Asymptotic Relative Efficiency Research Articles

On some ridge regression estimators: a nonparametric approach

This paper considers the R-estimation of the parameters of a multiple regression model when the design matrix is ill-conditioned. Accordingly, we introduce the ridge regression (RR) modification to the usual R-estimators and consider five RR R-estimators when it is suspected that the regression parameters may belong to a linear subspace of the parameter space. The regions of optimality of the proposed estimators are determined based on the quadratic risks. Asymptotic relative efficiency tables and risk graphs are provided for the numerical and graphical comparisons of the five estimators.

Least-modules estimates for spatial autoregression coefficients

For the process of spatial autoregression of order (1, 1), the least-modules estimates were found to be asymptotically normal and the relative asymptotic efficiency of these estimates with respect to least-squares estimates was calculated. Using computer simulation methods, the stability of least-modules estimates to observational overshoots was studied.

Rank-based testing in linear models with stable errors

Linear models with stable error densities are considered, and their local asymptotic normality with respect to the regression parameter is established. We use this result, combined with Le Cam's third lemma, to obtain local powers and asymptotic relative efficiencies for various classical rank tests (the regression and analysis of variance counterparts of the Wilcoxon, van der Waerden and median tests) under α-stable densities with various values of the skewness parameter and tail index. The same results are used to construct new rank tests, based on ‘stable scores’, achieving parametric optimality at specified stable densities. A Monte Carlo study is conducted to compare their finite-sample relative performances.

A class of simple distribution-free rank-based unit root tests

We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference density g , which need not coincide with the unknown actual innovation density f . The validity of these tests, in terms of exact finite-sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than stressing asymptotic optimality, however, we emphasize finite-sample performances, which also depend, quite heavily, on initial values. It appears that our rank-based tests significantly outperform the traditional Dickey–Fuller tests, as well as the more recent procedures proposed by Elliott et al. (1996), Ng and Perron (2001), and Elliott and Müller (2006), for a broad range of initial values and for heavy-tailed innovation densities. Thus, they provide a useful complement to existing techniques.

Asymptotic power of tests of normality under local alternatives

Seven tests of univariate normality are studied in view of their asymptotic power under local alternatives. The procedures under consideration are either based on the empirical skewness and/or kurtosis, including the popular Jarque–Bera statistic, as well as Cramér–von Mises, Anderson–Darling and Kolmogorov–Smirnov functionals of an empirical process with estimated parameters. The large-sample behavior of these test statistics under contiguous sequences is obtained; this allows for the computation of their associated local power curves and of their asymptotic relative efficiency in the light of a measure proposed by Berg and Quessy (2009). Comparisons are made under four classes of local alternatives, including those used by Thadewald and Büning (2007) in a recent Monte-Carlo power study. These theoretical results are related to empirical ones and many recommendations are formulated.

Planning Life Tests Based on Progressively Type-I Grouped Censored Data from the Weibull Distribution

In this article, we apply the simulated annealing algorithm to determine optimally spaced inspection times for the two-parameter Weibull distribution for any given progressive Type-I grouped censoring plan. We examine how the asymptotic relative efficiencies of the estimates are affected by the position of the monitoring points and the number of monitoring points used. A comparison of different inspection plans is made that will enable the user to select a plan for a specified quality goal. Using the same algorithm, we can also determine an optimal progressive Type-I grouped censoring plan when the inspection times and the expected proportions of total failures in the experiment are pre-fixed. Finally, we discuss the sample size and the acceptance constant of the progressively Type-I grouped censored reliability sampling plan when the optimal inspection times are used.

A QUASI‐LOCALLY MOST POWERFUL TEST FOR CORRELATION IN THE CONDITIONAL VARIANCE OF POSITIVE DATA

SummaryA test is derived for short‐memory correlation in the conditional variance of strictly positive, skewed data. The test is quasi‐locally most powerful (QLMP) under the assumption of conditionally gamma data. Analytical asymptotic relative efficiency calculations show that an alternative test, based on the first‐order autocorrelation coefficient of the squared data, has negligible relative power to detect correlation in the conditional variance. Finite‐sample simulation results confirm the poor performance of the squares‐based test for fixed alternatives, as well as demonstrating the poor performance of the test based on the first‐order autocorrelation coefficient of the raw (levels) data. The robustness of the QLMP test, both to misspecification of the conditional distribution and to misspecification of the dynamics, is also demonstrated using simulation. The test is illustrated using financial trade durations data.

