Estimation of two order restricted normal means with unknown and possibly unequal variances

Estimation of each of and linear functions of two order restricted normal means is considered when variances are unknown and possibly unequal. We replace unknown variances with sample variances and construct isotonic regression estimators, which we call in our paper the plug-in estimators, to estimate ordered normal means. Under squared error loss, a necessary and sufficient condition is given for the plug-in estimators to improve upon the unrestricted maximum likelihood estimators uniformly. As for the estimation of linear functions of ordered normal means, we also show that when variances are known, the restricted maximum likelihood estimator always improves upon the unrestricted maximum likelihood estimator uniformly, but when variances are unknown, the plug-in estimator does not always improve upon the unrestricted maximum likelihood estimator uniformly.     

Estimation of two ordered normal means when a covariance matrix is known

Estimation of two normal means with an order restriction is considered when a covariance matrix is known. It is shown that restricted maximum likelihood estimator (MLE) stochastically dominates both estimators proposed by Hwang and Peddada [Confidence interval estimation subject to order restrictions. Ann Statist. 1994;22(1):67–93] and Peddada et al. [Estimation of order-restricted means from correlated data. Biometrika. 2005;92:703–715]. The estimators are also compared under the Pitman nearness criterion and it is shown that the MLE is closer to ordered means than the other two estimators. Estimation of linear functions of ordered means is also considered and a necessary and sufficient condition on the coefficients is given for the MLE to dominate the other estimators in terms of mean squared error.     

A tow-stage procedure for testing homogeneity of means against ordered alternatives in analysis of variance with unequal variances

In this paper we present a two-stage sampling procedure for testing the equality of normal means against ordered alternatives in one-way analysis of variance with unequal unknown variances. A table of approximated percentiles needed for implementation is provided. Some Monte Carlo results for estimating the power of the proposed test statistic are presented.     

PITMAN NEARNESS COMPARISONS OF ESTIMATES OF TWO ORDERED NORMAL MEANS

Maximum likelihood estimates of ordered means of two normal distributions having common variance have been shown to be better than the usual maximum likelihood estimates (i.e. corresponding sample means) with respect to Pitman Nearness criterion. The maximum likelihood estimate of common variance taking into consideration the order restriction of the means is shown to have smaller mean square error than the unrestricted maximum likelihood estimate of the common variance. These two estimators have also been compared with respect to Pitman Nearness criterion.     

Estimation of cumulative incidence functions when the lifetime distributions are uniformly stochastically ordered

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of risk in the presence of other risks. Assume that the lifetime distributions of two populations are uniformly stochastically ordered. Since this ordering may not hold for the empiricals due to sampling variability, it is natural to estimate these distributions under this constraint. This will in turn affect the estimation of the CIFs. This article considers this estimation problem. We do not assume that the risk sets in the two populations are related, give consistent estimators of all the CIFs and study the weak convergence of the resulting processes. We also report the results of a simulation study that show that our restricted estimators outperform the unrestricted ones in terms of mean square error. A real life example is used to illustrate our theoretical results.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Proper Bayes minimax estimators of the normal mean matrix with common unknown variances

This paper addresses the problem of estimating a matrix of the normal means, where the variances are unknown but common. The approach to this problem is provided by a hierarchical Bayes modeling for which the first stage prior for the means is matrix-variate normal distribution with mean zero matrix and a covariance structure and the second stage prior for the covariance is similar to Jeffreys’ rule. The resulting hierarchical Bayes estimators relative to the quadratic loss function belong to a class of matricial shrinkage estimators. Certain conditions are obtained for admissibility and minimaxity of the hierarchical Bayes estimators.     

MODEL-BASED VARIANCE ESTIMATION IN SURVEYS WITH STRATIFIED CLUSTERED DESIGN

A model-based method for estimating the sampling variances of estimators of (sub-)population means, proportions, quantiles, and regression parameters in surveys with stratified clustered design is described and applied to a survey of US secondary education. The method is compared with the jackknife by a simulation study. The model-based estimators of the sampling variances have much smaller mean squared errors than their jackknife counterparts. In addition, they can be improved by incorporating information about the unknown parameters (variances) from external sources. A regression-based smoothing method for estimating the sampling variances of the estimators for a large number of subpopulation means is proposed. Such smoothing may be invaluable when subpopulations are represented in the sample by only few subjects.     

Optimal estimation for finite population mean in two-phase sampling

Using two-phase sampling scheme, we propose a general class of estimators for finite population mean. This class depends on the sample means and variances of two auxiliary variables. The minimum variance bound for any estimator in the class is provided (up to terms of ordern ⁻¹). It is also proved that there exists at least a chain regression type estimator which reaches this minimum. Finally, it is shown that other proposed estimators can reach the minimum variance bound, i.e. the optimal estimator is not unique.     

