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This book collects contributions written by well-known statisticians and econometricians to acknowledge Léopold Simar’s far-reaching scientific impact on Statistics and Econometrics throughout his career. The papers contained herein were presented at a conference in
Louvain-la-Neuve in May 2009 in honor of his retirement. The contributions cover a broad variety of issues surrounding frontier
estimation, which Léopold Simar has contributed much to over the past two decades, as well as related issues such as semiparametric regression and models for censored data.

This book collects contributions written by well-known statisticians and econometricians to acknowledge Léopold Simar’s far-reaching scientific impact on Statistics and Econometrics throughout his career. The papers contained herein were presented at a conference in
Louvain-la-Neuve in May 2009 in honor of his retirement. The contributions cover a broad variety of issues surrounding frontier
estimation, which Léopold Simar has contributed much to over the past two decades, as well as related issues such as semiparametric regression and models for censored data.



Chapter 1. Nadaraya’s Estimates for Large Quantiles and Free Disposal Support Curves

A new characterization of partial boundaries of a free disposal multivariate support, lying near the true support curve, is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya’s estimates of upper quantile-based frontiers. The benefits of the new class of partial boundaries are illustrated through simulated examples and a real data set, and both empirical and smoothed estimates are compared via Monte Carlo experiments. When the transformed distribution is attracted to the Weibull extreme-value type distribution, the smoothed estimator of the full frontier outperforms frankly the sample estimator in terms of both bias and Mean-Squared Error, under optimal bandwidth. In this domain of attraction, Nadaraya’s estimates of extreme quantiles might be superior to the sample versions in terms of MSE although they have a higher bias. However, smoothing seems to be useless in the heavy tailed case.
Abdelaati Daouia, Laurent Gardes, Stéphane Girard

Chapter 2. Production Efficiency versus Ownership: The Case of China

In this study, we explore the pattern of efficiency among enterprises in China’s 29 provinces across different ownership types in heavy and light industries and across different regions (coastal, central and western). We do so by performing a bootstrap-based analysis of group efficiencies (weighted and non-weighted), estimating and comparing densities of efficiency distributions, and conducting a bootstrapped truncated regression analysis. We find evidence of interesting differences in efficiency levels among various ownership groups, especially for foreign and local ownership, which have different patterns for light and heavy industries.
Alice Shiu, Valentin Zelenyuk

Chapter 3. Nonparametric Frontier Estimation from Noisy Data

A new nonparametric estimator of production frontiers is defined and studied when the data set of production units is contaminated by measurement error. The measurement error is assumed to be an additive normal random variable on the input variable, but its variance is unknown. The estimator is a modification of the m-frontier, which necessitates the computation of a consistent estimator of the conditional survival function of the input variable given the output variable. In this paper, the identification and the consistency of a new estimator of the survival function is proved in the presence of additive noise with unknown variance. The performance of the estimator is also studied using simulated data.
Maik Schwarz, Sébastien Van Bellegem, Jean-Pierre Florens

Chapter 4. Estimating Frontier Cost Models Using Extremiles

In the econometric literature on the estimation of production technologies, there has been considerable interest in estimating so called cost frontier models that relate closely to models for extreme non-standard conditional quantiles (Aragon et al. Econ Theor 21:358–389, 2005) and expected minimum input functions (Cazals et al. J Econometrics 106:1–25, 2002). In this paper, we introduce a class of extremile-based cost frontiers which includes the family of expected minimum input frontiers and parallels the class of quantile-type frontiers. The class is motivated via several angles, which reveals its specific merits and strengths. We discuss nonparametric estimation of the extremile-based costs frontiers and establish asymptotic normality and weak convergence of the associated process. Empirical illustrations are provided.
Abdelaati Daouia, Irène Gijbels

Chapter 5. Panel Data, Factor Models, and the Solow Residual

In this paper we discuss the Solow residual (Solow, Rev. Econ. Stat. 39:312–320, 1957) and how it has been interpreted and measured in the neoclassical production literature and in the complementary literature on productive efficiency. We point out why panel data are needed to measure productive efficiency and innovation and thus link the two strands of literatures. We provide a discussion on the various estimators used in the two literatures, focusing on one class of estimators in particular, the factor model. We evaluate in finite samples the performance of a particular factor model, the model of Kneip, Sickles, and Song (A New Panel Data Treatment for Heterogeneity in Time Trends, Econometric Theory, 2011), in identifying productive efficiencies. We also point out that the measurement of the two main sources of productivity growth, technical change and technical efficiency change, may be not be feasible in many empirical settings and that alternative survey based approaches offer advantages that have yet to be exploited in the productivity accounting literature.
Alois Kneip, Robin C. Sickles

