Two population classification in Gaussian process

The problem is to classify an individual into one of two populations based on an observation on the individual which follows a stationary Gaussian process and the populations are two distinct time points. Plug-in likelihood ratio rules are considered using samples from the process. The distribution of associated classification statistics are derived. For the special case when the mis-classification probabilities are equal, the nature of dependence between the population distributions on the probability of correct classification is studied. Lower bounds and iterative method of evaluation of the optimal correlation between the populations are obtained.     

基于平均自下而上时间的两种分类方法的比较

内容提要：诸如疾病分类系统的预后预测和分类方法,常可用于帮助进行临床管理决策。同一疾病总体常可得到多种分类方法,因此有必要比较这些方法以确定最优分类,或者寻找不逊于最优分类的替代方法。本文基于约束平均寿命引入分离度指标来度量分类方法的预后分类效率,这个指标可用来比较以生存时间为结局的两种分类方法的功效,特别是用于非劣性和等效性检验。我们给出了基于配对数据的两个分离度的估计与检验方法。模拟结果提示,检验方法在适当的样本量条件下能够控制第一类错误,两个实例表明在医学临床中的应用。     

Classification into two normal populations with a common mean and unequal variances

The problem of classification into two univariate normal populations with a common mean is considered. Several classification rules are proposed based on efficient estimators of the common mean. Detailed numerical comparisons of probabilities of misclassifications using these rules have been carried out. It is shown that the classification rule based on the Graybill-Deal estimator of the common mean performs the best. Classification rules are also proposed for the case when variances are assumed to be ordered. Comparison of these rules with the rule based on the Graybill-Deal estimator has been done with respect to individual probabilities of misclassification.     

Regularization through variable selection and conditional MLE with application to classification in high dimensions

It is often the case that high-dimensional data consist of only a few informative components. Standard statistical modeling and estimation in such a situation is prone to inaccuracies due to overfitting, unless regularization methods are practiced. In the context of classification, we propose a class of regularization methods through shrinkage estimators. The shrinkage is based on variable selection coupled with conditional maximum likelihood. Using Stein's unbiased estimator of the risk, we derive an estimator for the optimal shrinkage method within a certain class. A comparison of the optimal shrinkage methods in a classification context, with the optimal shrinkage method when estimating a mean vector under a squared loss, is given. The latter problem is extensively studied, but it seems that the results of those studies are not completely relevant for classification. We demonstrate and examine our method on simulated data and compare it to feature annealed independence rule and Fisher's rule.     

Proportion estimation in mixtures with covariates

Linear maps of a single unclassified observation are used to estimate the mixing proportion in a mixture of two populations with homogeneous variances in the presence of covariates. with complete knowledge of the parameters of the individual populations, the linear map for which the estimator is unbiased and has minimum variance amongst all similar estimators can be determined. Plug-in estimator based on independent training samples from the component populations can be constructed and is asymptotically equivalent to Cochran's classification statistic V* for covariate classification; see Memon and Okamoto (1970). Under normality assumptions, asymptotic expansion of the distribution of the plug-in estimator is available. In the absence of covariates, our estimator reduces to that suggested by Walker (1980) who has investigated the problem based on information on large unclassified samples from a mixture of two populations with heterogeneous variances. In contrast, distribution of Walker's estimator seems intractable in moderate sample sizes even with normality assumption.     

Exact Likelihood Inference for Two Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of two competing products with regard to their reliability. In this article, we consider two exponential populations and when joint progressive Type-II censoring is implemented on the two samples. We then derive the moment generating functions and the exact distributions of the maximum likelihood estimators (MLEs) of the mean lifetimes of the two exponential populations under such a joint progressive Type-II censoring. We then discuss the exact lower confidence bounds, exact confidence intervals, and simultaneous confidence regions. Next, we discuss the corresponding approximate results based on the asymptotic normality of the MLEs as well as those based on the Bayesian method. All these confidence intervals and regions are then compared by means of Monte Carlo simulations with those obtained from bootstrap methods. Finally, an illustrative example is presented in order to illustrate all the methods of inference discussed here.     

Classification rules for two parameter exponential populations under order restrictions on parameters

In this paper, we suggest classification procedures of an observation into one of two exponential populations assuming a known ordering between population parameters. We propose classification rules when either location or scale parameters are ordered. Some of these classification rules under ordering are better than usual classification rules with respect to the expected probability of correct classification. We also derive likelihood ratio-based classification rules. Comparison of these classification rules has been done using Monte Carlo simulations.     

Certain nonparametric classification rules and their asymptotic efficiencies

Two nonparametric classification rules for e-univariace populations are proposed, one in which the probability of correct classification is a specified number and the other in which one has to evaluate the probability of correct classification. In each case the classification is with respect to the Chernoff and Savage (1958) class of statistics, with possible specialization to populations having different location shifts and different changes of scale. An optimum property, namely the consistency of the classification procedure, is established for the second rule, when the distributions are either fixed or “near” in the Pitman sense and are tending to a common distribution at a specified rate. A measure of asymptotic efficiency is defined for the second rule and its asymptotic efficiency based on the Chernoff-Savage class of statistics relative to the parametric competitors ie the case of location shifts and scale changes is shown to be equal to the analogous Pitman efficiency.     

Classification of Observations into One of Two Artificially Dichotomized Classes by Using a Normal Screening Variable

This article considers a problem of normal based two group classification when the groups are artificially dichotomized by a screening variable. Each group distribution is derived and the best regions for the classification are obtained. These derivations yield yet another classification rule. The rule is studied from several aspects such as the distribution of the rule, the optimal error rate, and the testing of a hypothesis. This article gives relationships among these aspects along with the investigation of the performance of the rule. The classification method and ideas are illustrated in detail with two examples.     

