Doubly non-central Fisher \(F\) distribution

class ctx.dist_fisher_f_2nc(m, n, lambda1, lambda2)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The doubly non-central Fisher \(F\) distribution is a continuous probability distribution with \(m>0\) and \(n>0\) degrees of freedom, noncentrality parameters \(\lambda_1 \ge 0\) and \(\lambda_2 \ge 0\), and the support interval \((0, +\infty)\). See also Butler and Paolella [173], Chou [186], Chattamvelli and Jones [182].

dist_fisher_f_2nc.pdf(x)

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following a doubly non-central Fisher F distribution:

\[\text{pdf}_X(x) = f_{F''}(x;n_1, n_2;\lambda_1,\lambda_2) = \sum_{i=0}^{\infty} \omega_{i,\lambda_2} s_{i,n} f_{F'}(s_{i,n} x;n+2i,\lambda_1),\]

where \(f_{F'}(\cdot)\) denotes the PDF of the singly noncentral \(F\)-distribution, and

\[\omega_{i,\lambda_2} = \frac{\exp(-\lambda_2/2)(\lambda_2/2)^i}{i!} \quad \text{and} \quad s_{i,n}=\sqrt{\frac{n+2i}{n}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", fisher_f_2nc(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_fisher_f_2nc.cdf(x)

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following a doubly non-central Fisher F distribution:

\[\text{cdf}_X(x) = F_{F''}(x;n_1, n_2;\lambda_1,\lambda_2) = \sum_{i=0}^{\infty} \omega_{i,\theta} s_{i,n} F_{F'}(s_{i,n_2} x;n_1+2i,n_2;\lambda_1),\]

where \(F_{F'}(\cdot)\) denotes the CDF of the singly noncentral \(F\)-distribution, and

\[\omega_{i,\lambda_2} = \frac{\exp(-\lambda_2/2)(\lambda_2/2)^i}{i!} \quad \text{and} \quad s_{i,n}=\sqrt{\frac{n+2i}{n}}.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", fisher_f_2nc(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_fisher_f_2nc.sf(x)

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following a doubly non-central Fisher F distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x) = \int_{x}^{\infty} \text{pdf}_X(x) \mathrm{d} t.\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", fisher_f_2nc(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_fisher_f_2nc.qtf(q)

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following a doubly non-central Fisher F distribution:

There is no known explicit form for the quantile function \(\text{cdf}^{-1}_X(x)\): It is computed using Newton iterations with starting values from a central \(F\) approximation.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", fisher_f_2nc(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_fisher_f_2nc.isf(q)

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following a doubly non-central Fisher F distribution:

\[\text{isf}_X(q) = \text{qtf}_X(1-q) = \text{cdf}^{-1}_X(1-q).\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", fisher_f_2nc(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_fisher_f_2nc.c_x(t)

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following a doubly non-central Fisher F distribution:

\[C_X(t) = \int_{0}^{\infty} e^{i tx} \text{pdf}_X(x) \mathrm{d} x\]

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", fisher_f_2nc(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_fisher_f_2nc.m_x(t)

Returns NaN, since the moment generating function does not exist.

dist_fisher_f_2nc.k_x(t, k=0)

Returns NaN, since the cumulant generating function does not exist.

dist_fisher_f_2nc.moments(k)

Returns the first \(j\) raw moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following a doubly non-central Fisher F distribution. The rth moment only exists for \(n_2 > 2r\), and is given by

\[\mu'_X(r) = \left(\frac{n_2}{n_1}\right)^{r} \Gamma(\tfrac{1}{2}n_1+r) \Gamma(\tfrac{1}{2}n_2-r) \times {}_1\widetilde{F}_1(-r; \tfrac{1}{2}n_1; -\tfrac{1}{2}\lambda_1) \times {}_1\widetilde{F}_1(r; \tfrac{1}{2}n_2; -\tfrac{1}{2}\lambda_2),\]

where \({}_1\widetilde{F}_1(a,b;z)\) denotes Kummer’s regularized confluent hypergeometric function.

