Half-Student-t distribution

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Story

The Half-Student-t distribution is a Student-t distribution truncated to only have nonzero probability density for values greater than or equal to the location of the peak.

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Parameters

The Half-Student-t distribution is peaked at its location parameter $\mu$. The peak’s width is dictated by scale parameter $\sigma$, which is positive. Finally, the shape parameter, called “degrees of freedom,” is $\nu$. This last parameter imparts the distribution with a heavy tail for small $\nu$.

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Support

The Half-Student-t distribution is supported on the set of all real numbers that are greater than or equal to $\mu$, that is on $[\mu ,\mathrm{\infty })$.

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Probability density function

$$\begin{array}{r}f(y;\nu ,\mu ,\sigma )=\frac{2\mathrm{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu }{2}\right)\sqrt{\pi \nu {\sigma }^2}}{(1+\frac{(y-\mu {)}^2}{\nu {\sigma }^2})}^{-\frac{\nu +1}{2}}.\end{array}$$

Note that the distribution is only supported for $y\ge \mu$.

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Cumulative distribution function

$$\begin{array}{r}\begin{array}{c}F(y;\nu ,\mu ,\sigma )=\{\begin{array}{cll}{\displaystyle 1-I_{\nu /x^2+\nu }(\frac{\nu }{2},\frac{1}{2})}& & y\ge \mu ,\\ 0& & \text{otherwise},\end{array}\end{array}\end{array}$$

where $x=(y-\mu )/\sigma$ and $I_x(a,b)$ denotes the regularized incomplete beta function.

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Moments

Mean: $\mu +2\sigma \sqrt{\frac{\nu }{\pi }}\frac{\mathrm{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu }{2}\right)(\nu -1)}$ for $\nu >1$, otherwise undefined.

Variance: ${\sigma }^2(\frac{\nu }{\nu -2}-\frac{4\nu }{\pi (\nu -1{)}^2}{\left(\frac{\mathrm{\Gamma }\left(\frac{\nu +1}{2}\right)}{\mathrm{\Gamma }\left(\frac{\nu }{2}\right)}\right)}^2)$ for $\nu >2$. If $1<\nu <2$, then the variance is infinite. If $\nu \le 1$, the variance is undefined.

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Usage

Package	Syntax
NumPy	mu + sigma * np.abs(rng.standard_t(nu))
SciPy sampling	mu + np.abs(scipy.stats.t.rvs(nu, 0, sigma))
Distributions.jl	mu .+ sigma .* rand(truncated(TDist(nu); lower=mu), x)
Stan sampling	real<lower=mu> y; y ~ student_t(nu, mu, sigma)
Stan rng	real<lower=mu> y; y = mu + abs(student_t(nu, 0, sigma))

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Related distributions

• The Student-t distribution, distribution is obviously closely related. The Half-Student-t is simply a truncated Student-t distribution where only values at the peak or to its right have nonzero probability density.

• In the $\nu \to \mathrm{\infty }$ limit, the Half-Student-t distribution becomes a Half-Normal distribution.

• The Half-Cauchy distribution distribution is a special case of the Half-Student-t distribution in which $\nu =1$.

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Notes

• In Stan, a Half-Student-t is defined by putting a lower bound of $\mu$ on the variable and then using a Student-t distribution with location parameter $\mu$.

• Only the standard Student-t distribution ($\mu =0$ and $\sigma =1$) is available in NumPy. You can still draw out of the Student-t distribution by performing a transformation on the samples out of the standard Student-t distribution, as shown in the usage, above. The same is true for Distributions.jl. Shown is an example where $x$ random numbers are drawn from the distribution and transformed using broadcasting.

• The Half-Student-t distribution is not available in SciPy. To compute the PDF for $y\ge \mu$, use 2 * scipy.stats.t.pdf(y, nu, mu, sigma). To compute the CDF for $y\ge \mu$, use 2 * scipy.stats.t.cdf(y, nu, mu, sigma) - 1.

• The Half-Student-t distribution with $\mu =0$ is a useful prior for nonnegative parameters. The parameter $\nu$ of the Half-Student-t can be tuned to allow for a heavy tail ($\nu$ close to 1) or a light tail (large $\nu$).

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PDF and CDF plots

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Links

• Wikipedia

• Numpy

• Scipy

• Distributions.jl

• Stan