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基于多元 M 估计的鞍点近似与检验AI 解读

We consider multidimensional M-functional parameters defined by expectations of score functions associated with multivariate M-estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters. We propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the M-estimates. This statistic is analogous to the log likelihood ratio in the parametric case. We show that this statistic is…

经验鞍点与自助法尾部近似的小相对误差概率界AI 解读

To obtain test probabilities based on empirical approximations to the distribution of a Studentized function of a mean, we need the approximations to be accurate with sufficiently high probability. In particular, when these test probabilities are small it is best to consider relative errors. Here we show that in the case of univariate standardized means and in the general case of tests based on smooth functions of means, the empirical approximations have asymptotically small…

自助法鞍点近似AI 解读

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors.

自助法鞍点近似AI 解读

We compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation. We show that, on bounded sets, these empirical saddlepoint approximations achieve second order relative errors uniformly. We also consider the relative errors for larger deviations. It follows that the studentized-t bootstrap p-value and the coverage of the bootstrap confidence interval have second order relative errors.

简单随机抽样下比率估计的条件化调整AI 解读

The usual ratio estimates for a population mean use information on an auxiliary variable to improve the estimation. The standard randomization analysis, however, does not take enough account of the observations on the auxiliary variable, so estimates and estimated variates can be conditionally biased unless the sample is balanced with respect to the auxiliary variable. I propose a method of adjustment for the bias of the estimate and its variance based only on the assumption…

简单随机抽样下比率估计的条件化调整AI 解读

The usual ratio estimates for a population mean use information on an auxiliary variable to improve the estimation. The standard randomization analysis, however, does not take enough account of the observations on the auxiliary variable, so estimates and estimated variates can be conditionally biased unless the sample is balanced with respect to the auxiliary variable. I propose a method of adjustment for the bias of the estimate and its variance based only on the assumption…

卡方与似然比拟合优度检验的效率AI 解读

The classical problem of choice of number of classes in testing goodness of fit is considered for a class of alternatives, for the chi-square and likelihood ratio statistics. Pitman and Bahadur efficiencies are used to compare the two statistics and also to analyse the effect for each statistic of changing the number of classes for the case where the number of classes increases asymptotically with the number of observations. Overall, the results suggest that if the class of…

置换检验功效的 Edgeworth 展开AI 解读

A randomization model for a two-sample situation with additive treatment effects is considered. Edgeworth expansions for the power of the usual permutation test are derived, under some conditions on the unit errors, from previously obtained expansions under the null hypothesis of no treatment effect. A general error structure is considered and conditions for the validity of the expansions for both conditional and unconditional power are examined. The results are shown to…

置换检验与置信区间的鞍点近似AI 解读

Summary Asymptotic approximations for the tail probabilities of certain statistics with complex moment generating functions are obtained by the usual methods of large deviation theory. It is pointed out that this approximation should be adequate over the entire range but of particular use for extreme tails. Approximations to the significance levels for one and two sample permutation tests are obtained and these tests are inverted to obtain approximate confidence intervals.…

多样本置换检验的渐近展开AI 解读

Let $V_n$ be the standardized sum of squares of the means of $r + 1$ random samples of sizes $s_0, s_1, \cdots, s_r$, where $n = s_0 + s_1 + \cdots + s_r$, taken without replacement from $n$ numbers. Then using an approximation to the characteristic function of the means, an asymptotic expansion is obtained for the distribution of $V_n$ with first term being the distribution function of $\chi^2_r$ and with error of approximation generally of smaller order than $1/n$. When…

有限总体抽样分布的渐近展开AI 解读

An asymptotic expansion is obtained for the distribution function of the standardized mean of a sample of $s$ observations taken randomly without replacement from a finite population of $n$ numbers. The expansion is given to order $1/n$ and agrees with the formal Edgeworth expansion. The proof of the result is obtained using an approximation to the characteristic function of the standardized sum.

随机化模型下的协方差分析AI 解读

Summary The randomization model for the analysis of covariance in a randomized block design is considered. Asymptotic properties of tests and estimates are obtained and it is shown that if the number of replicates is large, then the usual results concerning tests are approximately true. The large sample power of the tests are obtained and it is shown that using the concomitant variate increases the power.