A multivariate parallelogram and its application to multivariate trimmed means

This paper introduces a multivariate parallelogram that can play the role of the univariate quantile in the location model, and uses it to define a multivariate trimmed mean. It assesses the asymptotic efficiency of the proposed multivariate trimmed mean by its asymptotic variance and by Monte Carlo simulation.     

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.     

Comparison of estimation methods for the finite population mean in simple random sampling: symmetric super-populations

In this paper, a new estimator combined estimator (CE) is proposed for estimating the finite population mean ¯ Y _N in simple random sampling assuming a long-tailed symmetric super-population model. The efficiency and robustness properties of the CE is compared with the widely used and well-known estimators of the finite population mean ¯ Y _N by Monte Carlo simulation. The parameter estimators considered in this study are the classical least squares estimator, trimmed mean, winsorized mean, trimmed L-mean, modified maximum-likelihood estimator, Huber estimator (W24) and the non-parametric Hodges–Lehmann estimator. The mean square error criteria are used to compare the performance of the estimators. We show that the CE is overall more efficient than the other estimators. The CE is also shown to be more robust for estimating the finite population mean ¯ Y _N , since it is insensitive to outliers and to misspecification of the distribution. We give a real life example.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

Book Reviews

Books reviewed:
Philip Hans Franses & Dick van Dijk, Non-linear Time Series Models in Empirical Finance
Herbert Spirer, Louise Spirer & A.J. Jaffe, Misused Statistics
Deborah J. Bennett, Randomness
C.E. Linneborg, Data Analysis by Resampling: Concepts and Applications
I. Clark and W.V. Harper, Practical Geostatistics 2000     

EMPIRICAL EVIDENCE FOR ADAPTIVE CONFIDENCE INTERVALS AND IDENTIFICATION OF OUTLDSRS USING METHODS OF TRIMMING

In an attempt to apply robust procedures, conventional t-tables are used to approximate critical values of a Studentized t-statistic which is formed from the ratio of a trimmed mean to the square root of a suitably normed Winsorized sum of squared deviations. It is shown here that the approximation is poor if the proportion of trimming is chosen to depend on the data. Instead a data dependent alternative is given which uses adaptive trimming proportions and confidence intervals based on trimmed likelihood statistics. Resulting statistics have high efficiency at the normal model, proper coverage for confidence intervals, yet retain breakdown point one half. Average lengths of confidence intervals are competitive with those of recent Studentized confidence intervals based on the biweight over a range of underlying distributions. In addition, the adaptive trimming is used to identify potential outliers. Evidence in the form of simulations and data analysis support the new adaptive trimming approach.     

Quantity quantiles linear regression

We show that the definition of the θth sample quantile as the solution to a minimization problem introduced by Koenker and Bassett (Econometrica 46(1):33–50, 1978) can be easily extended to obtain an analogous definition for the θth sample quantity quantile widely investigated and applied in the Italian literature. The key point is the use of the first-moment distribution of the variable instead of its distribution function. By means of this definition we introduce a linear regression model for quantity quantiles and analyze some properties of the residuals. In Sect. 4 we show a brief application of the methodology proposed. This research was partially supported by Fondo d’Ateneo per la Ricerca anno 2005—Università degli Studi di Milano-Bicocca. The paper is the result of the common work of the authors; in particular M. Zenga has written Sects. 1 and 5 while P. Radaelli has written the remaining sections.     

Two-Stage Welsh's Trimmed Mean for the Simultaneous Equations Model

This paper discusses the large sample theory of the two-stage Welsh's trimmed mean for the limited information simultaneous equations model. Besides having asymptotic normality, this trimmed mean, as the two-stage least squares estimator, is a generalized least squares estimator. It also acts as a robust Aitken estimator for the simultaneous equations model. Examples illustrate real data analysis and large sample inferences based on this trimmed mean.     

