Order Restricted Statistical Inference Research Articles

Order-restricted statistical inference and optimal censoring scheme for Gompertz distribution based on joint type-II progressive censored data

This paper considers the order-restricted statistical inference for two populations based on the joint type-II progressive censoring scheme. The lifetime distributions of the two populations are supposed to follow the Gompertz distribution with the same shape parameter but different scale parameters. The maximum likelihood estimates of the unknown parameters are derived by employing the Newton-Raphson algorithm and the expectation-maximization algorithm, respectively. The Fisher information matrix is then employed to construct asymptotic confidence intervals. For Bayes estimation, we assume an ordered Beta-Gamma prior for the scale parameters and a Gamma prior for the common shape parameter. Bayes estimations and the highest posterior density credible intervals for unknown parameters under two different loss functions are obtained with the importance sampling technique. To evaluate the performance of order-restricted inference, extensive Monte Carlo simulations are performed and two air-conditioning systems datasets are used to illustrate the proposed inference methods. In addition, the results are compared with the case when there is no order restriction on the parameters. Finally, the optimal censoring scheme is obtained by four optimality criteria.

Reliability estimation and order-restricted inference based on joint type-II progressive censoring scheme with application to splashing data in atomization process

Censoring commonly occurs in real-world scenarios, either intentionally or unintentionally. In addition, in engineering and applied studies, two or more populations are required to perform the experiment. In this article, we develop the order-restricted ( OR) statistical inference of two inverted exponentiated Rayleigh ( IERayleigh) populations based on the joint type-II progressive censoring scheme ( J-IIProgCS). We also consider the problem of the stress-strength reliability Re = P( X> Y) estimation under J-IIProgCS. The process of estimating the parameters with order restriction ( PaR-OR) and Re are performed using the maximum likelihood (ML) and Bayesian methods. On the presumption that the gamma priors are independent, the Bayesian estimate of reliability model is obtained using the hybrid Gibbs within Metropolis-Hastings ( G-MeHa) algorithm. Moreover, with assumption the ordered Beta-Gamma prior for the shape parameters and the Gamma prior for the common scale parameter, the Bayes estimates of the parameters with order restriction are evaluated. The classical confidence intervals and the Bayesian credible interval are created. A detailed numerical illustration using simulated data sets is carried out. A real engineering example related to splashing phenomenon is considered for illustrative purposes.

Isotonic design for phase I cancer clinical trials with late-onset toxicities

ABSTRACT This article addresses the problem of identifying the maximum tolerated dose (MTD) in Phase I dose-finding clinical trials with late-onset toxicities. The main design challenge is how best to adaptively allocate study participants to tolerable doses when the evaluation window for the toxicity endpoint is long relative to the accrual rate of new participants. We propose a new design framework based on order-restricted statistical inference that addresses this challenge in sequential dose assignments. We illustrate the proposed method on real data from a Phase I trial of bortezomib in lymphoma patients and apply it to a Phase I trial of radiotherapy in prostate cancer patients. We conduct extensive simulation studies to compare our design’s operating characteristics to existing published methods. Overall, our proposed design demonstrates good performance relative to existing methods in allocating participants at and around the MTD during the study and accurately recommending the MTD at the study conclusion.

Multivariate one-sided testing in matched observational studies as an adversarial game

Summary We present a multivariate one-sided sensitivity analysis for matched observational studies, appropriate when the researcher has specified that a given causal mechanism should manifest itself in effects on multiple outcome variables in a known direction. The test statistic can be thought of as the solution to an adversarial game, where the researcher determines the best linear combination of test statistics to combat nature’s presentation of the worst-case pattern of hidden bias. The corresponding optimization problem is convex, and can be solved efficiently even for reasonably sized observational studies. Asymptotically, the test statistic converges to a chi-bar-squared distribution under the null, a common distribution in order-restricted statistical inference. The test attains the largest possible design sensitivity over a class of coherent test statistics, and facilitates one-sided sensitivity analyses for individual outcome variables while maintaining familywise error control through its incorporation into closed testing procedures.

