Automatic Block-Length Selection for the Dependent Bootstrap

We review the different block bootstrap methods for time series, and present them in a unified framework. We then revisit a recent result of Lahiri [Lahiri, S. N. (1999b). Theoretical comparisons of block bootstrap methods, Ann. Statist. 27:386-404] comparing the different methods and give a corrected bound on their asymptotic relative efficiency; we also introduce a new notion of finite-sample “attainable” relative efficiency. Finally, based on the notion of spectral estimation via the flat-top lag-windows of Politis and Romano [Politis, D. N., Romano, J. P. (1995). Bias-corrected nonparametric spectral estimation. J. Time Series Anal. 16:67-103], we propose practically useful estimators of the optimal block size for the aforementioned block bootstrap methods. Our estimators are characterized by the fastest possible rate of convergence which is adaptive on the strength of the correlation of the time series as measured by the correlogram.     

Bootstrap mean squared error of a small-area EBLUP

Concerning the estimation of linear parameters in small areas, a nested-error regression model is assumed for the values of the target variable in the units of a finite population. Then, a bootstrap procedure is proposed for estimating the mean squared error (MSE) of the EBLUP under the finite population setup. The consistency of the bootstrap procedure is studied, and a simulation experiment is carried out in order to compare the performance of two different bootstrap estimators with the approximation given by Prasad and Rao [Prasad, N.G.N. and Rao, J.N.K., 1990, The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85, 163–171.]. In the numerical results, one of the bootstrap estimators shows a better bias behavior than the Prasad–Rao approximation for some of the small areas and not much worse in any case. Further, it shows less MSE in situations of moderate heteroscedasticity and under mispecification of the error distribution as normal when the true distribution is logistic or Gumbel. The proposed bootstrap method can be applied to more general types of parameters (linear of not) and predictors.     

A FINE-TUNED ESTIMATOR OF A GENERAL CONVERGENCE RATE

A general rate estimation method based on the in‐sample evolution of appropriately chosen diverging/converging statistics has recently been proposed by D.N. Politis [C. R. Acad. Sci. Paris, Ser. I, vol. 335, pp. 279–282, 2002] and T. McElroy & D.N. Politis [Ann. Statist., vol. 35, pp. 1827–1848, 2007]. In this paper, we show how a modification of the original estimators achieves a competitive rate of convergence. The modified estimators require the choice of a tuning parameter; an optimal such choice is generally a non‐trivial problem in practice. Some discussion to that effect is given, as well as a small simulation study in a heavy‐tailed setting.     

Extrapolation of subsampling distribution estimators: The i.i.d. and strong mixing cases

Politis & Romano (1994) proposed a general subsampling methodology for the construction of large‐sample confidence regions for an arbitrary parameter under minimal conditions. Nevertheless, the subsampling distribution estimators may sometimes be inefficient (in the case of the sample mean of i.i.d. data, for instance) as compared to alternative estimators such as the bootstrap and/or the asymptotic normal distribution (with estimated variance). The authors investigate here the extent to which the performance of subsampling distribution estimators can be improved by interpolation and extrapolation techniques, while at the same time retaining the robustness property of consistent distribution estimation even in nonregular cases; both i.i.d. and weakly dependent (mixing) observations are considered.     

Bias Correction in the Dynamic Panel Data Model with a Nonscalar Disturbance Covariance Matrix

Abstract

Approximation formulae are developed for the bias of ordinary and generalized Least Squares Dummy Variable (LSDV) estimators in dynamic panel data models. Results from Kiviet [Kiviet, J. F. (1995), on bias, inconsistency, and efficiency of various estimators in dynamic panel data models, J. Econometrics68:53–78; Kiviet, J. F. (1999), Expectations of expansions for estimators in a dynamic panel data model: some results for weakly exogenous regressors, In: Hsiao, C., Lahiri, K., Lee, L‐F., Pesaran, M. H., eds., Analysis of Panels and Limited Dependent Variables, Cambridge: Cambridge University Press, pp. 199–225] are extended to higher‐order dynamic panel data models with general covariance structure. The focus is on estimation of both short‐ and long‐run coefficients. The results show that proper modelling of the disturbance covariance structure is indispensable. The bias approximations are used to construct bias corrected estimators which are then applied to quarterly data from 14 European Union countries. Money demand functions for M1, M2 and M3 are estimated for the EU area as a whole for the period 1991: I–1995: IV. Significant spillovers between countries are found reflecting the dependence of domestic money demand on foreign developments. The empirical results show that in general plausible long‐run effects are obtained by the bias corrected estimators. Moreover, finite sample bias, although of moderate magnitude, is present underlining the importance of more refined estimation techniques. Also the efficiency gains by exploiting the heteroscedasticity and cross‐correlation patterns between countries are sometimes considerable.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

