Package 'dng'

Generalized hypergeometric function

Description

Evaluate generalized hypergeometric series used by the split-t gradient calculations.

Usage

ghypergeo(a, b, z, k)

Arguments

a	matrix of upper hypergeometric parameters.
b	matrix of lower hypergeometric parameters.
z	vector of hypergeometric function arguments.
k	maximum number of hypergeometric series terms. Non-positive values use the package default.

Value

A one-column numeric matrix of generalized hypergeometric function values. Rows of a, rows of b, and values of z are recycled to the output length.

See Also

gsplitt()

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Split-normal distribution

Description

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

Usage

dsplitn(x, mu, sigma, lmd, logarithm)

gsplitn(y, par, parCaller, denscaller)

psplitn(q, mu, sigma, lmd)

qsplitn(p, mu, sigma, lmd)

rsplitn(n, mu, sigma, lmd)

Arguments

x	vector of quantiles.
mu	vector of location parameter. (The mode of the density)
sigma	vector of standard deviations.
lmd	vector of skewness parameters (>0). If is 1, reduced to symmetric normal distribution.
logarithm	logical; if TRUE, probabilities p are given as log(p).
y	vector of quantiles for gradient evaluation.
par	list with mu, sigma, and lmd parameter vectors.
parCaller	character scalar naming the parameter to differentiate: "mu", "sigma", or "lmd".
denscaller	character vector selecting gradients to compute. Use "u" for the CDF gradient and "d" for the log-density gradient.
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-normal distribution, y~N($\mu$, ' $\sigma$, $\lambda$), which has density:

$1/(1+\lambda)\sigma ' \sqrt(2/\pi) exp{-(y-\mu)*2/2\sigma^2}, if y<=\mu$

'

$1/(1+\lambda)\sigma \sqrt(2/\pi) exp{-(y-\mu)*2/2\sigma^2 \lambda^2}, ' if y>\mu$

where $\sigma>0$ and $\lambda>0$. The Split-normal ' distribution reduce to normal distribution when $\lambda=1$.

Value

dsplitn gives the density; psplitn gives the percentile; qsplitn gives the quantile; and rsplitn gives the random variables. gsplitn returns a list with elements u and d containing gradients of the CDF and log-density. 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

• gsplitn(): Gradients for the split-normal CDF and log-density.

• psplitn(): Percentile for the split-normal distribution.

• qsplitn(): Quantile for the split-normal distribution.

• rsplitn(): Randon variables from the split-normal distribution.

Author(s)

Feng Li, Jiayue Zeng

References

Villani, M., & Larsson, R. (2006) The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis. Sveriges Riksbank Working Paper Series, No. 175.

See Also

splitn_mean(), splitn_var(),splitn_skewness() and splitn_kurtosis() for numerical characteristics of the split-normal distribution.

Examples

n <- 3
mu <- c(0,1,2)
sigma <- c(1,2,3)
lmd <- c(1,2,3)

q0 <- rsplitn(n, mu, sigma, lmd)
d0 <- dsplitn(q0, mu, sigma, lmd, logarithm = FALSE)
p0 <- psplitn(q0, mu, sigma, lmd)
q1 <- qsplitn(p0,mu, sigma, lmd)
all.equal(q0, q1)

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Moments of the split normal distribution

Description

Computing the mean, variance, skewness and kurtosis for the split-normal distribution.

Usage

splitn_kurtosis(lmd)

splitn_mean(mu, sigma, lmd)

splitn_skewness(sigma, lmd)

splitn_var(sigma, lmd)

Arguments

lmd	vector of skewness parameters (>0). If is 1, reduce to normal distribution.
mu	vector of location parameter. (The mode of the density)
sigma	vector of standard deviations.

Value

splitn_mean gives the mean. splitn_var gives the variance. splitn_skewness gives the skewness. splitn_kurtosis gives the kurtosis. (splitn_mean, splitn_var,splitn_skeness and splitn_kurtosis are all vectors.

Functions

• splitn_kurtosis(): Kurtosis for the split-normal distribution.

• splitn_skewness(): Skewness for the split-normal distribution.

• splitn_var(): Variance for the split-normal distribution.

Author(s)

Feng Li, Jiayue Zeng

References

Villani, M., & Larsson, R. (2006) The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis. Sveriges Riksbank Working Paper Series, No. 175.

See Also

psplitn() dsplitn() qsplitn() and rsplitn() for the split-normal distribution.

Examples

mu <- c(0,1,2)
sigma <- c(0.5,1,2)
lmd <- c(1,2,3)

mean0 <- splitn_mean(mu, sigma, lmd)
var0 <- splitn_var(sigma, lmd)
skewness0 <- splitn_skewness(sigma, lmd)
kurtosis0 <- splitn_kurtosis(lmd)

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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)

gsplitt(y, par, parCaller, denscaller)

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).
y	vector of quantiles for gradient evaluation.
par	list with mu, df, phi, and lmd parameter vectors.
parCaller	character scalar naming the parameter to differentiate: "mu", "df", "phi", or "lmd".
denscaller	character vector selecting gradients to compute. Use "u" for the CDF gradient and "d" for the log-density gradient.
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 $\nu$>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.

Value

dsplitt gives the density; psplitt gives the percentile; qsplitt gives the quantile; and rsplitt gives the random variables. gsplitt returns a list with elements u and d containing gradients of the CDF and log-density. 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

• gsplitt(): Gradients for the split-t CDF and log-density.

• 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.

See Also

splitt_mean(), splitt_var(),splitt_skewness() and splitt_kurtosis() for numerical characteristics of 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)

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Moments of the split-t distribution

Description

Computing the mean, variance, skewness and kurtosis for the split student-t distribution.

Usage

splitt_kurtosis(df, phi, lmd)

splitt_mean(mu, df, phi, lmd)

splitt_skewness(df, phi, lmd)

splitt_var(df, phi, lmd)

Arguments

df	degrees of freedom (> 0, can be non-integer). df = Inf is allowed.
phi	vector of scale parameters (> 0).
lmd	vector of skewness parameters (> 0). If is 1, reduced to symmetric student t distribution.
mu	vector of location parameter. (The mode of the density)

Value

splitt_mean gives the mean. splitt_var gives the variance. splitt_skewness gives the skewness. splitt_kurtosis gives the kurtosis. (splitt_mean, splitt_var,splitt_skeness and splitt_kurtosis are all vectors.)

Invalid arguments will result in return value NaN, with a warning.

Functions

• splitt_kurtosis(): Kurtosis for the split-t distribution.

• splitt_skewness(): Skewness for the split-t distribution.

• splitt_var(): Variance for 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.

See Also

dsplitt(), psplitt(), qsplitt() and rsplitt() for the split-t distribution.

Examples

mu <- c(0,1,2)
df <- rep(10,3)
phi <- c(0.5,1,2)
lmd <- c(1,2,3)

mean0 <- splitt_mean(mu, df, phi, lmd)
var0 <- splitt_var(df, phi, lmd)
skewness0 <- splitt_skewness(df, phi, lmd)
kurtosis0 <- splitt_kurtosis(df, phi, lmd)

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