Bivariate Normal Population Research Articles

Confidence estimation based on data from independent studies.

The problem of finding confidence intervals based on data from several independent studies or experiments is considered. A general method of finding confidence intervals by inverting a combined test is proposed. The combined tests considered are the Fisher test, the weighted inverse normal test, the inverse chi-square test and the inverse Cauchy test. The method is illustrated for finding confidence intervals for a common mean of several normal populations, common correlation coefficient of several bivariate normal populations, common coefficient of variation, common mean of several lognormal populations, and for a common mean of several gamma populations. For each case, the confidence intervals based on the combined tests are compared with the other available approximate confidence intervals with respect to coverage probability and precision. R functions to compute all confidence intervals are provided in a supplementary file. The methods are illustrated using several practical examples.

Estimation after selection from bivariate normal population with application to poultry feeds data

In many practical situations, it is often desired to select a population (treatment, product, technology, etc.) from a choice of several populations on the basis of a particular characteristic that associated with each population, and then estimate the characteristic associated with the selected population. The present paper is focused on estimating a characteristic of the selected bivariate normal population, using a LINEX loss function. A natural selection rule is used for achieving the aim of selecting the best bivariate normal population. Some natural-type estimators and Bayes estimator (using a conjugate prior) of a parameter of the selected population are presented. An admissible subclass of equivariant estimators, using the LINEX loss function, is obtained. Further, a sufficient condition for improving the competing estimators is derived. Using this sufficient condition, several estimators improving upon the proposed natural estimators are obtained. Further, an application of the derived results is provided by considering the poultry feeds data. Finally, a comparative study on the competing estimators of a parameter of the selected population is carried-out using simulation.

Computation of tolerance ellipses for bivariate and trivariate normal populations

We consider the computation of the critical constants c of a p-content -confidence tolerance ellipse for bivariate and trivariate normal populations which are probably the most used multivariate normal distributions in applications. There are only approximation methods available in the statistical literature for computing c. Without knowing the accurate value of c, it is impossible to assess the accuracy of the approximation methods. In this paper, a new method is given to compute c. Although the method is also based on Monte-Carlo simulation, it allows c to be computed as accurate as required if the number of simulation replications is sufficiently large. The R codes provided allow easy implementation of the method. Furthermore, the method allows the accuracy of the available approximation methods for bivariate and trivariate normal distributions to be assessed.

Combining independent tests for a common parameter of several continuous distributions: a new test and power comparisons

The problem of testing a common parameter of several independent continuous populations is considered. Among all tests, Fisher’s combined test is the most popular one and is routinely used in applications. In this article, we propose an alternative method of combining the p-values of independent tests using chi-square scores, referred to as the inverse chi-square test. The proposed test is as simple as other existing tests. We compare the powers of the combined tests for (i) testing a common mean of several normal populations, (ii) testing the common coefficient of variation of several normal populations, (iii) testing the common correlation coefficient of several bivariate normal populations, (iv) testing the common mean of several lognormal populations and (v) testing the common mean of several gamma distributions. Our comparison studies indicate that the inverse chi-square test is a better alternative combined test with good power properties. An illustrative example with real-world data is given for each problem.

Objective priors for common correlation coefficient in bivariate normal populations

Various objective priors have been defined for the common correlation coefficient concerning several bivariate normal populations. In this paper, the proposed approach relies on the asymptotic matching of coverage probabilities corresponding to Bayesian credible intervals considering the corresponding frequentist ones. In the present paper, we focus on several matching criteria including quantile matching, distribution function matching, highest posterior density matching, and matching via inversion of test statistics. In addition, we consider reference priors for different groups of ordering. The proposed methods are investigated and compared between each other in terms of a frequentist coverage probability and then, they are illustrated through a simulation study and two real data examples.

Noninformative priors for the ratio of variabilities in a bivariate normal population

In this paper, we consider some objective priors for the ratio of variabilities in a bivariate normal distribution. We develop the first and second order matching priors and reference priors. We obtain that the second order matching prior matches the alternative coverage probabilities up to the same order. It is also an HPD matching priors. It turns out that the derived reference priors do not satisfy a second order matching criterion. The simulation result shows that the second order matching prior performs better than reference priors based on matching the target coverage probabilities in a frequentist sense. Finally, we show that the second order matching prior and reference priors produce confidence sets with an expected length shorter than the Cox and Reid adjustment.

