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(lscale = "loge", nrfs = 1/3 + 0.01,
                oim.mean = TRUE, zero = NULL)
    cens.rayleigh(lscale = "loge", 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` | Details at `CommonVGAMffArguments`.

### 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.

### 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, crit = "coef")
    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))

[Package VGAM version 1.0-3 Index]