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两样本问题中似然比序的统计推断AI 解读

We obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered. We extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators. We also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is…

受限权重下次序约束推断分布的界AI 解读

SUMMARY Level probabilities, which are central to the theory of testing order restricted hypotheses, are intractable in some applications. Thus, approximations for them and the associated distributions are of considerable importance. The level probabilities depend on weights which in some situations are monotonic. For the simple linear and simple tree orderings, we study the improvements in the equal-weights and pattern approximations which arise when the weights are…

受限权重下序约束推断相关分布的界AI 解读

SUMMARY Level probabilities, which are central to the theory of testing order restricted hypotheses, are intractable in some applications. Thus, approximations for them and the associated distributions are of considerable importance. The level probabilities depend on weights which in some situations are monotonic. For the simple linear and simple tree orderings, we study the improvements in the equal-weights and pattern approximations which arise when the weights are…

多个总体一致随机序的统计推断AI 解读

Stochastic ordering between probability distributions is a widely studied concept. It arises in numerous settings and has useful applications. Since it is often easy to make value judgments when such orderings exist, it is desirable to recognize their occurrence and to model distributional structure under such orderings. Unfortunately, the necessary theory for statistical inference procedures has not been developed for many problems involving stochastic ordering and this…

序约束统计推断AI 解读

Isotonic Regression. Tests of Ordered Hypotheses: The Normal Means Case. Tests of Ordered Hypotheses: Generalizations of the Likelihood Ratio Tests and Other Procedures. Inferences about a Set of Multinomial Parameters. Duality. Inferences Regarding Distributions Subject to Shape Restrictions. Conditional Expectation Given a sigma-lattice: Projections in a More General Setting. Complements. Tables. Bibliography and Related References. Index. Symbols and Acronyms.

用多重对比比较若干处理与对照AI 解读

A problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard. We consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control. A number of statistical procedures have been proposed for this situation, of which the best known is Dunnett's (1955) multiple…

椭圆等高密度的概率不等式及其在序约束推断中的应用AI 解读

Anderson (1955) established the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric, convex set. A similar result is developed for integrals of elliptically contoured, unimodal densities over translates of an asymmetric, convex set in certain directions related to the set. This result is used to establish some monotonicity properties of the power functions of the likelihood ratio tests for determining whether or not a vector of normal…

两个嵌套但斜交的序约束假设的似然比检验AI 解读

Cadoret, Woolson, and Winokur (1977) considered two theories regarding the genetic makeup of patients suffering from unipolar affective disorder. These two theories imply nested but oblique order restrictions on the parameters of a statistical model. A likelihood ratio test for these two restrictions is studied and found to be dominated by a test that neglects some of the information in the model.

摘要暂缺

序约束推断中层级概率及相关分布的近似AI 解读

The use of much of the distribution theory developed for order restricted inference has been limited by the lack of algorithms for the ‘level probabilities’. An approximation for these, which accounts for the pattern of large and small ‘weights’, is developed. This approximation and the equal weights approximation are examined. Both approximations appear to be reasonable for weight sets having a moderate amount of variability. The quality of the equal weights approximation,…

Likelihood ratio statistics for (i) testing the homogeneity of a collection of multinomial parameters against the alternative which accounts for the restriction that those parameters are starshaped (cf. Shaked, Ann. Statist., 1979), and for (ii) testing the null hypothesis that this parameter vector is starshaped, are considered. For both tests the asymptotic distribution of the test statistic under the null hypothesis is a version of the Chi-bar-square distribution.…

两个及以上自变量的保序回归算法AI 解读

Algorithms for solving the isotonic regression problem in more than one dimension are difficult to implement because of the large number of lower sets present or because they involve search techniques which require a significant amount of checking and readjustment. Here a new algorithm for solving this problem based on a simple iterative technique is proposed and shown to converge to the correct solution.