Modeling semicontinuous longitudinal data with order constraints.

Semicontinuous longitudinal data are characterized by within-subjects repeated measurements that either indicate absence of abnormality or reflect different amount of abnormality. Joint models for semicontinuous longitudinal data have been increasingly receiving attention in the literature. Such models permit flexible characterization of covariates-outcome associations. Order-restricted statistical inference has been well established in the literature but has not yet been applied to joint models for semicontinuous longitudinal data. We incorporate general order-restricted inference into the general joint models for semicontinuous longitudinal data previously proposed. We develop computational methods to address general order restrictions. Through simulations and a real-data example, we demonstrate the advantages of order-restricted inference in terms of increased power in hypothesis testing and increased precision in parameter estimation.

A geometric interpretation of the characteristic polynomial of reflection arrangements

We consider projections of points onto fundamental chambers of finite real reflection groups. Our main result shows that for groups of types $A_n$, $B_n$, and $D_n$, the coefficients of the characteristic polynomial of the reflection arrangement are proportional to the spherical volumes of the sets of points that are projected onto faces of a given dimension. We also provide strong evidence that the same connection holds for the exceptional, and thus all, reflection groups. These results naturally extend those of De Concini and Procesi, Stembridge, and Denham, which establish the relationship for $0$-dimensional projections. This work is also of interest to the field of order-restricted statistical inference, where projections of random points play an important role.

Information criterion-based clustering with order-restricted candidate profiles in short time-course microarray experiments

BackgroundTime-course microarray experiments produce vector gene expression profiles across a series of time points. Clustering genes based on these profiles is important in discovering functional related and co-regulated genes. Early developed clustering algorithms do not take advantage of the ordering in a time-course study, explicit use of which should allow more sensitive detection of genes that display a consistent pattern over time. Peddada et al. [1] proposed a clustering algorithm that can incorporate the temporal ordering using order-restricted statistical inference. This algorithm is, however, very time-consuming and hence inapplicable to most microarray experiments that contain a large number of genes. Its computational burden also imposes difficulty to assess the clustering reliability, which is a very important measure when clustering noisy microarray data.ResultsWe propose a computationally efficient information criterion-based clustering algorithm, called ORICC, that also takes account of the ordering in time-course microarray experiments by embedding the order-restricted inference into a model selection framework. Genes are assigned to the profile which they best match determined by a newly proposed information criterion for order-restricted inference. In addition, we also developed a bootstrap procedure to assess ORICC's clustering reliability for every gene. Simulation studies show that the ORICC method is robust, always gives better clustering accuracy than Peddada's method and saves hundreds of times computational time. Under some scenarios, its accuracy is also better than some other existing clustering methods for short time-course microarray data, such as STEM [2] and Wang et al. [3]. It is also computationally much faster than Wang et al. [3].ConclusionOur ORICC algorithm, which takes advantage of the temporal ordering in time-course microarray experiments, provides good clustering accuracy and is meanwhile much faster than Peddada's method. Moreover, the clustering reliability for each gene can also be assessed, which is unavailable in Peddada's method. In a real data example, the ORICC algorithm identifies new and interesting genes that previous analyses failed to reveal.

Order restricted statistical inference for scale parameters based on sequential order statistics

In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of- n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework.

A One Sided Test Based on Sample Quasi Ranges

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F i (x) = F[(x − μ i )/θ i ], i = 1,…, k (k ≥ 2). Here μ i (−∞ < μ i < ∞) is the location parameter, θ i (θ i > 0) is the scale parameter and F(⋅) is any absolutely continuous cdf, i.e., F i (⋅) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H 0 ≔ θ1 = · = θ k against the simple ordered alternative H A ≔ θ1 ≤ · ≤ θ k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X i1,…, X in be a random sample of size n from the population π i and R ir = X i:n−r − X i:r+1, r = 0, 1,…, [n/2] − 1 be the sample quasi range corresponding to this random sample, where X i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W r = max1≤i<j≤k (R jr /R ir ). The test is reject H 0 for large values of W r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ j /θ i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.

