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尺度未知的指数数据的 Kolmogorov–Smirnov 型检验及相关问题AI 解读

Let D^ , D^+ and D^− denote Kolmogorov-Smirnov typo one-sample statistics to test good ness of fit in the presence of unknown nuisance parameters; then the distributions of D^ , D^+ and D^− depend on the population sampled and the estimator used. Simulation has been the primary tool for studying these statistics. Recently, Durbin obtained the distributions of D^ , D^+ and D^− in terms of a Fourier transform for a wide class of underlying populations, and produced explicit…

非独立同分布情形下经验分布函数的若干性质AI 解读

For $N = 1, 2, \cdots$ let $X_{1N}, X_{2N}, \cdots, X_{NN}$ be independent rv's having continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. For the set $X_{1N}, X_{2N}, \cdots, X_{NN}$, let us denote by $X_{1:N} \leqq X_{2:N} \leqq \cdots \leqq X_{N:N}$ the order statistics, by $\mathbb{F}_N$ the empirical df and by $\bar{F}_N$ the averaged df, i.e. $\bar{F}_N(x) = N^{-1}\sum^N_{n=1} F_{nN}(x)$ for $x\epsilon(-\infty, \infty)$. It is shown that for each $\varepsilon > 0$ there…

Conditions to ensure the asymptotic normality of rank statistics having a scores generating function with finitely many jumps are obtained. These conditions are derived by studying rank statistics with a scores generating function $J$ such that $J(u) = 1$ or $0$ as $u \geqq s$ or $u < s$ for a fixed $s, 0 < s < 1$. No differentiability conditions are imposed on the underlying distribution functions at the jump points of the scores generating function.

正态变量二次型的级数展开AI 解读

A differential operator is defined and applied to the gamma density function. By formal mathematical manipulation the resulting function can be identified with linear combinations of independent gamma variates and central and noncentral definite quadratic forms in independent normal variates. Laguerre series expansions in which the choice of parameters is of a general nature are given. These Laguerre series and chi-square, power and Edgeworth series are then compared in the…

三角阵观测点下积分回归估计的一致性AI 解读

Let $\mu$ be a continuous mean regression function defined on $U$, the unit cube in $N$-dimensional Euclidean space. Let $F$ be a distribution function with support in $U$, and let $M$ denote the indefinite integral of $\mu$ with respect to $F$. This paper provides consistency results, including rates of convergence, for a certain estimator of $M$ in the case that the $n$th estimate is based on observations at points $\mathbf{t}_{n1},\cdots, \mathbf{t}_{nn}$ of $U$. The…

非对称情形下位置参数的稳健估计AI 解读

Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let $F$ be a distribution function that is governed by the standard normal density on the set $\lbrack - d, d\rbrack$ and is otherwise arbitrary. Let $X_1,\cdots, X_n$ be a random sample from $F(x - \theta)$, where $\theta$ is the unknown location parameter. If $\psi$ is in a class of continuous skew-symmetric…

对称分布的估计AI 解读

One estimate of a symmetric location cumulative distribution function F{x - θ) is obtained by symmetrizing the sample cumulative distribution function with respect to the estimated centre of symmetry θ. We prove an earlier conjecture that if θ is asymptotically efficient, then the symmetrized estimated cumulative distribution function is uniformly superior to the sample distribution function.

生物测定中 Spearman 型方差与百分位数估计量的性质AI 解读

Theoretical results for the Spearman-type variance estimator in the quantal response bioassay problem are determined when the mean of the underlying tolerance distribution is known, and also when it is unknown. For an experiment with infinitely many dose levels, the estimator is proved to be finite with probability one, and its mean and variance are given. Allowing the dose interval to approach zero, the estimator is shown to be asymptotically normal and asymptotically…

随机删失数据的 Cramér–von Mises 统计量AI 解读

The asymptotic distributions of camér-von Mises type statistics based on the productlimit estimate of the distribution function of a certain class of randomly censored observations are derived; the asymptotic significance points of the statistics for various degrees of censoring are given. The statistics are also partitioned into orthogonal components in the manner of Durbin & Knott (1972). The asymptotic powers of the statistics and their components against normal mean and…

删失数据的最小二乘回归AI 解读

The analysis for a standard linear regression model is extended to the case of data randomly censored on the right. The slope and intercept estimators are weighted linear combinations of the uncensored observations where the weights are derived from the Kaplan-Meier product-limit estimator of a distribution function. Some distribution theory for the slope estimator is given. For illustration the estimators are applied to the Stanford heart transplant data.

