Univariate Normal Populations Research Articles

Shrinkage Testimator for the Common Mean of Several Univariate Normal Populations

The challenge of combining two unbiased estimators is a common occurrence in applied statistics, with significant implications across diverse fields such as manufacturing quality control, medical research, and the social sciences. Despite the widespread relevance of estimating the common population mean μ, this task is not without its challenges. A particularly intricate issue arises when the variations within populations are unknown or possibly unequal. Conventional approaches, like the two-sample t-test, fall short in addressing this problem as they assume equal variances among the two populations. When there exists prior information regarding population variances (σi2,i=1,2), with the consideration that σ12 and σ22 might be equal, a hypothesis test can be conducted: H0:σ12=σ22 versus H1:σ12≠σ22. The initial sample is utilized to test H0, and if we fail to reject H0, we gain confidence in incorporating our prior knowledge (after testing) to estimate the common mean μ. However, if H0 is rejected, indicating unequal population variances, the prior knowledge is discarded. In such cases, a second sample is obtained to compensate for the loss of prior knowledge. The estimation of the common mean μ is then carried out using either the Graybill–Deal estimator (GDE) or the maximum likelihood estimator (MLE). A noteworthy discovery is that the proposed preliminary testimators, denoted as μ^PT1 and μ^PT2, exhibit superior performance compared to the widely used unbiased estimators (GDE and MLE).

Estimating error rate of classification into several normal populations under equal mean restriction

Classification of observation into several univariate normal populations is considered when the population means are unknown but equal. Plug-in Bayes classification rules based on different estimators of the common mean are proposed for k populations. When the variances are ordered, the rule based on the Graybill–Deal estimator is compared with another rule. We prove the consistency property of the classification rules. Confidence intervals of conditional error rate are derived for two and three populations. Under the assumption of ordered variances, Bayes estimator of the ratio of variances is derived to use as a plug-in estimator for classification. We derive estimators of the parameters of mixture densities associated with two normal populations with a common mean and propose classification rules for mixture distribution. An extensive simulation is performed to compare different rules and interval estimators of the conditional error rates.

A New Approach to Constructing Tolerance Limits on Order Statistics in Future Samples Coming from a Normal Distribution

Although the concept of statistical tolerance limits has been well recognized for long time, surprisingly, it seems that their applications remain still limited. Analytic formulas for the tolerance limits are available in only simple cases, for example, for the upper or lower tolerance limit for a univariate normal population. Thus it becomes necessary to use new or innovative approaches which will allow one to construct tolerance limits on future order statistics for many populations. In this paper, a new approach to constructing lower and upper tolerance limits on order statistics in future samples is proposed. Attention is restricted to invariant families of distributions under parametric uncertainty. The approach used here emphasizes pivotal quantities relevant for obtaining tolerance factors and is applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. It does not require the construction of any tables and is applicable whether the past data are complete or Type II censored. The proposed approach requires a quantile of the F distribution and is conceptually simple and easy to use. For illustration, the normal distribution is considered. The discussion is restricted to one-sided tolerance limits. A practical example of finding a warranty assessment of image quality is given.

Admissibility of the Usual Confidence Set for the Mean of a Univariate or Bivariate Normal Population: The Unknown Variance Case

Summary In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi. This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post model selection or post shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location–scale model.

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

ABSTRACTThe 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.

New Riemannian Priors on the Univariate Normal Model

The current paper introduces new prior distributions on the univariate normal model, with the aim of applying them to the classification of univariate normal populations. These new prior distributions are entirely based on the Riemannian geometry of the univariate normal model, so that they can be thought of as “Riemannian priors”. Precisely, if {pθ ; θ ∈ Θ} is any parametrization of the univariate normal model, the paper considers prior distributions G( θ - , γ) with hyperparameters θ - ∈ Θ and γ > 0, whose density with respect to Riemannian volume is proportional to exp(−d2(θ, θ - )/2γ2), where d2(θ, θ - ) is the square of Rao’s Riemannian distance. The distributions G( θ - , γ) are termed Gaussian distributions on the univariate normal model. The motivation for considering a distribution G( θ - , γ) is that this distribution gives a geometric representation of a class or cluster of univariate normal populations. Indeed, G( θ - , γ) has a unique mode θ - (precisely, θ - is the unique Riemannian center of mass of G( θ - , γ), as shown in the paper), and its dispersion away from θ - is given by γ. Therefore, one thinks of members of the class represented by G( θ - , γ) as being centered around θ - and lying within a typical distance determined by γ. The paper defines rigorously the Gaussian distributions G( θ - , γ) and describes an algorithm for computing maximum likelihood estimates of their hyperparameters. Based on this algorithm and on the Laplace approximation, it describes how the distributions G( θ - , γ) can be used as prior distributions for Bayesian classification of large univariate normal populations. In a concrete application to texture image classification, it is shown that this leads to an improvement in performance over the use of conjugate priors.

