Maxwell Regression Family Function

Estimating the parameter of the Maxwell distribution by maximum likelihood estimation.

maxwell(link = "loglink", zero = NULL, parallel = FALSE,
        type.fitted = c("mean", "percentiles", "Qlink"),
        percentiles = 50)

Arguments

link

Parameter link function applied to \(a\), which is called the parameter rate. See Links for more choices and information; a log link is the default because the parameter is positive. More information is at CommonVGAMffArguments.

zero, parallel

See CommonVGAMffArguments.

type.fitted, percentiles

See CommonVGAMffArguments for information. Using "Qlink" is for quantile-links in VGAMextra.

Details

The Maxwell distribution, which is used in the area of thermodynamics, has a probability density function that can be written $$f(y;a) = \sqrt{2/\pi} a^{3/2} y^2 \exp(-0.5 a y^2)$$ for \(y>0\) and \(a>0\). The mean of \(Y\) is \(\sqrt{8 / (a \pi)}\) (returned as the fitted values), and its variance is \((3\pi - 8)/(\pi a)\).

Value

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

References

von Seggern, D. H. (1993). CRC Standard Curves and Surfaces, Boca Raton, FL, USA: CRC Press.

Author

T. W. Yee

Note

Fisher-scoring and Newton-Raphson are the same here. A related distribution is the Rayleigh distribution. This VGAM family function handles multiple responses. This VGAM family function can be mimicked by poisson.points(ostatistic = 1.5, dimension = 2).

See also

Maxwell, rayleigh, poisson.points.

Examples

mdata <- data.frame(y = rmaxwell(1000, rate = exp(2)))
fit <- vglm(y ~ 1, maxwell, mdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 2.2364372
#> Iteration 2: coefficients = 1.9797996
#> Iteration 3: coefficients = 2.0076038
#> Iteration 4: coefficients = 2.0079975
#> Iteration 5: coefficients = 2.0079976
coef(fit, matrix = TRUE)
#>             loglink(rate)
#> (Intercept)      2.007998
Coef(fit)
#>     rate 
#> 7.448388