Chapter 6. Asymptotic Properties of Some Non-Parametric Hyperbolic Efficiency Estimators

A hyperbolic measure of technical efficiency was proposed by Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 1985) wherein efficiency is measured by the simultaneous maximum, feasible reduction in input quantities and increase in output quantities. In cases where returns to scale are not constant, the non-parametric data envelopment analysis (DEA) estimator of hyperbolic efficiency cannot be written as a linear program; consequently, the measure has not been used in empirical studies except where returns to scale are constant, allowing the estimator to be computed by linear programming methods. This paper develops an alternative estimator of the hyperbolic measure proposed by Fare et al. (The Measurement of Efficiency of Production, Kluwer-Nijhoff Publishing, Boston, 1985). Statistical consistency and rates of convergence are established for the new estimator. A numerical procedure allowing computation of the original estimator is provided, and this estimator is also shown to be consistent, with the same rate of convergence as the new estimator. In addition, an unconditional, hyperbolic order-m efficiency estimator is developed by extending the ideas of Cazals et al. (J. Econometric. 106:1–25, 2002). Asymptotic properties of this estimator are also given.
Paul W. Wilson

Chapter 7. Explaining Efficiency in Nonparametric Frontier Models: Recent Developments in Statistical Inference

The explanation of efficiency differentials is an essential step in any frontier analysis study that aims to measure and compare the performance of decision making units. The conditional efficiency measures that have been introduced in recent years (Daraio and Simar, J. Prod. Anal. 24:93–121, 2005) represent an attractive alternative to two-step approaches, to handle external environmental factors, avoiding additional assumptions such as the separability between the input-output space and the space of external factors. Although affected by the curse of dimensionality, nonparametric estimation of conditional measures of efficiency eliminates any potential specification issue associated with parametric approaches. The nonparametric approach requires, however, estimation of a nonstandard conditional distribution function which involves smoothing procedures, and therefore the estimation of a bandwidth parameter. Recently, Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) proposed a data driven procedure for selecting the optimal bandwidth based on a general result obtained by Hall et al. (J. Am. Stat. Assoc. 99(486):1015–1026, 2004) for estimating conditional probability densities. The method employs least squares cross-validation (LSCV) to determine the optimal bandwidth with respect to a weighted integrated squared error (WISE) criterion.This paper revisits some of the recent advances in the literature on handling external factors in the nonparametric frontier framework. Following the Bădin et al. (Eur. J. Oper. Res. 201(2):633–640, 2010) approach, we provide a detailed description of optimal bandwidth selection in nonparametric conditional efficiency estimation, when mixed continuous and discrete external factors are available. We further propose an heterogeneous bootstrap which allows improving the detection of the impact of the external factors on the production process, by computing pointwise confidence intervals on the ratios of conditional to unconditional efficiency measures.We illustrate these extensions through some simulated data and an empirical application using the sample of U.S. mutual funds previously analyzed in Daraio and Simar (J. Prod. Anal. 24:93–121, 2005; Eur. J. Oper. Res. 175(1):516–542, 2006; Advanced Robust and Nonparametric Methods in Efficiency Analysis: Methodology and Applications, Springer, New York, 2007a).
Luiza Bădin, Cinzia Daraio

Chapter 8. Estimation of a General Parametric Location in Censored Regression

Consider the random vector (X, Y ), where Y represents a response variable and X an explanatory variable. The response Y is subject to random right censoring, whereas X is completely observed. Let m(x) be a conditional location function of Y given X = x. In this paper we assume that m( ⋅) belongs to some parametric class \(\mathcal{M} =\{ {m}_{\theta } : \theta \in \Theta \}\) and we propose a new method for estimating the true unknown value θ0. The method is based on nonparametric imputation for the censored observations. The consistency and asymptotic normality of the proposed estimator are established.
Cédric Heuchenne, Ingrid Van Keilegom

