Optimal Binning Strategies Under Squared Error Loss in Selective Assembly with a Tolerance Constraint

Selective assembly is an effective approach for improving the quality of a product assembled from two types of components when the quality characteristic is the clearance between the mating components. In this article, optimal binning strategies under squared error loss in selective assembly when the clearance is constrained by a tolerance parameter are discussed. Conditions for a set of constrained optimal partition limits are given, and uniqueness of this set is shown for the case when the dimensional distributions of the two components are identical and strongly unimodal. Some numerical results are reported that compare constrained optimal partitioning, unconstrained optimal partitioning, and equal width partitioning.     

Characterization of an optimal matrix estimator under convex loss function

Characterization of an optimal vector estimator and an optimal matrix estimator are obtained. In each case appropriate convex loss functions are considered. The results are illustrated through the problems of simultaneous unbiased estimation, simultaneous equivariant estimation and simultaneous unbiased prediction. Further an optimality criterion is proposed for matrix unbiased estimation and it is shown that the matrix unbiased estimation of a matrix parametric function and the minimum variance unbiased estimation of its components are equivalent.     

Optimal equivariant estimator with respect to convex loss function

Consider a family of probability distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. We illustrate this theorem under some conditions on convex loss function.     

Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles

Abstract

In this paper, we study Pareto-optimal reinsurance policies from the perspectives of an insurer and a reinsurer, assuming reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. By geometric approach, we determine the forms of the optimal policies among two classes of ceded loss functions, the class of increasing convex ceded loss functions and the class that the constraints on both ceded and retained loss functions are relaxed to increasing functions. Then we demonstrate the applicability of our results by giving the parameters of the optimal ceded loss functions under Dutch premium principle and Wang’s premium principle.     

A Loss Function Model for the Restoration of Grey Level Images

Common loss functions used for the restoration of grey scale images include the zero–one loss and the sum of squared errors. The corresponding estimators, the posterior mode and the posterior marginal mean, are optimal Bayes estimators with respect to their way of measuring the loss for different error configurations. However, both these loss functions have a fundamental weakness: the loss does not depend on the spatial structure of the errors. This is important because a systematic structure in the errors can lead to misinterpretation of the estimated image. We propose a new loss function that also penalizes strong local sample covariance in the error and we discuss how the optimal Bayes estimator can be estimated using a two-step Markov chain Monte Carlo and simulated annealing algorithm. We present simulation results for some artificial data which show improvement with respect to small structures in the image.     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.     

Some limitation on stein's phenomenon

We show that Stein's phenomenon is impossible for the following decision setting: the parameter space is compact, the loss function is the sum of coordinatewise strictly convex continuous loss functions, and the probability density function having the same support for every parameter θ is continuous in θ. Therefore, any coordinatewise admissible decision rule is admissible.     

A necessary and sufficient condition for an estimator to be optimal

A necessary condition for an estimator δ₀ to be optimal in a class Δ of estimators is derived. Thic condition becomes also sufficient when the loss function is convex.     

Signatures in Markov-Modulated Processes

□ In recent years, signatures are widely used for analysis of coherent systems consisting of unreliable components. If component lifetimes are independent and identically distributed, then system lifetime distribution function is a convex combination of distribution functions of order statistics for component lifetimes. Coefficients of this convex combination are called signatures. This article considers the case when a system operates in a so-called random environment, i.e., component failure rates are jointly modulated by a finite-state continuous-time Markov chain. In this model, component lifetimes remain exchangeable. An expression for distribution function of time to system failure is derived. Here, a crucial role is played by an elaborated procedure of deriving a distribution function of order statistics for system component lifetimes. A numerical example illustrates the suggested approach and analyzes the influence of random environment on the distribution function of system lifetime.     

Laplace Error Penalty‐based Variable Selection in High Dimension

We propose the Laplace Error Penalty (LEP) function for variable selection in high‐dimensional regression. Unlike penalty functions using piecewise splines construction, the LEP is constructed as an exponential function with two tuning parameters and is infinitely differentiable everywhere except at the origin. With this construction, the LEP‐based procedure acquires extra flexibility in variable selection, admits a unified derivative formula in optimization and is able to approximate the L₀ penalty as close as possible. We show that the LEP procedure can identify relevant predictors in exponentially high‐dimensional regression with normal errors. We also establish the oracle property for the LEP estimator. Although not being convex, the LEP yields a convex penalized least squares function under mild conditions if p is no greater than n. A coordinate descent majorization‐minimization algorithm is introduced to implement the LEP procedure. In simulations and a real data analysis, the LEP methodology performs favorably among competitive procedures.     

Functional quasi-likelihood regression models with smooth random effects

Summary. We propose a class of semiparametric functional regression models to describe the influence of vector-valued covariates on a sample of response curves. Each observed curve is viewed as the realization of a random process, composed of an overall mean function and random components. The finite dimensional covariates influence the random components of the eigenfunction expansion through single-index models that include unknown smooth link and variance functions. The parametric components of the single-index models are estimated via quasi-score estimating equations with link and variance functions being estimated nonparametrically. We obtain several basic asymptotic results. The functional regression models proposed are illustrated with the analysis of a data set consisting of egg laying curves for 1000 female Mediterranean fruit-flies (medflies).     

A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared.

