splitt function

Split-t distribution

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

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.

Returns

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.

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

where,

is the kernel of a student $t$ density with variance $\phi ^2\nu/(\nu-2)$ and

is the normalization constant.

Functions

• psplitt: Percentile for the split-t distribution.
• qsplitt: Quantile for the split-t distribution.
• rsplitt: Randon variables from the split-t distribution.

Examples

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)

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.

Author(s)

Feng Li, Jiayue Zeng