Data driven smooth tests for composite hypotheses

Abstract

The classical problem of testing goodness-of-fit of a parametric family is reconsidered. A new test for this problem is proposed and investigated. The new test statistic is a combination of the smooth test statistic and Schwarz's selection rule. More precisely, as the sample size increases, an increasing family of exponential models describing departures from the null model is introduced and Schwarz's selection rule is presented to select among them. Schwarz's rule provides the "right" dimension given by the data, while the smooth test in the "right" dimension finishes the job. Theoretical properties of the selection rules are derived under null and alternative hypotheses. They imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative.

Data driven efficient score tests for Poissonity

“…. Easy calculations show that regularity conditions (R1)-(R4) in [5] for P are satisfied. So, when ψ j 's are twice differentiable and…”

“…for some positive c and nonnegative m, it follows from Theorem 3.1 in [5] that under H 0 , N k converges in distribution to the chi-square distribution with k degrees of freedom.…”

“…and F (y, λ) = y π 0 (λ) for y ∈ [0, 1). Hence, to test H 0 we can simply apply results of Inglot et al [5].…”

“…In the present note we propose data driven efficient score tests for testing Poissonity which are a direct application of the general construction of data driven goodness-of-fit tests for composite hypotheses studied in Inglot et al [5]. Our construction is valid for any family of discrete distributions concentrated on nonnegative integers.…”

Dependence and independence: Structure and inference

“…According to the modified Schwarz's rule (similar to the one presented in Inglot et al), 29 a ''diagonal'' test statistic TS2 ¼ T S2 can be obtained where the order is chosen as S2 ¼ arg min k fT k À k logðnÞ ! T j À j logðnÞ, 1 j, k d ðnÞg and d(n) is a sequence of numbers, d ðnÞ !…”