rayleigh {VGAM}R Documentation

Rayleigh Distribution Family Function

Description

Estimating the parameter of the Rayleigh distribution by maximum likelihood estimation. Right-censoring is allowed.

Usage

rayleigh(link = "loge", earg=list(), nrfs=1/3+0.01)
crayleigh(link ="loge", earg=list(), expected=FALSE)

Arguments

link Parameter link function applied to the parameter a. See Links for more choices. A log link is the default because a is positive.
earg List. Extra argument for the link. See earg in Links for general information.
nrfs Numeric, of length one, with value in [0,1]. Weighting factor between Newton-Raphson and Fisher scoring. The value 0 means pure Newton-Raphson, while 1 means pure Fisher scoring. The default value uses a mixture of the two algorithms, and retaining positive-definite working weights.
expected Logical. For censored data only, FALSE means the Newton-Raphson algorithm, and TRUE means Fisher scoring.

Details

The Rayleigh distribution, which is used in physics, has a probability density function that can be written

f(y) = y*exp(-0.5*(y/a)^2)/a^2

for y>0 and a>0. The mean of Y is a * sqrt(pi / 2) and its variance is a^2 (4-pi)/2.

The VGAM family function crayleigh handles right-censored data (the true value is greater than the observed value). To indicate which type of censoring, input extra = list(rightcensored = vec2) where vec2 is a logical vector the same length as the response. If the component of this list is missing then the logical values are taken to be FALSE. The fitted object has this component stored in the extra slot.

Value

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

Warning

The theory behind the argument expected is not fully complete.

Note

A related distribution is the Maxwell distribution.

Author(s)

T. W. Yee

References

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

See Also

Rayleigh, riceff, maxwell.

Examples

n = 1000; a = exp(2)
ystar = rrayleigh(n, a=a)
fit = vglm(ystar ~ 1, rayleigh, trace=TRUE, crit="c")
fitted(fit)[1:5]
mean(ystar)
coef(fit, matrix=TRUE)
Coef(fit)

# Censored data
U = runif(n, 5, 15)
y = pmin(U, ystar)
## Not run: 
par(mfrow=c(1,2)); hist(ystar); hist(y)
## End(Not run)
extra = list(rightcensored = ystar > U)
fit = vglm(y ~ 1, crayleigh, trace=TRUE, extra=extra)
table(fit@extra$rightcen)
coef(fit, matrix=TRUE)
fitted(fit)[1:4,]

[Package VGAM version 0.7-7 Index]