Estimation of location extremes within general families of scale mixtures

In this paper, we study the estimation of the minimum and maximum location parameters, respectively, representing the minimum guaranteed lifetime of series and parallel systems of components, within a general class of scale mixtures. The conditional or underlying distribution has only the primary restriction of being a location-scale family with positive support. The mixing distribution is also quite general in that we only assume that it has positive support and finite second moment. For demonstrative purposes several special cases are highlighted such as the gamma, inverse-Gaussian, and discrete mixture. Various estimators, including bootstrap bias corrected estimators, are compared with respect to both mean-squared-error and Pitman's measure of closeness.     

Resampling-based bias-corrected time series prediction

In this paper, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study.     

Robustness and Corrections for Sample Size Adaptation Strategies Based on Effect Size Estimation

A study on the robustness of the adaptation of the sample size for a phase III trial on the basis of existing phase II data is presented—when phase III is lower than phase II effect size. A criterion of clinical relevance for phase II results is applied in order to launch phase III, where data from phase II cannot be included in statistical analysis. The adaptation consists in adopting the conservative approach to sample size estimation, which takes into account the variability of phase II data. Some conservative sample size estimation strategies, Bayesian and frequentist, are compared with the calibrated optimal γ conservative strategy (viz. COS) which is the best performer when phase II and phase III effect sizes are equal. The Overall Power (OP) of these strategies and the mean square error (MSE) of their sample size estimators are computed under different scenarios, in the presence of the structural bias due to lower phase III effect size, for evaluating the robustness of the strategies. When the structural bias is quite small (i.e., the ratio of phase III to phase II effect size is greater than 0.8), and when some operating conditions for applying sample size estimation hold, COS can still provide acceptable results for planning phase III trials, even if in bias absence the OP was higher.

Main results concern the introduction of a correction, which affects just sample size estimates and not launch probabilities, for balancing the structural bias. In particular, the correction is based on a postulation of the structural bias; hence, it is more intuitive and easier to use than those based on the modification of Type I or/and Type II errors. A comparison of corrected conservative sample size estimation strategies is performed in the presence of a quite small bias. When the postulated correction is right, COS provides good OP and the lowest MSE. Moreover, the OPs of COS are even higher than those observed without bias, thanks to higher launch probability and a similar estimation performance. The structural bias can therefore be exploited for improving sample size estimation performances. When the postulated correction is smaller than necessary, COS is still the best performer, and it also works well. A higher than necessary correction should be avoided.     

Assessing the bias due to non-coverage of residential movers in the German Microcensus Panel: an evaluation using data from the Socio-Economic Panel

The German Microcensus (MC) is a large scale rotating panel survey over three years. The MC is attractive for longitudinal analysis over the entire participation duration because of the mandatory participation and the very high case numbers (about 200000 respondents). However, as a consequence of the area sampling that is used for the MC, residential mobility is not covered and consequently statistical information at the new residence is lacking in the MC sample. This raises the question whether longitudinal analyses, like transitions between labour market states, are biased and how different methods perform that promise to reduce such a bias. Similar problems occur also for other national Labour Force Surveys (LFS) which are rotating panels and do not cover residential mobility, see Clarke and Tate (2002). Based on data of the German Socio-Economic Panel (SOEP), which covers residential mobility, we analysed the effects of missing data of residential movers by the estimation of labour force flows. By comparing the results from the complete SOEP sample and the results from the SOEP, restricted to the non-movers, we concluded that the non-coverage of the residential movers can not be ignored in Rubin’s sense. With respect to correction methods we analysed weighting by inverse mobility scores and log-linear models for partially observed contingency tables. Our results indicate that weighting by inverse mobility scores reduces the bias to about 60% whereas the official longitudinal weights obtained by calibration result in a bias reduction of about 80%. The estimation of log-linear models for non-ignorable non-response leads to very unstable results.     

Canonical Correlation Analysis Using Small Number of Samples

Canonical correlation assesses the relationship between two groups of variables. Although it has been a useful tool in a wide variety of research areas, it is not well known that weaker canonical correlations require larger sample sizes to be correctly inferred. In this article, we investigate small sample bias in canonical correlation analysis and apply the jackknife bias correction to the estimation of canonical correlations. We use bootstrap samples to obtain a better confidence interval for the jackknife canonical correlation estimator.     

The Extensively Corrected Score for Measurement Error Models

In measurement error problems, two major and consistent estimation methods are the conditional score and the corrected score. They are functional methods that require no parametric assumptions on mismeasured covariates. The conditional score requires that a suitable sufficient statistic for the mismeasured covariate can be found, while the corrected score requires that the object score function can be estimated without bias. These assumptions limit their ranges of applications. The extensively corrected score proposed here is an extension of the corrected score. It yields consistent estimations in many cases when neither the conditional score nor the corrected score is feasible. We demonstrate its constructions in generalized linear models and the Cox proportional hazards model, assess its performances by simulation studies and illustrate its implementations by two real examples.     

