R: Birnbaum-Saunders Distribution Family Function

bisa {VGAM} | R Documentation

## 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.

### Details

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.

### Value

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

### Author(s)

T. W. Yee

### 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.

### See Also

`pbisa`, `inv.gaussianff`, `CommonVGAMffArguments`.

### Examples

    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)

[Package VGAM version 1.1-3 Index]