On the Exact Two-Sided Tolerance Intervals for Univariate Normal Distribution and Linear Regression

Statistical tolerance intervals are another tool for making statistical inference on anunknown population. The tolerance interval is an interval estimator based on the resultsof a calibration experiment, which can be asserted with stated confidence level 1 ? ,for example 0.95, to contain at least a specified proportion 1 ? , for example 0.99, ofthe items in the population under consideration. Typically, the limits of the toleranceintervals functionally depend on the tolerance factors. In contrast to other statisticalintervals commonly used for statistical inference, the tolerance intervals are used relativelyrarely. One reason is that the theoretical concept and computational complexity of thetolerance intervals is significantly more difficult than that of the standard confidence andprediction intervals.In this paper we present a brief overview of the theoretical background and approachesfor computing the tolerance factors based on samples from one or several univariate normal(Gaussian) populations, as well as the tolerance factors for the non-simultaneousand simultaneous two-sided tolerance intervals for univariate linear regression. Such toleranceintervals are well motivated by their applicability in the multiple-use calibrationproblem and in construction of the calibration confidence intervals. For illustration, wepresent examples of computing selected tolerance factors by the implemented algorithmin MATLAB.

Testing Equality of Variances for Multiple Univariate Normal Populations

To test for equality of variances in independent random samples from multiple univariate normal populations, the test of first choice would usually be the likelihood ratio test, the Bartlett test. This test is known to be powerful when normality can be assumed. Here two Wald tests of equality of variances are derived. The first test compares every variance with every other variance and was announced in Mather and Rayner (2002), but no proof was given there. The second test is derived from a quite different model using orthogonal contrasts, but is identical to the first. This second test statistic is similar to one given in Rippon and Rayner (2010), for which no empirical assessment has been given. These tests are compared with the Bartlett test in size and power. The Bartlett test is known to be nonrobust to the normality assumption, as is the orthogonal contrasts test. To deal with this difficulty an analogue of the new test is given. An indicative empirical assessment shows that it is more robust than t...

Generalised Score and Wald Tests

The generalised score and Wald tests are described and related to their nongeneralised versions. Two interesting applications are discussed. In the first a new test for the Behrens-Fisher problem is derived. The second is testing homogeneity of variances from multiple univariate normal populations.

A generalized [formula omitted]-value approach to inference on common mean

Statistical inference for the common mean of several univariate normal populations has many useful applications in practice. In this paper, we develop a generalized p -value approach to derive tests and confidence intervals for the common mean based on several well-known estimates of the common mean. The case of a multivariate normal distribution with a common component mean is also addressed and a generalized p -value approach based on the maximum likelihood estimate of the common mean is developed. We also report results of a simulation study showing powers and coverage probabilities.

Consistent estimation of the minimum normal mean under the tree-order restriction

Let ( X ij ∣ j = 1 , … , n i ( s ) , i = 0 , 1 , … , s ) be independent observations from s + 1 univariate normal populations, with X ij ∼ N ( μ i , σ 2 ) . The tree-order restriction ( μ 0 ⩽ μ i , i = 1 , … , s ) arises naturally when comparing a treatment ( μ 0 ) to several controls ( μ 1 , … , μ s ). When the sample sizes and population means and variances are equal and fixed, the maximum likelihood-based estimator (MLBE) of μ 0 is negatively biased and diverges to - ∞ a.s. as s → ∞ , leading some to assert that maximum likelihood may “fail disastrously” in order-restricted estimation. By viewing this problem as one of estimating a target parameter μ 0 in the presence of an increasing number of nuisance parameters μ 1 , … , μ s , however, this behavior is reminiscent of the classical Neyman–Scott example. This suggests an alternative formulation of the problem wherein the sample size n 0 ( s ) for the target parameter increases with s. Here the MLBE of μ 0 is either consistent or admits a bias-reducing adjustment, depending on the rate of increase of n 0 ( s ) . The consistency of an estimator due to Cohen and Sackrowitz [2002. Inference for the model of several treatments and a control. J. Statist. Plann. Inference 107, 89–101] is also discussed.