Improved estimation of quantiles of two normal populations with common mean and ordered variances

Abstract

Estimation of quantiles from two normal populations is considered under the assumption of common mean and ordered variances. Several new estimators have been proposed using certain estimators of the common mean, including the plug-in type restricted MLE. A sufficient condition for improving equivariant estimators is proved and as a result improved estimators are derived. The percentage of risk improvements for each of the improved estimators have been computed numerically, which are quite significant. All the improved estimators have been compared numerically using Monte-Carlo simulation method. Finally, recommendations have been made for the use of estimators in practice.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

Optimal domain estimation under summation restriction

The domain estimators that do not sum up to the population total (estimated or known) are considered. In order to achieve their additivity, the theory of the general restriction (GR)-estimator [Knottnerus P., 2003. Sample Survey Theory: Some Pythagorean Perspectives. Springer, New York] is used. The elaborated domain GR-estimators are optimal, they have the minimum variance in a class of estimators that satisfy summation restriction. Furthermore, their variances are smaller than the variances of the corresponding initial domain estimators. The variance/covariance formulae of the domain GR-estimators are explicitly given.The ratio estimators as representatives of the non-additive domain estimators are considered. Their design-based covariance matrix, being crucial for the GR-estimator, is presented. Its structure simplifies under certain assumptions on sampling design (and population model). The corresponding simpler forms of the domain GR-estimators are elaborated as well. The hypergeometric [Traat I., Ilves M., 2007. The hypergeometric sampling design, theory and practice. Acta Appl. Math. 97, 311–321] and the simple random sampling designs are considered in more detail. The results are illustrated in a simulation study where the optimal domain estimator displays its superiority among other meaningful domain estimators. It is noteworthy that due to the imposed restrictions also these other estimators, though not optimal, can be much more precise than the initial estimators.     

Estimation of the common mean of two univariate normal populations

The problem of estimating the common mean μ of two univariate normal populations with unknown and unequal variances is considered from a decision-theoretic point of view. We restrict our attention to an appropriate class C and its three subclasses C_0C1C2of un-biased estimates of μ. We consider the usual estimate μ⁰ of μ which is the weighted linear combination of the sample means with weights as reciprocals of the sample variances. Its admissibility in C₀ and extended admissibility in C is proved. Admissible estimates in C₁ and C₂are also obtained.The loss is always assumed to be squared error. The question of admissibility of μ⁰ in the class of all estimators is still open.     

New types of shrinkage estimators of Poisson means under the normalized squared error loss

In estimating p( ? 2) independent Poisson means, Clevenson and Zidek (1975) have proposed a class of estimators that shrink the unbiased estimator to the origin and dominate the unbiased one under the normalized squared error loss. This class of estimators was subsequently enlarged in several directions. This article deals with the problem and proposes new classes of dominating estimators using prior information pertinently. Dominance is shown by partitioning the sample space into disjoint subsets and averaging the loss difference over each subset. Estimation of several Poisson mean vectors is also discussed. Further, simultaneous estimation of Poisson means under order restriction is treated and estimators which dominate the isotonic regression estimator are proposed for some types of order restrictions.     

Comparison of dispersions of mixtures of ordered distributions

In the paper we consider mixtures of unknown stochastically ordered distribution functions according to known mixing distribution functions. We provide optimal lower and upper bounds on ratios of general dispersion measures of such mixtures. The bounds do not depend on the particular form of dispersion measure. We present applications of the results in reliability theory, insurance mathematics, Bayesian statistics, and regression analysis.     

Asymptotic variances of maximum likelihood estimator for the correlation coefficient from a BVN distribution with one variable subject to censoring

This paper deals with the problem of estimating the Pearson correlation coefficient when one variable is subject to left or right censoring. In parallel to the classical results on the Pearson correlation coefficient, we derive a workable formula, through tedious computation and intensive simplification, of the asymptotic variances of the maximum likelihood estimators in two cases: (1) known means and variances and (2) unknown means and variances. We illustrate the usefulness of the asymptotic results in experimental designs.     

Variance estimation in the analysis of microarray data

Summary.  Microarrays are one of the most widely used high throughput technologies. One of the main problems in the area is that conventional estimates of the variances that are required in the t -statistic and other statistics are unreliable owing to the small number of replications. Various methods have been proposed in the literature to overcome this lack of degrees of freedom problem. In this context, it is commonly observed that the variance increases proportionally with the intensity level, which has led many researchers to assume that the variance is a function of the mean. Here we concentrate on estimation of the variance as a function of an unknown mean in two models: the constant coefficient of variation model and the quadratic variance–mean model. Because the means are unknown and estimated with few degrees of freedom, naive methods that use the sample mean in place of the true mean are generally biased because of the errors-in-variables phenomenon. We propose three methods for overcoming this bias. The first two are variations on the theme of the so-called heteroscedastic simulation–extrapolation estimator, modified to estimate the variance function consistently. The third class of estimators is entirely different, being based on semiparametric information calculations. Simulations show the power of our methods and their lack of bias compared with the naive method that ignores the measurement error. The methodology is illustrated by using microarray data from leukaemia patients.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

SEMIPARAMETRIC COMPARISON OF REGRESSION CURVES VIA NORMAL LIKELIHOODS

Härdle & Marron (1990) treated the problem of semiparametric comparison of nonparametric regression curves by proposing a kernel-based estimator derived by minimizing a version of weighted integrated squared error. The resulting estimators of unknown transformation parameters are n-consistent, which prompts a consideration of issues. of optimality. We show that when the unknown mean function is periodic, an optimal nonparametric estimator may be motivated by an elegantly simple argument based on maximum likelihood estimation in a parametric model with normal errors. Strikingly, the asymptotic variance of an optimal estimator of θ does not depend at all on the manner of estimating error variances, provided they are estimated n-consistently. The optimal kernel-based estimator derived via these considerations is asymptotically equivalent to a periodic version of that suggested by Härdle & Marron, and so the latter technique is in fact optimal in this sense. We discuss the implications of these conclusions for the aperiodic case.