Linear trimmed means for the linear regression with AR(1) errors model

For the linear regression with AR(1) errors model, the robust generalized and feasible generalized estimators of Lai et al. (2003) of regression parameters are shown to have the desired property of a robust Gauss Markov theorem. This is done by showing that these two estimators are the best among classes of linear trimmed means. Monte Carlo and data analysis for this technique have been performed.     

Nonparametric estimation of mean residual quantile function under right censoring

In this paper, we develop non-parametric estimation of the mean residual quantile function based on right-censored data. Two non-parametric estimators, one based on the empirical quantile function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.     

Random gradient boosting for predicting conditional quantiles

Gradient Boosting (GB) was introduced to address both classification and regression problems with great power. People have studied the boosting with L2 loss intensively both in theory and practice. However, the L2 loss is not proper for learning distributional functionals beyond the conditional mean such as conditional quantiles. There are huge amount of literatures studying conditional quantile prediction with various methods including machine learning techniques such like random forests and boosting. Simulation studies reveal that the weakness of random forests lies in predicting centre quantiles and that of GB lies in predicting extremes. Is there an algorithm that enjoys the advantages of both random forests and boosting so that it can perform well over all quantiles? In this article, we propose such a boosting algorithm called random GB which embraces the merits of both random forests and GB. Empirical results will be presented to support the superiority of this algorithm in predicting conditional quantiles.     

A varying coefficient approach to estimating hedonic housing price functions and their quantiles

The varying coefficient (VC) model introduced by Hastie and Tibshirani [26 T. Hastie and R. Tibshirani, Varying-coefficient models, J. R. Statist. Soc. (Ser. B) 55 (1993), pp. 757–796.[Web of Science ®] , [Google Scholar]] is arguably one of the most remarkable recent developments in nonparametric regression theory. The VC model is an extension of the ordinary regression model where the coefficients are allowed to vary as smooth functions of an effect modifier possibly different from the regressors. The VC model reduces the modelling bias with its unique structure while also avoiding the ‘curse of dimensionality’ problem. While the VC model has been applied widely in a variety of disciplines, its application in economics has been minimal. The central goal of this paper is to apply VC modelling to the estimation of a hedonic house price function using data from Hong Kong, one of the world's most buoyant real estate markets. We demonstrate the advantages of the VC approach over traditional parametric and semi-parametric regressions in the face of a large number of regressors. We further combine VC modelling with quantile regression to examine the heterogeneity of the marginal effects of attributes across the distribution of housing prices.     

Missing data in clinical trials: control‐based mean imputation and sensitivity analysis

In some randomized (drug versus placebo) clinical trials, the estimand of interest is the between‐treatment difference in population means of a clinical endpoint that is free from the confounding effects of “rescue” medication (e.g., HbA1c change from baseline at 24 weeks that would be observed without rescue medication regardless of whether or when the assigned treatment was discontinued). In such settings, a missing data problem arises if some patients prematurely discontinue from the trial or initiate rescue medication while in the trial, the latter necessitating the discarding of post‐rescue data. We caution that the commonly used mixed‐effects model repeated measures analysis with the embedded missing at random assumption can deliver an exaggerated estimate of the aforementioned estimand of interest. This happens, in part, due to implicit imputation of an overly optimistic mean for “dropouts” (i.e., patients with missing endpoint data of interest) in the drug arm. We propose an alternative approach in which the missing mean for the drug arm dropouts is explicitly replaced with either the estimated mean of the entire endpoint distribution under placebo (primary analysis) or a sequence of increasingly more conservative means within a tipping point framework (sensitivity analysis); patient‐level imputation is not required. A supplemental “dropout = failure” analysis is considered in which a common poor outcome is imputed for all dropouts followed by a between‐treatment comparison using quantile regression. All analyses address the same estimand and can adjust for baseline covariates. Three examples and simulation results are used to support our recommendations.     

