Optimal asymmetric classification procedures for interval-screened normal data

Statistical methods for an asymmetric normal classification do not adapt well to the situations where the population distributions are perturbed by an interval-screening scheme. This paper explores methods for providing an optimal classification of future samples in this situation. The properties of the screened population distributions are considered and two optimal regions for classifying the future samples are obtained. These developments yield yet other rules for the interval-screened asymmetric normal classification. The rules are studied from several aspects such as the probability of misclassification, robustness, and estimation of the rules. The investigation of the performance of the rules as well as the illustration of the screened classification idea, using two numerical examples, is also considered.     

Sampling strategy for optimal classification into one of two correlated normal populations

A unit ω is to be classified into one of two correlated homoskedastic normal populations by linear discriminant function known as W classification statistic [T.W. Anderson, An asymptotic expansion of the distribution of studentized classification statistic, Ann. Statist. 1 (1973), pp. 964–972; T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd edn, Wiley, New York, 1984; G.J. Mclachlan, Discriminant Analysis and Statistical Pattern Recognition, John Wiley and Sons, New York, 1992]. The two populations studied here are two different states of the same population, like two different states of a disease where the population is the population of diseased patient. When a sample unit is observed in both the states (populations), the observations made on it (which form a pair) become correlated. A training sample is unbalanced when not all sample units are observed in both the states. Paired and also unbalanced samples are natural in studies related to correlated populations. S. Bandyopadhyay and S. Bandyopadhyay [Choosing better training sample for classifying an individual into one of two correlated normal populations, Calcutta Statist. Assoc. Bull. 54(215–216) (2003), pp. 167–180] studied the effect of unbalanced training sample structure on the performance of W statistics in the univariate correlated normal set-up for finding optimal sampling strategy for a better classification rate. In this study, the results are extended to the multivariate case with discussion on application in real scenario.     

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.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

Robustness of locally most powerful invariant test for normal mixture model in control and treatment populations

Two independent samples from control with N(μ₁, σ²) and treatment with pN(μ₁, σ²) + (1 − p)N(μ₂, σ²) are considered. A locally most powerful invariant test for testing H₀: μ₁ = μ₂ against H₁ : μ₂ > μ₁, where σ² > 0, 0 < p < 1 are unknown, is obtained. Also, the robustness of the test statistic on the lines of Kariya and Sinha (Robustness of Statistical Tests (1989). Academic Press, New York) is studied.     

Bayesian two-stage optimal design for mixture models

In this paper, a Bayesian two-stage D–D optimal design for mixture experimental models under model uncertainty is developed. A Bayesian D-optimality criterion is used in the first stage to minimize the determinant of the posterior variances of the parameters. The second stage design is then generated according to an optimalityprocedure that collaborates with the improved model from the first stage data. The results show that a Bayesian two-stage D–D-optimal design for mixture experiments under model uncertainty is more efficient than both the Bayesian one-stage D-optimal design and the non-Bayesian one-stage D-optimal design in most situations. Furthermore, simulations are used to obtain a reasonable ratio of the sample sizes between the two stages.     

An online classification EM algorithm based on the mixture model

Mixture model-based clustering is widely used in many applications. In certain real-time applications the rapid increase of data size with time makes classical clustering algorithms too slow. An online clustering algorithm based on mixture models is presented in the context of a real-time flaw-diagnosis application for pressurized containers which uses data from acoustic emission signals. The proposed algorithm is a stochastic gradient algorithm derived from the classification version of the EM algorithm (CEM). It provides a model-based generalization of the well-known online k-means algorithm, able to handle non-spherical clusters. Using synthetic and real data sets, the proposed algorithm is compared with the batch CEM algorithm and the online EM algorithm. The three approaches generate comparable solutions in terms of the resulting partition when clusters are relatively well separated, but online algorithms become faster as the size of the available observations increases.     

Random search algorithm for optimal mixture experimental design

It is well known that it is difficult to obtain an accurate optimal design for a mixture experimental design with complex constraints. In this article, we construct a random search algorithm which can be used to find the optimal design for mixture model with complex constraints. First, we generate an initial set by the Monte-Carlo method, and then run the random search algorithm to get the optimal set of points. After that, we explain the effectiveness of this method by using two examples.     

