VGAM (version 1.1-14)

rayleigh: Rayleigh Regression Family Function

Description

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

Usage

rayleigh(lscale = "loglink", nrfs = 1/3 + 0.01,
            oim.mean = TRUE, zero = NULL, parallel = FALSE,
            type.fitted = c("mean", "percentiles", "Qlink"),
            percentiles = 50)
cens.rayleigh(lscale = "loglink", oim = TRUE)

Value

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

rrvglm

and vgam.

Details

The Rayleigh distribution, which is used in physics, has a probability density function that can be written $$f(y) = y \exp(-0.5 (y/b)^2)/b^2$$ for \(y > 0\) and \(b > 0\). The mean of \(Y\) is \(b \sqrt{\pi / 2}\) (returned as the fitted values) and its variance is \(b^2 (4-\pi)/2\).

The VGAM family function cens.rayleigh 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.

The VGAM family function rayleigh handles multiple responses.

References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

See Also

Rayleigh, genrayleigh, riceff, maxwell, weibullR, poisson.points, simulate.vlm.

Examples

nn <- 1000; Scale <- exp(2)
rdata <- data.frame(ystar = rrayleigh(nn, scale = Scale))
fit <- vglm(ystar ~ 1, rayleigh, data = rdata, trace = TRUE)
head(fitted(fit))
with(rdata, mean(ystar))
coef(fit, matrix = TRUE)
Coef(fit)

# Censored data
rdata <- transform(rdata, U = runif(nn, 5, 15))
rdata <- transform(rdata, y = pmin(U, ystar))
if (FALSE)  par(mfrow = c(1, 2))
hist(with(rdata, ystar)); hist(with(rdata, y)) 
extra <- with(rdata, list(rightcensored = ystar > U))
fit <- vglm(y ~ 1, cens.rayleigh, data = rdata, trace = TRUE,
            extra = extra, crit = "coef")
table(fit@extra$rightcen)
coef(fit, matrix = TRUE)
head(fitted(fit))