A Modified Kolmogorov-Smirnov, Anderson-Darling, and Cramer-Von Mises Test for the Gamma Distribution with Unknown Location and Scale Parameters.

Goodness-of-fit tests are developed for the gamma distribution when the scale and location parameters are unspecified and must be estimated from the sample data. The Anderson-Darling, Cramer-von Mises, and the Kolmogorov-Smirnov statistics are used to develop a new test of fit for the three-parameter gamma distribution with unknown shape and location parameters. The critical values generated were obtained by a Monte Carlo procedure. For each value of n (sample size), 5000 sample sets were drawn from a gamma population whose shape is specified. The location and scale parameters are estimated from the data, and the three statistics are calculated based on the estimated distribution. The simulation was preformed for sample sizes n = 5, 10, 15...30 and shape parameters, K = .5, 1.0, 1.5...4.0. Using gamma distributions for shape equal to 1.5 and 4.0, the power of each test is investigated against tne alternative distributions for sample sizes n = 5, 15, and 30. In general both the Anderson-Darling and the Cramer-von Mises tests are more powerful than the Kolmogorov-Smirnov test. Except for the case where the alternative distribution is lognormal, the Cramer-von Mises test is the most powerful test. The functional relationship between the critical values of the Anderson-Darling, Cramer-von Mises, and Kolmogorov-Smirnov is also examined. A critical value for a shape parameter between 1.5 and 4.0 which is not included in the tables can then be easily derived from this functional relationship.