Model selection and parameter estimation of a multinomial logistic regression model

In the multinomial regression model, we consider the methodology for simultaneous model selection and parameter estimation by using the shrinkage and LASSO (least absolute shrinkage and selection operation) [R. Tibshirani, Regression shrinkage and selection via the LASSO, J. R. Statist. Soc. Ser. B 58 (1996), pp. 267–288] strategies. The shrinkage estimators (SEs) provide significant improvement over their classical counterparts in the case where some of the predictors may or may not be active for the response of interest. The asymptotic properties of the SEs are developed using the notion of asymptotic distributional risk. We then compare the relative performance of the LASSO estimator with two SEs in terms of simulated relative efficiency. A simulation study shows that the shrinkage and LASSO estimators dominate the full model estimator. Further, both SEs perform better than the LASSO estimators when there are many inactive predictors in the model. A real-life data set is used to illustrate the suggested shrinkage and LASSO estimators.     

Model selection and post estimation based on a pretest for logistic regression models

ABSTRACT

This article addresses the problem of parameter estimation of the logistic regression model under subspace information via linear shrinkage, pretest, and shrinkage pretest estimators along with the traditional unrestricted maximum likelihood estimator and restricted estimator. We developed an asymptotic theory for the linear shrinkage and pretest estimators and compared their relative performance using the notion of asymptotic distributional bias and asymptotic quadratic risk. The analytical results demonstrated that the proposed estimation strategies outperformed the classical estimation strategies in a meaningful parameter space. Detailed Monte-Carlo simulation studies were conducted for different combinations and the performance of each estimation method was evaluated in terms of simulated relative efficiency. The results of the simulation study were in strong agreement with the asymptotic analytical findings. Two real-data examples are also given to appraise the performance of the estimators.     

Doubly robust point and variance estimation in the presence of imputed survey data

Imputation is often used in surveys to treat item nonresponse. It is well known that treating the imputed values as observed values may lead to substantial underestimation of the variance of the point estimators. To overcome the problem, a number of variance estimation methods have been proposed in the literature, including resampling methods such as the jackknife and the bootstrap. In this paper, we consider the problem of doubly robust inference in the presence of imputed survey data. In the doubly robust literature, point estimation has been the main focus. In this paper, using the reverse framework for variance estimation, we derive doubly robust linearization variance estimators in the case of deterministic and random regression imputation within imputation classes. Also, we study the properties of several jackknife variance estimators under both negligible and nonnegligible sampling fractions. A limited simulation study investigates the performance of various variance estimators in terms of relative bias and relative stability. Finally, the asymptotic normality of imputed estimators is established for stratified multistage designs under both deterministic and random regression imputation. The Canadian Journal of Statistics 40: 259–281; 2012 © 2012 Statistical Society of Canada     

Estimation of a rare sensitive attribute in a stratified sample using Poisson distribution

This study proposes the estimators for the mean and its variance of the number of respondents who possessed a rare sensitive attribute based on stratified sampling schemes (stratified sampling and stratified double sampling). This study deals with the extension of the estimation reported in Land et al. [Estimation of a rare sensitive attribute using Poisson distribution, Statistics (2011), in press. DOI: 10.1080/02331888.2010.524300] using a Poisson distribution and an unrelated question randomized response model reported in Greenberg et al. [The unrelated question randomized response model: Theoretical framework, J. Amer. Statist. Assoc. 64 (1969), 520–539]. In the stratified sampling, the estimators are proposed when the parameter of the rare unrelated attribute is known and unknown. The variances of estimators using a proportional and optimum allocation are also suggested. The proposed estimators are evaluated using a relative efficiency comparing variances of the estimators reported in Land et al. depending on the parameters and the probability of selecting a question. We showed that our proposed methods have better efficiencies than Land et al.’s randomized response model in some conditions. When the sizes of stratified populations are not given, other estimators are suggested using a stratified double sampling. For the proportional allocation, the difference between two variances in the stratified sampling and the stratified double sampling is given with the known rare unrelated attribute.     

