R: Expectiles/Quantiles of the Scaled Student t Distribution...

Expectiles-sc.t2 {VGAM} | R Documentation

## Expectiles/Quantiles of the Scaled Student t Distribution with 2 Df

### Description

Density function, distribution function, and quantile/expectile function and random generation for the scaled Student t distribution with 2 degrees of freedom.

### Usage

    dsc.t2(x, location = 0, scale = 1, log = FALSE)
    psc.t2(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
    qsc.t2(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
    rsc.t2(n, location = 0, scale = 1)

### Arguments

`x, q` | Vector of expectiles/quantiles. See the terminology note below.
`p` | Vector of probabilities. These should lie in (0,1).
`n, log` | See `runif`.
`location, scale` | Location and scale parameters. The latter should have positive values. Values of these vectors are recyled.
`lower.tail, log.p` | Same meaning as in `pt` or `qt`.

### Details

A Student-t distribution with 2 degrees of freedom and a scale parameter of `sqrt(2)` is equivalent to the standard form of this distribution (called Koenker's distribution below). Further details about this distribution are given in `sc.studentt2`.

### Value

`dsc.t2(x)` gives the density function. `psc.t2(q)` gives the distribution function. `qsc.t2(p)` gives the expectile and quantile function. `rsc.t2(n)` gives n random variates.

### Author(s)

T. W. Yee and Kai Huang

### See Also

`dt`, `sc.studentt2`.

### Examples

    my.p <- 0.25; y <- rsc.t2(nn <- 5000)
    (myexp <- qsc.t2(my.p))
    sum(myexp - y[y <= myexp]) / sum(abs(myexp - y))  # Should be my.p
    # Equivalently:
    I1 <- mean(y <= myexp) * mean( myexp - y[y <= myexp])
    I2 <- mean(y >  myexp) * mean(-myexp + y[y >  myexp])
    I1 / (I1 + I2)  # Should be my.p
    # Or:
    I1 <- sum( myexp - y[y <= myexp])
    I2 <- sum(-myexp + y[y >  myexp])

    # Non-standard Koenker distribution
    myloc <- 1; myscale <- 2
    yy <- rsc.t2(nn, myloc, myscale)
    (myexp <- qsc.t2(my.p, myloc, myscale))
    sum(myexp - yy[yy <= myexp]) / sum(abs(myexp - yy))  # Should be my.p
    psc.t2(mean(yy), myloc, myscale)  # Should be 0.5
    abs(qsc.t2(0.5, myloc, myscale) - mean(yy))  # Should be 0
    abs(psc.t2(myexp, myloc, myscale) - my.p)  # Should be 0
    integrate(f = dsc.t2, lower = -Inf, upper = Inf,
              locat = myloc, scale = myscale)  # Should be 1

    y <- seq(-7, 7, len = 201)
    max(abs(dsc.t2(y) - dt(y / sqrt(2), df = 2) / sqrt(2)))  # Should be 0
    ## Not run:  plot(y, dsc.t2(y), type = "l", col = "blue", las = 1,
         ylim = c(0, 0.4), main = "Blue = Koenker; orange = N(0, 1)")
    lines(y, dnorm(y), type = "l", col = "orange")
    abline(h = 0, v = 0, lty = 2)
    ## End(Not run)

[Package VGAM version 1.1-3 Index]