Some Aspects of Goodness-of-Fit Tests in the Presence of Unknown Parameters

Abstract

<p>Goodness-of-fit testing of simple hypotheses has a long, well known history. When the Null Hypothesis specifies only the form of the Null Cumulative Distribution Function (CDF), with values for one or more parameters unspecified, the problem is not so clear cut. This project examines several methods of testing fit in the presence of unknown parameters. The methods, briefly described below, are all based on the Empirical Distribution Function (EDF).</p> <p>(1) The unknown parameters are estimated from the sample. Modified EDF statistics using these sample estimates, are computed, and compared to the significance points which have been obtained by computer simulation.</p> <p>(2) The unknown parameters are estimated from the sample. Transformations are applied to the observations to obtain transformed variates which, under the Null Hypothesis, are distributed as dependent uniform variates. These transforms are tested for uniformity by the EDF statistics.</p> <p>(3) A series of transformations is applied to the data to obtain transformed variates which under the Null Hypothesis follow a completely specified Distribution Function; the nuisance parameters have been eliminated. This new, simple hypothesis is then tested by the EDF statistics.</p> <p>(4) For various parameter values throughout the parameter space, the EDF statistics are computed. A region in the parameter space for acceptance of the corresponding simple hypotheses is determined.</p>