Birnbaum-Saunders Regression Family Function
Estimates the shape and scale parameters of the Birnbaum-Saunders distribution by maximum likelihood estimation.
bisa(lscale = "loglink", lshape = "loglink", iscale = 1,
ishape = NULL, imethod = 1, zero = "shape", nowarning = FALSE)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 |
iscale, ishape |
Initial values for a and b.
A |
imethod |
An integer with value |
zero |
Specifies which linear/additive predictor is modelled as intercept-only.
If used, choose one value from the set {1,2}.
See |
The (two-parameter) Birnbaum-Saunders distribution has a cumulative distribution function that can be written as
F(y;a,k) = pnorm[xi(y/b)/a]
where pnorm() is the
cumulative distribution function of a standard normal
(see pnorm),
xi(t) = t^(0.5) - t^(-0.5),
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*a/2).
and the variance is
a^2 b^2 (1 + (5/4)*a^2).
By default, eta1 = log(a) and
eta2 = 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.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
T. W. Yee
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.
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)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.