See also: https://mathworld.wolfram.com/NoncentralF-Distribution.html

See also: Paoella 2, page 358-360

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f_2nc(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_fisher_f_2nc.cumulants(k)

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following a doubly non-central Fisher F distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", fisher_f_2nc(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00

Additional methods: confidence intervals and sample size estimates

dist_fisher_f_2nc.nc_ci(alpha, beta)

Returns a confidence interval for the noncentrality parameter lambda

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", fisher_f_2nc(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

Recurrences: Doubly non-central Fisher F, recurrence pdf

ctx.fisher_f_nc2_pdf_recurrence(x, lambda, start_n1, start_n2, target_n1, target_n2)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

Applies a recurrence relation to calculate the cdf for different degrees of freedom for a given value of x. This is mostly useful when using asymptotic methods.

Let the density \(g_{m,n}\) be that of \(m/n\) times an \(F_{m,n}\) random variable. Let \(G_{m,n}(y)\) be its distribution function, and let \(g_{m,n}^{\lambda_1,\lambda_2}\) and \(G_{m,n}^{\lambda_1,\lambda_2}(y)\) be the density and distribution function of its doubly noncentral version (the distribution of \(\chi_m^2(\lambda_1)/\chi_n^2(\lambda_2)\)). Then the following recurrence relations hold (see Chattamvelli and Jones [182])

\[n\left[G_{m,n}^{\lambda}(y)-G_{m,n}^{\lambda}(y)\right] = -2g_{m,n}^{\lambda}(y)\]

\[\lambda_1(1+y) g_{m+4,n}^{\lambda_1,\lambda_2}(y) = [\lambda_1 y - m(1+y)]g_{m+2,n}^{\lambda_1,\lambda_2}(y) + y(m+n)g_{m,n}^{\lambda_1,\lambda_2}(y) + \lambda_2 g_{m,n+2}^{\lambda_1,\lambda_2}(y).\]

\[\lambda_2(1+y) g_{m,n+4}^{\lambda_1,\lambda_2}(y) = [\lambda_2 y - n(1+y)]g_{m+2,n}^{\lambda_1,\lambda_2}(y) + (m+n)g_{m,n}^{\lambda_1,\lambda_2}(y) + \lambda_1 g_{m+2,n}^{\lambda_1,\lambda_2}(y).\]

\[\lambda_1 g_{m+4,n-2}^{\lambda_1,\lambda_2}(y) = - m g_{m+2,n}^{\lambda_1,\lambda_2}(y) + n y g_{m,n+2}^{\lambda_1,\lambda_2}(y) + \lambda_2 g_{m+2,n}^{\lambda_1,\lambda_2}(y).\]

Recurrences: Doubly non-central Fisher F, recurrence cdf

ctx.fisher_f_nc2_cdf_recurrence(x, lambda, start_n1, start_n2, target_n1, target_n2)

where ctx is fpm, mpm, ipm, dec, gmp or apm.

Applies a recurrence relation to calculate the cdf for different degrees of freedom for a given value of x. This is mostly useful when using asymptotic methods.

\begin{eqnarray} \lambda_1(1+y) G_{m+6,n}^{\lambda_1,\lambda_2}(y) & = & [\lambda_1 y - (m+2-\lambda_1 )(1+y)]G_{m+4,n}^{\lambda_1,\lambda_2}(y) \\ & + & [(m+2)(1+y)+y(m+n-\lambda_1)]G_{m+2,n}^{\lambda_1,\lambda_2}(y) \nonumber \\ & - & y(m+n)G_{m,n}^{\lambda_1,\lambda_2}(y) + \lambda_2 y :cite:t:`G_{m+2,n+2}^{\lambda_1,\lambda_2}(y)-G_{m,n+2}^{\lambda_1,\lambda_2}(y)] \nonumber \end{eqnarray}