Estimating Ordered Means of a Bivariate Normal Distribution

SYNOPTIC ABSTRACTThe problem of estimating ordered parameters arises widely in biological, agricultural, economic, and reliability experiments. Consider a bivariate normal population with unknown mean (θ1, θ2), known variances, and known correlation coefficient ρ, where θ1 ⩽ θ2. The problem of estimation of (θ1, θ2) is studied when the loss function used is sum of squared errors. We have considered two cases: equal variances and unequal variances. In both cases, a class of minimax estimators is proposed. These estimators improve upon the usual estimators. A class of admissible estimators is obtained within this class. The minimaxity and admissibility of a generalized Bayes estimator is established. Finally, the risk performance of all the proposed estimators has been compared numerically.

Estimating the parameters of a selected bivariate normal population

This paper deals with estimating a characteristic of the selected bivariate normal population. A minimax estimator of the random parameter and some admissibility results are derived. A sufficient condition for inadmissibility is provided.

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.

Asymptotic properties of Pearson׳s rank-variate correlation coefficient in bivariate normal model

This paper establishes the asymptotic closed forms of the expectation and variance of the Pearson׳s rank-variate correlation coefficient (PRVCC) with respect to samples drawn from bivariate normal populations. The variance-stability features of Fisher׳s z-transform on PRVCC is also investigated under normal assumptions. To gain deeper insight into PRVCC, we further compare PRVCC to other two closely related correlation coefficients, namely, Pearson׳s product moment correlation coefficient (PPMCC) and Gini correlation (GC). Theoretical and simulation results reveal the advantages of PRVCC over PPMCC and/or GC in terms of mathematical tractability and smaller mean square error (MSE) under different circumstances. The newly found theoretical results along with other desirable properties enable PRVCC to be a useful alternative to the existing coefficients, especially when the samples follow distributions whose tails are heavier than that of normal distribution.

Estimating Common Mean of a Bivariate Normal Population with Order Restricted Variances

Let ( X1, Y1), ( X2, Y2),…,( X n, Y n) be a random sample of size n from a bivariate normal population with a common mean μ variances [Formula: see text] and [Formula: see text] and correlation coefficient ρ The estimation of the common mean μ has been considered when it is known a priori that the variances are ordered, that is, [Formula: see text] A new estimator has been proposed which dominates the maximum likelihood estimator (without restrictions) stochastically and also in terms of Pitman measure of closeness. Further an inadmissibility condition has been obtained for affine equivariant estimators when the variances are ordered. A numerical comparison of various estimators has been carried out.

Asymptotic Mean and Variance of Gini Correlation for Bivariate Normal Samples

This paper derives the asymptotic analytical forms of the mean and variance of the Gini correlation (GC) with respect to samples drawn from bivariate normal populations. The asymptotic relative efficiency (ARE) of the Gini correlation to Pearson's product moment correlation coefficient (PPMCC) is investigated under the normal assumptions. To gain further insight into GC, we also compare the Gini correlation to other two closely related correlation coefficients, namely, the order statistics correlation coefficient (OSCC) and Spearman's rho (SR). Theoretical and simulation results suggest that the performance of GC lies in between those of OSCC and SR when estimating the correlation coefficient of the bivariate normal population. The newly found theoretical results along with other desirable properties enable GC to be a useful alternative to the existing coefficients, especially when one wants to make a trade-off between the efficiency and robustness to monotone nonlinearity.

A General Procedure for Estimating the General Parameter Using Auxiliary Information in Presence of Measurement Errors

This article addresses the problem of estimating a family of general population parameter <TEX>${\theta}_{({\alpha},{\beta})}$</TEX> using auxiliary information in the presence of measurement errors. The general results are then applied to estimate the coefficient of variation <TEX>$C_Y$</TEX> of the study variable Y using the knowledge of the error variance <TEX>${\sigma}^2{_U}$</TEX> associated with the study variable Y, Based on large sample approximation, the optimal conditions are obtained and the situations are identified under which the proposed class of estimators would be better than conventional estimator. Application of the main result to bivariate normal population is illustrated.

Noninformative priors for the common mean in the bivariate normal distribution

In this paper, we consider some noninformative priors for the common mean in a bivariate normal population. We develop the first-order and second-order matching priors and reference priors. We find that the second-order matching prior is also an HPD matching prior, and matches the alternative coverage probabilities up to the second order. It turns out that derived reference priors do not satisfy a second-order matching criterion. Our simulation study indicates that the second-order matching prior performs better than the reference priors in terms of matching the target coverage probabilities in a frequentist sense. We also illustrate our results using real data.

