R: Scaled Student t Distribution with 2 df Family Function

sc.studentt2 {VGAM} | R Documentation

## Scaled Student t Distribution with 2 df Family Function

### Description

Estimates the location and scale parameters of a scaled Student t distribution with 2 degrees of freedom, by maximum likelihood estimation.

### Usage

    sc.studentt2(percentile = 50, llocation = "identitylink", lscale = "loge",
                 ilocation = NULL, iscale = NULL, imethod = 1, zero = "scale")

### Arguments

`percentile` | A numerical vector containing values between 0 and 100, which are the quantiles and expectiles. They will be returned as ‘fitted values’.
`llocation, lscale` | See `Links` for more choices, and `CommonVGAMffArguments`.
`ilocation, iscale, imethod, zero` | See `CommonVGAMffArguments` for details.

### Details

Koenker (1993) solved for the distribution whose quantiles are equal to its expectiles. Its canonical form has mean and mode at 0, and has a heavy tail (in fact, its variance is infinite).

The standard (“canonical”) form of this distribution can be endowed with a location and scale parameter. The standard form has a density that can be written as

f(z) = 2 / (4 + z^2)^(3/2)

for real y. Then z = (y-a)/b for location and scale parameters a and b > 0. The mean of Y is a. By default, eta1=a and eta2=log(b). The expectiles/quantiles corresponding to `percentile` are returned as the fitted values; in particular, `percentile = 50` corresponds to the mean (0.5 expectile) and median (0.5 quantile).

Note that if Y has a standard `dsc.t2` then Y = sqrt(2) * T_2 where T_2 has a Student-t distribution with 2 degrees of freedom. The two parameters here can also be estimated using `studentt2` by specifying `df = 2` and making an adjustment for the scale parameter, however, this VGAM family function is more efficient since the EIM is known (Fisher scoring is implemented.)

### Value

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

### Author(s)

T. W. Yee

### References

Koenker, R. (1993) When are expectiles percentiles? (solution) Econometric Theory, 9, 526–527.

### See Also

`dsc.t2`, `studentt2`.

### Examples

    set.seed(123); nn <- 1000
    kdata <- data.frame(x2 = sort(runif(nn)))
    kdata <- transform(kdata, mylocat = 1 + 3 * x2,
                              myscale = 1)
    kdata <- transform(kdata, y = rsc.t2(nn, loc = mylocat, scale = myscale))
    fit  <- vglm(y ~ x2, sc.studentt2(perc = c(1, 50, 99)), data = kdata)
    fit2 <- vglm(y ~ x2,    studentt2(df = 2), data = kdata)  # 'same' as fit

    coef(fit, matrix = TRUE)
    head(fitted(fit))
    head(predict(fit))

    # Nice plot of the results
    ## Not run:  plot(y ~ x2, data = kdata, col = "blue", las = 1,
         sub  = paste("n =", nn),
         main = "Fitted quantiles/expectiles using the sc.studentt2() distribution")
    matplot(with(kdata, x2), fitted(fit), add = TRUE, type = "l", lwd = 3)
    legend("bottomright", lty = 1:3, lwd = 3, legend = colnames(fitted(fit)),
           col = 1:3)
    ## End(Not run)

    fit@extra$percentile  # Sample quantiles

[Package VGAM version 1.0-3 Index]