Efficiency of nonparametric tests for scale shift at an unknown time point

Recently Hsieh considered three nonparametric tests for the scalechange problem and showed them to be asymptotically as efficient as their parametric competitors. Here the class of so called sum-type statistics is studied which contains the statistics of Hsieh. It is proved that max-type statistics can be constructed that are at least as efficient in the sense of Bahadur as the sum-type statistics.     

Block permutation principles for the change analysis of dependent data

We study an AMOC model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods. The theoretical results show that the original test statistics and their corresponding block permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then use laws of large numbers to obtain the permutation asymptotics conditionally on the given data.     

An Optimal Retrospective Change Point Detection Policy

Abstract.  Since the middle of the twentieth century, the problem of making inferences about the point in a surveyed series of observations at which the underlying distribution changes has been extensively addressed in the economics, biostatistics and statistics literature. Cumulative sum-type statistics have commonly been thought to play a central role in non-sequential change point detections. Alternatively, we present and examine an approach based on the Shiryayev–Roberts scheme. We show that retrospective change point detection policies based on Shiryayev–Roberts statistics are non-asymptotically optimal in the context of most powerful testing.     

Improving an estimator of Hsieh and Turnbull for the binormal ROC curve

The estimator of Hsieh and Turnbull (1996) for the binormal receiver operating characteristic (ROC) curve is extended from grouped to ungrouped data. The new estimator is shown to be consistent and asymptotically normally distributed, and simulation results show that it outperforms Hsieh and Turnbull's original estimator.     

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.     

Methods for missing covariates in logistic regression

Various methods have been suggested in the literature to handle a missing covariate in the presence of surrogate covariates. These methods belong to one of two paradigms. In the imputation paradigm, Pepe and Fleming (1991) and Reilly and Pepe (1995) suggested filling in missing covariates using the empirical distribution of the covariate obtained from the observed data. We can proceed one step further by imputing the missing covariate using nonparametric maximum likelihood estimates (NPMLE) of the density of the covariate. Recently Murphy and Van der Vaart (1998a) showed that such an approach yields a consistent, asymptotically normal, and semiparametric efficient estimate for the logistic regression coefficient. In the weighting paradigm, Zhao and Lipsitz (1992) suggested an estimating function using completely observed records after weighting inversely by the probability of observation. An extension of this weighting approach designed to achieve semiparametric efficient bound is considered by Robins, Hsieh and Newey (RHN) (1995). The two ends of each paradigm (NPMLE and RHN) attain the efficiency bound and are asymptotically equivalent. However, both require a substantial amount of computation. A question arises whether and when, in practical situations, this extensive computation is worthwhile. In this paper we investigate the performance of single and multiple imputation estimates, weighting estimates, semiparametric efficient estimates, and two new imputation estimates. Simulation studies suggest that the sample size should be substantially large (e.g. n=2000) for NPMLE and RHN to be more efficient than simpler imputation estimates. When the sample size is moderately large (n≤ 1500), simpler imputation estimates have as small a variance as semiparametric efficient estimates.     

Asymptotic relative efficiency of score tests in Weibull models with measurement errors

The main object of this paper is to discuss properties of the score statistics for testing the null hypothesis of no association in Weibull model with measurement errors. Three different score statistics are considered. The efficient score statistics, a naive score statistics obtained by replacing the unobserved true covariate with the observed one and a score statistics based on the corrected score statistics. It is shown that corrected and naive score statistics are equivalent and the asymptotic relative efficiency between naive and efficient score statistics is derived.     

Amemiya's generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

An L 2 Comparison Between the Bootstrap and the Empirical Edgeworth Expansion

The Bootstrap estimate for studentized statistics is more accurate than both the normal approximation and the two-term empirical Edgeworth expansion. In this article, it will be shown that the three-term empirical Edgeworth expansion for studentized statistics compares well with the bootstrap. It is also shown that the three-term Edgeworth expansion is superior to the bootstrap in some cases, using more efficient estimators than sample moments in the Edgeworth expansion, such as using maximum likelihood estimators in the one-parameter exponential family.     