\begin{eqnarray} [n(1+y)-\lambda_2] :cite:t:`G_{m,n+2}^{\lambda_1,\lambda_2}(y)-G_{m+2,n+2}^{\lambda_1,\lambda_2}(y) & = & (m+n) :cite:t:`G_{m+2,n}^{\lambda_1,\lambda_2}(y) - G_{m,n}^{\lambda_1,\lambda_2}(y)] \\ & + & \lambda_2(1+y)[G_{m+2,n}^{\lambda_1,\lambda_2}(y) - G_{m+2,n}^{\lambda_1,\lambda_2}(y)] \nonumber \\ & - & \lambda_1 :cite:t:`G_{m+2,n}^{\lambda_1,\lambda_2}(y) - G_{m+4,n}^{\lambda_1,\lambda_2}(y)] \nonumber \end{eqnarray}

\begin{eqnarray} (m+2) :cite:t:`G_{m+2,n}^{\lambda_1,\lambda_2}(y) & = & (\lambda_1 - m -2) G_{m+4,n}^{\lambda_1,\lambda_2}(y)- \lambda_1 G_{m+6,n}^{\lambda_1,\lambda_2}(y) \\ & + & y[nG_{m,n}^{\lambda_1,\lambda_2}(y)-nG_{m+2,n}^{\lambda_1,\lambda_2}(y) \nonumber \\ & - & \lambda_2 G_{m,n}^{\lambda_1,\lambda_2}(y)- \lambda_2 G_{m+2,n}^{\lambda_1,\lambda_2}(y)] \nonumber \end{eqnarray}

Approximations

ctx.fisher_f_nc2_ecf(x, m, n, lambda1, lambda2, results='cdf')

where ctx is ipm, dec, mpm, or gmp.

Calculates the Edgeworth approximation to the pdf, cdf and sf. See also: MathWorld [882], Paolella [479], page 358-360.

ctx.fisher_f_nc2_ecf_inv(q, m, n, lambda1, lambda2, results='qtf')

where ctx is ipm, dec, mpm, or gmp.

Calculates the Cornish-Fisher approximation to the qtf and isf.

ctx.fisher_f_nc2_spa(x, m, n, lambda1, lambda2, results='cdf')

where ctx is fpm, mpm, ipm, dec, gmp or apm.

Calculates the Luggannini-Rice saddlepoint approximation of the pdf, cdf and sf.

A random variable \(X\) follows a doubly noncentral \(F\)-distribution with \(n_1\) and \(n_2\) degrees of freedom and noncentrality parameters \(\lambda_1\) and \(\lambda_2\), if it is defined as \(X = (U_1/n_1) / (U_2/n_2)\), where \(U_1\) and \(U_2\) are independent with \(U_i \sim \chi^2(n_i,\lambda_i)\), the \(n_i\) are the degrees of freedom, and the \(\lambda_i\) are the noncentrality parameters of the noncentral \(\chi^2\) distributions.

The moment generating function and the cumulant generating function of \(X\) do not exist. However, the CDF of \(X\) can also be computed by writing

\[\text{cdf}_X(x) = \text{Pr}(X \le x) = \text{Pr} \left( \frac{n_2}{n_1} U_1 - x U_2 \le 0 \right) = \text{Pr} \left( Y_x \le 0 \right),\]

where \(Y_x\) is the so-defined linear combination of \(U_1\) and \(U_2\), and then using the inversion methods for weighted sums. In the following equations, we use these definitions:

\(l_1 = n_2 / n_1\), \(l_2 = -x\), \(v_1 = 1 / (1 - 2 s l_1)\), \(v_2 = 1 / (1 - 2 s l_2)\), \(g_1 = l_1 v_1\), \(g_2 = l_2 v_2\), \(c_j=1\) for \(j=1\) and \(c_j = 2(j-1)c_{j-1}\) for \(j>1\).

