Léopold Simar

Statistical inference in nonparametric frontier models: the state of the art

Efficiency scores of firms are measured by their distance to an estimated production frontier. The economic literature proposes several nonparametric frontier estimators based on the idea of enveloping the data (FDH and DEA-type estimators). Many have claimed that FDH and DEA techniques are non-statistical, as opposed to econometric approaches where particular parametric expressions are posited to model the frontier. We can now define a statistical model allowing determination of the statistical properties of the nonparametric estimators in the multi-output and multi-input case. New results provide the asymptotic sampling distribution of the FDH estimator in a multivariate setting and of the DEA estimator in the bivariate case. Sampling distributions may also be approximated by bootstrap distributions in very general situations. Consequently, statistical inference based on DEA/FDH-type estimators is now possible. These techniques allow correction for the bias of the efficiency estimators and estimation of confidence intervals for the efficiency measures. This paper summarizes the results which are now available, and provides a brief guide to the existing literature. Emphasizing the role of hypotheses and inference, we show how the results can be used or adapted for practical purposes.

Applied Multivariate Statistical Analysis

I Descriptive Techniques: Comparison of Batches.- II Multivariate Random Variables: A Short Excursion into Matrix Algebra.- Moving to Higher Dimensions.- Multivariate Distributions.- Theory of the Multinormal.- Theory of Estimation.- Hypothesis Testing.- III Multivariate Techniques: Regression Models.- Variable Selection.- Decomposition of Data Matrices by Factors.- Principal Components Analysis.- Factor Analysis.- Cluster Analysis.- Discriminant Analysis.- Correspondence Analysis.- Canonical Correlation Analysis.- Multidimensional Scaling.- Conjoint Measurement Analysis.- Applications in Finance.- Computationally Intensive Techniques.- IV Appendix: Symbols and Notations.- Data.

Nonparametric Frontier Estimation: A Robust Approach

Most nonparametric methods for estimating production frontiers (data envelopment analysis and free disposal hall (FDH)) are based on envelopment techniques. Statistical inference based on these estimators is available but, by construction, they are very sensitive to extreme values or outliers. We propose a nonparametric estimator, which is more robust to these extreme values. It is based on a concept of expected minimum input function (or expected maximal output function). We show how this function is related to the efficient frontier itself. The resulting estimator is related to the FDH estimator but it will not envelop all the data. The asymptotic theory is provided. Our approach includes the multiple input and multiple output cases

Estimating and bootstrapping Malmquist indices

This paper develops a consistent bootstrap estimation procedure for obtaining confidence intervals for Malmquist indices of productivity and their decompositions. Although the exposition is in terms of input-oriented indices, the techniques can he trivially extended to the output orientation. The bootstrap methodology is an extension of earlier work described in Simar and Wilson (1996). Some empirical examples are also given, using data on Swedish pharmacies.

Estimation and Inference in Nonparametric Frontier Models: Recent Developments and Perspectives

Nonparametric estimators are widely used to estimate the productive efficiency of firms and other organizations, but often without any attempt to make statistical inference. Recent work has provided statistical properties of these estimators as well as methods for making statistical inference, and a link between frontier estimation and extreme value theory has been established. New estimators that avoid many of the problems inherent with traditional efficiency estimators have also been developed; these new estimators are robust with respect to outliers and avoid the well-known curse of dimensionality. Statistical properties, including asymptotic distributions, of the new estimators have been uncovered. Finally, several approaches exist for introducing environmental variables into production models; both two-stage approaches, in which estimated efficiencies are regressed on environmental variables, and conditional efficiency measures, as well as the underlying assumptions required for either approach, are examined.

A note on the convergence of nonparametric dea estimators for production efficiency scores

Efficiency scores of production units are measured by their distance to an estimated production frontier. Nonparametric data envelopment analysis estimators are based on a finite sample of observed production units, and radial distances are considered.We investigate the consistency and the speed of convergence of these estimated efficiency scores ~or of the radial distances! in the very general setup of a multioutput and multi-input case. It is shown that the speed of convergence relies on the smoothness of the unknown frontier and on the number of inputs and outputs. Furthermore, one has to distinguish between the output- and the input-oriented cases.

Detecting outliers in frontier models: A simple approach

In frontier analysis, most of the nonparametric approaches (DEA, FDH) are based on envelopment ideas which suppose that with probability one, all the observed units belong to the attainable set. In these “deterministic” frontier models, statistical theory is now mostly available (Simar and Wilson, 2000a). In the presence of super-efficient outliers, envelopment estimators could behave dramatically since they are very sensitive to extreme observations. Some recent results from Cazals et al. (2002) on robust nonparametric frontier estimators may be used in order to detect outliers by defining a new DEA/FDH “deterministic” type estimator which does not envelop all the data points and so is more robust to extreme data points. In this paper, we summarize the main results of Cazals et al. (2002) and we show how this tool can be used for detecting outliers when using the classical DEA/FDH estimators or any parametric techniques. We propose a methodology implementing the tool and we illustrate through some numerical examples with simulated and real data. The method should be used in a first step, as an exploratory data analysis, before using any frontier estimation.

Non-parametric tests of returns to scale

This paper discusses various statistics for testing hypotheses regarding returns to scale in the context of non-parametric models of technical efficiency. In addition, the paper presents bootstrap estimation procedures which yield appropriate critical values for the test statistics. Evidence on the true sizes and power of the various proposed tests is obtained from Monte-Carlo experiments. This paper is an extension of earlier work in [Manage. Sci. 44 (1998) 49 J. Appl. Statist. 27 (2000b) 779]

Asymptotics And Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models

Nonparametric data envelopment analysis (DEA) estimators based on linear programming methods have been widely applied in analyses of productive efficiency. The distributions of these estimators remain unknown except in the simple case of one input and one output, and previous bootstrap methods proposed for inference have not been proved consistent, making inference doubtful. This paper derives the asymptotic distribution of DEA estimators under variable returns to scale. This result is used to prove consistency of two different bootstrap procedures (one based on subsampling, the other based on smoothing). The smooth bootstrap requires smoothing the irregularly bounded density of inputs and outputs and smoothing the DEA frontier estimate. Both bootstrap procedures allow for dependence of the inefficiency process on output levels and the mix of inputs in the case of input-oriented measures, or on input levels and the mix of outputs in the case of output-oriented measures.

The FDH estimator for productivity efficiency scores

In productivity analysis, the free disposal hull (FDH) is a nonparametric estimator for the production set, the set of inputs and outputs that are technically feasible. It is defined as the smallest free disposal set containing all observations in a sample of production units. One can then derive the production frontier and efficiency scores from the FDH. In the literature the method is used as if the FDH estimator were the true feasible set. However, assuming that individuals are drawn independently from a distribution where the support is the true production set, FDH efficiency scores are random variables. This paper investigates its stochastic properties