An estimator of the common mean of two normal populations

In this paper we consider the estimation of the common mean of two normal populations when the variances are unknown. If it is known that one specified variance is smaller than the other, then it is possible to modify the Graybill-Deal estimator in order to obtain a more efficient estimator. One such estimator is proposed by Mehta and Gurland (1969). We prove that this estimator is more efficient than the Graybill-Deal estimator under the condition that one variance is known to be less than the other.     

Discretisation for inference on normal mixture models

The problem of inference in Bayesian Normal mixture models is known to be difficult. In particular, direct Bayesian inference (via quadrature) suffers from a combinatorial explosion in having to consider every possible partition of n observations into k mixture components, resulting in a computation time which is O(k ⁿ). This paper explores the use of discretised parameters and shows that for equal-variance mixture models, direct computation time can be reduced to O(D ^k n ^k), where relevant continuous parameters are each divided into D regions. As a consequence, direct inference is now possible on genuine data sets for small k, where the quality of approximation is determined by the level of discretisation. For large problems, where the computational complexity is still too great in O(D ^k n ^k) time, discretisation can provide a convergence diagnostic for a Markov chain Monte Carlo analysis.     

On some optimal selection procedures for weibull populations

Consider k (k >(>)2) Weibull populations. We shall derive a method of constructing optimal selection procedures to select a subset of the k populations containing the best population which control the size of the selected subset and which maximises the minimum probability of making a correct selection. Procedures and results are derived for the case when sample sizes are unequal. Some tables and figures are given at the end of this paper.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

Estimation of scale parameters in mixture distributions

Simultaneous estimation of scale parameters is considered in mixture distributions under squared-error loss. A general class of estimators is obtained which dominates the componentwise best multiple estimators and the moment estimators. As special cases, improved estimators are obtained for the multivariate t-distribution and the p-variate Lomax distribution.     

Evaluating subject-level incremental values of new markers for risk classification rule

Suppose that we need to classify a population of subjects into several well-defined ordered risk categories for disease prevention or management with their “baseline” risk factors/markers. In this article, we present a systematic approach to identify subjects using their conventional risk factors/markers who would benefit from a new set of risk markers for more accurate classification. Specifically for each subgroup of individuals with the same conventional risk estimate, we present inference procedures for the reclassification and the corresponding correct re-categorization rates with the new markers. We then apply these new tools to analyze the data from the Cardiovascular Health Study sponsored by the US National Heart, Lung, and Blood Institute. We used Framingham risk factors plus the information of baseline anti-hypertensive drug usage to identify adult American women who may benefit from the measurement of a new blood biomarker, CRP, for better risk classification in order to intensify prevention of coronary heart disease for the subsequent 10 years.     

Linear discrimination for three known normal populations

A random vector is assumed to have one of three known multivariate normal distributions with equal covariance matrices. It is desired to separate the three distributions by means of a single linear discriminant function. Such a function can lead to a classification rule. The function whose classification rule minimizes the average of the three probabilities of misclassification is found. Also the function is found whose rule minimizes the maximum of the three probabilities of misclassification.     

Testing equality of variances for several normal populations

In this study, we develop a test based on computational approach for the equality of variances of several normal populations. The proposed method is numerically compared with the existing methods. The numeric results demonstrate that the proposed method performs very well in terms of type I error rate and power of test. Furthermore we study the robustness of the tests by using simulation study when the underlying data are from t, exponential and uniform distributions. Finally we analyze a real dataset that motivated our study using the proposed test.     

A nonparametric plug-in rule for selecting optimal block lengths for block bootstrap methods

In this paper, we consider the problem of empirical choice of optimal block sizes for block bootstrap estimation of population parameters. We suggest a nonparametric plug-in principle that can be used for estimating ‘mean squared error’-optimal smoothing parameters in general curve estimation problems, and establish its validity for estimating optimal block sizes in various block bootstrap estimation problems. A key feature of the proposed plug-in rule is that it can be applied without explicit analytical expressions for the constants that appear in the leading terms of the optimal block lengths. Furthermore, we also discuss the computational efficacy of the method and explore its finite sample properties through a simulation study.