Distribuzione asintotica della statistica V2 nel caso di stime non standard

Abstract

The asymptotic distribution of the godness-of-fit test statistic V², recently proposed by Dancelli (1993), is derived when estimators other than the standard maximum likelihood are employed. We prove that in these cases, under the null hypothesis, V² is non asymptotically distributed as a Chi-Square variate, but his distribution is that of a weighted sum of independent Chi-Square variates, the weights being represented by the non negative eingevalues of a particular matrix depending by the unknown parameters. A similar result was derived by Molinari (1977) for the classical Pearson's X². Numerical examples are also carried out with the log normal model and the method of moments estimator in order to evaluate the actual size of the V² and X² tests when referring to the usual X² tables. The results indicate that, when the number of the class intervals is sufficiently high (15 or more), the bias for the V² test is not so serious then that of the X² test.