52.2.3 Noncentral Student’s t Random Variable

Let \(ncp\) be the non-centrality parameter, \(n\) be the degrees of freedom for the non-central Student’s \(t\) random variable.

Then let \(X\) be a \({\it Normal}(n, ncp)\) and \(S^2\) be an independent \(\chi^2\) random variable with \(n\) degrees of freedom, the random variable

has a non-central Student’s \(t\) distribution with non-centrality parameter \(ncp\).

Function: pdf_noncentral_student_t (x,n,ncp) ¶

Returns the value at x of the density function of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>0\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

The pdf is

\[f(x; n, \mu) = \left[\sqrt{n} B\left({1\over 2}, {n\over 2}\right)\right]^{-1}\left(1+{x^2\over n}\right)^{-{(n+1)/2}} e^{-\mu^2/ 2} \bigg[A_n(x; \mu) + B_n(x; \mu)\bigg] \]

where

\[\eqalign{ A_n(x;\mu) &= {}_1F_1\left({n+1\over 2}; {1\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) \cr B_n(x;\mu) &= {\sqrt{2}\mu x \over \sqrt{x^2+n}} {\Gamma\left({n\over 2} + 1\right)\over \Gamma\left({n+1\over 2}\right)}\; {}_1F_1\left({n\over 2} + 1; {3\over 2}; {\mu^2 x^2\over 2\left(x^2+n\right)}\right) } \]

and \(\mu\) is the non-centrality parameter \(ncp\).

Sometimes an extra work is necessary to get the final result.

(%i1) load ("distrib")$

(%i2) expand(pdf_noncentral_student_t(3,5,0.1));
rat: replaced 0.01889822365046136 by 15934951/843198350 = 0.01889822365046136

rat: replaced -8.734356480209641 by -294697965/33740089 = -8.734356480209641

rat: replaced 4.136255165816327 by 51033443/12338079 = 4.136255165816332

rat: replaced 1.08061432164203 by 56754827/52520891 = 1.08061432164203

rat: replaced 0.0565127306411839 by 5608717/99246965 = 0.05651273064118384

rat: replaced -300.8069396896258 by -79782423/265228 = -300.8069396896256

rat: replaced 160.6269176184973 by 178374907/1110492 = 160.626917618497
                           7/2                         7/2
      0.04296414417400905 5      1.323650307289301e-6 5
(%o2) ------------------------ + -------------------------
         3/2   5/2                       sqrt(%pi)
        2    14    sqrt(%pi)
                                                              7/2
                                         1.94793720435093e-4 5
                                       + ------------------------
                                                   %pi

(%i3) float(%);
(%o3)                  0.02080593159405671

Categories: Package distrib ·

Function: cdf_noncentral_student_t (x,n,ncp) ¶

Returns the value at x of the distribution function of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>0\) degrees of freedom and noncentrality parameter \(ncp\). This function has no closed form and it is numerically computed.

(%i1) load ("distrib")$

(%i2) cdf_noncentral_student_t(-2,5,-5);
(%o2)                   0.995203009331975

Categories: Package distrib ·

Function: quantile_noncentral_student_t (q,n,ncp) ¶

Returns the q-quantile of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>0\) degrees of freedom and noncentrality parameter \(ncp\); in other words, this is the inverse of cdf_noncentral_student_t. Argument q must be an element of \([0,1]\). To make use of this function, write first load("distrib").

Categories: Package distrib ·

Function: mean_noncentral_student_t (n,ncp) ¶

Returns the mean of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>1\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

The mean is

\[E[X] = {\mu \sqrt{n}\; \Gamma\left(\displaystyle{n-1\over 2}\right) \over \sqrt{2}\;\Gamma\left(\displaystyle{n\over 2}\right)} \]

where \(\mu\) is the noncentrality parameter \(ncp\).

