naginterfaces.library.univar.ci_​binomial

naginterfaces.library.univar.ci_binomial(n, k, clevel)

ci_binomial computes a confidence interval for the parameter $p$ (the probability of a success) of a binomial distribution.

For full information please refer to the NAG Library document for g07aa

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g07/g07aaf.html

Parameters

nint

$n$, the number of trials.

kint

$k$, the number of successes.

clevelfloat

The confidence level, $(1-\alpha )$, for two-sided interval estimate. For example $clevel=0.95$ will give a $95\%$ confidence interval.

Returns

plfloat

The lower limit, $p_l$, of the confidence interval.

pufloat

The upper limit, $p_u$, of the confidence interval.

Raises

NagValueError

(errno $1$)

On entry, $clevel=⟨value⟩$.

Constraint: $0.0<clevel<1.0$.

(errno $1$)

On entry, $n=⟨value⟩$ and $k=⟨value⟩$.

Constraint: $n\ge k$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

When using the relationship with the gamma distribution the series to calculate the gamma probabilities has failed to converge.

Notes

Given the number of trials, $n$, and the number of successes, $k$, this function computes a $100(1-\alpha )\%$ confidence interval for $p$, the probability parameter of a binomial distribution with probability function,

$$\begin{array}{c}f\left(x\right)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{(1-p)}^{n-x}\text{,}x=0,1,\dots ,n\text{,}\end{array}$$

where $\alpha$ is in the interval $(0,1)$.

Let the confidence interval be denoted by [$p_l,p_u$].

The point estimate for $p$ is $\hat{p}=k/n$.

The lower and upper confidence limits $p_l$ and $p_u$ are estimated by the solutions to the equations;

$$\begin{array}{c}\sum _{x=k}^n\left(\begin{array}{c}n\\ x\end{array}\right)p_l^x{(1-p_l)}^{n-x}=\alpha /2\text{,}\end{array}$$

$$\begin{array}{c}\sum _{x=0}^k\left(\begin{array}{c}n\\ x\end{array}\right)p_u^x{(1-p_u)}^{n-x}=\alpha /2\text{.}\end{array}$$

Three different methods are used depending on the number of trials, $n$, and the number of successes, $k$.

1. If $max(k,n-k)<{10}^6$.

   The relationship between the beta and binomial distributions (see page 38 of Hastings and Peacock (1975)) is used to derive the equivalent equations,

   $$\begin{array}{c}\begin{array}{ccc}{p}_l& =& {\beta }_{k,n-k+1,\alpha /2}\text{,}\\ & & \\ p_u& =& {\beta }_{k+1,n-k,1-\alpha /2}\text{,}\end{array}\end{array}$$

   where ${\beta }_{a,b,\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the beta distribution with parameters $a$ and $b$. These beta deviates are computed using stat.inv_cdf_beta.

2. If $max(k,n-k)\ge {10}^6$ and $min(k,n-k)\le 1000$.

   The binomial variate with parameters $n$ and $p$ is approximated by a Poisson variate with mean $np$, see page 38 of Hastings and Peacock (1975).

   The relationship between the Poisson and ${\chi }^2$-distributions (see page 112 of Hastings and Peacock (1975)) is used to derive the following equations;

   $$\begin{array}{c}\begin{array}{ccc}{p}_l& =& \frac{1}{2n}{\chi }_{2k,\alpha /2}^2\text{,}\\ & & \\ p_u& =& \frac{1}{2n}{\chi }_{2k+2,1-\alpha /2}^2\text{,}\end{array}\end{array}$$

   where ${\chi }_{\delta ,\nu }^2$ is the deviate associated with the lower tail probability, $\delta$, of the ${\chi }^2$-distribution with $\nu$ degrees of freedom.

   In turn the relationship between the ${\chi }^2$-distribution and the gamma distribution (see page 70 of Hastings and Peacock (1975)) yields the following equivalent equations;

   $$\begin{array}{c}\begin{array}{ccc}{p}_l& =& \frac{1}{2n}{\gamma }_{k,2\text{;}\alpha /2}\text{,}\\ & & \\ p_u& =& \frac{1}{2n}{\gamma }_{k+1,2\text{;}1-\alpha /2}\text{,}\end{array}\end{array}$$

   where ${\gamma }_{\alpha ,\beta \text{;}\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$. These deviates are computed using stat.inv_cdf_gamma.

3. If $max(k,n-k)>{10}^6$ and $min(k,n-k)>1000$.

   The binomial variate with parameters $n$ and $p$ is approximated by a Normal variate with mean $np$ and variance $np(1-p)$, see page 38 of Hastings and Peacock (1975).

   The approximate lower and upper confidence limits $p_l$ and $p_u$ are the solutions to the equations;

   $$\begin{array}{c}\begin{array}{ccc}\frac{k-n{p}_l}{\sqrt{n{p}_l(1-p_l)}}& =& z_{1-\alpha /2}\text{,}\\ & & \\ & & \\ \frac{k-n{p}_u}{\sqrt{n{p}_u(1-p_u)}}& =& z_{\alpha /2}\text{,}\end{array}\end{array}$$

   where $z_{\delta }$ is the deviate associated with the lower tail probability, $\delta$, of the standard Normal distribution. These equations are solved using a quadratic equation solver (zeros.quadratic_real).

References

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth

Snedecor, G W and Cochran, W G, 1967, Statistical Methods, Iowa State University Press