Chapter 9. On Convex Boundary Estimation

Consider a convex set S of the form \(S =\{ (\mathbf{x},y) \in {\mathbb{R}}_{+}^{p} \times \{ {\mathbb{R}}_{+}\,\vert \,0 \leq y \leq g(\mathbf{x})\}\), where the function g stands for the upper boundary of the set S. Suppose that one is interested in estimating the set S (or equivalently, the boundary function g) based on a set of observations laid on S. Then one may think of building the convex-hull of the observations to estimate the set S, and the corresponding estimator of the boundary function g is given by the roof of the constructed convex-hull. In this chapter we give an overview of statistical properties of the convex-hull estimator of the boundary function g. Also, we discuss bias-correction and interval estimation with the convex-hull estimator.
Seok-Oh Jeong, Byeong U. Park

Chapter 10. The Skewness Issue in Stochastic Frontiers Models: Fact or Fiction?

Skewness plays an important role in the stochastic frontier model. Since the model was introduced by Aigner et al. (J. Econometric 6:21–37, 1977), Meeusen and van den Broeck (Int. Econ. Rev. 18:435–444, 1997), and Battese and Cora (Aust. J. Agr. Econ. 21:169–179, 1977), researchers have often found that the residuals estimated from these models displayed skewness in the wrong direction. In such cases applied researchers were faced with two main and often overlapping alternatives, either respecifying the model or obtaining a new sample, neither of which are particularly appealing due to inferential problems introduced by such data-mining approaches. Recently, Simar and Wilson (Econometric Rev. 29:62–98, 2010) developed a bootstrap procedure to address the skewness problem in finite samples. Their findings point to the latter alternative as potentially the more appropriate-increase the sample size. That is, the skewness problem is a finite sample one and it often arises in finite samples from a data generating process based on the correct skewness. Thus the researcher should first attempt to increase the sample size instead of changing the model specification if she finds the “wrong” skewness in her empirical analyses. In this chapter we consider an alternative explanation to the “wrong” skewness problem and offer a new solution in cases where this is not a mere finite sample fiction but also a fact. We utilize the Qian and Sickles (Stochastic Frontiers with Bounded Inefficiency, Rice University, Mimeo, 2008) model in which an upper bound to inefficiencies or a lower bound to efficiencies is specified based on a number of alternative one-sided bounded inefficiency distributions. We consider one of the set of specifications considered by Qian and Sickles (Stochastic Frontiers with Bounded Inefficiency, Rice University, Mimeo, 2008) wherein inefficiencies are assumed to be doubly-truncated normal. This allows the least square residuals to display skewness in both directions and nests the standard half-normal and truncated-normal inefficiency models. We show and formally prove that finding incorrect skewness does not necessarily indicate that the stochastic frontier model is misspecified in general. Misspecification instead may arise when the researcher considers the wrong distribution for the bounded inefficiency process. Of course if the canonical stochastic frontier model is the proper specification the residuals still may have the incorrect skew in finite samples but this problem goes away as sample size increases. This point was originally made in Waldman (Estimation in Economic Frontier Functions, Unpublished manuscript, University of North Carolina, Chapel Hill, 1977) and Olson et al. (J. Econometric. 13:67–82, 1980). We also conduct a limited set of Monte Carlo experiments that confirm our general findings. We show that “wrong” skewness can be a large sample issue. There is nothing inherently misspecified about the model were this to be found in large samples if one were to consider the bounded inefficiency approach. In this way the “wrong” skewness, while problematic in standard models, can become a property of samples drawn from distributions of bounded inefficiencies.
Pavlos Almanidis, Robin C. Sickles

Chapter 11. Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator

In semiparametric models it is a common approach to under-smooth the nonparametric functions in order that estimators of the finite dimensional parameters can achieve root-n consistency. The requirement of under-smoothing may result, as we show, from inefficient estimation methods or technical difficulties. Xia et al. (J. Roy. Statist. Soc. B. 64:363–410, 2002) proposed an adaptive method for the multiple-index model, called MAVE. In this chapter we further refine the estimation method. Under some conditions, our estimator of the single-index is asymptotically normal and most efficient in the semi-parametric sense. Moreover, we derive higher-order expansions for our estimator and use them to define an optimal bandwidth for the purposes of index estimation. As a result we obtain a practically more relevant method and we show its superior performance in a variety of applications.
Yingcun Xia, Wolfgang Karl Härdle, Oliver Linton


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