Empirical Saddlepoint Approximations for Multivariate M-estimators
Empirical Saddlepoint Approximations for Multivariate M-estimators
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多元 M 估计量的经验鞍点近似
DOI：
10.1111/j.2517-6161.1994.tb01980.x
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发表时间：
1994
期刊：
影响因子：
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通讯作者：
A. Welsh
中科院分区：
文献类型：
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作者：
E. Ronchetti;A. Welsh
在本文中，我们研究了用经验分布函数代替底层分布函数F来构造一般多元m估计量的密度fn的经验鞍点近似。我们得到了近似中误差项的显式形式，研究了估计量重整化的效果，进行了数值比较，并讨论了回归问题。统计量的抽样分布是评价统计量性质和构造基于统计量的推理程序的基本工具。抽样分布的密度可用于计算矩，发展近似值或计算分布函数，因此可用于构造检验或置信区间的各种尾部概率。当精确的密度难以处理时，可以用正态密度、Edgeworth展开或鞍点近似来近似（Daniels, 1954）。鞍点近似是由渐近展开式推导出来的，但即使在很小的样本中也常常是非常精确的。为了强调这一性质，Hampel（1973）将这些和相关技术描述为小样本渐近逼近。从后面的式(3)可以看出，这种近似的主要性质和它相对于Edgeworth展开式的主要优点是它总是非负的，相对误差一致为n-1阶。最近的评论见Reid (1988), Barndorff-Nielsen and Cox（1989）和Field and Ronchetti（1990）。这些近似的构造通常需要了解底层分布。然而，正如我们可以估计正态近似的方差或估计Edgeworth展开式中的项一样，我们也可以估计鞍点近似。在本文中，我们研究了鞍点近似估计的性质。我们的调查是出于三个考虑。首先，估计的近似对于比较给定数据集上的各种估计可能是有用的。如果抽样分布是非正态分布，至少在原则上，我们需要对整个分布进行比较，仅根据估计的渐近方差进行比较可能会产生误导。其次，估计近似值可用于评价包括正态近似值在内的较简单近似值的质量。最后，它可能对开发推理过程有用。方法如
SUMMARY In this paper, we investigate the use of the empirical distribution function in place of the underlying distribution function F to construct an empirical saddlepoint approximation to the density fn of a general multivariate M-estimator. We obtain an explicit form for the error term in the approximation, investigate the effect of renormalizing the estimator, carry out some numerical comparisons and discuss the regression problem. The sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic. The density of the sampling distribution may be used to compute moments, to develop approximations or to compute the distribution function and hence various tail prob- abilities which can be used to construct tests or confidence intervals. When the exact density is intractable, it may be possible to approximate it by a normal density, an Edgeworth expansion or a saddlepoint approximation (Daniels, 1954). The saddle- point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples. To highlight this property, Hampel (1973) described these and related techniques as small sample asymptotic approximations. As can be seen from equation (3) later, the main property of this approximation and its major advantage over Edgeworth expansions is that it is always non-negative and the relative error is uniformly of order n-1. For recent reviews see Reid (1988), Barndorff-Nielsen and Cox (1989) and Field and Ronchetti (1990). The construction of these approximations typically requires knowledge of the underlying distribution. However, just as we can estimate the variance of a normal approxima- tion or estimate the terms in an Edgeworth expansion, we can estimate the saddle- point approximation. In this paper, we examine the properties of an estimate of the saddlepoint approximation. Our investigation is motivated by three considerations. Firstly, the estimated approximation may be useful for comparing various estimators on a given data set. If the sampling distribution is non-normal, at least in principle, we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading. Secondly, the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima- tion. Finally, it may be useful for developing inference procedures. Methods like