bisa: Birnbaum-Saunders Distribution Family Function

Description

Estimates the shape and scale parameters of the Birnbaum-Saunders distribution by maximum likelihood estimation.

Usage

bisa(lshape = "loge", lscale = "loge",
     eshape = list(), escale = list(),
     ishape = NULL, iscale = 1, method.init = 1, zero = NULL)

Arguments

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

The (two-parameter) Birnbaum-Saunders distribution has a cumulative distribution function that can be written as

F (y; a, b) = Φ [ξ (y / b) / a]

where $\Phi(\cdot)$ is the cumulative distribution function of a standard normal (see pnorm), $\xi(t) = \sqrt{t} - 1 / \sqrt{t}$, $y > 0$, $a>0$ is the shape parameter, $b>0$ is the scale parameter. The mean of $Y$ (which is the fitted value) is $b(1 + a^2/2)$. and the variance is $a^2 b^2 (1 + \frac{5}{4}a^2)$. By default, $\eta_1=\log(a)$ and $\eta_2=\log(b)$ for this family function.

Note that $a$ and $b$ are orthogonal, i.e., the Fisher information matrix is diagonal. This family function implements Fisher scoring, and it is unnecessary to compute any integrals numerically.

References

Lemonte, A. J. and Cribari-Neto, F. and Vasconcellos, K. L. P. (2007) Improved statistical inference for the two-parameter Birnbaum-Saunders distribution. Computational Statistics & Data Analysis, 51, 4656--4681.

Birnbaum, Z. W. and Saunders, S. C. (1969) A new family of life distributions. Journal of Applied Probability, 6, 319--327.

Birnbaum, Z. W. and Saunders, S. C. (1969) Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability, 6, 328--347.

Engelhardt, M. and Bain, L. J. and Wright, F. T. (1981) Inferences on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation. Technometrics, 23, 251--256.

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, 2nd edition, Volume 2, New York: Wiley.

Examples

Run this code

x = runif(n <- 1000)
y = rbisa(n, shape=exp(-0.5+x), scale=exp(1.5))
fit = vglm(y ~ x, bisa(zero=2), trace=TRUE)
coef(fit, matrix=TRUE)

y = rbisa(n=1000, shape=exp(-0.5), scale=exp(0.5))
fit = vglm(y ~ 1, bisa, trace=TRUE)
hist(y, prob=TRUE, ylim=c(0,0.5), col="lightblue")
coef(fit, matrix=TRUE)
mean(y)
fitted(fit)[1:4]
x = seq(0, max(y), len=200)
lines(x, dbisa(x, Coef(fit)[1], Coef(fit)[2]), col="red", lwd=2)

Run the code above in your browser using DataLab

Last chance! 50% off unlimited learning