CLASSIFICATION, DISTRIBUTIONS AND ADMISSIBILITY IN AUTOREGRESSIVE PROCESS

The problem considered here is to classify a unit into one of two populations based on a vector of measurements on the unit. The observation vector is assumed to follow an auto-regressive process. Samples from the process are used to construct classification rules. The distributions of some classification statistics are obtained. The admissibility of some classification rules is established.     

Errors of misclassification associated with the inverse gaussion distribution

Errors of misclassification and their probabilities are studied for classification problems associated with univariate inverse Gaussian distributions. The effects of applying the linear discriminant function (LDF), based on normality, to inverse Gaussian populations are assessed by comparing probabilities (optimum and conditional) based on the LDF with those based on the likelihood ratio rule (LR) for the inverse Gaussian, Both theoretical and empirical results are presented     

From Conditional Independence to Conditionally Negative Association: Some Preliminary Results

Conditional moment estimates on the cumulative sum of conditionally independent random variables are derived, conditional prophet inequalities for conditionally independent random variables are established, a comparison theorem on conditional moment inequalities between conditionally independent and conditionally negatively associated random variables is established. As applications of these results, a conditional Rosenthal type inequality and two conditional Kolmogorov exponential inequalities for conditionally negatively associated random variables are obtained.     

Classification of an observation into linear manifolds,their closed subsets and closed convex subsets

In this paper, we generalize the notion of classification of an observation (sample), into one of the given n populations to the case where some or all of the populations into which the new observation is to be classified may be new but related in a simple way to the given n populations. The discussion is in the frame-work of the given set of observations obeying the usual multivariate general linear hypothesis model. The set ofpopulations into which the new observation may be classified could be linear manifolds of the parameter space or their closed subsets or closed convex subsets or a combination of them or simply t subsets of the parameter space each of which has a finite number of elements. In the last case alikelihood ratio procedure can be obtained easily. Classification procedures given here are based on Mahalanobis distance. Bonferroni lower bound estimate of the probability of correctly classifying an observation is given for the case when the covariance matrix is known or is estimated from a large sample. A numerical example relating to the classification procedures suggested her is given.     

Asymptotic Comparison of Two Discriminants Used in Normal Covariate Classification

In this paper, we examine the performance of Anderson's classification statistic with covariate adjustment in comparison with the usual Anderson's classification statistic without covariate adjustment in a two-population normal covariate classification problem. The same problem has been investigated using different methods of comparison by some authors. See the bibliography. The aim of this paper is to give a direct comparison based upon the asymptotic probabilities of misclassification. It is shown that for large equal sample size of a training sample from each population, Anderson's classification statistic with covariate adjustment and cut-off point equal to zero, has better performance.     

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.     

A Permutation and Combination-Based Solution on the Ranking of Multivariate Populations in the Case of Ordered Categorical Responses with Application to the Evaluation of the Indoor Environment

This article deals with the problem of classifying and ranking several multivariate populations of interests using the permutation and combination approach providing also an inferential validity of the procedure. The need to define an appropriate classification of populations, i.e., products, services, teaching courses, degree programs, and so on, is very common within many areas of applied research. Many times the populations of interest are multivariate in nature meaning that many aspects of that populations can be simultaneously observed on the same unit/subject. From a statistical point of view, when the response variable of interest is multivariate in nature, the problem may become quite difficult to cope with especially in case of ordered categorical responses, due to the large dimensionality of the parametric space. Nonparametric inference based on the NPC methodology however, allows us to overcome these limitations, without the need of referring to assume any specified random distribution.     

Information-based thinning of point processes and its application to shock models

Thinning of point processes is a useful operation that is implemented in various stochastic models. When the initial point process is the nonhomogeneous Poisson process (NHPP), the thinned processes are also nonhomogeneous Poisson processes independent of each other. The crucial assumption in deriving this result is that the corresponding classification of events is independent of all other events, including the history of the process. However, in practice, this classification is often dependent on the history. In our paper, we define and describe the thinned processes for the history-dependent case using different levels of available information. We also discuss the applications of the obtained general results to the corresponding shocks models.     

Bayesian Network Structure Learning with Permutation Tests

In literature there are several studies on the performance of Bayesian network structure learning algorithms. The focus of these studies is almost always the heuristics the learning algorithms are based on, i.e., the maximization algorithms (in score-based algorithms) or the techniques for learning the dependencies of each variable (in constraint-based algorithms). In this article, we investigate how the use of permutation tests instead of parametric ones affects the performance of Bayesian network structure learning from discrete data. Shrinkage tests are also covered to provide a broad overview of the techniques developed in current literature.     

A classification rule for ordered exponential populations

In this paper, we consider classification procedures for exponential populations when an order on the populations parameters is known. We define and study the behavior of a classification rule which takes into account the additional information and outperforms the likelihood-ratio-based rule when two populations are considered. Moreover, we study the behavior of this rule in each of the two populations and compare the misclassification probabilities with the classical ones. Type II censorship, which is usual in practice, is considered and results obtained. The performance for more than two populations is evaluated by simulation.     

Classification of Higher-order Data with Separable Covariance and Structured Multiplicative or Additive Mean Models

Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in higher-order data. Moreover, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the higher-order observations have a separable variance-covariance matrix and two different Kronecker product structures on the mean vector. In this article, we develop quadratic classification rules among g different populations where each individual has κth order (κ ≥2) measurements. We also provide the computational algorithms to compute the maximum likelihood estimates for the model parameters and eventually the sample classification rules.