bisa: Birnbaum-Saunders Distribution Family Function

Description

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

Usage

bisa(lscale = "loglink", lshape = "loglink", iscale = 1,
     ishape = NULL, imethod = 1, zero = "shape", nowarning = FALSE)

Arguments

nowarning

Logical. Suppress a warning? Ignored for VGAM 0.9-7 and higher.

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 imethod.

imethod

An integer with value 1 or 2 or 3 which specifies the initialization method. If failure to converge occurs try the other value, or else specify a value for ishape and/or iscale.

zero

Specifies which linear/additive predictor is modelled as intercept-only. If used, choose one value from the set {1,2}. See CommonVGAMffArguments for more details.

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 $Φ (\cdot)$ is the cumulative distribution function of a standard normal (see pnorm), $ξ (t) = \sqrt{t} - 1 / \sqrt{t}$ , $y > 0$ , $a > 0$ 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, $η_{1} = \log (a)$ and $η_{2} = \log (b)$ 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

# NOT RUN {
bdata1 <- data.frame(x2 = runif(nn <- 1000))
bdata1 <- transform(bdata1, shape = exp(-0.5 + x2), scale = exp(1.5))
bdata1 <- transform(bdata1, y = rbisa(nn, scale, shape))
fit1 <- vglm(y ~ x2, bisa(zero = 1), data = bdata1, trace = TRUE)
coef(fit1, matrix = TRUE)

# }
# NOT RUN {
bdata2 <- data.frame(shape = exp(-0.5), scale = exp(0.5))
bdata2 <- transform(bdata2, y = rbisa(nn, scale, shape))
fit <- vglm(y ~ 1, bisa, data = bdata2, trace = TRUE)
with(bdata2, hist(y, prob = TRUE, ylim = c(0, 0.5), col = "lightblue"))
coef(fit, matrix = TRUE)
with(bdata2, mean(y))
head(fitted(fit))
x <- with(bdata2, seq(0, max(y), len = 200))
lines(dbisa(x, Coef(fit)[1], Coef(fit)[2]) ~ x, data = bdata2,
      col = "orange", lwd = 2) 
# }

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