Choosing a robustness tuning parameter

A novel method is proposed for choosing the tuning parameter associated with a family of robust estimators. It consists of minimising estimated mean squared error, an approach that requires pilot estimation of model parameters. The method is explored for the family of minimum distance estimators proposed by [Basu, A., Harris, I.R., Hjort, N.L. and Jones, M.C., 1998, Robust and efficient estimation by minimising a density power divergence. Biometrika, 85, 549–559.] Our preference in that context is for a version of the method using the L ₂ distance estimator [Scott, D.W., 2001, Parametric statistical modeling by minimum integrated squared error. Technometrics, 43, 274–285.] as pilot estimator.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Asymptotic behaviour of M‐estimators in AR(p) models under nonstandard conditions

The authors derive the limiting distribution of M‐estimators in AR(p) models under nonstandard conditions, allowing for discontinuities in score and density functions. Unlike usual regularity assumptions, these conditions are satisfied in the context of L₁‐estimation and autoregression quantiles. The asymptotic distributions of the resulting estimators, however, are not generally Gaussian. Moreover, their bootstrap approximations are consistent along very specific sequences of bootstrap sample sizes only.     

Interval estimation of the joint survival function for successive duration times under left truncation and right censoring

In incident cohort studies, survival data often include subjects who have had an initiate event at recruitment and may potentially experience two successive events (first and second) during the follow-up period. Since the second duration process becomes observable only if the first event has occurred, left truncation and dependent censoring arise if the two duration times are correlated. To confront the two potential sampling biases, we propose two inverse-probability-weighted (IPW) estimators for the estimation of the joint survival function of two successive duration times. One of them is similar to the estimator proposed by Chang and Tzeng [Nonparametric estimation of sojourn time distributions for truncated serial event data – a weight adjusted approach, Lifetime Data Anal. 12 (2006), pp. 53–67]. The other is the extension of the nonparametric estimator proposed by Wang and Wells [Nonparametric estimation of successive duration times under dependent censoring, Biometrika 85 (1998), pp. 561–572]. The weak convergence of both estimators are established. Furthermore, the delete-one jackknife and simple bootstrap methods are used to estimate standard deviations and construct interval estimators. A simulation study is conducted to compare the two IPW approaches.     

Non parametric resampling for stationary Markov processes: The local grid bootstrap approach

A new resampling technique, referred as “local grid bootstrap” (LGB), based on nonparametric local bootstrap and applicable to a wide range of stationary general space Markov processes is proposed. This nonparametric technique resamples local neighborhoods defined around the true samples of the observed multivariate time serie. The asymptotic behavior of this resampling procedure is studied in detail. Applications to linear and nonlinear (in particular chaotic) simulated time series are presented, and compared to Paparoditis and Politis [2002. J. Statist. Plan. Inf. 108, 301–328] approach, referred as “local bootstrap” (LB) and developed in earlier similar works. The method shows to be efficient and robust even when the length of the observed time series is reasonably small.     

On tests of symmetry,marginal homogeneity and quasi-symmetry in two-way contingency tables based on minimum φ-divergence estimator with constraints

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ₂ for estimation and the best function φ₁ for testing.     

Estimating the Intercept in an Orthogonally Blocked Experiment when the Block Effects are Random

Abstract

For an orthogonally blocked experiment, Khuri [Khuri, A. I. (1992). Response surface models with random block effects. Technometrics 34:26–37] has shown that the ordinary least squares estimator, the generalized least squares estimator and the intra-block estimator of the factor effects in a response surface model with random block effects coincide. The ordinary least squares estimator ignores the blocks, whereas the generalized least squares and the intra-block estimators treat the block effects as random and fixed, respectively. As shown in this paper, the equivalence does not hold for the estimation of the intercept when the block sizes are heterogeneous. Practical examples are given to illustrate the theoretical results.     

Parameter Estimation in Conditional Heteroscedastic Models

Abstract

We study asymptotics of parameter estimates in conditional heteroscedastic models. The estimators considered are those obtained by minimizing certain functionals and those obtained by solving estimation equations. We establish consistency and derive asymptotic limit laws of the estimators. Condition under which the limit law is normal is studied. Further, bootstrap for these estimators is discussed. The limiting distribution of the estimators is not necessary always normal, and we present a real data example to illustrate this.     

Power-transformed linear regression on quantile residual life for censored competing risks data

ABSTRACT

This paper proposes a power-transformed linear quantile regression model for the residual lifetime of competing risks data. The proposed model can describe the association between any quantile of a time-to-event distribution among survivors beyond a specific time point and the covariates. Under covariate-dependent censoring, we develop an estimation procedure with two steps, including an unbiased monotone estimating equation for regression parameters and cumulative sum processes for the Box–Cox transformation parameter. The asymptotic properties of the estimators are also derived. We employ an efficient bootstrap method for the estimation of the variance–covariance matrix. The finite-sample performance of the proposed approaches are evaluated through simulation studies and a real example.