序约束推断中混合分布的界AI 解读

In testing hypotheses involving order restrictions on a collection of parameters, distributions arise which are mixtures of standard distributions. Since tractable expressions for the mixing proportions generally do not exist even for parameter collections of moderate size, the implementation of these tests may be difficult. Stochastic upper and lower bounds are obtained for such test statistics in a variety of these kinds of problems. These bounds are also shown to be…

多项总体间随机序关系的似然比检验AI 解读

Likelihood ratio tests concerning the parameters of two multinomial populations are discussed. A stochastic ordering restriction is considered as a one sided alternative to equality. The one and two sample tests for equality versus stochastic ordering and stochastic ordering versus all alternatives are derived and their large sample distributions are obtained. The large sample distributions are mixtures of chi-squared distributions. The tests developed provide discrete…

序约束统计推断算法与柯西中值性质AI 解读

Most algorithms in order restricted statistical inference express the estimates in terms of certain summary statistics computed from pooled samples. These algorithms may or may not yield optimal estimates depending on whether or not the Cauchy mean value property holds strictly for the summary statistics. In this paper a minimum lower sets algorithm, which holds generally, is described and used to prove the optimality of estimates described by a max-min formula.

指数族中序约束的似然比检验AI 解读

This paper considers likelihood ratio tests for testing hypotheses that a collection of parameters satisfy some order restriction. The first problem considered is to test a hypothesis specifying an order restriction on a collection of means of normal distributions. Equality of the means is the subhypothesis of the null hypothesis which yields the largest type I error probability (i.e., is least favorable). Furthermore, the distribution of $T = -\ln$ (likelihood ratio) is…

多项分布参数次序约束的支持与违背检验AI 解读

Likelihood-ratio statistics are considered for testing a simple null hypothesis on a collection of multinomial parameters against an order-restricted alternative and for testing an order restriction against all alternatives. For the former test, the asymptotic distribution of the test statistic under the null hypothesis is a version of the chi-bar-squared distribution. This extends the work of Chacko (1966). For the latter test, homogeneity is, asymptotically, the least…

分枝过程灭绝概率的保序估计AI 解读

Suppose X 1, X 2, ···, Xn is a random sample from a distribution which assigns all of its mass to the nonnegative integers. Let p j = P[X 1 = j]; j = 0, 1, ··· and let f(·) be the generating function of the sequence {p j }∞ j=0. An estimate of the smallest nonnegative root of f(s) = s which is based on isotonized estimates of p 0, p 1, ··· is considered. Results of a simulation study, almost sure convergence rates and the asymptotic distribution theory are discussed.

分枝过程灭绝概率的保序估计AI 解读

Suppose X 1, X 2, ···, Xn is a random sample from a distribution which assigns all of its mass to the nonnegative integers. Let p j = P[X 1 = j]; j = 0, 1, ··· and let f(·) be the generating function of the sequence {p j }∞ j=0. An estimate of the smallest nonnegative root of f(s) = s which is based on isotonized estimates of p 0, p 1, ··· is considered. Results of a simulation study, almost sure convergence rates and the asymptotic distribution theory are discussed.

广义保序回归的一致性AI 解读

Suppose $T$ is a partially ordered set and that associated with each $t$ in $T$ we have a distribution with "location parameter" $m(t)$. In this paper we discuss consistency properties of estimates of $m(\cdot)$ which are isotonic with respect to a partial order. The results extend results in the literature, some of which are contained in Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press,…

保序化柯西均值函数的范数缩减性质AI 解读

We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$…

摘要暂缺

随机序约束下随机变量的最大似然估计AI 解读

Brunk, Franck, Hanson and Hogg (1966) ("Maximum likelihood estimation of the distributions of two stochastically ordered random variables," J. Amer. Statist. Assoc. 61 1067-1080) found and studied maximum likelihood estimates of a pair of stochastically ordered distribution functions. In this paper we discuss a generalization of this problem in which we do not require the domain of these "distribution functions" to be the real line. We think of the order restriction we…

多维保序中位数回归AI 解读

We consider the partial order on the unit square; $s_1 = (x_1, y_1) \ll s_2 = (x_2, y_2)$ if and only if $x_i \leqq y_i$ for $i = 1, 2$, and say that a real-valued function $f$ is isotone if $s_1 \ll s_2$ implies that $f(s_1) \leqq f(s_2)$. Suppose that for each point, $s$, in the unit square we have a distribution with median $m(s)$ and $m(s)$ is isotone. In this paper we propose an isotone estimator for $m$ which we denote by $\hat{m}$ and give an algorithm for computing $\hat{m}$…