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经验分布在凸集类上的一致收敛
The empirical distribution function $P_n$ converges with probability 1 to a true distribution $P$ in $R^k$, uniformly over measurable convex sets, if and only if $P$ is a countable mixture of distributions, each of which is carried by a flat and gives zero probability to the relative boundaries of convex sets included in the flat.
无矩条件下停止时间指数有界的一般定理及应用
The stopping time $N$ of a sequential test based on an i.i.d. sequence $X, X_1, X_2, \cdots$ with common distribution $P$ is defined as the first integer $n \geqq 1$ such that $l_1 < L_n < l_2$ is violated, where $L_n$ is a statistic depending only on $X_1, \cdots, X_n$. If for all $l_1, l_2$ there exist constants $c > 0, \rho < 1$ such that $P(N > n) < c\rho^n, n = 1, 2, \cdots,$ then $N$ is called exponentially bounded under $P$. In the contrary case $P$ is termed…
以预测分布函数误差最小化估计指数分布分位数
We consider the problem of estimating the quantiles of a one-parameter exponential distribution. Using the estimated quantiles to predict the distribution function, and attempting to minimize mean square errors for these predicted probabilities rather than the quantiles themselves, we are led to estimators differing from the natural ones obtained by appropriate scaling of the estimated mean of the distribution. The form of these estimators is studied for complete and…
以预测分布函数误差最小化估计指数分布分位数
We consider the problem of estimating the quantiles of a one-parameter exponential distribution. Using the estimated quantiles to predict the distribution function, and attempting to minimize mean square errors for these predicted probabilities rather than the quantiles themselves, we are led to estimators differing from the natural ones obtained by appropriate scaling of the estimated mean of the distribution. The form of these estimators is studied for complete and…
极值分布的拟合优度检验
In this paper we present goodness of fit tests for the extreme value distribution, based on the empirical distribution function statistics W2, U2and A2. Asymptotic percentage points are given for each of the three statistics, for the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples.
关于 Wilks 分位数区间的分布无关置信区间
This paper investigates the distribution-free outer confidence interval for the quantile interval introduced by Wilks. Besides exact expressions and a recurrence formula some bounds are derived for the probability of correct coverage of the quantile interval which improve bounds known up to now, and which are tested in several examples. Asymptotic approximations are discussed and applications of the developed methods to other statistics based on order statistics are…
多元容斥公式与相依随机变量的次序统计量
A variant of the general multivariate inclusion-exclusion formula of Meyer (1969) is derived for the case where $K$ classes of events are considered and specific subsets of the events, one from each class, are related to one another by set inclusion. This result, in turn, yields a formula for the cumulative distribution function of any subset of order statistics from dependent random variables in terms of cumulative distribution functions of subsets of the unordered…
凹回归中的一致性
For each $t$ in some subinterval $T$ of the real line let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is concave. Let $t_1, t_2, \cdots$ be a sequence of points in $T$ and let $Y_1, Y_2, \cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. We consider estimators $m_n(t) = m_n(t; Y_1, \cdots, Y_n)$ which are concave in $t$ and which minimize $\sum^n_{i=1} \lbrack m_n(t_i; Y_1, \cdots, Y_n) - Y_i\rbrack^2$…
杜宾–沃森检验统计量分布函数的计算算法
IN REGRESSION ANALYSIS most empirical economists use the well-known Durbin-Watson (DW) procedure [1] to test the hypothesis of no autocorrelation among the disturbances of a linear regression model against the hypothesis of a first-order autocorrelation. The use of this procedure is compromised by the fact that it is a bounds text and, hence, cannot discriminate between the two competing hypotheses for a range of intermediate values of the test statistic. This shortcoming…
两种替代结构t统计量的分布
Two alternative structural t statistics for simultaneous equation models are derived. The statistics could be used to test hypotheses or construct confidence intervals for coefficients in econometric models. It is demonstrated that the exact distribution functions converge to Student's t-distribution function as the concentration parameter increases. Computations of the variance of the t-statistics indicate the degree of approximation afforded by the limiting distribution…
随机排序约束下的无分布检验
A two-sample distribution-free procedure is proposed for testing equal locations under a stochastic ordering restriction. The test statistic M is an estimator of P(X ≤ Y) based upon maximum likelihood estimators of stochastically ordered distribution functions. Some properties of the M test are developed and critical values are provided for selected sample sizes. A Monte Carlo power study indicates that the M test is more effective (for shift alternatives) than the…
利用删失数据构造均值的非参数置信区间
Summary General Kolmogorov-type non-parametric confidence contours for an unknown continuous distribution function are converted into non-parametric confidence intervals for the mean of the distribution. It is shown how these estimation procedures may be modified to enable censored data to be handled.
