Performs a chi-squared goodness-of-fit test.

The type double function is imsl_d_chi_squared_test.

The p-value for the goodness-of-fit chi-squared statistic.

The function imsl_f_chi_squared_test performs a chi-squared goodness-of-fit test that a random sample of observations is distributed according to a specified theoretical cumulative distribution. The theoretical distribution, which may be continuous, discrete, or a mixture of discrete and continuous distributions, is specified via the user-defined function user_proc_cdf. Because the user is allowed to give a range for the observations, a test conditional upon the specified range is performed.

Argument n_categories gives the number of intervals into which the observations are to be divided. By default, equiprobable intervals are computed by imsl_f_chi_squared_test, but intervals that are not equiprobable can be specified (through the use of optional argument IMSL_CUTPOINTS).

Regardless of the method used to obtain the cutpoints, the intervals are such that the lower endpoint is not included in the interval, while the upper endpoint is always included. If the cumulative distribution function has discrete elements, then user-provided cutpoints should always be used since imsl_f_chi_squared_test cannot determine the discrete elements in discrete distributions.

By default, the lower and upper endpoints of the first and last intervals are − ∞ and + ∞, respectively. If IMSL_BOUNDS is specified, the endpoints are defined by the user via the two arguments lower_bound and upper_bound.

A tally of counts is maintained for the observations in x as follows. If the cutpoints are specified by the user, the tally is made in the interval to which xi belongs using the endpoints specified by the user. If the cutpoints are determined by imsl_f_chi_squared_test, then the cumulative probability at xi, F(xi), is computed via the function user_proc_cdf. The tally for xi is made in interval number

If the expected count in any cell is less than 1, then a rule of thumb is that the chi-squared approximation may be suspect. A warning message to this effect is issued in this case, as well as when an expected value is less than 5.

On some platforms, imsl_f_chi_squared_test can evaluate the user-supplied function user_proc_cdf in parallel. This is done only if the function imsl_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables

The user must supply a function user_proc_cdf with calling sequence user_proc_cdf(y), that returns the value of the cumulative distribution function at any point y in the (optionally) specified range. Many of the cumulative distribution functions in Special Functions can be used for user_proc_cdf, either directly, if the calling sequence is correct, or indirectly, if, for example, the sample means and standard deviations are to be used in computing the theoretical cumulative distribution function.

This example illustrates the use of imsl_f_chi_squared_test on a randomly generated sample from the normal distribution. One-thousand randomly generated observations are tallied into 10 equiprobable intervals. The null hypothesis that the sample is from a normal distribution is specified by use of the imsl_f_normal_cdf (see Special Functions) as the hypothesized distribution function. In this example, the null hypothesis is not rejected.

In this example, some optional arguments are used for the data in the initial example.

In this example, a discrete Poisson random sample of size 1000 with parameter θ = 5.0 is generated via function imsl_f_random_poisson. In the call to imsl_f_chi_squared_test, function imsl_f_poisson_cdf is used as function user_proc_cdf.