Anderson studied the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric convex set. Restricting attention to elliptically contoured, unimodal densities, Mukerjee, Robertson and Wright weakened the assumption of symmetry on the set and obtained monotonicity properties of power functions, including unbiasedness, for some likelihood ratio tests in order restricted inference for the variance-known case. For elliptically contoured,…
Chu-in Charles Lee · Tim Robertson · F. T. Wright·刊期 1993-06 · 卷 80（2）·Biometrika已编入正刊
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…
CHU-IN CHARLES LEE · TIM ROBERTSON · F. T. WRIGHT·刊期 1993 · 卷 80（2）·Biometrika已编入正刊
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…
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.
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…
Hari Mukerjee · Tim Robertson · F. T. Wright·刊期 1986-12-01 · 卷 14（4）·The Annals of Statistics已编入正刊
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…
F. T. Wright · Tuan Tran·刊期 1985-08 · 卷 72（2）·Biometrika已编入正刊
序约束推断中层级概率的近似：简单树序AI 解读
Distribution theory for the likelihood ratio statistics for the comparison of several treatments with a control is discussed. These test statistics account for prior information that the treatments are at least as effective as the control. Their null distributions are mixtures of chi-squared or beta distributions and the mixing coefficients, which are ‘level probabilities’, are intractable for even a moderate number of treatments and unequal weights, which are typically the…
F. T. WRIGHT · TUAN TRAN·刊期 1985 · 卷 72（2）·Biometrika已编入正刊
序约束推断中层级概率的近似：简单树序AI 解读
Distribution theory for the likelihood ratio statistics for the comparison of several treatments with a control is discussed. These test statistics account for prior information that the treatments are at least as effective as the control. Their null distributions are mixtures of chi-squared or beta distributions and the mixing coefficients, which are ‘level probabilities’, are intractable for even a moderate number of treatments and unequal weights, which are typically the…
TIM ROBERTSON · F. T. WRIGHT·刊期 1983 · 卷 70（3）·Biometrika已编入正刊
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,…
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…
The maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature. Its asymptotic distribution at a point is related to a solution of the heat equation, and its rate of convergence to the underlying regression function is of order $n^{-1/3}$. This estimator can be modified by grouping adjacent observations and then "isotonizing" the corresponding means. It is shown that the resulting estimator has…
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…
The estimator proposed by Brunk for the indefinite integral of a regression function defined on the unit cube in $\beta$ dimensional Euclidean space is studied. It is shown to be strongly uniformly consistent if the errors satisfy a first moment type of condition and an almost sure rate of convergence of order $O((n/\log_2n)^{-1/2})$ is obtained.
An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on $r$ observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by $r^{1/3}$. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at…
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.
Inferences concerning the location and scale parameters of an exponential distribution are discussed for samples censored at a fixed value. The procedures considered are uniformly most powerful unbiased if the tests are performed with replacement and they seem to be quite reasonable when testing without replacement.
The maximum likelihood estimate of a nondecreasing regression function with independent, normal errors has been studied in detail in the literature. However, it is often constant over one or more intervals and for this reason may be unacceptable if the regression function is believed to be strictly increasing. Using simulation techniques, the least squares line, the maximum likelihood estimate, and a weighted average of these two are compared here. In each case considered,…
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,…
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 $\hat{\alpha}(\bullet)$…
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…
D. L. Hanson · Gordon Pledger · F. T. Wright·刊期 1973-05-01 · 卷 1（3）·The Annals of Statistics已编入正刊
单调回归的一致性研究AI 解读
For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. 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 $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and…
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 $\hat{m}$…
The maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature. Its asymptotic distribution at a point is related to a solution of the heat equation, and its rate of convergence to the underlying regression function is of order n−1/3. This estimator can be modified by grouping adjacent observations and then "isotonizing" the corresponding means. It is shown that the resulting estimator has…
The estimator proposed by Brunk for the indefinite integral of a regression function defined on the unit cube in β dimensional Euclidean space is studied. It is shown to be strongly uniformly consistent if the errors satisfy a first moment type of condition and an almost sure rate of convergence of order O((n/log2n)−1/2) is obtained.
An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on r observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by r1/3. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at…
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(⋅) 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,…
We consider functions α(∙) and α^(∙) on a finite set S which correspond to a function M(∙) 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). α^(∙) is isotone with respect to a partial ordering on S and is equal to α(∙) when α(∙) is isotone. It is shown that α^(∙)…
For each t in some subset T of N-dimensional Euclidean space let Ft be a distribution function with mean m(t). Suppose m(t) is non-decreasing in each of the coordinates of t. Let t1,t2,⋯ be a sequence of points in T and let Y1,Y2,⋯ be an independent sequence of random variables such that the distribution function of Yk is Ftk. Estimators m^n(t;Y1,⋯,Yn) of m(t) which are monotone in each coordinate of t and…
We consider the partial order on the unit square; s1=(x1,y1)≪s2=(x2,y2) if and only if xi≦yi for i=1,2, and say that a real-valued function f is isotone if s1≪s2 implies that f(s1)≦f(s2). 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 m^ and give an algorithm for computing m^…