splitt: Split-t distribution In dng: Distributions and Gradients

Description

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

Usage

 1 2 3 4 5 6 7 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.

Details

The random variable y follows a split-t distribution with ν>0 degrees of freedom, y~t(μ, φ, λ, ν), if its density function is of the form

C K(μ, φ, ν,)I(y≤qμ) + C K(μ, λ φ, ν)I(y>μ),

where,

K(μ, φ, ν,) =[ν/(ν+(y-μ)^2 /φ ^2)]^{(ν+1)/2}

is the kernel of a student t density with variance φ ^2ν/(ν-2) and

c = 2[(1+λ)φ (√ ν) Beta(ν/2,1/2)]^{-1}

is the normalization constant.

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.

Author(s)

Feng Li, Jiayue Zeng

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.