Skew Generalized T distribution functions

Functions
real	variance_adjusted_sgt (real sigma, real lambda, real p, real q)
vector	mean_centered_sgt (vector x, real sigma, real lambda, real p, real q)
real	mean_centered_sgt (real x, real sigma, real lambda, real p, real q)
real	mean_centered_sgt (real x, real sigma, real lambda, real q)
real	skew_generalized_t_lpdf (vector x, real mu, real sigma, real lambda, real p, real q)
real	skew_t_lpdf (vector x, real mu, real sigma, real lambda, real q)
real	generalized_t_lpdf (vector x, real mu, real sigma, real p, real q)
real	skew_generalized_t_lcdf (real x, real mu, real sigma, real lambda, real p, real q)

Detailed Description

From the sgt R package Carter Davis (2015). sgt: Skewed Generalized T Distribution Tree. R package version 2.0. https://CRAN.R-project.org/package=sgt

The Skewed Generalized T Distribution is a univariate 5-parameter distribution introuced by Theodossiou (1998) and known for its extreme flexibility. Special and limiting cases of the SGT distribution include the skewed generalized error distribution, the generalized t distribution introduced by McDonald and Newey (1988), the skewed t proposed by Hansen (1994), the skewed Laplace distribution, the generalized error distribution (also known as the generalized normal distribution), the skewed normal distribution, the student t distribution, the skewed Cauchy distribution, the Laplace distribution, the uniform distribution, the normal distribution, and the Cauchy distribution.

Hansen, B. E., 1994, Autoregressive Conditional Density Estimation, International Economic Review 35, 705-730.

Hansen, C., J. B. McDonald, and W. K. Newey, 2010, Instrumental Variables Estimation with Flexible Distribution sigma Journal of Business and Economic Statistics 28, 13-25.

McDonald, J. B. and W. K. Newey, 1988, Partially Adaptive Estimation of Regression Models via the Generalized t Distribution, Econometric Theory 4, 428-457.

Theodossiou, Panayioti sigma 1998, Financial Data and the Skewed Generalized T Distribution, Management Science 44, 1650-166

 
real variance_adjusted_sgt(real sigma, real lambda, real p, real q) {
  if (p * q <= 2) 
    reject("p * q must be > 2 found p * q = ", p * q);
  
  if (is_inf(q)) 
    return sigma
           * inv_sqrt((pi() * (1 + 3 * lambda ^ 2) * tgamma(3.0 / p)
                       - 16 ^ (1.0 / p) * lambda ^ 2 * (tgamma(1.0 / 2 + 1.0 / p)) ^ 2
                         * tgamma(1.0 / p))
                      / (pi() * tgamma(1.0 / p)));
  
  return sigma
         / (q ^ (1.0 / p)
            * sqrt((3 * lambda ^ 2 + 1) * (beta(3.0 / p, q - 2.0 / p) / beta(1.0 / p, q))
                   - 4 * lambda ^ 2 * (beta(2.0 / p, q - 1.0 / p) / beta(1.0 / p, q)) ^ 2));
}
 
vector mean_centered_sgt(vector x, real sigma, real lambda, real p, real q) {
  if (p * q <= 1) 
    reject("p * q must be > 1 found p * q = ", p * q);
  
  if (is_inf(q)) 
    return x + (2 ^ (2.0 / p) * sigma * lambda * tgamma(1.0 / 2 + 1.0 / p)) / sqrt(pi());
  
  return x
         + (2 * sigma * lambda * q ^ (1.0 / p) * beta(2 / p, q - 1.0 / p))
           / beta(1.0 / p, q);
}
 
real mean_centered_sgt(real x, real sigma, real lambda, real p, real q) {
  if (p * q <= 1) 
    reject("p * q must be > 1 found p * q = ", p * q);
  
  if (is_inf(q)) 
    return x + (2 ^ (2.0 / p) * sigma * lambda * tgamma(1.0 / 2 + 1.0 / p)) / sqrt(pi());
  
  return x
         + (2 * sigma * lambda * q ^ (1.0 / p) * beta(2 / p, q - 1.0 / p))
           / beta(1.0 / p, q);
}
 
real mean_centered_sgt(real x, real sigma, real lambda, real q) {
  if (q <= 1) 
    reject("q must be > 1 found q = ", q);
  
  if (is_inf(q)) 
    return x + (4 * sigma * lambda * tgamma(1.0 / 2 + 1.0)) / sqrt(pi());
  
