Inferences on the common mean of several normal populations under heteroscedasticity

In this paper, we consider the problem of making inferences on the common mean of several normal populations when sample sizes and population variances are possibly unequal. We are mainly concerned with testing hypothesis and constructing confidence interval for the common normal mean. Several researchers have considered this problem and many methods have been proposed based on the asymptotic or approximation results, generalized inferences, and exact pivotal methods. In addition, Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) proposed a parametric bootstrap (PB) approach for this problem based on the maximum likelihood estimators. We also propose a PB approach for making inferences on the common normal mean under heteroscedasticity. The advantages of our method are: (i) it is much simpler than the PB test proposed by Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) since our test statistic is not based on the maximum likelihood estimators which do not have explicit forms, (ii) inverting the acceptance region of test yields a genuine confidence interval in contrast to some exact methods such as the Fisher’s method, (iii) it works well in terms of controlling the Type I error rate for small sample sizes and the large number of populations in contrast to Chang and Pal (Comput Stat Data Anal 53:321–333, 2008) method, (iv) finally, it has higher power than recommended methods such as the Fisher’s exact method.     

Hypothesis testing for the common mean of two normal distributions in the presence of an indifference zone

Summary LetX ₁,...,X _m andY _t,...,Y be independent, random samples from populations which are N(θ,σ _x ² ) and N(θ,σ _y ² ), respectively, with all parameters unknown. In testingH ₀:θ=0 againstH ₁:θ≠0, thet-test based upon either sample is known to be admissible in the two-sample setting. If, however, one testsH ₀ againstH ₁:|θ|≧ε>0, with ε arbitrary, our main results show: (i) the construction of a test which is better than the particulart-test chosen, (ii) eacht-test is admissible under the invariance principle with respect to the group of scale changes, and (iii) there does not exist a test which simultaneously is better than botht-tests.     

On the common mean of several inverse Gaussian distributions based on a higher order likelihood method

An interval estimation method for the common mean of several heterogeneous inverse Gaussian (IG) populations is discussed. The proposed method is based on a higher order likelihood-based procedure. The merits of the proposed method are numerically compared with the signed log-likelihood ratio statistic, two generalized pivot quantities and the simple t-test method with respect to their expected lengths, coverage probabilities and type I errors. Numerical studies show that the coverage probabilities of the proposed method are very accurate and type I errors are close to the nominal level.05 even for very small samples. The methods are also illustrated with two examples.     

Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution

This paper deals with the problem of estimating the mean matrix in an elliptically contoured distribution with unknown scale matrix. The Laplace and inverse Laplace transforms of the density allow us not only to evaluate the risk function with respect to a quadratic loss but also to simplify expressions of Bayes estimators. Consequently, it is shown that generalized Bayes estimators against shrinkage priors dominate the unbiased estimator.     

A note on testing two-dimensional normal mean

Summary For the problem of testing a composite hypothesis with one-sided alternatives of the mean vector of a two-dimensional normal distribution, a characterization of similar tests is presented and an unbiased test dominating the likelihood ratio test is proposed. A sufficient condition for admissibility is given, which implies the result given by Cohen et al. (1983,Studies in Econometrics, Time Series and Multivariate Statistics, Academic Press): the admissibility of the likelihood ratio test.     

Asymptotic distributions in the testing and estimation of the missing-data multivariate normal linear patterned mean and correlation matrix

Techniques used by Szatrowski (1979, 1983) to solve the testing and estimation problem for linear patterned covariance are used to obtain results for the linear patterned correlation problem in the presence of missing data. Iterative algorithms are given for finding the maximum-likelihood estimates (MLE). Asymptotic distributions of the MLE and likelihood-ratio statistics (LRS) are obtained using the delta method.     

Estimation of a common multivariate normal mean vector

Let X ₁,..., X _N be independent observations from N _p (, ₁) and Y ₁,..., Y _N be independent observations from N _p (, ₂). Assume that X _i 's and Y _i 's are independent. An unbiased estimator of which dominates the sample mean % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbae% baaaa!3A32!\[\bar X\]for p1 under the loss function L(, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBga% qcaaaa!3B03!\[\hat \mu\]) = (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBga% qcaaaa!3B03!\[\hat \mu\]– )^–1 ₁(% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH8oqBga% qcaaaa!3B03!\[\hat \mu\]– ) is suggested. The exact risk (under L) of the new estimator is also evaluated.     

Estimation of a common mean of several univariate inverse Gaussian populations

The problem of estimating the common mean of k independent and univariate inverse Gaussian populations IG(, _i ), i=1,..., k with unknown and unequal 's is considered. The difficulty with the maximum likelihood estimator of is pointed out, and a natural estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] of along the lines of Graybill and Deal is proposed. Various finite sample properties and some decision-theoretic properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] are discussed.This research was partially supported by research grants #A3661 and #A3450 from NSERC of Canada.     

