Abstract
A class of pseudo distances is used to derive test statistics using transformed data or spacings for testing goodness-of-fit for parametric models. These statistics can be considered as density based statistics and expressible as simple functions of spacings. It is known that when the null hypothesis is simple, the statistics follow asymptotic normal distributions without unknown parameters. In this paper we emphasize results for the null composite hypothesis: the parameters can be estimated by a generalized spacing method (GSP) first which is equivalent to minimize a pseudo distance from the class which is considered; subsequently the estimated parameters are used to replace the parameters in the pseudo distance used for estimation; goodness-of-fit statistics for the composite hypothesis can be constructed and shown to have again an asymptotic normal distribution without unknown parameters. Since these statistics are related to a discrepancy measure, these tests can be shown to be consistent in general. Furthermore, due to the simplicity of these statistics and they come a no extra cost after fitting the model, they can be considered as alternative statistics to chi-square statistics which require a choice of intervals and statistics based on empirical distribution (EDF) using the original data with a complicated null distribution which might depend on the parametric family being considered and also might depend on the vector of true parameters but EDF tests might be more powerful against some specific models which are specified by the alternative hypothesis.
Highlights
Let X1, X n−1 be a sample of size n −1 from a continuous distributionF ∈{Fθ } and let X1 ≤ ≤ X n−1 be the order statistics and let the transformed ( ) data be defined as U= i (θ ) Fθ ( = Xi ),i 1, n −1, U(= i) (θ ) Fθ X= (i),i 1, n −1 ( ) ( ) and define Fθ X(n) = 1 and Fθ X(0) = 0
In this paper we emphasize results for the null composite hypothesis: the parameters can be estimated by a generalized spacing method (GSP) first which is equivalent to minimize a pseudo distance from the class which is considered; subsequently the estimated parameters are used to replace the parameters in the pseudo distance used for estimation; goodness-of-fit statistics for the composite hypothesis can be constructed and shown to have again an asymptotic normal distribution without unknown parameters
Due to the simplicity of these statistics and they come a no extra cost after fitting the model, they can be considered as alternative statistics to chi-square statistics which require a choice of intervals and statistics based on empirical distribution (EDF) using the original data with a complicated null distribution which might depend on the parametric family being considered and might depend on the vector of true parameters but EDF tests might be more powerful against some specific models which are specified by the alternative hypothesis
Summary
The approach used in this paper hopefully will unify estimation and model testing and facilitate the comparisons of these density based statistics with traditional EDF statistics and chi-square statistics which are more often used than these density based statistics We note that these statistics can be computed and their null asymptotic distribution is normal without unknown parameters which make it easy to use these statistics and comparing to the related chi-square statistics, these statistics do not need a choice of intervals and they come as by products when fitting models using the corresponding GSP methods.
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