(%i1) load ("distrib")$

(%i2) mean_noncentral_student_t(df,k);
                          df - 1
                    gamma(------) sqrt(df) k
                            2
(%o2)               ------------------------
                                     df
                       sqrt(2) gamma(--)
                                     2

Categories: Package distrib ·

Function: var_noncentral_student_t (n,ncp) ¶

Returns the variance of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>2\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

The variance is

\[V[X] = {n(\mu^2+1)\over n-2} - {n\mu^2\; \Gamma\left(\displaystyle{n-1\over 2}\right)^2 \over 2\Gamma\left(\displaystyle{n\over 2}\right)^2} \]

where \(\mu\) is the noncentrality parameter \(ncp\).

Categories: Package distrib ·

Function: std_noncentral_student_t (n,ncp) ¶

Returns the standard deviation of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>2\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

The standard deviation is

\[D[X] = \sqrt{{n(\mu^2+1)\over n-2} - {n\mu^2\; \Gamma\left(\displaystyle{n-1\over 2}\right)^2 \over 2\Gamma\left(\displaystyle{n\over 2}\right)^2}} \]

Categories: Package distrib ·

Function: skewness_noncentral_student_t (n,ncp) ¶

Returns the skewness coefficient of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>3\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

If \(U\) is a non-central Student’s \(t\) random variable with \(n\) degrees of freedom and a noncentrality parameter \(\mu\) , the skewness is

\[\eqalign{ SK[U] &= {\mu\sqrt{n}\,\Gamma\left({{n-1}\over{2}}\right) \over{\sqrt{2}\Gamma\left({{n }\over{2}}\right)\sigma^{3}}}\left({{n \left(2n+\mu^2-3\right)}\over{\left(n-3\right)\left(n-2\right)}} -2\sigma^2\right) \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2\, \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} } \]

Categories: Package distrib ·

Function: kurtosis_noncentral_student_t (n,ncp) ¶

Returns the kurtosis coefficient of a noncentral Student random variable \({\it nc\_t}(n, ncp)\) , with \(n>4\) degrees of freedom and noncentrality parameter \(ncp\). To make use of this function, write first load("distrib").

If \(U\) is a non-central Student’s \(t\) random variable with \(n\) degrees of freedom and a noncentrality parameter \(\mu\) , the kurtosis is

\[\eqalign{ KU[U] &= {\mu_4\over \sigma^4} - 3\cr \mu_4 &= {{\left(\mu^4+6\mu^2+3\right)n^2}\over{(n-4)(n-2)}} -\left({{n\left(3(3n-5)+\mu^2(n+1)\right) }\over{(n-3)(n-2)}}-3\sigma^2\right) F \cr \sigma^2 &= {{n\left(\mu^2+1\right)}\over{n-2}}-{{n \mu^2 \Gamma\left({{n-1}\over{2}}\right)^2}\over{2\Gamma\left({{n }\over{2}}\right)^2}} \cr F &= {n\mu^2\Gamma\left({n-1\over 2}\right)^2 \over 2\sigma^4\Gamma\left({n\over 2}\right)^2} } \]

Categories: Package distrib ·

Function: random_noncentral_student_t (n,ncp)
    random_noncentral_student_t (n,ncp,m) ¶

Returns a noncentral Student random variate \({\it nc\_t}(n, ncp)\) , with \(n>0\). Calling random_noncentral_student_t with a third argument m, a random sample of size m will be simulated.

The implemented algorithm is based on the fact that if X is a normal random variable \({\it Normal}(ncp, 1)\) and \(S^2\) is a \(\chi^2\) random variable with n degrees of freedom, \(\chi^2(n)\) , then

\[U={{X}\over{\sqrt{{S^2}\over{n}}}} \]

is a noncentral Student random variable with \(n\) degrees of freedom and noncentrality parameter \(ncp\), \({\it nc\_t}(n, ncp)\) .

To make use of this function, write first load("distrib").

Categories: Package distrib · Random numbers ·