Finite sample properties of ml and reml estimators in time series regression models with long memory noise

Finite sample properties of ML and REML estimators in time series regression models with fractional ARIMA noise are examined. In particular, theoretical approximations for bias of ML and REML estimators of the noise parameters are developed and their accuracy is assessed through simulations. The impact of noise parameter estimation on performance of t -statistics and likelihood ratio statistics for testing regression parameters is also investigated.     

Estimating and testing group effects from type i censored normal samples in experimental design

The maximum likelihood (ML) equations calculated from censored normal samples do not admit explicit solutions. A principle of modification is given and modified maximum likelihood (MML) equations, which admit explicit solutions, are defined. This approach makes it possible to tackle the hitherto unresolved problem of estimating and testing hypotheses about group-effects in one-way classification experimental designs based on Type I censored normal samples. The MML estimators of group-effects are obtained as explicit functions of sample observations and shown to be asymptotically identical with the ML estimators and hence BAN (best asymptotic normal) estimators. A statistic t is defined to test a linear contrast of group-effects and shown to be asymptotically normally distributed. A numerical example is presented which illustrates the procedure.     

On mitigating collinearity through mixtures

In linear models having near collinear columns of X, ridge and surrogate estimators often are used to mitigate collinearity. A new class of estimators is based on mixtures, either of X and a design minimal in an ordered class or of the Fisher information and a scalar matrix. Comparisons are drawn among choices for the mixing parameter, and the estimators are found to be admissible relative to ordinary least squares. Case studies demonstrate that selected mixture designs are perturbed from the original design to a lesser extent than are those of the surrogate method, while retaining reasonable efficiency characteristics.     

On Bayesian estimators in multistage binomial designs

A new class of Bayesian estimators for a proportion in multistage binomial designs is considered. Priors belong to the beta-J distribution family, which is derived from the Fisher information associated with the design. The transposition of the beta parameters of the Haldane and the uniform priors in fixed binomial experiments into the beta-J distribution yields bias-corrected versions of these priors in multistage designs. We show that the estimator of the posterior mean based on the corrected Haldane prior and the estimator of the posterior mode based on the corrected uniform prior have good frequentist properties. An easy-to-use approximation of the estimator of the posterior mode is provided. The new Bayesian estimators are compared to Whitehead's and the uniformly minimum variance estimators through several multistage designs. Last, the bias of the estimator of the posterior mode is derived for a particular case.     

Sequential D-optimal designs for generalized linear mixed models

We discuss the construction of D-optimal sequential designs for the analysis of longitudinal data or repeated measurements using generalized linear mixed models (GLMMs). We investigate the performance of the design through a simulation study, which indicates that the proposed design can be very successful in improving the efficiency of the ML estimators in GLMMs relative to some common competitors. Our simulations also suggest that the usual normal-theory inference procedures remain valid under the sequential sampling schemes. We also present an example using real data obtained from a clinical study.     

Optimal designs for tumor regrowth models

Most growth curves can only be used to model the tumor growth under no intervention. To model the growth curves for treated tumor, both the growth delay due to the treatment and the regrowth of the tumor after the treatment need to be taken into account. In this paper, we consider two tumor regrowth models and determine the locally D- and c-optimal designs for these models. We then show that the locally D- and c-optimal designs are minimally supported. We also consider two equally spaced designs as alternative designs and evaluate their efficiencies.     

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.     

Parallel, AA/BB, AB/BA and Balaam's design: efficient and maximin choices when testing the treatment effect in a mixed effects linear regression

When examining the effect of treatment A versus B, there may be a choice between a parallel group design, an AA/BB design, an AB/BA cross‐over and Balaam's design. In case of a linear mixed effects regression, it is examined, starting from a flexible function of the costs involved and allowing for subject dropout, which design is most efficient in estimating this effect. For no carry‐over, the AB/BA cross‐over design is most efficient as long as the dropout rate at the second measurement does not exceed 2ρ/(1 + ρ), ρ being the intraclass correlation. For steady‐state carry‐over, depending on the costs involved, the dropout rate and ρ, either a parallel design or an AA/BB design is most efficient. For types of carry‐over that allow for self carry‐over, interest is in the direct treatment effect plus the self carry‐over effect, with either an AA/BB or Balaam's design being most efficient. In case of insufficient knowledge on the dropout rate or ρ, a maximin strategy is devised: choose the design that minimizes the maximum variance of the treatment estimator. Such maximin designs are derived for each type of carry‐over. Copyright © 2012 John Wiley & Sons, Ltd.     

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

Testing for generalized linear mixed models with cluster correlated data under linear inequality constraints

Generalized linear mixed models (GLMMs) are often used for analyzing cluster correlated data, including longitudinal data and repeated measurements. Full unrestricted maximum likelihood (ML) approaches for inference on both fixed‐and random‐effects parameters in GLMMs have been extensively studied in the literature. However, parameter orderings or constraints may occur naturally in practice, and in such cases, the efficiency of a statistical method is improved by incorporating the parameter constraints into the ML estimation and hypothesis testing. In this paper, inference for GLMMs under linear inequality constraints is considered. The asymptotic properties of the constrained ML estimators and constrained likelihood ratio tests for GLMMs have been studied. Simulations investigated the empirical properties of the constrained ML estimators, compared to their unrestricted counterparts. An application to a recent survey on Canadian youth smoking patterns is also presented. As these survey data exhibit natural parameter orderings, a constrained GLMM has been considered for data analysis. The Canadian Journal of Statistics 40: 243–258; 2012 © 2012 Crown in the right of Canada     

Circular balanced repeated measurements designs

The concept of a circular design is defined and when proper balance for various effects is assumed, its universal optimality is proved over the class of all designs with the same set of parameters, Such designs are shown to minimize the variance of the best linear unbiased estimators of contrasts of residual and direct effects over the class of equireplicated designs. All models assume first order residual effects and are of a circular nature. The proofs are presented in a unified manner for several models at a time. They are based on certain matrix domination which occurs when parameters are eliminated from a linear modelj this latter fact is proved for a general linear model.     

