# 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.

`splitt_mean()`, `splitt_var()`,`splitt_skewness()` and `splitt_kurtosis()` for numerical characteristics of the Split-t distribution.
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```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) ```