Boost: Negative binomial distribution

Boost: Negative binomial distribution#

The following functions return the pmf, cdf, qtf or boost class of the negative binomial distribution with target for number of successful trials \(r > 0\) and success probability \(0 \le p \le 1\), and \(0 \le q \le 1\).

See also Wikipedia [1280], MathWorld [905], BoostMath [89], Ehrhardt [309] (3.9.23).

Ctx.negbinomial_pmf(k, r, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pmf}(x)\), the value of the probability mass function (Pmf) of the negative binomial distribution:

\[\text{pmf}(x) = \frac{\Gamma(k+r)}{k! \Gamma(r)} p^r (1-p)^k.\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("NegbinomPdf(x, a, b): ", NegbinomPdf(x, a, b))
>>> print ("dist_negbinomial(a, b).pdf(x): ", dist_negbinomial(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.negbinomial_cdf(k, r, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the negative binomial distribution:

\[\text{cdf}(x) = \sum_{j=0}^{k} \text{pmf}_X(j) = \text{ibeta}(r, k+1, p).\]

Here \(\text{ibeta}(\cdot)\) denotes the real normalised incomplete beta function (RealIBeta).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("NegbinomCdf(x, a, b): ", NegbinomCdf(x, a, b))
>>> print ("dist_negbinomial(a, b).cdf(x): ", dist_negbinomial(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.negbinomial_qtf(q, r, p)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the negative binomial distribution:

\[\text{qtf}(q) = \mathrm{ibeta\_invb}(r, p, q) - 1.\]

Here \(\mathrm{ibeta\_invb}(\cdot)\) denotes the inverse (on parameter b) of the real normalised incomplete beta function (RealIBetaInvb).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("NegbinomQtf(q, a, b): ", NegbinomQtf(q, a, b))
>>> print ("dist_negbinomial(a, b).qtf(q): ", dist_negbinomial(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_negbinom(r, p)#

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

The negative binomial distribution is a discrete (lattice) probability distribution with target for number of successful trials \(r > 0\) and success probability \(0 \le p \le 1\). See also Wikipedia [1280], MathWorld [905], BoostMath [89] , and Witkovský [1641], R (Statistical System) [559].

dist_negbinom.pmf(x)#

Returns \(\text{pmf}_X(x)\), the probability mass function (pmf) of a random variable \(X\), following an negative binomial distribution:

\[\text{pmf}_X(x) = \frac{\Gamma(n+k)}{K! \Gamma(n)} P^n (1-P)^k.\]

and \(f_{\text{Beta}}(\cdot)\) and \(F_{\text{Beta}}(\cdot)\) denote the PDF and CDF, respectively, of the central beta distribution. The following recursions are used for the PMF:

\[\text{Pr}(X=k+1 |n) = \frac{(n+k) (1-P)}{(k+1) } \text{Pr}(X=k |n)\]

\[\text{Pr}(X=k-1 |n) = \frac{k}{(n-k+1)(1-P)} \text{Pr}(X=k |n)\]

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

dist_negbinom.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an negative binomial distribution:

\[\text{cdf}_X(x) = \sum_{j=0}^{k} \text{pmf}_X(j) = F_{\text{Beta}}(1-p; r,k+1).\]

and \(f_{\text{Beta}}(\cdot)\) and \(F_{\text{Beta}}(\cdot)\) denote the PDF and CDF, respectively, of the central beta distribution.

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

dist_negbinom.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following an negative binomial distribution:

\[\text{sf}_X(x) = 1 - \text{cdf}_X(x).\]

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

dist_negbinom.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an negative binomial distribution:

\[\text{qtf}_X(q) = a-b \: \text{ln}\left((1-y)/y\right).\]

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

dist_negbinom.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following an negative binomial 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: ", negative_binomial(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_negbinom.g_x(t)#

Returns \(G_X(t)\), the probability generating function of a random variable \(X\), following an negative binomial distribution:

\[G_X(t) = \left(\frac{1}{P} - \frac{1-P}{P} t \right)^{-n}.\]

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

dist_negbinom.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an negative binomial distribution:

\[C_X(t) = \left(\frac{1}{P} - \frac{1-P}{P} e^{it} \right)^{-n}.\]

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

dist_negbinom.m_x(t)#

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an negative binomial distribution:

\[M_X(t) = \left(\frac{1}{P} - \frac{1-P}{P} e^{t} \right)^{-n}.\]

\[L_X(t) = \left(\frac{1}{P} - \frac{1-P}{P} e^{t} \right)^{-n}.\]

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

dist_negbinom.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an negative binomial distribution:

\[K_X(t) = -n \log \left(\frac{1}{P} - \frac{1-P}{P} e^t \right).\]

\(K_X(t)\), the cumulant generating function, and its \(j^{\text{th}}\) derivatives, \(K_X^{(j)}(t), j = 1 \ldots k\), of a random variable \(X\), following a negative binomial distribution, are defined as

\[K_X(t) = r \log \left(\frac{1-p}{1-p \cdot e^t} \right), \quad \text{for } t < -\log(p)\]

\[K_X^{(1)}(t) = \frac{p \cdot r \cdot e^x}{1-p \cdot e^x},\]

\[K_X^{(2)}(t) = \frac{p \cdot r \cdot e^x}{(1-p \cdot e^x)^2},\]

\[K_X^{(3)}(t) = \frac{p \cdot r \cdot e^x (p \cdot e^x +1)}{(1-p \cdot e^x)^3},\]

and for \(j \ge 4\) the derivatives are calculated by numerically differentiating \(K_X^{(3)}(t)\).

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

dist_negbinom.moments(k)#

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an negative binomial distribution (Wikipedia). The raw moments are calculated from the central moments.

\[\mu^{}_X(r) = \mu_2 \sum_{i=0}^{r-2} \binom{r-1}{i} \mu_i - \frac{1-P}{P} \sum_{i=0}^{r-2} \binom{r-1}{i} \mu_{i+1}, \quad r>2; \quad \mu_1 = n \frac{1-P}{P}, \quad \mu_2 = \frac{\mu_1}{P}.\]

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

dist_negbinom.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an negative binomial distribution. The cumulants are calculated from the moments.

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

Approximations

ctx.negbinom_ecf(k, r, p, results='cdf')#

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

Calculates the Edgeworth approximation to the pdf, cdf and sf

ctx.negbinom_ecf_inv(q, r, p, results='qtf')#

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

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

ctx.negbinom_spa(k, r, p, results='c')#

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

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

The saddlepoint is given by:

\[\hat{s}(x)= \log \left( \frac{k (1-P)}{(n-k) P} \right).\]

ctx.negbinom_spa_inv(q, r, p, results='qtf')#

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

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

Boost: Negative binomial distribution

Contents

Boost: Negative binomial distribution#