The gradient test. Another likelihood-based test (Q2798789)

The monograph deals with the so-called gradient test, which is a recently developed parametric large-sample test. The gradient statistic \(S_T\) for testing a point null hypothesis \(H_0:\theta=\theta_0\) against the alternative \(H_a:\theta\neq\theta_0\) for a \(p\)-dimensional parameter \(\theta\) was introduced in 2002 by George R. Terrell in the Proceedings of the 34th Symposium on the Interface of Computing Science And Statistics. Assuming an independent and identically distributed sample \(X_1=x_1,\dots,X_n=x_n\) from the distribution with parametric probability density function \(f(\cdot;\theta)\), where \(\theta\) is unknown, \(S_T\) is given by NEWLINE\[NEWLINE S_T =U(\theta_0)^\top \big(\hat{\theta}-\theta_0\big). NEWLINE\]NEWLINE In this, \(U\) denotes the score function associated with this parametric statistical model, and \(\hat{\theta}\) denotes the maximum likelihood estimator of \(\theta\) based on \(X_1 = x_1, \dots, X_n = x_n\).NEWLINENEWLINEIt can be shown that \(S_T\) has the same asymptotic (\(n \to \infty\)) null distribution as the traditionally used likelihood ratio, score, and Wald test statistics, respectively, which is the central chi-square distribution with \(p\) degrees of freedom. However, carrying out the asymptotic chi-square test as a gradient test based on \(S_T\) is computationally less involved, because it avoids the computation or the estimation, respectively, of the Fisher information matrix of the model.NEWLINENEWLINEChapter 1 introduces the general concept of the gradient test, with particular consideration of two specific lifetime models with potentially censored observations. In Chapter 2, the local power of the gradient test is investigated under various models by means of Edgeworth-type asymptotic expansions. Chapter 3 deals with a Bartlett-type corrected version of \(S_T\) in order to reduce the asymptotic order of the error of the chi-square approximation of the null distribution of the gradient test. Finally, Chapters 4 and 5 are concerned with robust modifications of \(S_T\) in order to cope with model misspecification and sample contamination.