  return x + (2 * sigma * lambda * q * beta(2, q - 1.0)) / beta(1.0, q);
}
 
real skew_generalized_t_lpdf(vector x, real mu, real sigma, real lambda, real p, real q) {
  if (sigma <= 0) 
    reject("sigma must be > 0 found sigma = ", sigma);
  
  if (lambda >= 1 || lambda <= -1) 
    reject("lambda must be between (-1, 1) found lambda = ", lambda);
  
  if (p <= 0) 
    reject("p must be > 0 found p = ", p);
  
  if (q <= 0) 
    reject("q must be > 0 found q = ", q);
  
  int N = num_elements(x);
  real out = 0;
  real sigma_adj = variance_adjusted_sgt(sigma, lambda, p, q);
  
  if (is_inf(q) && is_inf(p)) 
    return uniform_lpdf(x | mu - sigma_adj, mu + sigma_adj);
  
  vector[N] r = mean_centered_sgt(x, sigma_adj, lambda, p, q) - mu;
  vector[N] s;
  
  for (n in 1 : N) 
    s[n] = r[n] < 0 ? -1 : 1;
  
  if (is_inf(q) && !is_inf(p)) {
    out = sum((abs(r) ./ (sigma_adj * (1 + lambda * s))) ^ p);
    return log(p) - log(2) - log(sigma_adj) - lgamma(1.0 / p) - out;
  } else {
    out = sum(log1p(abs(r) ^ p ./ (q * sigma_adj ^ p * pow(1 + lambda * s, p))));
  }
  
  return N * (log(p) - log2() - log(sigma_adj) - log(q) / p - lbeta(1.0 / p, q))
         - (1.0 / p + q) * out;
}
 
real skew_t_lpdf(vector x, real mu, real sigma, real lambda, real q) {
  return skew_generalized_t_lpdf(x | mu, sigma, lambda, 2, q);
}
 
real generalized_t_lpdf(vector x, real mu, real sigma, real p, real q) {
  return skew_generalized_t_lpdf(x | mu, sigma, 0, p, q);
}
 
real skew_generalized_t_lcdf(real x, real mu, real sigma, real lambda, real p, real q) {
  if (sigma <= 0) 
    reject("sigma must be > 0 found sigma = ", sigma);
  
  if (lambda >= 1 || lambda <= -1) 
    reject("lambda must be between (-1, 1) found lambda = ", sigma);
  
  if (p <= 0) 
    reject("p must be > 0 found p = ", p);
  
  if (q <= 0) 
    reject("q must be > 0 found q = ", q);
  
  if (is_inf(x) && x < 0) 
    return 0;
  
  if (is_inf(x) && x > 0) 
    return 1;
  
  real sigma_adj = variance_adjusted_sgt(sigma, lambda, p, q);
  real x_cent = mean_centered_sgt(x, sigma_adj, lambda, p, q);
  real r = x_cent - mu;
  real lambda_new;
  real r_new;
  
  if (r > 0) {
    lambda_new = -lambda;
    r_new = -r;
  } else {
    lambda_new = lambda;
    r_new = r;
  }
  
  if (!is_inf(p) && is_inf(q) && !is_inf(x)) 
    return log_sum_exp([log1m(lambda_new) + log2(),
                        log(lambda_new - 1) + log2()
                        + beta_lcdf((r_new / (sigma * (1 + lambda_new))) ^ p | p, 1)]);
  if (is_inf(p) && !is_inf(x)) 
    return uniform_lcdf(x | mu, sigma);
  
  if (is_inf(x) && x < 0) 
    return 0;
  
  if (is_inf(x) && x > 0) 
    return 1;
  
  return log_sum_exp([log1m(lambda_new) + log2(),
                      log(lambda_new - 1) + log2()
                      + beta_lcdf(1.0
                                  / (1 + q * (sigma * (1 - lambda_new) / (-r_new)) ^ p) | 1.0
                                                                    / p, q)]);
}
 

Function Documentation

generalized_t_lpdf()

real generalized_t_lpdf	(	vector	x,
		real	mu,
		real	sigma,
		real	p,
		real	q
	)

The Generalized T distribution

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

x	Vector
mu	Real
sigma	Real \(\in (0, \infty)\) scale parameter
q	Real \(\in (0, \infty)\) kurtosis parameter

Returns

log probability

mean_centered_sgt() [1/3]

real mean_centered_sgt	(	real	x,
		real	sigma,
		real	lambda,
		real	p,
		real	q
	)

mean_centered_sgt() [2/3]

real mean_centered_sgt	(	real	x,
		real	sigma,
		real	lambda,
		real	q
	)

mean_centered_sgt() [3/3]

vector mean_centered_sgt	(	vector	x,
		real	sigma,
		real	lambda,
		real	p,
		real	q
	)

Skew Generalized T Center Mean

Centers the mean around mu, otherwise mu is the mode.

