A theoretical analysis of the power of biased coin designs

Outlining some recently obtained results of Hu and Rosenberger [2003. Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98, 671–678] and Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835] on the relationship between sequential randomized designs and the power of the usual statistical procedures for testing the equivalence of two competing treatments, the aim of this paper is to provide theoretical proofs of the numerical results of Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835]. Furthermore, we prove that the Adjustable Biased Coin Design [Baldi Antognini A., Giovagnoli, A., 2004. A new “biased coin design” for the sequential allocation of two treatments. J. Roy. Statist. Soc. Ser. C 53, 651–664] is uniformly more powerful than the other “coin” designs proposed in the literature for any sample size.     

Domains of study and poststratification

Sugden and Smith (2002. J. Statist. Plann. Inference 102, 25–38) investigated conditions under which exact linear unbiased estimators of linear estimands, and also exact quadratic unbiased estimators of quadratic estimands, could be constructed under the randomisation approach. In this paper the method is applied to domains of study and extended to poststratified estimators of finite population totals. The resulting estimators generalise some of those in Doss et al. (1979. J. Statist. Plann. Inference 3, 235–247). Some further properties of these estimators are explored.     

On extreme order statistics from heterogeneous Weibull variables

In this paper, we focus on stochastic comparisons of extreme order statistics from heterogeneous independent/interdependent Weibull samples. Specifically, we study extreme order statistics from Weibull distributions with (i) common shape parameter but different scale parameters, and (ii) common scale parameter but different shape parameters. Several new comparison results in terms of the likelihood ratio order, reversed hazard rate order and usual stochastic order are studied in those scenarios. The results established here strengthen and generalize some of the results known in the literature including Khaledi and Kochar [Weibull distribution: some stochastic comparisons. J Statist Plann Inference. 2006;136:3121–3129], Fang and Zhang [Stochastic comparisons of series systems with heterogeneous Weibull components. Statist Probab Lett. 2013;83:1649–1653], Torrado [Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters. J Korean Statist Soc. 2015;44:68–76] and Torrado and Kochar [Stochastic order relations among parallel systems from Weibull distributions. J Appl Probab. 2015;52:102–116]. Some numerical examples are also provided for illustration.     

Tests for assessment of agreement using probability criteria

For the assessment of agreement using probability criteria, we obtain an exact test, and for sample sizes exceeding 30, we give a bootstrap-t$t$ test that is remarkably accurate. We show that for assessing agreement, the total deviation index approach of Lin [2000. Total deviation index for measuring individual agreement with applications in laboratory performance and bioequivalence. Statist. Med. 19, 255–270] is not consistent and may not preserve its asymptotic nominal level, and that the coverage probability approach of Lin et al. [2002. Statistical methods in assessing agreement: models, issues and tools. J. Amer. Statist. Assoc. 97, 257–270] is overly conservative for moderate sample sizes. We also show that the nearly unbiased test of Wang and Hwang [2001. A nearly unbiased test for individual bioequivalence problems using probability criteria. J. Statist. Plann. Inference 99, 41–58] may be liberal for large sample sizes, and suggest a minor modification that gives numerically equivalent approximation to the exact test for sample sizes 30 or less. We present a simple and accurate sample size formula for planning studies on assessing agreement, and illustrate our methodology with a real data set from the literature.     

A note on new confidence intervals for the difference between two proportions based on an Edgeworth expansion

Zhou and Qin [2004. New intervals for the difference between two independent binomial proportions. J. Statist. Plann. Inference 123, 97–115; 2005. A new confidence interval for the difference between two binomial proportions of paired data. J. Statist. Plann. Inference 128, 527–542] “new confidence intervals” for the difference between two treatment proportions exhibit a severe lack of invariance property that is a compelling reason not to use them.     

A mixed randomized response model

To implement the privacy problem with the Moors (J. Amer. Statist. Assoc. 66 (1971) 627) model, Mangat et al. (Commun. Statist. Theory Methods 26(3) (1997) 243) and Singh et al. (J. Statist. Plann. Inference 83 (2000) 243) presented several strategies as an alternative to Moors model, but their models may lose a large portion of data information and require a high cost to obtain confidentiality of the respondents. Our proposed model has the advantage of simplicity over the previous models while protecting confidentiality. We extend the proposed model to stratified sampling.     

Bayesian sample size calculations for binomial experiments

This paper proposes new methodology for calculating the optimal sample size when a hypothesis test between two binomial proportions is conducted. The problem is addressed from the Bayesian point of view. Following the formulation by DasGupta and Vidakovic (1997, J. Statist. Plann. Inference 65, 335–347), the posterior risk is determined and set not to exceed a prespecified bound. A second constraint deals with the likelihood of data not satisfying the bound on the risk. The cases when the two proportions are equal to a fixed or to a random value are examined.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

Mutiple three-decision procedure for comparing several exponential treatments with the best control

In this paper, the three-decision procedures to classify p treatments as better than or worse than one control, proposed for normal/symmetric probability models [Bohrer, Multiple three-decision rules for parametric signs. J. Amer. Statist. Assoc. 74 (1979), pp. 432–437; Bohrer et al., Multiple three-decision rules for factorial simple effects: Bonferroni wins again!, J. Amer. Statist. Assoc. 76 (1981), pp. 119–124; Liu, A multiple three-decision procedure for comparing several treatments with a control, Austral. J. Statist. 39 (1997), pp. 79–92 and Singh and Mishra, Classifying logistic populations using sample medians, J. Statist. Plann. Inference 137 (2007), pp. 1647–1657]; in the literature, have been extended to asymmetric two-parameter exponential probability models to classify p(p≥1) treatments as better than or worse than the best of q(q≥1) control treatments in terms of location parameters. Critical constants required for the implementation of the proposed procedure are tabulated for some pre-specified values of probability of no misclassification. Power function of the proposed procedure is also defined and a common sample size necessary to guarantee various pre-specified power levels are tabulated. Optimal allocation scheme is also discussed. Finally, the implementation of the proposed methodology is demonstrated through numerical example.     

A goodness-of-fit test with nuisance parameters: numerical performance

Jurečková and Sen (J. Statist. Plann. Inference 91 (2000) 377–397) proposed goodness-of-fit tests for models admitting nuisance location or nuisance location and scale parameters, based on the difference of two estimators of the location parameter, that are asymptotically first-order equivalent iff the null hypothesis is true. We illustrate here the numerical performance of these tests on simulated data, demonstrating their applicability to practical problems. Comparisons are also made with the well-known Shapiro–Wilk goodness-of-fit test.     

Controlling Bayes directional false discovery rate in random effects model

Starting with a decision theoretic formulation of simultaneous testing of null hypotheses against two-sided alternatives, a procedure controlling the Bayesian directional false discovery rate (BDFDR) is developed through controlling the posterior directional false discovery rate (PDFDR). This is an alternative to Lewis and Thayer [2004. A loss function related to the FDR for random effects multiple comparison. J. Statist. Plann. Inference 125, 49–58.] with a better control of the BDFDR. Moreover, it is optimum in the sense of being the non-randomized part of the procedure maximizing the posterior expectation of the directional per-comparison power rate given the data, while controlling the PDFDR. A corresponding empirical Bayes method is proposed in the context of one-way random effects model. Simulation study shows that the proposed Bayes and empirical Bayes methods perform much better from a Bayesian perspective than the procedures available in the literature.     

Nonadditivity in loglinear models using φ-divergences and MLEs

In this paper three families of test statistics for testing nonadditivity in loglinear models are presented under the assumption of either Poisson, multinomial, or product-multinomial sampling. These new families are based on the φ-divergence measures. The standard method for testing nonadditivity is used, i.e., the two-stage tests procedure. In this procedure the parameters are first estimated using an additive model and then the estimates are treated as known constants for the second stage of the procedure. These test statistics, which are asymptotically chi-squared, generalize the likelihood ratio test for this problem given by Christensen and Utts (J. Statist. Plann. Inference 33 (1992) 333). An example and a simulation study are included.     

On matrix variance inequalities

Olkin and Shepp [2005, A matrix variance inequality. J. Statist. Plann. Inference 130, 351-358] presented a matrix form of Chernoff's inequality for Normal and Gamma (univariate) distributions. We extend and generalize this result, proving Poincaré-type and Bessel-type inequalities, for matrices of arbitrary order and for a large class of distributions.     

Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law

When measuring units are expensive or time consuming, while ranking them is relatively easy and inexpensive, it is known that ranked set sampling (RSS) is preferable to simple random sampling (SRS). Many authors have suggested several extensions of RSS. As a variation, Al-Saleh and Al-Kadiri [Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp. 205–212] introduced double ranked set sampling (DRSS) and it was extended by Al-Saleh and Al-Omari [Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp. 273–286] to multistage ranked set sampling (MSRSS). The entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of information required on the average to determine the value of a (discrete) r.v.. In this work, we discuss entropy estimation in RSS design and aforementioned extensions and compare the results with those in SRS design in terms of bias and root mean square error (RMSE). Motivated by the above observed efficiency, we continue to investigate entropy-based goodness-of-fit test for the inverse Gaussian distribution using RSS. Critical values for some sample sizes determined by means of Monte Carlo simulations are presented for each design. A Monte Carlo power analysis is performed under various alternative hypotheses in order to compare the proposed testing procedure with the existing methods. The results indicate that tests based on RSS and its extensions are superior alternatives to the entropy test based on SRS.     

The equivalence of the mid p-value and the expected p-value for testing equality of two balanced Binomial proportions

This paper considers the problem of testing equality between two independent binomial proportions. Hwang and Yang (Statist. Sinica 11 (2001) 807) apply the Neyman–Pearson fundamental lemma and the estimated truth approach to derive optimal procedures, named expected p-values. This p-value has been shown to be identical to the mid p-value in Lancaster (J. Amer. Statist. Assoc. (1961) 223) for the one-sided test. For the two-sided test, the paper proves the usual two-sided mid p-value is identical to the expected p-value in the balanced sample case.     

Order restricted statistical inference for scale parameters based on sequential order statistics

In the situation of a multi-sample experiment consisting of differently equipped sequential k-out-of-n systems, scale parameters of underlying distributions from a general location-scale family of distributions are estimated under an order restriction. In each sample, the case of missing the smallest observations is included. Moreover, based on a profile likelihood a homogeneity test against an ordered alternative is proposed and analyzed. This work extends an approach of Bhattacharya [2007. Testing for ordered failure rates under general progressive censoring. J. Statist. Plann. Inference 137, 1775–1786] in the progressive Type-II censoring framework.     

Optimal estimating function for estimation and prediction in semi-parametric models

In a seminal paper, Godambe [1985. The foundations of finite sample estimation in stochastic processes. Biometrika 72, 419–428.] introduced the ‘estimating function’ approach to estimation of parameters in semi-parametric models under a filtering associated with a martingale structure. Later, Godambe [1987. The foundations of finite sample estimation in stochastic processes II. Bernoulli, Vol. 2. V.N.V. Science Press, 49–54.] and Godambe and Thompson [1989. An extension of quasi-likelihood Estimation. J. Statist. Plann. Inference 22, 137–172.] replaced this filtering by a more flexible conditioning. Abraham et al. [1997. On the prediction for some nonlinear time-series models using estimating functions. In: Basawa, I.V., et al. (Eds.), IMS Selected Proceedings of the Symposium on Estimating Functions, Vol. 32. pp. 259–268.] and Thavaneswaran and Heyde [1999. Prediction via estimating functions. J. Statist. Plann. Inference 77, 89–101.] invoked the theory of estimating functions for one-step ahead prediction in time-series models. This paper addresses the problem of simultaneous estimation of parameters and multi-step ahead prediction of a vector of future random variables in semi-parametric models by extending the inimitable approach of 13 and 14. The proposed technique is in conformity with the paradigm of the modern theory of estimating functions leading to finite sample optimality within a chosen class of estimating functions, which in turn are used to get the predictors. Particular applications of the technique give predictors that enjoy optimality properties with respect to other well-known criteria.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.     

On reciprocal symmetry

On the positive half line, there are two natural, and complementary, analogues of the single notion of symmetry of distributions on the real line. One is the R-symmetry recently proposed and investigated by Mudholkar and Wang [2007. IG-symmetry and R-symmetry: interrelations and applications to the inverse Gaussian theory. J. Statist. Plann. Inference 137, 3655–3671]; the other is the ‘log-symmetry’ investigated here. Log-symmetry can be thought of either in terms of a random variable having the same distribution as its reciprocal or as ordinary symmetry of the distribution of the logged random variable. Various properties, analogies, comparisons and consequences are investigated.     

Asymptotic linearity of serial and nonserial multivariate signed rank statistics

Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered in Hallin and Paindaveine (Ann. Statist. 30 (2002a) 1103; Bernoulli 8 (2002b) 787 Ann. Statist. 32 (2004), to appear) and Oja and Paindaveine (J. Statist. Plann. Inference (2004), to appear). These statistics, which are multivariate versions of classical signed rank statistics, involve (i) multivariate signs based either on (pseudo-)Mahalanobis residuals, or on a modified version (absolute interdirections) of Randles's interdirections, and (ii) a concept of ranks based either on (pseudo-)Mahalanobis distances or on lift-interdirections.