As an application to statistics, a new family of probability densities on the sphere is defined in terms of cost-convex functions. The log-concave property of the likelihood function follows from the inequality.     

Sequential Estimation of a Continuous Distribution Function from Delayed Observations

The problem of sequentially estimating a continuous distribution function is considered in the case when the observations become available at random times. A certain class of sequential estimation procedures which are composed of optimal stopping time and sequential minimum risk invariant estimator of a continuous distribution function is obtained under a nonparametric invariant loss function and with the observation cost determined by a convex function of the moment of stopping and the number of observations up to this moment.     

Projection estimators of Pickands dependence functions

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme‐value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite‐dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite‐dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.     

综合评价中一种凹性指数型功效函数

目前主流功效函数多为凸性,在处理社会经济数据集中常见的右偏样本时效果难以令人满意。通过系统探讨凸性和凹性功效函数各自的适用特征,指出凹性函数在应用中有其必要性。归纳并使用偏度、区分度、P-P图三种方法作为分布形态的评判标准,比较了常见功效函数对指标原始数据分布形态的调整作用。在分析基础上,提出一种改进的凹性指数功效函数,能有效地处理右偏数据,且相比使用对数预处理的凸性功效函数更具适用性与便利性。     

Best monotone M‐estimators

The author shows how to find M‐estimators of location whose generating function is monotone and which are optimal or close to optimal. It is easy to identify a consistent sequence of estimators in this class. In addition, it contains simple and efficient approximations in cases where the likelihood function is difficult to obtain. In some neighbourhoods of the normal distribution, the loss of efficiency due to the approximation is quite small. Optimal monotone M‐estimators can also be determined in cases when the underlying distribution is known only up to a certain neighbourhood. The author considers the e‐contamination model and an extension thereof that allows the distributions to be arbitrary outside compact intervals. His results also have implications for distributions with monotone score functions. The author illustrates his methodology using Student and stable distributions.     

Neighborhood graphs,stripes and shadow plots for cluster visualization

Centroid-based partitioning cluster analysis is a popular method for segmenting data into more homogeneous subgroups. Visualization can help tremendously to understand the positions of these subgroups relative to each other in higher dimensional spaces and to assess the quality of partitions. In this paper we present several improvements on existing cluster displays using neighborhood graphs with edge weights based on cluster separation and convex hulls of inner and outer cluster regions. A new display called shadow-stars can be used to diagnose pairwise cluster separation with respect to the distribution of the original data. Artificial data and two case studies with real data are used to demonstrate the techniques.     

Conditional probability of correct selection under the continuum partition with applications

The problem of selecting the best population from among a finite number of populations in the presence of uncertainty is a problem one faces in many scientific investigations, and has been studied extensively, Many selection procedures have been derived for different selection goals. However, most of these selection procedures, being frequentist in nature, don't tell how to incorporate the information in a particular sample to give a data-dependent measure of correct selection achieved for this particular sample. They often assign the same decision and probability of correct selection for two different sample values, one of which actually seems intuitively much more conclusive than the other. The methodology of conditional inference offers an approach which achieves both frequentist interpret ability and a data-dependent measure of conclusiveness. By partitioning the sample space into a family of subsets, the achieved probability of correct selection is computed by conditioning on which subset the sample falls in. In this paper, the partition considered is the so called continuum partition, while the selection rules are both the fixed-size and random-size subset selection rules. Under the distributional assumption of being monotone likelihood ratio, results on least favourable configuration and alpha-correct selection are established. These re-sults are not only useful in themselves, but also are used to design a new sequential procedure with elimination for selecting the best of k Binomial populations. Comparisons between this new procedure and some other se-quential selection procedures with regard to total expected sample size and some risk functions are carried out by simulations.     

Optimal Binning Strategies under Squared Error Loss in Selective Assembly with Measurement Error

Selective assembly is an effective approach for improving a quality of a product assembled from two types of components, when the quality characteristic is the clearance between the mating components. Mease et al. (2004 Mease , D. , Nair , V. N. , Sudjianto , A. ( 2004 ). Selective assembly in manufacturing: statistical issues and optimal binning strategies . Technometrics 46 : 165 – 175 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) have extensively studied optimal binning strategies under squared error loss in selective assembly, especially for the case when two types of component dimensions are identically distributed. However, the presence of measurement error in component dimensions has not been addressed. Here we study optimal binning strategies under squared error loss when measurement error is present. We give the equations for the optimal partition limits minimizing expected squared error loss, and show that the solution to them is unique when the component dimensions and the measurement errors are normally distributed. We then compare the expected losses of the optimal binning strategies with and without measurement error for normal distribution, and also evaluate the influence of the measurement error.     

Optimal distribution-free confidence bands for a distribution function

Distribution-free confidence bands for a distribution function are typically obtained by inverting a distribution-free hypothesis test. We propose an alternate strategy in which the upper and lower bounds of the confidence band are chosen to minimize a narrowness criterion. We derive necessary and sufficient conditions for optimality with respect to such a criterion, and we use these conditions to construct an algorithm for finding optimal bands. We also derive uniqueness results, with the Brunn–Minkowski Inequality from the theory of convex bodies playing a key role in this work. We illustrate the optimal confidence bands using some galaxy velocity data, and we also show that the optimal bands compare favorably to other bands both in terms of power and in terms of area enclosed.