Rao and Wu's re-scaling bootstrap modified to achieve extended coverages

Horvitz and Thompson's (HT) [1952. A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc. 47, 663–685] well-known unbiased estimator for a finite population total admits an unbiased estimator for its variance as given by [Yates and Grundy, 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. B 15, 253–261], provided the parent sampling design involves a constant number of distinct units in every sample to be chosen. If the design, in addition, ensures uniform non-negativity of this variance estimator, Rao and Wu [1988. Resampling inference with complex survey data. J. Amer. Statist. Assoc. 83, 231–241] have given their re-scaling bootstrap technique to construct confidence interval and to estimate mean square error for non-linear functions of finite population totals of several real variables. Horvitz and Thompson's estimators (HTE) are used to estimate the finite population totals. Since they need to equate the bootstrap variance of the bootstrap estimator to the Yates and Grundy's estimator (YGE) for the variance of the HTE in case of a single variable, i.e., in the linear case the YG variance estimator is required to be positive for the sample usually drawn.     

Minimum Hellinger distance estimation for randomized play the winner design

Response-adaptive designs in clinical trials incorporate information from prior patient responses in order to assign better performing treatments to the future patients of a clinical study. An example of a response adaptive design that has received much attention in recent years is the randomized play the winner design (RPWD). Beran [1977. Minimum Hellinger distance estimates for parametric models. Ann. Statist. 5, 445–463] investigated the problem of minimum Hellinger distance procedure (MHDP) for continuous data and showed that minimum Hellinger distance estimator (MHDE) of a finite dimensional parameter is as efficient as the MLE (maximum likelihood estimator) under a true model assumption. This paper develops minimum Hellinger distance methodology for data generated using RPWD. A new algorithm using the Monte Carlo approximation to the estimating equation is proposed. Consistency and asymptotic normality of the estimators are established and the robustness and small sample performance of the estimators are illustrated using simulations. The methodology when applied to the clinical trial data conducted by Eli-Lilly and Company, brings out the treatment effect in one of the strata using the frequentist techniques compared to the Bayesian argument of Tamura et al [1994. A case study of an adaptive clinical trialin the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89, 768–776].     

Minimax estimation of normal precisions via expansion estimators

In this paper, the simultaneous estimation of the precision parameters of k normal distributions is considered under the squared loss function in a decision-theoretic framework. Several classes of minimax estimators are derived by using the chi-square identity, and the generalized Bayes minimax estimators are developed out of the classes. It is also shown that the improvement on the unbiased estimators is characterized by the superharmonic function. This corresponds to Stein's [1981. Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9, 1135–1151] result in simultaneous estimation of normal means.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

Conditionally unbiased estimation in the normal setting with unknown variances

To efficiently and completely correct for selection bias in adaptive two-stage trials, uniformly minimum variance conditionally unbiased estimators (UMVCUEs) have been derived for trial designs with normally distributed data. However, a common assumption is that the variances are known exactly, which is unlikely to be the case in practice. We extend the work of Cohen and Sackrowitz (Statistics & Probability Letters, 8(3):273-278, 1989), who proposed an UMVCUE for the best performing candidate in the normal setting with a common unknown variance. Our extension allows for multiple selected candidates, as well as unequal stage one and two sample sizes.     

Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

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.     

Analysis of dose-response in flexible dose titration clinical studies

Assessing dose-response from flexible-dose clinical trials (e.g., titration or dose escalation studies) is challenging and often problematic due to the selection bias caused by 'titration-to-response'. We investigate the performance of a dynamic linear mixed-effects (DLME) model and marginal structural model (MSM) in evaluating dose-response from flexible-dose titration clinical trials via simulations. The simulation results demonstrated that DLME models with previous exposure as a time-varying covariate may provide an unbiased and efficient estimator to recover exposure-response relationship from flexible-dose clinical trials. Although the MSM models with independent and exchangeable working correlations appeared to be able to recover the right direction of the dose-response relationship, it tended to over-correct selection bias and overestimated the underlying true dose-response. The MSM estimators were also associated with large variability in the parameter estimates. Therefore, DLME may be an appropriate modeling option in identifying dose-response when data from fixed-dose studies are absent or a fixed-dose design is unethical to be implemented.     

