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

Routine CHIGF 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 a user‑defined FUNCTION. Because the user is allowed to specify a range for the observations, a test that is conditional upon the specified range is performed.

∣NCAT∣ gives the number of intervals into which the observations are to be divided. These intervals can be specified via the vector CUTP, which contains the cutpoints (or endpoints) for the intervals. Or if NCAT is negative, equiprobable intervals computed by CHIGF can be used. Regardless of the method used to obtain them, the intervals are such that the lower endpoint is not included in the interval while the upper endpoint is always included. The user should determine the cutpoints when the cumulative distribution function has discrete elements since CHIGF cannot determine them in this case. Regardless of how the cutpoints are determined, the lower endpoint of the first interval is specified by RNGE(1) when RNGE(1) ≠ RNGE(2) and is given as minus machine infinity otherwise. The upper endpoint of the last interval is defined similarly.

Routine CHIGF tallies the observations in X as follows. If the cutpoints are determined by CHIGF, then the cumulative probability at xi, F(xi), is computed via function CDF. The tally for xi is made in interval number ⌊mF (x) + 1⌋, where m = ∣NCAT∣ and ⌊ ⌋ is the function that takes the greatest integer that is no larger than the argument of the function. 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. Thus, if the computer time required to calculate the cumulative distribution function is large, user‑specified cutpoints may be preferred in order to reduce the total computing time.

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.

The user must supply a function CDF with calling sequence CDF(Y), which returns the value of the cumulative distribution function at any point Y in the range of the distribution. The supplied function must be declared in an EXTERNAL statement in the calling program. Many of the IMSL cumulative distribution functions in Chapter 17, “Probability Distribution Functions and Inverses” can be used for 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 CDF.

Informational errors

In this example, a discrete binomial random sample of size 1000 with binomial parameter p = 0.3 and binomial sample size 5 is generated via routine RNBIN (see Chapter 18, “Random Number Generation”). Routine RNSET is first used to set the seed. One call to CHIGF is made. Routine BINDF (see Chapter 17, “Probability Distribution Functions and Inverses”) is used to compute the CDF.

This example illustrates the use of CHIGF on a randomly generated sample from the normal distribution. One thousand randomly generated observations are tallied into 10 equiprobable intervals. Twelve calls to CHIGF are made. The first call is solely for initialization since IDO = 1 and NROW = 0. The next 10 calls tally the data, 100 observations at a time, with IDO = 2 and NROW = 100. The last call is for wrap up only since IDO = 3 and NROW = 0. All twelve calls could have been replaced with one call to CHIGF with IDO = 0 and NROW = 1000. X would need to be of length 1000 if one call were used. In this example, the null hypothesis is not rejected.