任意分组、删失与截断数据的经验分布函数
Summary This paper is concerned with the non-parametric estimation of a distribution function F, when the data are incomplete due to grouping, censoring and/or truncation. Using the idea of self-consistency, a simple algorithm is constructed and shown to converge monotonically to yield a maximum likelihood estimate of F. An application to hypothesis testing is indicated.
若干收入不平等测度的大样本理论
THE MEASUREMENT of economic inequality is a timely and important topic. Often the Gini index or the entire Lorenz curve is used; however, the relative mean deviation (or Pietra ratio) has been used by Schutz [9] and Budd [1] to study United States data. Eltet6 and Frigyes [2] developed new measures to aid in their analysis of Hungary's income distribution, and Kondor [6] has shown that these new indices are related to the relative mean deviation. In order to draw valid…
指数随机变量线性组合的倒数矩
The reciprocal moments of a linear combination of exponential variates are derived. The resultant formula is used to obtain the moments of quantile and other similar estimators for the shape parameter of a Pareto distribution. The calculations show that such estimators are more biased and less “precise” than indicated by Monte Carlo estimates of the moments previously published. The general formula for the reciprocal moments is shown to be potentially useful in linear models…
正态总体分位数的置信带
Given the general linear regression model , where y is N X 1, X of full rank is N X (q + 1), is (q + 1) X 1, and e is N(0, σ2I), Steinhorst and Bowden [5], using the theory of Roy's maximum root test statistics, set confidence bands for , of the normal population with mean and variance σ2, k being the pth quantile of the standard normal distribution, I fixed or specified but unknown and p known. However, narrower bands may be obtained using the exact test statistic,…
分位数区间的无分布置信区间
Wilks [6] introduced distribution-free confidence intervals based on order statistics for quantile intervals. For the rth and sth order statistics in a sample of size n (1 ≤ r < s ≤ n), upper and lower bounds and an exact expression for the confidence coefficient are derived. Numerical examples comparing the bounds and the exact solution are provided. A connection with distribution-free tolerance intervals is indicated.
正态总体分位数的置信带
Given the general linear regression model , where y is N X 1, X of full rank is N X (q + 1), is (q + 1) X 1, and e is N(0, σ2I), Steinhorst and Bowden [5], using the theory of Roy's maximum root test statistics, set confidence bands for , of the normal population with mean and variance σ2, k being the pth quantile of the standard normal distribution, I fixed or specified but unknown and p known. However, narrower bands may be obtained using the exact test statistic,…
非满秩正态回归中同时外推的一个判据
In recent work by O'Reilly, a necessary and sufficient condition for the existence of an unbiased estimate of the distribution function of a "future" observation was given. The result was obtained under the condition that the model had full rank. Here, the result is generalized to any number of future observations and the full rank condition is relaxed. The corresponding uniformly minimum variance unbiased estimator is identified from the density estimates given by Ghurye…
Kendall 秩相关系数整数值形式的渐近展开与局部极限定理
In the present paper, an integer-valued version $(T_N)$ of the Kendall rank correlation coefficient is considered. Under the hypothesis of independence, a local limit theorem with the Edgeworth expansion for $T_N$ is proved and an asymptotic expansion of the distribution function of $T_N$ is derived.
分布函数的经验贝叶斯估计
A sequence of empirical Bayes estimators is defined for estimating a distribution function. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior. Exact risk expressions are derived and the rate, at which the overall expected loss approaches the minimum Bayes risk, is exhibited. The empirical Bayes approach, based on the Dirichlet process, is also applied to the problem of estimating the mean of a distribution.
单调百分位数回归
Suppose that for each number $t$ in [0, 1] there is a distribution with distribution function $F_t(\bullet)$ which has $p$th percentile $\xi(t)$. Consider the problem of estimating $\xi(\bullet)$ under the assumption that $\xi(\bullet)$ is monotone. An estimator which is analogous to the median regression estimator considered in Cryer, Robertson, Wright and Casady (1972), is studied. Asymptotic properties including consistency and law of the iterated logarithm results are…
基于分组数据的洛伦兹曲线与基尼指数插值
ECONOMISTS OFTEN SUMMARIZE the income distribution by the Lorenz curve and Gini index. A variety of parametric methods (e.g., [1 and 8]) have been developed to estimate these measures from the grouped income data governments provide (e.g., [3 and 12]). Previously, one of the authors developed a distribution-free approach [5] which yielded accurate bounds on the Gini index. While analogous bounds on the Lorenz curve can be obtained [5 and 10], the resulting curve is not…