Statistical inference for α-series process with the inverse Gaussian distribution

Statistical inferences for the geometric process (GP) are derived when the distribution of the first occurrence time is assumed to be inverse Gaussian (IG). An α-series process, as a possible alternative to the GP, is introduced since the GP is sometimes inappropriate to apply some reliability and scheduling problems. In this study, statistical inference problem for the α-series process is considered where the distribution of first occurrence time is IG. The estimators of the parameters α, μ, and σ² are obtained by using the maximum likelihood (ML) method. Asymptotic distributions and consistency properties of the ML estimators are derived. In order to compare the efficiencies of the ML estimators with the widely used nonparametric modified moment (MM) estimators, Monte Carlo simulations are performed. The results showed that the ML estimators are more efficient than the MM estimators. Moreover, two real life datasets are given for application purposes.     

Efficient estimators for adaptive stratified sequential sampling

In stratified sampling, methods for the allocation of effort among strata usually rely on some measure of within-stratum variance. If we do not have enough information about these variances, adaptive allocation can be used. In adaptive allocation designs, surveys are conducted in two phases. Information from the first phase is used to allocate the remaining units among the strata in the second phase. Brown et al. [Adaptive two-stage sequential sampling, Popul. Ecol. 50 (2008), pp. 239–245] introduced an adaptive allocation sampling design – where the final sample size was random – and an unbiased estimator. Here, we derive an unbiased variance estimator for the design, and consider a related design where the final sample size is fixed. Having a fixed final sample size can make survey-planning easier. We introduce a biased Horvitz–Thompson type estimator and a biased sample mean type estimator for the sampling designs. We conduct two simulation studies on honey producers in Kurdistan and synthetic zirconium distribution in a region on the moon. Results show that the introduced estimators are more efficient than the available estimators for both variable and fixed sample size designs, and the conventional unbiased estimator of stratified simple random sampling design. In order to evaluate efficiencies of the introduced designs and their estimator furthermore, we first review some well-known adaptive allocation designs and compare their estimator with the introduced estimators. Simulation results show that the introduced estimators are more efficient than available estimators of these well-known adaptive allocation designs.     

Simultaneous confidence intervals in the analysis of orthogonal saturated designs

We present the first known method of constructing exact simultaneous confidence intervals for the analysis of orthogonal, saturated factorial designs. Given m independent, normally distributed, unbiased estimators of treatment contrasts, if there is an independent chi-squared estimator of error variance, then simultaneous confidence intervals based on the Studentized maximum modulus distribution are exact under all parameter configurations. In this paper, an analogous method is developed for the case of an orthogonal saturated design, for which the treatment contrasts are independently estimable but there is no independent estimator of error variance. Lacking an independent estimator of the error variance, the smallest sums of squares of effect estimators are pooled. The simultaneous confidence intervals are based on a probability inequality, for which the simultaneous confidence coefficient is achieved in the null case.     

Maximum likelihood estimation of the parameters of the bivariate binomial distribution

We consider the problem of maximum likelihood estimation of the parameters of the bxvariate binomial distribution, In the statistical literature, this problem is solved when the observed sample is available in the form of a 2x2 contingency table, that is, with all four cell fre quencies given,, The present paper provides a solution for this problem when only the marginal totals of the 2x2 table are observed, which is the natural set-up in a bivariate sampling situation.. Thus, based on a sample [(X_i,Y_i:), i = 1, …, k] from a bivariate binomial population, we derive maximum likelihood (ML) estimators for the two marginal parameters p₁,p₂: and the covariance parameter p₁₁: It. turns out that the ML estimators for P₁: and P₂: are expressed explicitly in terms of the sample values, whereas the ML estimator for p₁₁: can only be obtained numerically by iterative methods Two nu merical illustrations are also presented     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

Statistical properties of Rechtschaffner designs

Rechtschaffner designs are saturated designs of resolution V   in which main effects and two-factor interactions are estimable if three-factor and higher order interactions are negligible. Statistical properties of Rechtschaffner designs are studied in this paper. Best linear unbiased estimators of main effects and two-factor interactions are given explicitly and asymptotic properties of correlations between these estimators are studied as well. It is shown that designs recommended by Rechtschaffner [1967. Saturated fractions of 2ⁿ$2^n$ and 3ⁿ$3^n$ factorial designs, Technometrics 9, 569–576] are not only A-optimal but also D-optimal. Comparisons of Rechtschaffner designs with other A- and D-optimal designs of resolution V are also discussed.     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

Optimum designs for estimating EDp based on raw optical density data

For raw optical density (ROD) data, such as those generated in biological assays employing an ELISA plate reader, ED_p-optimal designs are identified for a family of homogeneous non-linear models with two parameters. In every case, the theoretical ED_p-optimal design is a design with one or two support points. These theoretical optimal designs might not be suitable for many practical applications. To overcome this shortcoming, we have specified ED_p-optimal designs within the class of k-point equally spaced and uniform designs. The efficiency robustness of these designs with respect to initial nominal values of the parameters have been investigated.