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IMSL_CHISQCDF

# IMSL_CHISQCDF

The IMSL_CHISQCDF function evaluates the chi-squared distribution or non-central chi-squared distribution. Using a keyword the inverse of these distributions can be computed.

## If Two Input Arguments Are Used

The IMSL_CHISQCDF function evaluates the distribution function, F, of a chi-squared random variable with Î½ = df. Then:

where Î(Â·) is the gamma function. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

For Î½ > 65, IMSL_CHISQCDF uses the Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.17) to the normal distribution, and IMSL_NORMALCDF function is used to evaluate the normal distribution function.

For Î½ â¤ 65, IMSL_CHISQCDF uses series expansions to evaluate the distribution function. If x < max(Î½ / 2, 26), IMSL_CHISQCDF uses the series 6.5.29 in Abramowitz and Stegun (1964); otherwise, it uses the asymptotic expansion 6.5.32 in Abramowitz and Stegun.

If INVERSEis specified, the IMSL_CHISQCDF function evaluates the inverse distribution function of a chi-squared random variable with Î½ = df and with probability p. That is, it determines x, such that:

where Î(Â·) is the gamma function. The probability that the random variable takes a value less than or equal to x is p.

For Î½ < 40, IMSL_CHISQCDF uses bisection (if Î½ â¤ 2 or p > 0.98) or regula falsi to find the expression for which the chi-squared distribution function is equal to p.

For 40 â¤ Î½ < 100, a modified Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.18) to the normal distribution is used. The IMSL_NORMALCDF function is used to evaluate the inverse of the normal distribution function. For Î½ â¥ 100, the ordinary Wilson-Hilferty approximation (Abramowitz and Stegun 1964, Equation 26.4.17) is used.

## If Three Input Arguments Are Used

The IMSL_CHISQCDF function evaluates the distribution function of a non-central chi-squared random variable with df degrees of freedom and non-centrality parameter delta, that is, with v = df, Î» = delta, and x = chisq:

where Î(Â·) is the gamma function. This is a series of central chi-squared distribution functions with Poisson weights. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

The non-central chi-squared random variable can be defined by the distribution function above, or alternatively and equivalently, as the sum of squares of independent normal random variables. If Yi have independent normal distributions with means Âµi and variances equal to one and:

then X has a non-central chi-squared distribution with n degrees of freedom and non-centrality parameter equal to:

With a non-centrality parameter of zero, the non-central chi-squared distribution is the same as the chi-squared distribution.

The IMSL_CHISQCDF function determines the point at which the Poisson weight is greatest, and then sums forward and backward from that point, terminating when the additional terms are sufficiently small or when a maximum of 1000 terms have been accumulated. The recurrence relation 26.4.8 of Abramowitz and Stegun (1964) is used to speed the evaluation of the central chi-squared distribution functions.

If INVERSE is specified, IMSL_CHISQCDF evaluates the inverse distribution function of a non-central chi-squared random variable with df degrees of freedom and non-centrality parameter delta; that is, with P = chisq, v = df, and Î» = delta, it determines c0 (= IMSL_CHISQCDF(chisq, df, delta)), such that:

where Î(Â·) is the gamma function. The probability that the random variable takes a value less than or equal to c0 is P.