In this paper we discuss testing the hypothesis that a random sample can be modeled by a gamma distribution for the case of a known or unknown shape parameter. We introduce a new test, one based on the modified Greenwood statistic, and compare this with two presently available tests, Anderson-Darling and Locke. In this paper we emphasize how to use each of the three tests, and, based on the results of a simulation study, we recommend under what conditions each should be used.
Keywords: Anderson Darling Test, Greenwood Statistic, Gamma Distribution, Probability Integral Transformation.
by SAMUEL S. SHAPIRO and LING CHEN, Florida International University, Miami, FL 33199
The gamma distribution is used in many areas of engineering, the sciences, and business. Such applications include queuing problems, reliability assessment, inventory control, computer evaluations, and biological studies where the time of occurrence of an event depends on the occurrence of a series of independent sub-events whose occurrence times are identically distributed exponential variables. When the gamma distribution has an integer shape parameter, it is sometimes called the Erlang distribution. In spite of the many uses of this model there have been few distributional assessment procedures developed. In this paper we describe procedures for testing the hypothesis that the sample was drawn from a gamma distribution with unknown scale parameter for the cases where the shape parameter is known or unknown. The shape parameter known situation arises when the number of sub-events needed to activate the occurrence is fixed and known. Thus when evaluating a computer buffer which batches six messages before transmitting them to the processor, the time between receipt of batches has a gamma distribution with a shape parameter of six if the times between message arrivals are independent, identically distributed (i.i.d.) exponential variates. The shape parameter unknown case would occur, for example, in the study of the lifetime of a component which fails after k shocks where the times between shocks are i.i.d. exponential variables and the value of k is unknown.
The purpose of this paper is two-fold. The first is to describe some procedures which are currently available, and the second is to introduce a new procedure which can be used to test for a gamma hypothesis. The situations where each of these procedures should be used are specified.
One of the current methods of evaluating the composite (shape and scale parameters unknown case) gamma hypothesis is due to Locke (1976) which makes use of the unique property of the gamma distribution that if X and Y are independent, non-degenerate random variables, then X/Y and X+Y are independent if and only if they are gamma distributed. Tests for the uniform distribution can be used to assess the gamma hypothesis if the parameters are known. (Such procedures are known as empirical distribution tests which include the Kolmogorov-Smirnov test (see Kolmogorov (1933) or D'Agostino and Stephens (1986)), Anderson-Darling test (1952), and the Greenwood statistic (1946)). These tests use the cumulative probability distribution function transformation to convert the data, when the null distribution is correct, to the uniform distribution and then test for the uniform distribution. When the parameters are known, percentiles of the test statistic do not depend on the null distribution; however, if the parameters are unknown and must be estimated, then these procedures must be revised. Some of the revised procedures can be found in D'Agostino and Stephens (1986). In this paper we present an adaptation of the Greenwood statistic for use when the shape and scale parameters are unknown.
In the second section three tests for the gamma distribution are described and in the third the new test is presented. The fourth section contains a power study comparing the test procedures, and in the fifth section we discuss some examples which illustrate use of the tests. The final section summarizes the results of the paper, and we make recommendations as to which situation each of the tests is best suited.
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