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

sigma	Real \(\in (0, \infty)\) scale parameter
lambda	Real \(-1 < \lambda < 1\)
p	Real \(\in (0, \infty)\) kurtosis parameter
q	Real \(\in (0, \infty)\) kurtosis parameter

Exceptions

reject

if \( pq \leq 1 \)

Returns

rescaled x

skew_generalized_t_lcdf()

real skew_generalized_t_lcdf	(	real	x,
		real	mu,
		real	sigma,
		real	lambda,
		real	p,
		real	q
	)

Skew Generalized T log cumulative density function

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

x	Real
mu	Real
sigma	Real \(\in (0, \infty)\) scale parameter
lambda	Real \(-1 < \lambda < 1\)
p	Real \(\in (0, \infty)\) kurtosis parameter
q	Real \(\in (0, \infty)\) kurtosis parameter

Returns

log probability

skew_generalized_t_lpdf()

real skew_generalized_t_lpdf	(	vector	x,
		real	mu,
		real	sigma,
		real	lambda,
		real	p,
		real	q
	)

The Skewed Generalized T distribution is defined as

\[ f_{SGT}(x; \mu, \sigma, \lambda, p, q) = \frac{p}{2 v \sigma q^{1/p} B(\frac{1}{p},q) \left(\frac{| x-\mu + m |^p}{q (v \sigma)^p (\lambda sign(x-\mu + m)+1)^p}+1\right)^{\frac{1}{p}+q}} \]

where \(B\) is the beta function, \( \mu \) is the location parameter, \(\sigma > 0\) is the scale parameter, \(-1 < \lambda < 1\) is the skewness parameter, and \(p > 0\) and \(q > 0\) are the parameters that control the kurtosis. \(m\) and \(v\) are not parameter sigma
but functions of the other parameters that are used here to scale or shift the distribution appropriately to match the various parameterizations of this distribution.

In the original parameterization Theodossiou of the skewed generalized t distribution,

\[ m = \frac{2 v \sigma \lambda q^{\frac{1}{p}} B(\frac{2}{p},q-\frac{1}{p})}{B(\frac{1}{p},q)} \]

and

\[ v = \frac{q^{-\frac{1}{p}}}{\sqrt{ (3 \lambda^2 + 1) \frac{ B ( \frac{3}{p}, q - \frac{2}{p} )}{B (\frac{1}{p}, q )} -4 \lambda^2 \frac{B ( \frac{2}{p}, q - \frac{1}{p} )^2}{ B (\frac{1}{p}, q )^2}}}. \]

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

x	Vector
mu	Real
sigma	Real \(\in (0, \infty)\) scale parameter
lambda	Real \(-1 < \lambda < 1\)
p	Real \(\in (0, \infty)\) kurtosis parameter
q	Real \(\in (0, \infty)\) kurtosis parameter

Returns

log probability

skew_t_lpdf()

real skew_t_lpdf	(	vector	x,
		real	mu,
		real	sigma,
		real	lambda,
		real	q
	)

The Skewed T distribution

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

x	Vector
mu	Real
sigma	Real \(\in (0, \infty)\) scale parameter
lambda	Real \(-1 < \lambda < 1\)
q	Real \(\in (0, \infty)\) kurtosis parameter

Returns

log probability

variance_adjusted_sgt()

real variance_adjusted_sgt	(	real	sigma,
		real	lambda,
		real	p,
		real	q
	)

Skew Generalized T Rescale Sigma to be Variance

Copyright

Sean Pinkney, 2021

Author

Sean Pinkney

Parameters

sigma	Real \(\in (0, \infty)\) scale parameter
lambda	Real \(-1 < \lambda < 1\)
p	Real \(\in (0, \infty)\) kurtosis parameter
q	Real \(\in (0, \infty)\) kurtosis parameter

Exceptions

reject

if \( pq \leq 2 \)

Returns

rescaled sigma