Bayesian estimation for a constrained bivariate normal population

In this paper we derive Bayes estimates of the parameters of a bivariate normal population under the constraint of either a common mean or a common variance. Some environmental and medical applications are indicated. Results of a simulation study based on a small sample size to compare the maximum likelihood estimates and the Bayes estimates indicate that they are mostly equivalent.

A mass function based on correlation coefficient and its application

A discrete distribution involving a product of two gamma functions has been proposed. It arises naturally in connection with the distribution of sample correlation coefficient based on a bivariate normal population. The first four factorial moments, raw moments, three corrected moments, coefficient of skewness and kurtosis have been derived. Some illustrations have been provided to show how the product moments of sample variances and correlation can be derived by exploiting the moments of the new distribution. These product moments are important for correlation analysis, covariance analysis, intra-sire regression and inference in bivariate normal population with a common mean.

Asymptotic Properties of Order Statistics Correlation Coefficient in the Normal Cases

We have previously proposed a novel order statistics correlation coefficient (OSCC), which possesses some desirable advantages when measuring linear and monotone nonlinear associations between two signals. However, the understanding of this new coefficient is far from complete. A lot of theoretical questions, such as the expressions of its distribution and moments, remain to be addressed. Motivated by this unsatisfactory situation, in this paper we prove that for samples drawn from bivariate normal populations, the distribution of OSCC is asymptotically equivalent to that of the Pearson's product moment correlation coefficient (PPMCC). We also reveal its close relationships with the other two coefficients, namely, Gini correlation (GC) and Spearman's rho (SR). Monte Carlo simulation results agree with the theoretical findings.

Confidence interval estimation of a common correlation coefficient

This paper presents a generalized variable approach for confidence interval estimation of a common correlation coefficient from several independent samples drawn from bivariate normal populations. This approach can provide one-sided bounds and two-sided confidence intervals with satisfying coverage probabilities regardless of the number of samples, sample sizes and magnitude of the common correlation coefficient while the large sample approach can be very liberal for one-sided bounds. The large sample approach generally performs well for two-sided confidence interval estimation.

From Unit Root to Stein’s Estimator to Fisher’s k Statistics: If You Have a Moment, I Can Tell You More

Any general textbook that discusses moment generating functions (MGFs) shows how to obtain a moment of positive-integer order via differentiation, although usually the presented examples are only illustrative, because the corresponding moments can be calculated in more direct ways. It is thus somewhat unfortunate that very few textbooks discuss the use of MGFs when it becomes the simplest, and sometimes the only, approach for analytic calculation and manipulation of moments. Such situations arise when we need to evaluate the moments of ratios and logarithms, two of the most common transformations in statistics. Such moments can be obtained by differentiating and integrating a joint MGF of the underlying untransformed random variables in appropriate ways. These techniques are examples of multivariate Laplace transform methods and can also be derived from the fact that moments of negative orders can be obtained by integrating an MGF. This article reviews, extends and corrects various results scattered in the literature on this joint-MGF approach, and provides four applications of independent interest to demonstrate its power and beauty. The first application, which motivated this article, is for the exact calculation of the moments of a well-known limiting distribution under the unit-root AR(1) model. The second, which builds on Stigler’s Galtonian perspective, reveals a straightforward, non-Bayesian constructive derivation of the Stein estimator, as well as convenient expressions for studying its risk and bias. The third finds an exceedingly simple bound for the bias of a sample correlation from a bivariate normal population, namely the magnitude of the relative bias is not just of order n−1, but actually is bounded above by n−1 for all sample sizes n≥2. The fourth tackles the otherwise intractable problem of studying the finite-sample optimal bridge in the context of bridge sampling for computing normalizing constants. A by-product of the joint-MGF approach is that positive-order fractional moments can be easily obtained from an MGF without invoking the concept of fractional differentiation, a method used by R. A. Fisher in his study of k statistics 45 years before it reappeared in the probability literature.

Choosing a Better Training Sample for Classifying an Individual into One of Two Correlated Normal Populations

The problem is to classify a unit into one of two populations which jointly constitute a bivariate normal population N2 ( μ1 , μ2, σ 2, σ 2, ρ]. Unlike the usual situation, the populations are not independent but correlated . A matched training sample consists of paired observations only. Earlier studies on classification in correlated setup considered the use of either matched training sample or independent training sample (set of only un-paired observations from each of the two populations). We consider classification problem in correlated setup hased on unbalanced training sample consisting of paired observations as well as single observations from each population. It is observed that probability of correct classification (PCC) of maximum likelihood (ML) rule based on unbalanced training sample is always higher than the minimum of the PCCs based on matched and independent training samples. Moreover, in some cases, it is maxiwnm among the three.