Enumerating and testing conjoint measurement models

A Monte Carlo procedure was used to generate three types of 3×3×2 conjoint measurement structures, each type having an additive (respectively, distributive, dual-distributive) simple polynomial representation. The Monte Carlo results illustrate the restrictiveness of the axioms of joint independence (respectively, distributive cancellation, dual-distributive cancellation), and show how close each comes to characterizing the relevant simple polynomial representation. An empirical example illustrates how the generated measurement structures aid in the selection and testing of axiomatic models using order-restricted statistical inference.

A Monotonicity of Moments Concerned with Order Restricted Statistical Inference

In the order restricted statistical inference problem, moments of the distance between a true parameter and the least square estimate are non-decreasing when the true parameter moves along a half line from an initial point in the null space. This follows from “stochastically larger” property of the distance.

A weighted geometric regularity from order restricted statistical inference

In Euclideank-space, the cone of vectors x = (x 1,x 2,...,x k ) satisfyingx 1 ≤x 2 ≤ ... ≤x k and\(\sum\nolimits_{j = 1}^k {x_j } = 0\) is generated by the vectorsv j = (j −k, ...,j −k,j, ...,j) havingj −k’s in its firstj coordinates andj’s for the remainingk −j coordinates, for 1 ≤j <k. In this equal weights case, the average angle between v i and v j over all pairs (i, j) with 1 ≤i <j <k is known to be 60°. This paper generalizes the problem by considering arbitrary weights with permutations.

Inference for Lorenz curve orderings

In this paper we consider the issue of performing statistical inference for Lorenz curve orderings. This involves testing for an ordered relationship in a multivariate context and making comparisons among more than two population distributions. Our approach is to frame the hypotheses of interest as sets of linear inequality constraints on the vector of Lorenz curve ordinates, and apply order-restricted statistical inference to derive test statistics and their sampling distributions. We go on to relate our results to others which have appeared in recent literature, and use Monte Carlo analysis to highlight their respective properties and comparative performances. Finally, we discuss in general terms the issue and problems of framing hypotheses, and testing them, in the context of the study of income inequality, and suggest ways in which the distributional analyst could best proceed, illustrating with empiricalexamples.

Uniform designs for mixture-amount experiments and for mixture experiments under order restrictions

With order statistics of the uniform distribution on [0, l], exponential and beta distributions, a stochastic representation is obtained for the uniform distribution over various domains, where A-type domains are closely associated with reliability growth analysis, order restricted statistical inference and isotonic regression theory, V-type domains are connected with the mixture-amount experiments, and T-type domains are well related to mixture experiments. With these stochastic representations, the corresponding uniform distribution and number-theoretic nets can be generated. This approach seems to be new and is called order statistics method. Some examples on reliability growth analysis and experimental design are presented.

On maximum likelihood estimation of cell probabilities in 2 × k contingency tables under negative dependence restrictions with various sampling schemes

It is well known that dependence concepts are related to concepts of various types of stochastic orderings. Using estimation techniques developed for order restricted statistical inference we find maximum likelihood estimates of cell probabilities of 2 × k contingency table under negative dependence restrictions. Four types of sampling schemes, namely single multinomial, product multinomial, binomial and Poisson models, are considered.

A Matrix Approach to Finding a Set of Generators and Finding the Polar (Dual) of a Class of Polyhedral Cones

