VGAM (version 0.7-1)

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",
     ishape = NULL, iscale = 1, method.init = 1,
     fsmax=9001, zero = NULL)

Arguments

lscale, lshape

Parameter link functions applied to the shape and scale parameters ($a$ and $b$ below). See Links for more choices. A log link is the default for both because they are positive.

iscale, ishape

Initial values for $a$ and $b$. A NULL means an initial value is chosen internally using method.init.

method.init

An integer with value 1 or 2 which specifies the initialization method. If failure to converge occurs try the other value, or else specify a value for ishape and/or iscale.

fsmax

Integer. If the formula is an intercept-only or if the number of observations $n$ is less than fsmax then Fisher scoring is used (recommended), else a BFGS quasi-Newton update formula for the working weight matrices is used.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The default is none of them. If used, choose one value from the set {1,2}.

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 $$F(y;a,b) = \Phi[ \xi(y/b)/a]$$ 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.

References

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.

See Also

pbisa, inv.gaussianff.

Examples

y = rbisa(n=1000, shape=exp(-0.5), scale=exp(0.5))
fit1 = vglm(y ~ 1, bisa, trace=TRUE)
coef(fit1, matrix=TRUE)
mean(y)
fitted(fit1)[1:4]

hist(y, prob=TRUE)
x = seq(0, max(y), len=200)
lines(x, dbisa(x, Coef(fit1)[1], Coef(fit1)[2]), col="red")