Reduced-Variance Methods for Detectability Correction of Population Abundance

Detectability issues create uncertainty in field surveys of animal and plant populations. Detectability correction is one method employed to deal with this problem when there is reasonable certainty that detectability is roughly constant with time or in different areas. Two new reduced-variance estimators of detectability are introduced and evaluated for the case of using a detectability correction for new areas that are surveyed only once. The new estimates are unbiased or nearly unbiased and produce population estimates with smaller variance than the Lincoln–Petersen estimate.     

Efficiency of two classes of stochastic restricted almost unbiased type principal component estimators in linear regression model

In this paper, we introduce two new classes of estimators called the stochastic restricted almost unbiased ridge-type principal component estimator (SRAURPCE) and the stochastic restricted almost unbiased Liu-type principal component estimator (SRAURPCE) to overcome the well-known multicollinearity problem in linear regression model. For the two cases when the restrictions are true and not true, necessary and sufficient conditions for the superiority of the proposed estimators are derived and compared, respectively. Furthermore, a Monte Carlo simulation study and a numerical example are given to illustrate the performance of the proposed estimators.     

Multiple inverse sampling in post-stratification with subpopulation sizes unknown: a solution for quota sampling

We extend traditional inverse sampling to multiple case. We then modify the multiple inverse sampling design to a version with taking a simple random sample at the beginning similar to Chang et al (J. Statist. Plan. Inference 69 (1998) 209) and a truncated version similar to Chang et al (J. Statist. Plan. Inference 76 (1999) 215). Using Murthy (Sankhya 18 (1957) 379) we develop their unbiased estimators and their unbiased variance estimators. These unbiased estimators can also be applied to a frequently used sampling scheme called quota sampling by practitioners. The multiple inverse sampling may be viewed as an improved version of quota sampling in some sense. We show that our estimators for estimating the proportions (weights) of subpopulations are more efficient and robust than available estimators using a small simulation study.     

A comparison of estimation techniques for the three parameter pareto distribution

This paper compares minimum distance estimation with best linear unbiased estimation to determine which technique provides the most accurate estimates for location and scale parameters as applied to the three parameter Pareto distribution. Two minimum distance estimators are developed for each of the three distance measures used (Kolmogorov, Cramer‐von Mises, and Anderson‐Darling) resulting in six new estimators. For a given sample size 6 or 18 and shape parameter 1(1)4, the location and scale parameters are estimated. A Monte Carlo technique is used to generate the sample sets. The best linear unbiased estimator and the six minimum distance estimators provide parameter estimates based on each sample set. These estimates are compared using mean square error as the evaluation tool. Results show that the best linear unbaised estimator provided more accurate estimates of location and scale than did the minimum estimators tested.     

Some unbiased estimators versus mean per unit and ratio estimators in finite population sample surveys

We present some unbiased estimators at the population mean in a finite population sample surveys with simple random sampling design where information on an auxiliary variance x positively correlated with the main variate y is available. Exact variance and unbiased estimate of the variance are computed for any sample size. These estimators are compared for their precision with the mean per unit and the ratio estimators. Modifications of the estimators are suggested to make them more precise than the mean per unit estimator or the ratio estimator regardless of the value of the population correlation coefficient between the variates x and y. Asymptotic distribution of our estimators and confidnece intervals for the population mean are also obtained.     

