Limit Distribution of the Mean Square Deviation of the Gasser–Müller Nonparametric Estimate of the Regression Function

Asymptotic distribution of the mean square deviation of the Gasser–Müller estimate of the regression curve is investigated. The testing of hypotheses on regression function is considered. The asymptotic behaviour of the power of the proposed criteria under contiguous alternatives is studed.

Conditional likelihood and power: higher-order asymptotics

This paper considers, in the presence of a nuisance parameter, a very large class of tests that includes the conditional and the usual versions of the likelihood ratio (LR), Rao’s and Wald’s tests. Under contiguous alternatives and orthogonal parametrization, the power functions of the conditional and the usual versions of these tests have been compared and, in particular, it is seen that the power functions of the conditional versions, unlike those of the usual versions, are identical, up to the second-order, with the power functions of the corresponding tests with known nuisance parameter. An optimality property of the conditional LR test, in terms of second-order local maximinity, has been established. A test, optimal in the sense of third-order average power under contiguous alternatives, has been proposed. A weaker optimality property of Rao’s test, in terms of third-order average power, has also been indicated.

Asymptotic distributions of weighted pontograms under contiguous alternatives

Let {N(x), x ≥ 0} be a non-homogeneous, also called non-stationary, Poisson process with density function λ(x), x ≥ 0. We consider the problem of testing the null hypothesis of λ(x) having a constant value against the alternative that λ(x) is a function of x.