nakagami {VGAM} | R Documentation

## Nakagami Distribution Family Function

### Description

Estimation of the two parameters of the Nakagami distribution by maximum likelihood estimation.

### Usage

    nakagami(lshape = "loge", lscale = "loge",
             eshape=list(), escale=list(), ishape = NULL, iscale = 1)

### Arguments

`lshape, lscale` | Parameter link function applied to the shape and scale parameters. Log links ensure they are positive. See `Links` for more choices.
`eshape, escale` | List. Extra argument for each of the links. See `earg` in `Links` for general information.
`ishape, iscale` | Optional initial values for the shape and scale parameters. For `ishape`, a `NULL` value means it is obtained in the `initialize` slot based on the value of `iscale`. For `iscale`, assigning a `NULL` means a value is obtained in the `initialize` slot, however, setting another numerical value is recommended if convergence fails or is too slow.

### Details

The Nakagami distribution, which is useful for modelling wireless systems such as radio links, can be written

2 * (shape/scale)^shape * y^(2*shape-1) * exp(-shape*y^2/scale) / gamma(shape)

for y > 0, shape > 0, scale > 0. The mean of Y is sqrt(scale/shape) * gamma(shape+0.5) / gamma(shape) and these are returned as the fitted values. By default, the linear/additive predictors are eta1=log(shape) and eta2=log(scale). Fisher scoring is implemented.

### Value

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

### Note

The Nakagami distribution is also known as the Nakagami-m distribution, where m=shape here. Special cases: m=0.5 is a one-sided Gaussian distribution and m=1 is a Rayleigh distribution. The second moment is E(Y^2)=m.

If Y has a Nakagami distribution with parameters shape and scale then Y^2 has a gamma distribution with shape parameter shape and scale parameter scale/shape.

### Author(s)

T. W. Yee

### References

Nakagami, M. (1960) The m-distribution: a general formula of intensity distribution of rapid fading, pp.3–36 in: Statistical Methods in Radio Wave Propagation. W. C. Hoffman, Ed., New York: Pergamon.

### See Also

`rnaka`, `gamma2`, `rayleigh`.

### Examples

    n = 1000; shape = exp(0); Scale = exp(1)
    y = sqrt(rgamma(n, shape=shape, scale=Scale/shape))
    fit = vglm(y ~ 1, nakagami, trace=TRUE, crit="c")
    y = rnaka(n, shape=shape, scale=Scale)
    fit = vglm(y ~ 1, nakagami(iscale=3), trace=TRUE)
    head(fitted(fit))
    mean(y)
    coef(fit, matrix=TRUE)
    (Cfit = Coef(fit))
    ## Not run:
    hist(sy <- sort(y), prob=TRUE, main="", xlab="y", ylim=c(0,0.6))
    lines(sy, dnaka(sy, shape=Cfit[1], scale=Cfit[2]), col="red")
    ## End(Not run)

[Package VGAM version 0.7-9 Index]