Minimax estimators of the multinormal mean: Autoregressive priors

Empirical Bayes estimators are given for the mean of a k-dimensional normal distribution, k ≥ 3. We assume that y ～ N_k(θ, V₁), V₁ = diag(v_i), v_i known (i = 1, 2,…, k); also, θ ～ N_k(0, V₂) ? V₂ defined by one or more unknown parameters. Of particular interest is V₂ generated by an autoregressive process. A recent result of Efron and Morris is used to obtain necessary and sufficient conditions for the minimaxity of our estimators. Practical sufficient conditions (for minimaxity) are obtained by exploiting the structure of V₂. Another result shows that our estimators have good Bayesian properties. Estimates of the exact size of Pearson's chi-square test are given in an example; the autoregressive prior is very natural in this situation.     

Bayesian priors based on a parameter transformation using the distribution function

One of the tasks of the Bayesian consulting statistician is to elicit prior information from his client who may be unfamiliar with parametric statistical models. In some cases it may be more illuminating to base a prior distribution for parameter on the transformed version F(/), where F is the data distribution function and v is a designated reference value, rather than on directly. This approach is outlined and explored in various directions to assess its implications. Some applications are given, including general linear regression and transformed linear models.     

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69-78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660-671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318-1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.     

Two testing problems relating the real and complex multivariate normal distributions

The first problem considered is that of testing for the reality of the covariance matrix of a p-dimensional complex normal distribution, while the second is that of testing that a 2p-dimensional real normal distribution has a p-dimensional complex structure. Both problems are reduced by invariance to their maximal invariant statistics, and the null and non-null distributions of these are obtained. Complete classes of unbiased, invariant tests are described for both problems, the locally most powerful invariant tests are obtained, and the admissibility of the likelihood ratio tests is established.     

The effects of nonnormality on asymptotic distributions of some likelihood ratio criteria for testing covariance structures under normal assumption

This paper examines asymptotic distributions of the likelihood ratio criteria, which are proposed under normality, for several hypotheses on covariance matrices when the true distribution of a population is a certain nonnormal distribution. It is well known that asymptotic distributions of test statistics depend on the fourth moments of the true population's distribution. We study the effects of nonnormality on the asymptotic distributions of the null and nonnull distributions of likelihood ratio criteria for covariance structures.     

Minimax estimation of the mean of spherically symmetric distributions under general quadratic loss

For X one observation on a p-dimensional (p ≥ 4) spherically symmetric (s.s.) distribution about θ, minimax estimators whose risks dominate the risk of X (the best invariant procedure) are found with respect to general quadratic loss, L(δ, θ) = (δ − θ)′ D(δ − θ) where D is a known p × p positive definite matrix. For C a p × p known positive definite matrix, conditions are given under which estimators of the form δ_a,r,C,D(X) = (I − (ar(|X|²)) D^−1/2CD^1/2 |X|⁻²)X are minimax with smaller risk than X. For the problem of estimating the mean when n observations X₁, X₂, …, X_n are taken on a p-dimensional s.s. distribution about θ, any spherically symmetric translation invariant estimator, δ(X₁, X₂, …, X_n), with have a s.s. distribution about θ. Among the estimators which have these properties are best invariant estimators, sample means and maximum likelihood estimators. Moreover, under certain conditions, improved robust estimators can be found.     

Proper bayes minimax estimators for a multivariate normal mean with unknown common variance under a convex loss function

Summary Let X ∼ N_p(μ,σ²I_p) and let s/σ² ∼ χ _n ² , independent ofX, where μ and σ² are unknown. This paper considers the estimation of μ (by δ) relative to a convex loss function given by (δ−μ)′[(1−α)I_p/σ²+αQ](δ−μ)/[(1−α)p/σ²+α tr (Q)], whereQ is a knownp×p diagonal matrix and 0≦α≦1. Two classes of minimax estimators are obtained for μ whenp≦3; the first is a new result and the second is a generalization of a result of Strawderman (1973,Ann. Statist.,1, 1189–1194). A proper Bayes estimator is also obtained which is shown to satisfy the conditions of the second class of minimax estimators. The paper concludes by discussing the estimation of μ relative to another convex loss function. This work was supported by the Army, Navy and Air Force under Office of Naval Research Contract No. N00014-80-C-0093. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

Bayesian predictive densities based on superharmonic priors for the 2-dimensional Wishart model

Bayesian predictive densities for the 2-dimensional Wishart model are investigated. The performance of predictive densities is evaluated by using the Kullback–Leibler divergence. It is proved that a Bayesian predictive density based on a prior exactly dominates that based on the Jeffreys prior if the prior density satisfies some geometric conditions. An orthogonally invariant prior is introduced and it is shown that the Bayesian predictive density based on the prior is minimax and dominates that based on the right invariant prior with respect to the triangular group.