}{
(\alpha')^2 (\alpha + \beta)^2},
the variance of y_2 is
\frac{(\beta')^2 + 2 \alpha \beta + \alpha^2 }{
(\beta')^2 (\alpha + \beta)^2 },
the covariance of y_1 and y_2 is
\frac{\alpha' \beta' - \alpha \beta }{
\alpha' \beta' (\alpha + \beta)^2}.
Author(s)
T. W. Yee
References
Freund, J. E. (1961).
A bivariate extension of the exponential distribution.
Journal of the American Statistical Association,
56, 971–977.
See Also
exponential.
Examples
fdata <- data.frame(y1 = rexp(nn <- 1000, rate = exp(1)))
fdata <- transform(fdata, y2 = rexp(nn, rate = exp(2)))
fit1 <- vglm(cbind(y1, y2) ~ 1, freund61, fdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
vcov(fit1)
head(fitted(fit1))
summary(fit1)
# y1 and y2 are independent, so fit an independence model
fit2 <- vglm(cbind(y1, y2) ~ 1, freund61(indep = TRUE),
data = fdata, trace = TRUE)
coef(fit2, matrix = TRUE)
constraints(fit2)
pchisq(2 * (logLik(fit1) - logLik(fit2)), # p-value
df = df.residual(fit2) - df.residual(fit1),
lower.tail = FALSE)
lrtest(fit1, fit2) # Better alternative
VGAM documentation built on Sept. 19, 2023, 9:06 a.m.
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View source: R/family.bivariate.R
| freund61 | R Documentation |
Freund's (1961) Bivariate Extension of the Exponential Distribution
Description
Estimate the four parameters of the Freund (1961) bivariate extension of the exponential distribution by maximum likelihood estimation.
Usage
freund61(la = "loglink", lap = "loglink", lb = "loglink",
lbp = "loglink", ia = NULL, iap = NULL, ib = NULL,
ibp = NULL, independent = FALSE, zero = NULL)
Arguments
la, lap, lb, lbp |
Link functions applied to the (positive)
parameters |
ia, iap, ib, ibp |
Initial value for the four parameters respectively. The default is to estimate them all internally. |
independent |
Logical.
If |
zero |
A vector specifying which
linear/additive predictors are modelled as intercepts only.
The values can be from the set {1,2,3,4}.
The default is none of them.
See |
Details
This model represents one type of bivariate extension
of the exponential
distribution that is applicable to certain problems,
in particular,
to two-component systems which can function if one of
the components
has failed. For example, engine failures in
two-engine planes, paired
organs such as peoples' eyes, ears and kidneys.
Suppose y_1 and y_2 are random variables
representing the lifetimes of
two components A and B
in a two component system.
The dependence between y_1 and y_2
is essentially such that the failure of the B component
changes the parameter of the exponential life distribution
of the A component from \alpha to
\alpha', while the failure of
the A component
changes the parameter of the exponential life distribution
of the B component from \beta to
\beta'.
The joint probability density function is given by
f(y_1,y_2) = \alpha \beta' \exp(-\beta' y_2 -
(\alpha+\beta-\beta')y_1)
for 0 < y_1 < y_2, and
f(y_1,y_2) = \beta \alpha' \exp(-\alpha' y_1 -
(\alpha+\beta-\alpha')y_2)
for 0 < y_2 < y_1.
Here, all four parameters are positive, as well
as the responses
y_1 and y_2.
Under this model, the probability that component A
is the first to fail is
\alpha/(\alpha+\beta).
The time to the first failure is distributed as an
exponential distribution with rate
\alpha+\beta. Furthermore, the
distribution of the time from first failure to failure
of the other component is a mixture of
Exponential(\alpha') and
Exponential(\beta') with proportions
\beta/(\alpha+\beta)
and \alpha/(\alpha+\beta)
respectively.
The marginal distributions are, in general, not exponential.
By default, the linear/additive predictors are
\eta_1=\log(\alpha),
\eta_2=\log(\alpha'),
\eta_3=\log(\beta),
\eta_4=\log(\beta').
A special case is when \alpha=\alpha'
and \beta=\beta', which means that
y_1 and y_2 are independent, and
both have an ordinary exponential distribution with means
1 / \alpha and 1 / \beta
respectively.
Fisher scoring is used, and the initial values correspond to the MLEs of an intercept model. Consequently, convergence may take only one iteration.
Value
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm
and vgam.
Note
To estimate all four parameters, it is necessary to have some
data where y_1<y_2 and y_2<y_1.
The response must be a two-column matrix, with columns
y_1 and y_2.
Currently, the fitted value is a matrix with two columns; the
first column has values
(\alpha'+\beta)/(\alpha' (\alpha+\beta))
for the mean of y_1,
while the second column has values
(\beta'+\alpha)/(\beta' (\alpha+\beta))
for the mean of y_2.
The variance of y_1 is
\frac{(\alpha')^2 + 2 \alpha \beta + \beta^2}{
(\alpha')^2 (\alpha + \beta)^2},
the variance of y_2 is
\frac{(\beta')^2 + 2 \alpha \beta + \alpha^2 }{
(\beta')^2 (\alpha + \beta)^2 },
the covariance of y_1 and y_2 is
\frac{\alpha' \beta' - \alpha \beta }{
\alpha' \beta' (\alpha + \beta)^2}.
Author(s)
T. W. Yee
References
Freund, J. E. (1961). A bivariate extension of the exponential distribution. Journal of the American Statistical Association, 56, 971–977.
See Also
exponential.
Examples
fdata <- data.frame(y1 = rexp(nn <- 1000, rate = exp(1)))
fdata <- transform(fdata, y2 = rexp(nn, rate = exp(2)))
fit1 <- vglm(cbind(y1, y2) ~ 1, freund61, fdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
vcov(fit1)
head(fitted(fit1))
summary(fit1)
# y1 and y2 are independent, so fit an independence model
fit2 <- vglm(cbind(y1, y2) ~ 1, freund61(indep = TRUE),
data = fdata, trace = TRUE)
coef(fit2, matrix = TRUE)
constraints(fit2)
pchisq(2 * (logLik(fit1) - logLik(fit2)), # p-value
df = df.residual(fit2) - df.residual(fit1),
lower.tail = FALSE)
lrtest(fit1, fit2) # Better alternative
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