An Automatic Formula for the Second-Order Approximation of the Distributions of Test Statistics under Contiguous Alternatives
本文推广了LeCam第三引理，利用二阶Edgeworth近似，为邻近备择下检验统计量的分布给出自动公式，并应用于局部功效比较，适用于时间序列和非同分布情形。
Asymptotic distributions of test statistics under the alternative are important from the point of view of their power properties. When the limiting distributions of test statistics are specified under the hypothesis in a certain sense, LeCam's third lemma enables us to give their limiting distributions under contiguous alternatives. In this paper we generalize LeCam's third lemma by using the second-order Edgeworth approximation. For a class 3 of test statistics, this enables us to give an automatic formula of the second-order Edgeworth expansion of the distribution of any test statistic u1 E E under contiguous alternatives if the second-order asymptotic distribution of u, is specified under the hypothesis. We apply this automatic formula to the second-order local power comparisons in F. Then we get a unified result if we modify ul E a to be second-order asymptotically unbiased. Although our results are applicable to i.i.d. case, the main intention lies in time series analysis and non-identically distributed case. Two examples will be given. One is a Gaussian ARMA process, and the other is a nonlinear regression model.