Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

Pooling means under uncertain prior information with application to discrete distributions

The problem of pooling means is considered based on two samples in presence of the uncertain prior information that these samples are taken from possibly identical populations. Two discrete models, Poisson and binomial are considered in particular. Three estimators, i.e. the unrestricted estimator, shrinkage restricted estimator and estimators based on preliminary test are proposed. Their asymptotic mean squared errors are derived and compared. It is demonstrated via asymptotic results that the range of the parameter space in which shrinkage preliminary test estimator dominates the unrestricted estimator is wider than that of the usual preliminary test estimator. A Monte Carlo study for Poisson model is presented to compare the performance of the estimators for small samples.     

Estimation of regression coefficient from survey data based on test of significance

The ordinary least squares (OLS)estimator of regression coeffecient is implicitly based on I.I.D.assumption, which is rarely satisfied by survey data. Many approaches are proposed in the literature which can be classified in two broad categories as model based and design consistent.Du Mouchel and Duncan (1983) proposed a test statistic λwhich helps in testing the ignorability of sampling weights.In this article a preliminary test estimator based on λ is proposed. The model based properties of this estimator has been invetigated theoritically where as to study the design based properties simulation approach is adopted. It has been observed that the proposed estimator is a better cimpromise between model based and randomization based inferential frame work.     

Modified stationarity tests with improved power in small samples

In a recent paper Kwiatkowski et al. (1992) propose the so-called KPSS statistic for testing the null hypothesis of stationarity against the alternative of a unit root process. The statistic employs a spectral estimator which can be shown to diverge with increasing sample size, given the alternative is true. Here, we suggest a modified spectral estimator which is shown to stabilize for moving average models. It is shown that this test statistic uniformly outperforms the KPSS statistic in an MA(1) model. Furthermore, a two-step nonparametric correction procedure is suggested, giving a test statistic with similar asymptotic properties as the original KPSS statistic. However, in small samples this correction performs better especially in detecting large random walk components. This paper was written while the author was a post-doctoral fellow at the University of Amsterdam. The author likes to thank Peter Boswijk, Inge van den Doel, Noud van Giersbergen and Jan F.Kiviet for their help during that time. Moreover, I would like to thank an anonymous referee for a number of helpful comments.     

An Approach for Estimating Causal Effects in Randomized Trials with Noncompliance

We developed methods for estimating the causal risk difference and causal risk ratio in randomized trials with noncompliance. The developed estimator is unbiased under the assumption that biases due to noncompliance are identical between both treatment arms. The biases are defined as the difference or ratio between the expectations of potential outcomes for a group that received the test treatment and that for the control group in each randomly assigned group. Although the instrumental variable estimator yields an unbiased estimate under a sharp null hypothesis but may yield a biased estimate under a non-null hypothesis, the bias of the developed estimator does not depend on whether this hypothesis holds. Then the estimate of the causal effect from the developed estimator may have a smaller bias than that from the instrumental variable estimator when the treatment effect exists. There is not yet a standard method for coping with noncompliance, and thus it is important to evaluate estimates under different assumptions. The developed estimator can serve this purpose. Its application to a field trial for coronary heart disease is provided.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

Preliminary test and Bayes Estimation of a Location Parameter Under Blinex Loss

In this article, the preliminary test estimator is considered under the BLINEX loss function. The problem under consideration is the estimation of the location parameter from a normal distribution. The risk under the null hypothesis for the preliminary test estimator, the exact risk function for restricted maximum likelihood and approximated risk function for the unrestricted maximum likelihood estimator, are derived under BLINEX loss and the different risk structures are compared to one another both analytically and computationally. As a motivation on the use of BLINEX rather than LINEX, the risk for the preliminary test estimator under BLINEX loss is compared to the risk of the preliminary test estimator under LINEX loss and it is shown that the LINEX expected loss is higher than BLINEX expected loss. Furthermore, two feasible Bayes estimators are derived under BLINEX loss, and a feasible Bayes preliminary test estimator is defined and compared to the classical preliminary test estimator.     

