dng (version 0.2.1)

splitt: Split-t distribution

Description

Density, distribution function, quantile function and random generation for the normal distribution for the split student-t distribution.

Usage

dsplitt(x, mu, df, phi, lmd, logarithm)
psplitt(q, mu, df, phi, lmd)
qsplitt(p, mu, df, phi, lmd)
rsplitt(n, mu, df, phi, lmd)

Arguments

x

vector of quantiles.

mu

vector of location parameter. (The mode of the density)

df

degrees of freedom (> 0, can be non-integer). df = Inf is also allowed.

phi

vector of scale parameters (>0).

lmd

vector of skewness parameters (>0). If is 1, reduced to the symmetric student t distribution.

logarithm

logical; if TRUE, probabilities p are given as log(p).

q

vector of quantiles.

p

vector of probability.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Value

dsplitt gives the density; psplitt gives the percentile; qsplitt gives the quantile; and rsplitt gives the random variables. Invalid arguments will result in return value NaN, with a warning.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Functions

• psplitt: Percentile for the split-t distribution.

• qsplitt: Quantile for the split-t distribution.

• rsplitt: Randon variables from the split-t distribution.

Details

The random variable y follows a split-t distribution with \(\nu\)>0 degrees of freedom, y~t(\(\mu\), \(\phi\), \(\lambda\), \(\nu\)), if its density function is of the form

$$C K(\mu, \phi, \nu,)I(y\leq\mu) + C K(\mu, \lambda \phi, \nu)I(y>\mu), $$ where, $$K(\mu, \phi, \nu,) =[\nu/(\nu+(y-\mu)^2 /\phi ^2)]^{(\nu+1)/2} $$ is the kernel of a student \(t\) density with variance \(\phi ^2\nu/(\nu-2)\) and $$c = 2[(1+\lambda)\phi (\sqrt \nu) Beta(\nu/2,1/2)]^{-1} $$is the normalization constant.

References

Li, F., Villani, M., & Kohn, R. (2010). Flexible modeling of conditional distributions using smooth mixtures of asymmetric student t densities. Journal of Statistical Planning & Inference, 140(12), 3638-3654.

See Also

splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of the Split-t distribution.

Examples

# NOT RUN {
n <- 3
mu <- c(0,1,2)
df <- rep(10,3)
phi <- c(0.5,1,2)
lmd <- c(1,2,3)

q0 <- rsplitt(n, mu, df, phi, lmd)
d0 <- dsplitt(q0, mu, df, phi, lmd, logarithm = FALSE)
p0 <- psplitt(q0, mu, df, phi, lmd)
q1 <- qsplitt(p0,mu, df, phi, lmd)
all.equal(q0, q1)

# }