Information in the sample covariate distribution in prevalent cohorts

Methods of estimation and inference about survival distributions based on length-biased samples are well-established. Comparatively little attention has been given to the assessment of covariate effects in the context of length-biased samples, but prevalent cohort studies often have this objective. We show that, like the survival distribution, the covariate distribution from a prevalent cohort study is length-biased, and that this distribution may contain parametric information about covariate effects on the survival time. As a result, a likelihood based on the joint distribution of the survival time and the covariates yields estimates of covariate effects which are at least as efficient as estimates arising from a traditional likelihood which conditions on covariate values in the length-biased sample. We also investigate the empirical bias of estimators arising from a joint likelihood when the population covariate distribution is misspecified. The asymptotic relative efficiencies and empirical biases under model misspecification are assessed for both proportional hazards and accelerated failure time models. The various methods considered are applied in an illustrative analysis of risk factors for death following onset of dementia using data collected in the Canadian Study of Health and Aging.

Relative Efficiency of Trend Tests with Misspecified Genetic Models in Stratified Analyses of Case-Control or Cohort Data

Background/Aims: Genetic single-nucleotide polymorphism (SNP) data are often analyzed using trend tests that rely on a specific assumption about the way that disease frequency varies across genotypes, but the validity of this assumption is not typically known. We explore the relative efficiency of trend tests in which the assumed model may or may not correspond to the true genetic model. Methods: We derive formulae for the asymptotic relative efficiencies (AREs) comparing tests that assume different genetic models. We consider both unstratified and stratified tests, using both case-control and cohort data. We illustrate these formulae using realistic parameters and compare the calculated AREs to simulated relative efficiencies in finite samples. Results: The AREs are identical for unstratified tests using case-control and cohort data, but differ for stratified tests. Loss of efficiency can be substantial, given specific combinations of high-risk allele frequencies, disease frequencies, and assumed versus actual genetic models. Given reasonably large sample sizes, asymptotic calculations align well with finite sample simulations of relative efficiency. Conclusions: ARE is a useful estimate of the relative efficiency of statistics using different underlying genetic models. ARE calculations reveal that additive gene doses, which are most commonly used, lead to large losses in power in some settings.

Assessment of Misclassification in a Binary Response: Recovering Information on Clinically Significant Cataract Prevalence from Cataract Surgery Data in Atomic-bomb Survivors

Cataract surgery results when a patient decides to undergo lens surgery following a diagnosis of a clinically significant cataract (CSC). Because the presence of a CSC is generally latent and unobserved, a person might not receive cataract surgery even if the person has a CSC. This misclassification needs to be adjusted in the statistical analysis of CSC so as to reduce the bias in the parameter estimation. Following Magder and Hughes (1997) and using the cataract surgery data on atomic-bomb survivors at the Radiation Effects Research Foundation, we used this method for estimating the prevalence of CSC in a linear logistic dose response model taking account of the sensitivity and/or specificity of the decision for lens surgery. The estimated sensitivity was 0.385 (95% CI: 0.268, 0.517) and the estimated specificity was perfect. The odds ratio estimate for the radiation dose response changed from 1.39 (95% CI: 1.24, 1.55) to 1.58 (95% CI: 1.26, 1.98) when allowing for the imperfect sensitivity. A large sample simulation study with a continuous covariate was conducted, assuming either imperfect sensitivity or imperfect specificity, to investigate the performance of the method. Results indicated that the parameter estimates are almost correct. We calculated the asymptotic relative efficiency (ARE) for a simple logistic regression slope estimate and showed that the ARE depends only on the values of slope and intercept parameters.

A Class of Simple Distribution-Free Rank-Based Unit Root Tests

We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a , which needs not coincide with the unknown actual innovation density . The validity of these tests, in terms of exact finite sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than asymptotic optimality, however, we emphasize finite-sample performances, which, quite heavily, also depend on initial values. It appears that our rank-based tests significantly outperform the traditional Dickey-Fuller tests, as well as the more recent procedures proposed by Elliot, Rothenberg, and Stock (1996), Ng and Perron (2001), and Elliott and Muller (2006), for a broad range of initial values and for heavy-tailed innovation densities. As such, they provide a useful complement to existing techniques.

Testing Exponentiality Against NBUL Alternatives Using Positive and Negative Fractional Moments

In this note a moment inequality for the New is Better than Used in Laplace ordering (NBUL) class of lifetime distributions is derived. A simple test statistic for testing exponentiality against NBUL alternatives is constructed. The test statistic is proved to be asymptotically normal and consistent. Critical values obtained by a Monte Carlo simulation are tabulated for different sample sizes. The asymptotic relative efficiency of the test is also studied.

Spearman's footrule and Gini's gamma: a review with complements

The scattered literature on Spearman's footrule and Gini's gamma is surveyed. The following topics are covered: finite-sample moments and asymptotic distribution under independence; large-sample distribution under arbitrary alternatives; asymptotic relative efficiency for testing independence; consistent asymptotic variance estimation through the jackknife; multivariate generalisations and uses. Complementary results and an extensive bibliography are provided, along with several original illustrations.

A multivariate Wilcoxon regression estimate

Based on the multivariate spatial rank function introduced by Möttönen and Oja [(1995), ‘Multivariate Spatial Sign and Rank Methods’, Journal of Nonparametric Statistics, 5, 201–213] and Möttönen et al. [(1997), ‘On the Efficiency of Multivariate Spatial Sign and Rank Tests’, Annals of Statistics, 25, 542–552], an extension of the univariate Wilcoxon regression estimate to multivariate linear models is proposed and studied. For both of the cases covariates are deterministic and i.i.d. random: we show that the proposed estimate is consistent and asymptotically normal under some appropriate assumptions. We have demonstrated that the asymptotic relative efficiency of the new regression estimate is the same as that of the generalised multivariate Hodges–Lehmann location estimates proposed by Chaudhuri [(1992), ‘Multivariate Location Estimation Using Extension of R-estimates Through U-statistics Type Approach’, Annals of Statistics, 20, 897–916] (with m=2); thus it possesses high efficiency. Simulations show that it also performs very well in the finite sample data. While the estimate is only rotation invariant, a version that is affine invariant is proposed as well.

Dual divergence estimators and tests: Robustness results

The class of dual ϕ -divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.

Estimation and Testing in Elliptical Functional Measurement Error Models

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.

Comparison of three semiparametric methods for estimating dependence parameters in copula models

Three semiparametric methods for estimating dependence parameters in copula models are compared, namely maximum pseudo-likelihood estimation and the two method-of-moment approaches based on the inversion of Spearman’s rho and Kendall’s tau. For each of these three asymptotically normal estimators, an estimator of their asymptotic (co)variance is stated in three different situations, namely the bivariate one-parameter case, the multivariate one-parameter case and the multivariate multiparameter case. An extensive Monte Carlo study is carried out to compare the finite-sample performance of the three estimators under consideration in these three situations. In the one-parameter case, it involves up to six bivariate and four-variate copula families, and up to five levels of dependence. In the multiparameter case, attention is restricted to trivariate and four-variate normal and t copulas. The maximum pseudo-likelihood estimator appears as the best choice in terms of mean square error in all situations except for small and weakly dependent samples. It is followed by the method-of-moment estimator based on Kendall’s tau, which overall appears to be significantly better than its analogue based on Spearman’s rho. The simulation results are complemented by asymptotic relative efficiency calculations. The numerical computation of Spearman’s rho, Kendall’s tau and their derivatives in the case of copula families for which explicit expressions are not available is also investigated.

A method for resolving ties in asymptotic relative efficiency

This article presents a method for determining which of two tests may offer an advantage in power when their asymptotic relative efficiencies (ARE) are the same. The method maintains some of the desirable properties of ARE, such as ease of calculation and singular result based upon asymptotic properties of the tests. However, using higher-order Taylor expansions of the test statistics’ means, the procedure offers additional insight into the finite sample size behavior of the tests. Two examples demonstrate the method.

Formula omitted]-optimality of unequal versus equal cluster sizes for mixed effects linear regression analysis of randomized trials with clusters in one treatment arm

The efficiency loss due to varying cluster sizes in trials where treatments induce clustering of observations in one of the two treatment arms is examined. Such designs may arise when comparing group therapy to a condition with only medication or a condition not involving any kind of treatment. For maximum likelihood estimation in a mixed effects linear regression, asymptotic relative efficiencies ( RE) of unequal versus equal cluster sizes in terms of the D -criterion and D s -criteria are derived. A Monte Carlo simulation for small sample sizes shows these asymptotic REs to be very accurate for the D s -criterion of the fixed effects and rather accurate for the D -criterion. Taylor approximations of the asymptotic REs turn out to be accurate and can be used to predict the efficiency loss when planning a trial. The RE usually will be more than 0.94 and, when planning sample sizes, multiplying both the number of clusters in one arm and the number of persons in the other arm by 1/ RE is the most cost-efficient way of regaining the efficiency loss.

The Deflation-Based FastICA Estimator: Statistical Analysis Revisited

This paper provides a rigorous statistical analysis of the deflation-based FastICA estimator, where the independent components (ICs) are extracted sequentially. The focus is on two aspects of the estimator: robustness against outliers as measured by the influence function (IF) and on its asymptotic relative efficiency (ARE) as measured by the ratio of the asymptotic variance of the FastICA w.r.t. the optimal maximum likelihood estimator (MLE). The derived compact closed-form expression of the IF reveals the vulnerability of the FastICA estimator to outliers regardless of the used nonlinearity. A cautionary finding is that even a moderate observation towards certain directions can render the estimator deficient in the sense that its separation performance degrades worse than a plain guess. The IF allows the derivation of a compact closed-form expression for the asymptotic covariance matrix of the FastICA estimator and subsequently its asymptotic relative efficiencies (AREs). The ARE figures calculated for some selected source distributions illustrate the fact that the order which the ICs are found is crucial as the accuracy of the previously extracted components can dominantly affect the accuracy of the successive deflation stages.