A note on the Zhang omnibus test for normality based on the Q statistic

A discussion about the estimators proposed by Zhang (1999) for the true standard deviation σof a normal distribution is presented. Those estimators, called by Zhang q 1 and q 2 , are functions of the expected values of the order statistics from a standard normal distribution and they were the basis of the Q statistic used in the derivation of a new test for normality proposed by Zhang. Although the type I error and the power of the test was discussed by Zhang, no study was performed to test the reliability of q 1 and q 2 as estimators of σ. In this paper, it is shown that q 1 is a very poor estimator for σespecially when σis large. On the other hand, the estimator q 2 has a performance very similar to the well-known sample standard deviation S. When some correlation is introduced among the sample units it can be seen that the estimator q 1 is much more affected than the estimators q 2 and S.     

A large-scale monte carlo study of the buckley-james estimator with censored data

Estimation in the presence of censoring is an important problem. In the linear model, the Buckley-James method proceeds iteratively by estimating the censored values than re-estimating the regression coeffi- cients. A large-scale Monte Carlo simulation technique has been developed to test the performance of the Buckley-James (denoted B-J) estimator. One hundred and seventy two randomly generated data sets, each with three thousand replications, based on four failure distributions, four censoring patterns, three sample sizes and four censoring rates have been investigated, and the results are presented. It is found that, except for Type I1 censoring, the B-J estimator is essentially unbiased, even when the data sets with small sample sizes are subjected to a high censoring rate. The variance formula suggested by Buckley and James (1979) is shown to be sensitive to the failure distribution. If the censoring rate is kept constant along the covariate line, the sample variance of the estimator appears to be insensitive to the censoring pattern with a selected failure distribution. Oscillation of the convergence values associated with the B-J estimator is illustrated and thoroughly discussed.     

A nonparametric measure of interclass correlation

A nonparametric measure of interclass correlation is considered and its unbiased estimator and a test based on the estimator are studied. Hie measure is an analogue of the Kendall's measure of dependence. It is shown that the variance of the estimator is small and the information loss of the test based on the estimator is not serious relative to a standard parametric test in the sense of the Pitman asymptotic relative efficiency. Furthermore, the approximate variance of the estimator is given in the normal model.     

Estimation in second order branching processes with application to swine flu data

Abstract

This paper discusses inferential issues related to estimation of offspring mean and variance in a second order branching process, when both the offspring distributions are assumed to have identical mean and variance. Estimating equation approach is used to find the estimator of the offspring mean and the fact that a second order branching process model can be modeled as an autoregressive process is utilized to obtain the estimator of the offspring variance. Both the estimators are shown to be consistent and asymptotically normal. The second order branching process model is applied to H1N1 data for Pune, India, and Mexico and is found to be a suitable model. The estimates obtained from this model are used to compute the proportion of vaccination required for elimination of the disease.     

Pseudo‐R2 statistics under complex sampling

Model summaries based on the ratio of fitted and null likelihoods have been proposed for generalised linear models, reducing to the familiar R² coefficient of determination in the Gaussian model with identity link. In this note I show how to define the Cox–Snell and Nagelkerke summaries under arbitrary probability sampling designs, giving a design‐consistent estimator of the population model summary. It is also shown that for logistic regression models under case–control sampling the usual Cox–Snell and Nagelkerke R² are not design‐consistent, but are systematically larger than would be obtained with a cross‐sectional or cohort sample from the same population, even in settings where the weighted and unweighted logistic regression estimators are similar or identical. Implementation of the new estimators is straightforward and code is provided in R.     

Testing the Equality of the Noncentrality Parameters of Two Noncentral t-Distributions with Identical Degrees of Freedom

The problem of testing the equality of the noncentrality parameters of two noncentral t-distributions with identical degrees of freedom is considered, which arises from the comparison of two signal-to-noise ratios for simple linear regression models. A test procedure is derived that is guaranteed to maintain Type I error while having only minimal amounts of conservativeness, and comparisons are made with several other approaches to this problem based on variance stabilizing transformations. The new procedure derived in this article is shown to have good properties and will be useful for practitioners.     

Estimation of Several Intraclass Correlation Coefficients

An intraclass correlation coefficient observed in several populations is estimated. The basis is a variance-stabilizing transformation. It is shown that the intraclass correlation coefficient from any elliptical distribution should be transformed in the same way. Four estimators are compared. An estimator where the components in a vector consisting of the transformed intraclass correlation coefficients are estimated separately, an estimator based on a weighted average of these components, a pretest estimator where the equality of the components is tested and then the outcome of the test is used in the estimation procedure, and a James-Stein estimator which shrinks toward the mean.     

A Simple Approach to Inference in Random Coefficient Models

Random coefficient regression models have been used to analyze cross-sectional and longitudinal data in economics and growth-curve data from biological and agricultural experiments. In the literature several estimators, including the ordinary least squares and the estimated generalized least squares (EGLS), have been considered for estimating the parameters of the mean model. Based on the asymptotic properties of the EGLS estimators, test statistics have been proposed for testing linear hypotheses involving the parameters of the mean model. An alternative estimator, the simple mean of the individual regression coefficients, provides estimation and hypothesis-testing procedures that are simple to compute and teach. The large sample properties of this simple estimator are shown to be similar to that of the EGLS estimator. The performance of the proposed estimator is compared with that of the existing estimators by Monte Carlo simulation.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

Pooling reliability functions

In this article large sample pooling procedures for reliability functions of an exponential life testing model is considered. Asymptotic properties of shrinkage estimation procedure subsequent to preliminary tests are developed. It is shown that the proposed estimator possesses substantially snakker asymptotic mean squared error than the usual estimator in a region of the lparameter space. Relative efficiencies of the purposed estimators to the usual estimators are obtained and recommendations of the level of the preliminary tests are provided. Relative dominance picture of the estimators is presented. It is shown that the proposed estimator provides a wider dominance range over usual estimator than the usual preliminary test estimator. More importantly, the size of the preliminary test is meaningful. Simulation studies is also carried out to appraise the performance of the estimators when samples are small.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

Practical small sample inference for single lag subset autoregressive models

We propose a method for saddlepoint approximating the distribution of estimators in single lag subset autoregressive models of order one. By viewing the estimator as the root of an appropriate estimating equation, the approach circumvents the difficulty inherent in more standard methods that require an explicit expression for the estimator to be available. Plots of the densities reveal that the distributions of the Burg and maximum likelihood estimators are nearly identical. We show that one possible reason for this is the fact that Burg enjoys the property of estimation equation optimality among a class of estimators expressible as a ratio of quadratic forms in normal random variables, which includes Yule–Walker and least squares. By inverting a two-sided hypothesis test, we show how small sample confidence intervals for the parameters can be constructed from the saddlepoint approximations. Simulation studies reveal that the resulting intervals generally outperform traditional ones based on asymptotics and have good robustness properties with respect to heavy-tailed and skewed innovations. The applicability of the models is illustrated by analyzing a longitudinal data set in a novel manner.     

Estimating Distributions with Increasing Failure Rate in an Imperfect Repair Model

A failed system is repaired minimally if after failure, it is restored to the working condition of an identical system of the same age. We extend the nonparametric maximum likelihood estimator(MLE) of a systems lifetime distribution function to test units that are known to have an increasing failure rate. Such items comprise a significant portion of working components in industry. The order-restricted MLE is shown to be consistent. Similar results hold for the Brown-Proschan imperfect repair model, which dictates that a failed component is repaired perfectly with some unknown probability, and is otherwise repaired minimally. The estimators derived are motivated and illustrated by failure data in the nuclear industry. Failure times for groups of emergency diesel generators and motor-driven pumps are analyzed using the order-restricted methods. The order-restricted estimators are consistent and show distinct differences from the ordinary MLEs. Simulation results suggest significant improvement in reliability estimation is available in many cases when component failure data exhibit the IFR property.