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正依赖对分类数据卡方检验的影响AI 解读

SUMMARY We introduce a general definition of positive dependence for finite-state processes, and show that if successive observations are positively dependent, all tests that are asymptotically equivalent to standard Pearson chi-squared tests have asymptotic null distributions stochastically larger than those obtained under the usual independence assumptions. Ignoring positive dependence therefore leads to too frequent rejection of null hypotheses. This qualitative…

依赖性对卡方与经验分布拟合检验的影响AI 解读

Suppose that a test of fit to a parametric family of distributions is employed, with critical points determined from the limiting null distribution of the test statistic for IID observations. It is shown that if the observations are in fact a stationary process satisfying a positive dependence condition, the test will reject a true null hypothesis too often. This result is established for a broad class of chi squared and empiric df tests, including the Pearson,…

观测依赖对卡方拟合检验的影响AI 解读

Tests of fit that assume i.i.d. observations may be affected by dependence among the observations. This effect is studied for the Pearson chi squared test when the data form a stationary stochastic process. The general form of asymptotic distribution theory under the null hypothesis is outlined and examples are given. For testing fit to a specified normal law, it is shown that when observations come from a quite general class of Gaussian stationary processes, positive…

II 型删失数据的卡方拟合优度检验AI 解读

The theory of general chi-square statistics for testing fit to parametric families of distributions is extended to samples censored at sample quantiles. Data-dependent cells with sample quantiles as cell boundaries are employed. Asymptotic distribution theory is given for statistics in which unknown parameters are estimated by estimators asymptotically equivalent to linear combinations of functions of order statistics. Emphasis is placed on obtaining statistics having a…

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广义逆、Wald 方法与卡方拟合检验的构造AI 解读

Wald's method constructs test statistics having chi-squared limiting distributions from estimators having nonsingular multivariate normal limiting distributions, using the inverse of the covariance matrix of the limiting distribution. This method is here extended to singular multivariate normal limiting distributions by use of generalized inverses. The generalized Wald's method is then applied to give a quite general solution to the problem of constructing chi-squared type…

最近邻密度估计量的相合性性质AI 解读

Let $X_1, X_2,\cdots$ be $R^p$-valued random variables having unknown density function $f$. If $K$ is a density on the unit sphere in $R^p, \{k(n)\}$ a sequence of positive integers such that $k(n) \rightarrow \infty$ and $k(n) = o(n)$, and $R(k, z)$ is the distance from a point $z$ to the $k(n)$th nearest of $X_1,\cdots, X_n$, then $f_n(z) = (nR(k, z)^p)^{-1} \sum K((z - X_i)/R(k, z))$ is a nearest neighbor estimator of $f(z).$ When $K$ is the uniform kernel, $f_n$ is an…

一般卡方拟合检验的统一大样本理论AI 解读

We present a unified large-sample theory of general chi-squared tests of fit under composite null hypotheses and Pitman alternatives. The statistics are quadratic forms in the standardized cell frequencies, and we allow random cells, $k$-variate observations from not necessarily continuous distributions, and quite general estimates of unknown parameters. Generalizations of published results on a number of specific chi-squared tests follow.

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关于 Srinivasan 拟合优度检验的一则说明AI 解读

It is shown that for testing goodness of fit to the normal and exponontial families, a goodness-of-fit-test proposed by Srinivasan is asymptotically equivalent to a standard test of Kolmogorov-Smirnov type under the null hypothesis and under all nontrivial alternatives. This result appears to be true in general, though no proof is known.

关于 Srinivasan 拟合优度检验的札记AI 解读

It is shown that for testing goodness of fit to the normal and exponontial families, a goodness-of-fit-test proposed by Srinivasan is asymptotically equivalent to a standard test of Kolmogorov-Smirnov type under the null hypothesis and under all nontrivial alternatives. This result appears to be true in general, though no proof is known.

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