On a Comparison of Several Competing Estimates of a Univariate Normal Mean by the Multiple Criteria Decision-Making Method

In this article we consider the problem of estimation of the mean of a univariate normal population with an unknown variance when uncertain nonsample prior information about the mean is available. We compare four estimators of the mean, including pretest and shrinkage estimators. The performances of the estimators are compared based on the multiple criteria decision making (MCDM) procedure in order to find the best estimator.

Biases of the maximum likelihood and Cohen–Sackrowitz estimators for the tree-order model

Consider s + 1 univariate normal populations with common variance σ 2 and means μ i , i = 0 , 1 , … , s , constrained by the tree-order restrictions μ i ⩾ μ 0 , i = 1 , 2 , … , s . For certain sequences μ 0 , μ 1 , … the maximum likelihood-based estimator (MLBE) of μ 0 diverges to - ∞ as s → ∞ and its bias is unbounded. By contrast, the bias of an alternative estimator of μ 0 proposed by Cohen and Sackrowitz (J. Statist. Plan. Infer. 107 (2002) 89–101) remains bounded. In this note the biases of the MLBEs of the other components μ 1 , μ 2 , … are studied and compared to the biases of the corresponding Cohen–Sackrowitz estimators (CSE). Unlike the MLBE of μ 0 , the MLBEs of μ i for i ⩾ 1 , are asymptotically unbiased in most cases. By contrast, the CSEs of μ i , i = 1 , 2 , … , s more often have nonzero asymptotic bias.

4. Sequential Methods for Biomedical Applications

In this paper, we consider a mean-slippage detecting problem in univariate normal populations. By using Robbins' inequality, we construct some invariant confidence sequences for slippage of normal means when the common variance is unknown.

On the comparison of the pre-test and shrinkage estimators for the univariate normal mean

The estimation of the mean of an univariate normal population with unknown variance is considered when uncertain non-sample prior information is available. Alternative estimators are denned to incorporate both the sample as well as the non-sample information in the estimation process. Some of the important statistical properties of the restricted, preliminary test, and shrinkage estimators are investigated. The performances of the estimators are compared based on the criteria of unbiasedness and mean square error in order to search for a ‘best’ estimator. Both analytical and graphical methods are explored. There is no superior estimator that uniformly dominates the others. However, if the non-sample information regarding the value of the mean is close to its true value, the shrinkage estimator over performs the rest of the estimators.

Estimation of the Common Mean of Two Normal Populations Using Two Stage Sampling

We exhibit the superiority of the Graybill- Deal estimator for estimating the common mean of two univariate normal populations with unequal variances, under a two stage sampling scheme. Some properties of the two-stage Graybill-Deal estimator are discussed.

An Alternative Approach to the Anova Problem

When the hypothesis of equality of means of several univariate homoscedastic normal populations from which independent samples are available, is rejected on the basis of the ANOV A F-test, the populations have to be divided into disjoint groups such that within each group the populations are all same. For any such division a natural index of correctness of the grouping is defined and a procedure is proposed such that it leads to the choice of a division for which the true index does not fall below a stipulated threshold at a given level of confidence. The procedure is illustrated through some numerical examples.

On a Class of Admissible Estimators of the Common Mean of Two Univariate Normal Populations

The purpose of this article is to extend a result of Sinha and Mouqadem ( Commun. Stal. Theo. Meth. 11, 1982, 1603-1614), and present a class of admissible estimators of the common mean of two univariate normal populations with unknown unequal variances. An extension of tbis result in the case of two-stage procedures is also briefly discussed.

Estimation of mixing proportions in the presence of autoregressively correlated training data:the case of two univariate normal populations

This paper considers the estimation of mixing proportions when, in addition to the mixture sample, there are available autoregressively dependent data of known origin from each of the classes which make up the mixture population. An asymptotic variance of the usual discriminant analysis (or confusion matrix) estimator of the mixing proportion is obtained under this condition of correlated training data. The maximum likelihood estimator of the mixing proportion and the associated asymptotic variance are also obtained. A simulation experiment is used to investigate the behaviour of these estimators

Combining Independent Tests in Linear Models

Abstract A class of independent linear models is considered, where the variances of the different models are unequal but they have a common vector parameter θ occurring in their mean vectors. For testing the hypothesis H 0: θ = 0, some exact test procedures are derived that combine the information from all the models. The procedures are extensions of the test suggested by Cohen and Sackrowitz for recovering interblock information in balanced incomplete block designs. Simulation results on the power of the new tests and some standard tests are reported in the context of (1) testing the hypothesis concerning the common mean of two univariate normal populations and (2) recovering interblock information in a block design that is not a balanced incomplete block design. The numerical results indicate that the new tests have excellent performance in terms of power compared to some standard tests.