rayleigh: Rayleigh Regression Family Function
In VGAM: Vector Generalized Linear and Additive Models

rayleigh

R Documentation

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)

Arguments

`lscale`	Parameter link function applied to the scale parameter `b`. See `Links` for more choices. A log link is the default because `b` is positive.
`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.
`oim.mean`	Logical, used only for intercept-only models. `TRUE` means the mean of the OIM elements are used as working weights. If `TRUE` then this argument has top priority for working out the working weights. `FALSE` means use another algorithm.
`oim`	Logical. For censored data only, `TRUE` means the Newton-Raphson algorithm, and `FALSE` means Fisher scoring.
`zero`, `parallel`	Details at `CommonVGAMffArguments`.
`type.fitted`, `percentiles`	See `CommonVGAMffArguments` for information. Using `"Qlink"` is for quantile-links in VGAMextra.

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.

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 oim is not fully complete.

Note

The poisson.points family function is more general so that if ostatistic = 1 and dimension = 2 then it coincides with rayleigh. Other related distributions are the Maxwell and Weibull distributions.

Author(s)

T. W. Yee

References

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

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))
## Not run:  par(mfrow = c(1, 2))
hist(with(rdata, ystar)); hist(with(rdata, y)) 
## End(Not run)
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))

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.