\(K_{Y_x}(s)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_{Y_x}^{(j)}(s), j = 1 \ldots k\), of a random variable \(Y_x\), are defined as:

\[K_{Y_x}(s) = \tfrac{1}{2} n_1 \log(v_1) + \tfrac{1}{2} n_2 \log(v_2) + s (\lambda_1 g_1 + \lambda_2 g_2)\]

\[K_{Y_x}^{(j)}(s) = c_j \left( g_1^j \left( n_1 + j \lambda_1 v_1 \right) + g_2^j \left( n_2 + j \lambda_2 v_2 \right) \right).\]

saddlepoint central

Returns the solution \(\hat{s}\) of the saddlepoint equation \(K_{Y_x}^{(1)}(\hat{s})=0\), which can be obtained in closed form

\[\hat{s_0} = \frac{n_1(f-1)}{2f(n_1+n_2}\]

saddlepoint noncentral

Returns the solution \(\hat{s}\) of the saddlepoint equation \(K_{Y_x}^{(1)}(\hat{s})=0\), which can be obtained in closed form

\[\hat{s_1} = \left( x n_1(n_1+2n_2+\lambda)-n_1 n_2 - \sqrt{n_1 a_2}\right) / a_1, \quad \text{where } a_1 = 4n_2 x (n1+n2),\]

\[a_2 = x^2n_1^3+2x^2 n_1^2 \lambda + 2n_1^2 x n_2 + 4x^2 n_1n_2 \lambda + n_1 \lambda^2 x^2 + 2n_1 \lambda x n_2 + n_2^2 n_1 + 4x n_2^2 \lambda.\]

saddlepoint doubly noncentral

The solution \(\hat{s}\) of the saddlepoint equation \(K_{Y_x}^{(1)}(\hat{s})=0\), which can be obtained in closed form

\[\hat{s_2} = -2 p \cos \left(\tfrac{1}{3} \arccos(- \tfrac{1}{2}q p^{-3}) + \tfrac{1}{3} \pi \right ) - a_2, \quad \text{where }\]

\(p^2 = |3 a_2^2 - a_1| / 3\), \(q = a_2 (2 a_2^2 - a_1) + a_0,\) \(a = 8 x^2 n_2^2 (n_1 + n_2)\), \(a_0 = (x \lambda_2 n_1^2 - (1 - x) n_1^2 n_2 - n_1 n_2 \lambda_1) / a\),

\(a_1 = (2 (n_2^2 n_1 + n_1^2 n_2 x^2) - 4 x n_1 n_2 (n_1 + n_2 + \lambda_1 + \lambda_2)) / a\),

\(a_2 = (8 x (1 - x) n_1 n_2^2 + 4 x (n_2^3 + \lambda_2 n_2^2 - n_1^2 n_2 x - n_1 n_2 \lambda_1 x)) / (3 a)\).

While the PDF of \(X\) cannot be obtained directly from the above, it can be obtained from the CDF a follows: Let the density \(g_{m,n}\) be that of \(m/n\) times an \(F_{m,n}\) random variable. Let \(G_{m,n}(y)\) be its distribution function, and let \(g_{m,n}^{\lambda_1,\lambda_2}\) and \(G_{m,n}^{\lambda_1,\lambda_2}(y)\) be the density and distribution function of its doubly noncentral version (the distribution of \(\chi_m^2(\lambda_1)/\chi_n^2(\lambda_2)\)). Then the following recurrence relations hold citep{Chattamvelli_1995}

\[n\left[G_{m,n+2}^{\lambda}(y)-G_{m-2,n+2}^{\lambda}(y)\right] = -2g_{m,n}^{\lambda}(y)\]

ctx.fisher_f_nc2_spa_inv(q, m, n, lambda1, lambda2, results='cdf')

where ctx is fpm, mpm, ipm, dec, gmp or apm.

Calculates the inverse Jensen saddlepoint approximation of the qtf and isf.