Previous article Next article A Matrix Approach to Finding a Set of Generators and Finding the Polar (Dual) of a Class of Polyhedral ConesCarolyn Pillers DoblerCarolyn Pillers Doblerhttps://doi.org/10.1137/S0895479891199690PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractA set of generators for a polyhedral cone is found by using a new representation for the polar (dual) cone. Examples from order-restricted statistical inference are chosen to illustrate this method.[1] Josef Stoer and , Christoph Witzgall, Convexity and optimization in finite dimensions. I, Die Grundlehren der mathematischen Wissenschaften, Band 163, Springer-Verlag, New York, 1970ix+293 44:3707 0203.52203 CrossrefGoogle Scholar[2] A. W. Marshall, , B. W. Walkup and , R. J. B. Wets, Order-preserving functions: Applications to majorization and order statistics, Pacific J. Math., 23 (1967), 569–584 36:2756 0153.38502 CrossrefISIGoogle Scholar[3] T. Robertson, , F. T. 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Dykstra, Dual convex cones of order restrictions with applicationsInequalities in statistics and probability (Lincoln, Neb., 1982), IMS Lecture Notes Monogr. Ser., Vol. 5, Inst. Math. Statist., Hayward, CA, 1984, 228–235 86j:52014 0486.90068 CrossrefGoogle Scholar[12] Tim Robertson, On testing symmetry and unimodalityAdvances in order restricted statistical inference (Iowa City, Iowa, 1985), Lecture Notes in Statist., Vol. 37, Springer, Berlin, 1986, 231–248 88b:62038 0607.62019 CrossrefGoogle ScholarKeywordspolyhedral conegeneratorspolar conedual coneorder-restricted statistical inference Previous article Next article FiguresRelatedReferencesCited byDetails A note on computing the smallest conic singular valueJournal of Computational and Applied Mathematics, Vol. 340 Cross Ref On multivariate order statistics. Application to ranked set samplingComputational Statistics & Data Analysis, Vol. 53, No. 12 Cross Ref Analysis of multinomial models under inequality constraints: Applications to measurement theoryJournal of Mathematical Psychology, Vol. 53, No. 1 Cross Ref A complete description of cones and polytopes including hypervolumes of all facets of a polytope3 July 2007 | International Journal of Mathematical Education in Science and Technology, Vol. 38, No. 1 Cross Ref Interpreting linear systems of equalities and inequalities. Application to the water supply problem1 January 2006 | Numerical Linear Algebra with Applications, Vol. 13, No. 5 Cross Ref A new test for stochastic order of ordered multinomial populationsStatistics & Probability Letters, Vol. 76, No. 10 Cross Ref The Γ-algorithm and some applicationsInternational Journal of Mathematical Education in Science and Technology, Vol. 35, No. 3 Cross Ref Optimization by Direct Search: New Perspectives on Some Classical and Modern MethodsTamara G. Kolda, Robert Michael Lewis, and Virginia Torczon4 August 2006 | SIAM Review, Vol. 45, No. 3AbstractPDF (1718 KB)Obtaining simultaneous solutions of linear subsystems of inequalities and dualsLinear Algebra and its Applications, Vol. 346, No. 1-3 Cross Ref A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context17 February 2012 | Journal of the American Statistical Association, Vol. 93, No. 443 Cross Ref Volume 15, Issue 3| 1994SIAM Journal on Matrix Analysis and Applications History Submitted:28 May 1991Accepted:03 February 1993Published online:31 July 2006 InformationCopyright © 1994 © Society for Industrial and Applied MathematicsKeywordspolyhedral conegeneratorspolar conedual coneorder-restricted statistical inferenceMSC codes52B9915A39PDF Download Article & Publication DataArticle DOI:10.1137/S0895479891199690Article page range:pp. 796-803ISSN (print):0895-4798ISSN (online):1095-7162Publisher:Society for Industrial and Applied Mathematics

Semi-infinite programming duality for order restricted statistical inference models

The equivalence of multinomial maximum likelihood and the isotonic projection problem: $$\inf \left\{ {\sum\limits_{i = 1}^n {p_i \ln (p_i /r_i )|p \in K,a convex} {\mathbf{ }}subset of the probability vectors{\mathbf{ }}p} \right\}$$ can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relationships of convex analysis, all taking place over the real numbers.

Order Restricted Statistical Inference.

Order Restricted Statistical Inference.

Order Restricted Statistical Inference.

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.

Order Restricted Statistical Inference.

Order Restricted Statistical Inference.