Amemiya‘s generalized least squares and tests of overidentification in simultaneous equation models with qualitative or limited dependent variables

Amemiya's generalized least squares method for the estimation of simultaneous equation modeis with qualitative or limited dependent variables is known to be efficient relative to many popular two stage estimators. This note points out that test statistics for overidentification restrictions can be obtained as by-products of Amerniya's generalized least squares procedure. Amemiya's procedure is shown to be a minimum chisquare method. The Amemiya procedure is valuable both for efficient estimation and for model evaluation of such models.     

Progressive Variance Control Charts for Monitoring Process Dispersion

In a process, the deviation from location or scale parameters affects the quality of the process and waste resources. So it is essential to monitor such processes for possible changes due to any assignable causes. Control charts are the most famous tool used to meet this intention. It is useless to monitor process location until the assurance that process dispersion is in-control. This study proposes some new two-sided memory control charts named as progressive variance (PV) control charts which are based on sample variance to monitor changes in process dispersion assuming normality of quality characteristic to be monitored. Simulation studies are made, and an example is discussed to evaluate the performance of the proposed charts. The comparison of the proposed chart is made with exponentially weighted moving average- and cumulative sum-type charts for process dispersion. The study shows that performance of the proposed charts are uniformly better than its competitors for detecting positive shifts while for detecting negative shift in the variance their performance is better for small shifts and reasonably good for moderated shifts.     

Hypothesis Testing in Functional Comparative Calibration Models

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.     

Minimum variance rectangular designs for U-statistics

We consider a certain class of rectangular designs for incomplete U-statistics based on Latin squares and show it to be optimal with respect to the minimal variance criterion. We also show it to be asymptotically efficient when compared with the corresponding complete statistics, as well as uniformly more efficient than the random subset selection. We provide the necessary and sufficient conditions for the existence of our design and give some examples of applications.     

Influences of misspecification on asymptotic relative efficiency of coefficients of determination: Application to agriculture

Violation of correct specification may cause some undesirable results such as biased logistic regression coefficients and less efficient test statistics. In this paper, asymptotic relative efficiency (ARE) of various coefficients of determination in misspecified binary logistic regression models is investigated. Seven types of misspecification have been included. ARE of test statistics for exponential and Weibull distributions as a method of calculating optimal cutpoints is derived to demonstrate misspecification. Theoretical relationships between coefficients of determination have also been analyzed. Extensive simulations using bootstrap method and a real data application reveal more efficient one under various modeling scenarios.     

Efficient score tests for heteroskedasticity in micro-economics

This paper develops an artificial regression that can be employed to obtain efficient score test statistics for heteroskedasticity in the context of a variety of micro-econometric models     

Optimal designs for copula models

Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well.     

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.     

A partial score test for difference among heterogeneous populations

In event time data analysis, comparisons between distributions are made by the logrank test. When the data appear to contain crossing hazards phenomena, nonparametric weighted logrank statistics are usually suggested to accommodate different-weighted functions to increase the power. However, the gain in power by imposing different weights has its limits since differences before and after the crossing point may balance each other out. In contrast to the weighted logrank tests, we propose a score-type statistic based on the semiparametric-, heteroscedastic-hazards regression model of Hsieh [2001. On heteroscedastic hazards regression models: theory and application. J. Roy. Statist. Soc. Ser. B 63, 63–79.], by which the nonproportionality is explicitly modeled. Our score test is based on estimating functions derived from partial likelihood under the heteroscedastic model considered herein. Simulation results show the benefit of modeling the heteroscedasticity and power of the proposed test to two classes of weighted logrank tests (including Fleming–Harrington's test and Moreau's locally most powerful test), a Renyi-type test, and the Breslow's test for acceleration. We also demonstrate the application of this test by analyzing actual data in clinical trials.     

On the generation of subsets of order statistics

A modification of Schucany's (1972) method for generating order statistics is given which allows for efficient generation of certain subsets of order statistics.     

Local Power Analyses of Goodness-of-fit Tests for Copulas

Abstract.  The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramér–von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramér–von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods.