freund61: Freund's (1961) Bivariate Extension of the Exponential...

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freund61R 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. 18, 2024, 9:09 a.m.