bisa {VGAM} | R Documentation |
Estimates the shape and scale parameters of the Birnbaum-Saunders distribution by maximum likelihood estimation.
bisa(lshape = "loge", lscale = "loge", ishape = NULL, iscale = 1, method.init = 1, fsmax=9001, zero = NULL)
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}.
|
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.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
If the formula is an intercept-only or n is sufficiently small, then this family function implements Fisher scoring. This involves computing an integral numerically. Fisher scoring is generally recommended here provided the integrals can be computed successfully and it does not take too long.
For n large and non-intercept-only formulas the BFGS quasi-Newton
update formula for the working weight matrices is used by default.
This is more numerically fraught.
Additionally, the estimated variance-covariance matrix may be inaccurate
or simply wrong! The standard errors must be therefore treated with
caution; these are computed in functions such as vcov()
and
summary()
.
T. W. Yee
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.
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] ## Not run: hist(y, prob=TRUE) x = seq(0, max(y), len=200) lines(x, dbisa(x, Coef(fit1)[1], Coef(fit1)[2]), col="red") ## End(Not run)