Bootstrapping Regression Parameters in Multivariate Survival Analysis

Bootstrap methods are proposed for estimating sampling distributions and associated statistics for regression parameters in multivariate survival data. We use an Independence Working Model (IWM) approach, fitting margins independently, to obtain consistent estimates of the parameters in the marginal models. Resampling procedures, however, are applied to an appropriate joint distribution to estimate covariance matrices, make bias corrections, and construct confidence intervals. The proposed methods allow for fixed or random explanatory variables, the latter case using extensions of existing resampling schemes (Loughin,1995), and they permit the possibility of random censoring. An application is shown for the viral positivity time data previously analyzed by Wei, Lin, and Weissfeld (1989). A simulation study of small-sample properties shows that the proposed bootstrap procedures provide substantial improvements in variance estimation over the robust variance estimator commonly used with the IWM. This revised version was published online in July 2006 with corrections to the Cover Date.     

Selection bias for treatments with positive Phase 2 results

In drug development, treatments are most often selected at Phase 2 for further development when an initial trial of a new treatment produces a result that is considered positive. This selection due to a positive result means, however, that an estimator of the treatment effect, which does not take account of the selection is likely to over‐estimate the true treatment effect (ie, will be biased). This bias can be large and researchers may face a disappointingly lower estimated treatment effect in further trials. In this paper, we review a number of methods that have been proposed to correct for this bias and introduce three new methods. We present results from applying the various methods to two examples and consider extensions of the examples. We assess the methods with respect to bias of estimation of the treatment effect and compare the probabilities that a bias‐corrected treatment effect estimate will exceed a decision threshold. Following previous work, we also compare average power for the situation where a Phase 3 trial is launched given that the bias‐corrected observed Phase 2 treatment effect exceeds a launch threshold. Finally, we discuss our findings and potential application of the bias correction methods.     

Statistical inference for varying-coefficient models with error-prone covariates

Motivated by an application, we consider the statistical inference of varying-coefficient regression models in which some covariates are not observed, but ancillary variables are available to remit them. Due to the attenuation, the usual local polynomial estimation of the coefficient functions is not consistent. We propose a corrected local polynomial estimation for the unknown coefficient functions by calibrating the error-prone covariates. It is shown that the resulting estimators are consistent and asymptotically normal. In addition, we develop a wild bootstrap test for the goodness of fit of models. Some simulations are conducted to demonstrate the finite sample performances of the proposed estimation and test procedures. An example of application on a real data from Duchenne muscular dystrophy study is also illustrated.     

A simulation study of balanced and antithetic bootstrap resampling methods

An extensive simulation study is conducted to compare the performance between balanced and antithetic resampling for the bootstrap in estimation of bias, variance, and percentiles when the statistic of interest is the median, the square root of the absolute value of the mean, or the median absolute deviations from the median. Simulation results reveal that balanced resampling provide better efficiencies in most cases; however, antithetic resampling is superior in estimating bias of the median. We also investigate the possibility of combining an existing efficient bootstrap computation of Efron (1990) with balanced or antithetic resampling for percentile estimation. Results indicate that the combination method does indeed offer gains in performance though the gains are much more dramatic for the bootstrap t statistic than for any of the three statistics of interest as described above.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Estimation of a population size through capture-mark-recapture method: a comparison of various point and interval estimators

This article deals with the estimation of a fixed population size through capture-mark-recapture method that gives rise to hypergeometric distribution. There are a few well-known and popular point estimators available in the literature, but no good comprehensive comparison is available about their merits. Apart from the available estimators, an empirical Bayes (EB) estimator of the population size is proposed. We compare all the point estimators in terms of relative bias and relative mean squared error. Next, two new interval estimators – (a) an EB highest posterior distribution interval and (b) a frequentist interval estimator based on a parametric bootstrap method, are proposed. The comparison is then carried among the two proposed interval estimators and interval estimators derived from the currently available estimators in terms of coverage probability and average length (AL). Based on comprehensive numerical results, we rank and recommend the point estimators as well as interval estimators for practical use. Finally, a real-life data set for a green treefrog population is used as a demonstration for all the methods discussed.     

Improved point estimation for the Kumaraswamy distribution

We develop nearly unbiased estimators for the Kumaraswamy distribution proposed by Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 46 (1980), pp. 79–88], which has considerable attention in hydrology and related areas. We derive modified maximum-likelihood estimators that are bias-free to second order. As an alternative to the analytically bias-corrected estimators discussed, we consider a bias correction mechanism based on the parametric bootstrap. We conduct Monte Carlo simulations in order to investigate the performance of the corrected estimators. The numerical results show that the bias correction scheme yields nearly unbiased estimates.     

Boundary and Bias Correction in Kernel Hazard Estimation

A new class of local linear hazard estimators based on weighted least square kernel estimation is considered. The class includes the kernel hazard estimator of Ramlau-Hansen (1983), which has the same boundary correction property as the local linear regression estimator (see Fan & Gijbels, 1996). It is shown that all the local linear estimators in the class have the same pointwise asymptotic properties. We derive the multiplicative bias correction of the local linear estimator. In addition we propose a new bias correction technique based on bootstrap estimation of additive bias. This latter method has excellent theoretical properties. Based on an extensive simulation study where we compare the performance of competing estimators, we also recommend the use of the additive bias correction in applied work.     

Bias Reduction through First-order Mean Correction, Bootstrapping and Recursive Mean Adjustment

Standard methods of estimation for autoregressive models are known to be biased in finite samples, which has implications for estimation, hypothesis testing, confidence interval construction and forecasting. Three methods of bias reduction are considered here: first-order bias correction, FOBC, where the total bias is approximated by the O(T^-1) bias; bootstrapping; and recursive mean adjustment, RMA. In addition, we show how first-order bias correction is related to linear bias correction. The practically important case where the AR model includes an unknown linear trend is considered in detail. The fidelity of nominal to actual coverage of confidence intervals is also assessed. A simulation study covers the AR(1) model and a number of extensions based on the empirical AR(p) models fitted by Nelson & Plosser (1982). Overall, which method dominates depends on the criterion adopted: bootstrapping tends to be the best at reducing bias, recursive mean adjustment is best at reducing mean squared error, whilst FOBC does particularly well in maintaining the fidelity of confidence intervals.     

Boundary corrected cubic smoothing splines

Smoothing splines are known to exhibit a type of boundary bias that can reduce their estimation efficiency. In this paper, a boundary corrected cubic smoothing spline is developed in a way that produces a uniformly fourth order estimator. The resulting estimator can be calculated efficiently using an O(n) algorithm that is designed for the computation of fitted values and associated smoothing parameter selection criteria. A simulation study shows that use of the boundary corrected estimator can improve estimation efficiency in finite samples. Applications to the construction of asymptotically valid pointwise confidence intervals are also investigated .     

Evaluating bias correction in weighted proportional hazards regression

Often in observational studies of time to an event, the study population is a biased (i.e., unrepresentative) sample of the target population. In the presence of biased samples, it is common to weight subjects by the inverse of their respective selection probabilities. Pan and Schaubel (Can J Stat 36:111–127, 2008) recently proposed inference procedures for an inverse selection probability weighted (ISPW) Cox model, applicable when selection probabilities are not treated as fixed but estimated empirically. The proposed weighting procedure requires auxiliary data to estimate the weights and is computationally more intense than unweighted estimation. The ignorability of sample selection process in terms of parameter estimators and predictions is often of interest, from several perspectives: e.g., to determine if weighting makes a significant difference to the analysis at hand, which would in turn address whether the collection of auxiliary data is required in future studies; to evaluate previous studies which did not correct for selection bias. In this article, we propose methods to quantify the degree of bias corrected by the weighting procedure in the partial likelihood and Breslow-Aalen estimators. Asymptotic properties of the proposed test statistics are derived. The finite-sample significance level and power are evaluated through simulation. The proposed methods are then applied to data from a national organ failure registry to evaluate the bias in a post-kidney transplant survival model.     

Iterative Bias Correction of the Cross‐Validation Criterion

Abstract. The cross‐validation (CV) criterion is known to be asecond‐order unbiased estimator of the risk function measuring the discrepancy between the candidate model and the true model, as well as the generalized information criterion (GIC) and the extended information criterion (EIC). In the present article, we show that the 2kth‐order unbiased estimator can be obtained using a linear combination from the leave‐one‐out CV criterion to the leave‐k‐out CV criterion. The proposed scheme is unique in that a bias smaller than that of a jackknife method can be obtained without any analytic calculation, that is, it is not necessary to obtain the explicit form of several terms in an asymptotic expansion of the bias. Furthermore, the proposed criterion can be regarded as a finite correction of a bias‐corrected CV criterion by using scalar coefficients in a bias‐corrected EIC obtained by the bootstrap iteration.     

An exhaustive numerical assessment of alternative unconditional tests of a binary treatment effect

For testing the effectiveness of a treatment on a binary outcome, a bewildering range of methods have been proposed. How similar are all these tests? What are their theoretical strengths and weaknesses? Which are to be recommended and what is a coherent basis for deciding? In this paper, we take seven standard but imperfect tests and apply three different methods of adjustment to ensure size control: maximization (M), restricted maximization (B) and bootstrap/estimation (E). Across a wide conditions, we compute exact size and power of the 7 basic and 21 adjusted tests. We devise two new measures of size bias and intrinsic power, and employ novel graphical tools to summarise a huge set of results. Amongst the 7 basic tests, Liebermeister’s test best controls size but can still be conservative. Amongst the adjusted tests, E-tests clearly have the best power and results are very stable across different conditions.     

Bootstrap-based bias corrections for INAR count time series

Integer-valued autoregressive (INAR) processes form a very useful class of processes suitable to model time series of counts. Several practically relevant estimators based on INAR data are known to be systematically biased away from their population values, e.g. sample autocovariances, sample autocorrelations, or the dispersion index. We propose to do bias correction for such estimators by using a recently proposed INAR-type bootstrap scheme that is tailor-made for INAR processes, and which has been proven to be asymptotically consistent under general conditions. This INAR bootstrap allows an implementation with and without parametrically specifying the innovations' distribution. To judge the potential of corresponding bias correction, we compare these bootstraps in simulations to several competitors that include the AR bootstrap and block bootstrap. Finally, we conclude with an illustrative data application.