对称分布的分布函数估计AI 解读

SUMMARY This paper considers the problem of estimating the distribution function of a symmetric distribution when the centre of symmetry is unknown. An estimator is proposed which is shown to be asymptotically better than the empirical distribution function for well-known distributions including the normal, double exponential and Cauchy.

删失数据的修正 Cramér–von Mises 统计量AI 解读

Cramér-von Mises type statistics are modified so that they can be used to test the goodness of fit of a censored sample, when the distribution function is known completely. Asymptotic percentage points are given, small-sample applications are discussed and asymptotic power considered.

未知尺度指数分布的 Kolmogorov–Smirnov 型检验及相关问题AI 解读

SUMMARY: Let D, D⁺ and D⁻ denote Kolmogorov-Smirnov typo one-sample statistics to test good ness of fit in the presence of unknown nuisance parameters; then the distributions of D, D⁺ and D⁻ depend on the population sampled and the estimator used. Simulation has been the primary tool for studying these statistics. Recently, Durbin obtained the distributions of D, D⁺ and D⁻ in terms of a Fourier transform for a wide class of underlying populations, and produced explicit…

对称中心未知时的对称分布函数估计AI 解读

This paper considers the problem of estimating the distribution function of a symmetric distribution when the centre of symmetry is unknown. An estimator is proposed which is shown to be asymptotically better than the empirical distribution function for well-known distributions including the normal, double exponential and Cauchy.

利用两个次序统计量估计 Weibull 分布参数AI 解读

Summary The use of a computationally simple statistic for estimating, from small samples, the important reciprocal-shape parameter, γ, of the Weibull distribution is described. The estimator uses two suitably selected order statistics from a sample of N failure times. The choice of the two has been made to minimize the variance of the estimator; however, concise formulae are given to permit the use of any two order statistics in the case of censored samples. The statistic is…

非参数模型的描述统计（二）：位置AI 解读

Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, $R$-estimators and $M$-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and…

分布尾部推断的一种简单通用方法AI 解读

A simple general approach to inference about the tail behavior of a distribution is proposed. It is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference. Results are particularly simple for distributions of the Zipf type, i.e., where $G(y) = 1 - Cy^{-\alpha}$ for large $y$. The methods of inference are based upon an evaluation of the conditional likelihood for the parameters…

威布尔及其他受限分布族中次序统计量线性函数的矩界AI 解读

We give two different sets of bounds for the expected values of range, midrange and some other linear functions of order statistics from certain restricted families of distributions. The bounds are obtained by using Jensen's inequality on the expected values of some functions of the distribution function F which are convex (concave) strictly increasing. The bounds could be very useful, since exact values for the moments of these statistics, in general, are difficult to…

终身收入估计的一项更正AI 解读

A reexamination is made of the formula for the variance of lifetime income in a stochastic process income distribution model. The formula in the received literature is shown to introduce a significant upward bias in estimates of this variance.

利用分组数据界定分布矩AI 解读

This article derives bounds on a wide class of moments of the underlying distribution from data grouped in intervals when the mean of each group is given. The extremal distributions are constructed yielding the upper and lower bounds to moments of the form ∫h(x)dF(x), where h(x) is convex and F(x) is concave. To illustrate the results, bounds on the variance and Gini index are presented for two theoretical distributions.

关于若干贝叶斯非参数估计的一则说明AI 解读

With respect to a general quadratic loss function, the Bayes rule for the mean of a probability distribution of unknown form is obtained, in the class of linear functions of the sample. The associated Bayes risk is also obtained. A number of recent results in the literature are shown to be direct corollaries of this result, and applications are given for the empirical distribution function of the sample.

非中心卡方分布与正态分布的若干关系AI 解读

Cumulative distribution functions of noncentral ohi-squared distributions with odd degrees of freedom are expressed as functions of the distribution function and probability density function of the standard normal distribution. From these relationships, the probabilities can be easily computed. Some key words: Noncentral chi-squared distribution; Normal distribution. 1.

受随机扰动与尺度因子影响的经验分布函数弱收敛AI 解读

The weak convergence of empirical distribution functions subject to random perturbations and scale factors to a Gaussian process is established. This result is used to study the efficiencies of tests based on spacings in goodness-of-fit problems.

非线性回归模型中位置参数似然比检验的功效AI 解读

The Likelihood Ratio Test statistic, T, is considered for the hypothesis H: θ = θ0 against A: θ ≠ θ0 in the nonlinear regression model y = f(x, θ) + e with normal errors and unknown variance. The distribution function of a random variable X such that n · (T — X) converges in probability to zero is derived. Using X to approximate T, the power of the Likelihood Ratio Test is tabulated for selected sample sizes and departures from the null hypothesis. The adequacy of the…