ON EDGEWORTH EXPANSIONS IN THE MIXTURE CASES
ON EDGEWORTH EXPANSIONS IN THE MIXTURE CASES
复制标题
混合情况下的Edgeworth展开
DOI：
10.1214/aos/1176347029
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
Kesar Singh
中科院分区：
文献类型：
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作者：
G. Babu;Kesar Singh
其中'表示转置。在格点的情况下，每个单变量总体将其整个质量分配给可数个等距点，展开是不同的，并且仅对于一类受限的集合A已知显式形式。考虑一个单变量统计量的形式H（Zn），对于光滑函数H。Bhattacharya和Ghosh（1978）使用Zn的展开式来建立Vn[H（Zn）- H（E（Zn））]的形式Edgeworth展开式在非格点情况下的有效性。在格的情况下，一个显式形式的扩展通常是不可用的非线性单变量统计。例如，参见Yarnold（1972），其中当A是椭球时，给出了Pn（A）的显式展开。本文考虑H（Zn）的Edgeworth展开式，当X为格点分布，Y为连续分布时.在统计实验中确实会出现这样的情况。作为一个简单的例子，X可以是植物或动物的年龄，Y可以是它的重量; H（Zn）= YJXn。在本文中，我们建立了在一项Edgeworth展开中X的晶格特征不需要校正因子。在一个任期之后，我们目前还不知道答案。我们建立上述事实的方法如下：设q是具有有限三阶矩的对称随机变量，其特征函数在紧区间外为零。它首先表明，在
where ' denotes transpose. In the lattice case, where each univariate population assigns its entire mass to countably many equidistant points, the expansion is different and an explicit form is known only for a restricted class of sets A. Consider an univariate statistic of the form H(Zn), for a smooth function H. Bhattacharya and Ghosh (1978) used the expansion for Zn to establish the validity of the formal Edgeworth expansion for Vn[H(Zn) - H(E(Zn))] in the nonlattice case. In the lattice case, an explicit form of the expansion is typically not available for nonlinear univariate statistics. See Yarnold (1972) for instance, where an explicit expansion for Pn(A) is given, when A is an ellipsoid. In this paper we consider Edgeworth expansions for H(Zn), when X has a lattice distribution and Y has a continuous distribution. Situations like this do arise in statistical experiments. As a simple example, X could be the age of a plant or animal in years and Y could be its weight; H(Zn) = YJXn. In this article we establish that no correction factor is needed for the lattice character of X in the one-term Edgeworth expansion. Beyond one term, the answer is not known to us at present. Our approach for establishing the above mentioned fact is as follows: Let q be a symmetric random variable having finite third moment and whose characteristic function vanishes outside a compact interval. It is first shown that in the