On sign-based regression quantiles

A sign-based (SB) approach suggests an alternative criterion for quantile regression fit. The SB criterion is a piecewise constant function, which often leads to a non-unique solution. We compare the mid-point of this SB solution with the least absolute deviations (LAD) method and describe asymptotic properties of SB estimators under a weaker set of assumptions as compared with the assumptions often used with the generalized method of moments. Asymptotic properties of LAD and SB estimators are equivalent; however, there are finite sample differences as we show in simulation studies. At small to moderate sample sizes, the SB procedure for modelling quantiles at longer tails demonstrates a substantially lower bias, variance, and mean-squared error when compared with the LAD. In the illustrative example, we model a 0.8-level quantile of hospital charges and highlight finite sample advantage of the SB versus LAD.     

Testing for Granger-causality in quantiles

This paper proposes a consistent parametric test of Granger-causality in quantiles. Although the concept of Granger-causality is defined in terms of the conditional distribution, most articles have tested Granger-causality using conditional mean regression models in which the causal relations are linear. Rather than focusing on a single part of the conditional distribution, we develop a test that evaluates nonlinear causalities and possible causal relations in all conditional quantiles, which provides a sufficient condition for Granger-causality when all quantiles are considered. The proposed test statistic has correct asymptotic size, is consistent against fixed alternatives, and has power against Pitman deviations from the null hypothesis. As the proposed test statistic is asymptotically nonpivotal, we tabulate critical values via a subsampling approach. We present Monte Carlo evidence and an application considering the causal relation between the gold price, the USD/GBP exchange rate, and the oil price.     

Using power transformations when approximating quantiles

Estimators are obtained tor quantiles of survival distributions. This is accomplished by approximating Lritr distribution of the transtorrneri data, where the transformation used is that of Box and Cox (1964). The normal approximation as in Box and Cox and, in addition, the extreme value approximation are considered. More generally, to use the methods given, the approximating distribution must come from a location-scale family. For some commonly used survival random variables T the performance of the above approximations are evaluated in terms of the ratio of the true quantiles of T to the estimated one, in the long run. This performance is also evaluated for lower quantiles using simulated lognormai, Weibull and gamma data. Several examples are given to illustrate the methodology herein, including one with actual data.     

Jackknife method for intermediate quantiles

Quantile function plays an important role in statistical inference, and intermediate quantile is useful in risk management. It is known that Jackknife method fails for estimating the variance of a sample quantile. By assuming that the underlying distribution satisfies some extreme value conditions, we show that Jackknife variance estimator is inconsistent for an intermediate order statistic. Further we derive the asymptotic limit of the Jackknife-Studentized intermediate order statistic so that a confidence interval for an intermediate quantile can be obtained. A simulation study is conducted to compare this new confidence interval with other existing ones in terms of coverage accuracy.     

Bias reduction for high quantiles

High quantile estimation is of importance in risk management. For a heavy-tailed distribution, estimating a high quantile is done via estimating the tail index. Reducing the bias in a tail index estimator can be achieved by using either the same order or a larger order of number of the upper order statistics in comparison with the theoretical optimal one in the classical tail index estimator. For the second approach, one can either estimate all parameters simultaneously or estimate the first and second order parameters separately. Recently, the first method and the second method via external estimators for the second order parameter have been applied to reduce the bias in high quantile estimation. Theoretically, the second method obviously gives rise to a smaller order of asymptotic mean squared error than the first one. In this paper we study the second method with simultaneous estimation of all parameters for reducing bias in high quantile estimation.     

Estimating quantiles of several normal populations with a common mean

Suppose we have k( ? 2) normal populations with a common mean and possibly different variances. The problem of estimation of quantile of the first population is considered with respect to a quadratic loss function. In this paper, we have generalized the inadmissibility results obtained by Kumar and Tripathy (2011 Kumar, S., Tripathy, M.R. (2011). Estimating quantiles of normal populations with a common mean. Commun. Stat. - Theory Methods 40:2719–2736.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) for k = 2 to a general k( ? 2). Moreover, a massive simulation study has been done in order to numerically compare the risk values of various proposed estimators for the cases k = 3 and k = 4 and recommendations are made for the use of estimators under certain situations.     

Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.