Strawderman-type estimators for a scale parameter with application to the exponential distribution

It is shown that Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique for estimating the variance of a normal distribution can be extended to estimating a general scale parameter in the presence of a nuisance parameter. Employing standard monotone likelihood ratio-type conditions, a new class of improved estimators for this scale parameter is derived under quadratic loss. By imposing an additional condition, a broader class of improved estimators is obtained. The dominating procedures are in form analogous to those in Strawderman [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198]. Application of the general results to the exponential distribution yields new sufficient conditions, other than those of Brewster and Zidek [1974. Improving on equivariant estimators. Ann. Statist. 2, 21–38] and Kubokawa [1994. A unified approach to improving equivariant estimators. Ann. Statist. 22, 290–299], for improving the best affine equivariant estimator of the scale parameter. A class of estimators satisfying the new conditions is constructed. The results shed new light on Strawderman's [1974. Minimax estimation of powers of the variance of a normal population under squared error loss. Ann. Statist. 2, 190–198] technique.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Combining estimates from multiple early studies to obtain estimates of response: using shrinkage estimates to obtain estimates of response

Development of anti-cancer therapies usually involve small to moderate size studies to provide initial estimates of response rates before initiating larger studies to better quantify response. These early trials often each contain a single tumor type, possibly using other stratification factors. Response rate for a given tumor type is routinely reported as the percentage of patients meeting a clinical criteria (e.g. tumor shrinkage), without any regard to response in the other studies. These estimates (called maximum likelihood estimates or MLEs) on average approximate the true value, but have variances that are usually large, especially for small to moderate size studies. The approach presented here is offered as a way to improve overall estimation of response rates when several small trials are considered by reducing the total uncertainty.The shrinkage estimators considered here (James-Stein/empirical Bayes and hierarchical Bayes) are alternatives that use information from all studies to provide potentially better estimates for each study. While these estimates introduce a small bias, they have a considerably smaller variance, and thus tend to be better in terms of total mean squared error. These procedures provide a better view of drug performance in that group of tumor types as a whole, as opposed to estimating each response rate individually without consideration of the others. In technical terms, the vector of estimated response rates is nearer the vector of true values, on average, than the vector of the usual unbiased MLEs applied to such trials.     

On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over‐shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second‐order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson‐gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.     

Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data

When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.     

Optional versus compulsory randomized response techniques in complex surveys

In estimating the proportion of people bearing a sensitive attribute in a community, to mitigate possible evasive answer biases, Warner (J. Amer. Statist. Assoc. 60 (1965) 63) introduced a technique of randomized response (RR) in human surveys, by way of protecting individual privacy. Chaudhuri and Mukerjee (Calcutta Statist. Assoc. Bull. 34 (1985) 225; Randomized Response: Theory and Techniques, Marcel Dekker, New York) presented a modification allowing a direct response (DR) option to whom the attribute does not appear to be stigmatizing enough. Warner himself and many of his followers restrict the application of their RR devices to surveys with selection exclusively by ‘simple random sampling with replacement’. Chaudhuri (J. Statist. Plann. Inference 34 (2001a) 37; Pakistan J. Statist. 17 (3) (2001b) 259; Calcutta Statist. Assoc. Bull. 52 (205–208) (2002) 315) showed the efficacy of some of these devices when sample selection is by general unequal probabilities possibly even without replacement. Here, we present theories for unbiased estimation of the proportion along with unbiased estimation of the variances of the estimators when ‘compulsory’ or ‘optional’ RR's are gathered from persons sampled with varying probabilities. Gains in efficiency by allowing DR option rather than RR compulsion are illustrated numerically through simulation from data.     

Bayesian analysis of vector-autoregressive models with noninformative priors

In this paper, we investigate the properties of Bayes estimators of vector autoregression (VAR) coefficients and the covariance matrix under two commonly employed loss functions. We point out that the posterior mean of the variances of the VAR errors under the Jeffreys prior is likely to have an over-estimation bias. Our Bayesian computation results indicate that estimates using the constant prior on the VAR regression coefficients and the reference prior of Yang and Berger (Ann. Statist. 22 (1994) 1195) on the covariance matrix dominate the constant-Jeffreys prior estimates commonly used in applications of VAR models in macroeconomics. We also estimate a VAR model of consumption growth using both constant-reference and constant-Jeffreys priors.