Christine Thomas-Agnan

Reproducing kernel Hilbert spaces in probability and statistics

1 Theory.- 2 RKHS AND STOCHASTIC PROCESSES.- 3 Nonparametric Curve Estimation.- 4 Measures And Random Measures.- 5 Miscellaneous Applications.- 6 Computational Aspects.- 7 A Collection of Examples.- to Sobolev spaces.- A.l Schwartz-distributions or generalized functions.- A.1.1 Spaces and their topology.- A.1.2 Weak-derivative or derivative in the sense of distributions.- A.1.3 Facts about Fourier transforms.- A.2 Sobolev spaces.- A.2.1 Absolute continuity of functions of one variable.- A.2.2 Sobolev space with non negative integer exponent.- A.2.3 Sobolev space with real exponent.- A.2.4 Periodic Sobolev space.- A.3 Beppo-Levi spaces.

NONPARAMETRIC FRONTIER ESTIMATION: A CONDITIONAL QUANTILE-BASED APPROACH

In frontier analysis, most of the nonparametric approaches (free disposal hull [FDH], data envelopment analysis [DEA]) are based on envelopment ideas, and their statistical theory is now mostly available. However, by construction, they are very sensitive to outliers. Recently, a robust nonparametric estimator has been suggested by Cazals, Florens, and Simar (2002, Journal of Econometrics 1, 1–25). In place of estimating the full frontier, they propose rather to estimate an expected frontier of order m. Similarly, we construct a new nonparametric estimator of the efficient frontier. It is based on conditional quantiles of an appropriate distribution associated with the production process. We show how these quantiles are interesting in efficiency analysis. We provide the statistical theory of the obtained estimators. We illustrate with some simulated examples and a frontier analysis of French post offices, showing the advantage of our estimators compared with the estimators of the expected maximal output frontiers of order m.We thank J.P. Florens for helpful discussions and C. Cazals for providing the post office data set. We also are very grateful to the referees for useful suggestions.

Explaining the pattern of regional unemployment: The case of the Midi-Pyrénées region

Abstract. Unemployment rates vary widely at the sub-regional level. We seek to explain why such variation occurs, using data for 174 districts in the Midi-Pyrénées region of France for 1990–1991. A set of explanatory variables is derived from theory and the voluminous literature. The best model includes a correction for spatially autocorrelated errors. Unemployment rates are higher in urban areas and, where per capita income is higher, are consistent with the view that unemployment differences largely reflect variations in “amenities.” Along with a lack of evidence of housing market rigidities, these suggest that subregional variations in unemployment are not mainly the result of labor market disequilibrium.

Smooth Conditional Distribution Function and Quantiles under Random Censorship

We consider a nonparametric random design regression model in which the response variable is possibly right censored. The aim of this paper is to estimate the conditional distribution function and the conditional α-quantile of the response variable. We restrict attention to the case where the response variable as well as the explanatory variable are unidimensional and continuous. We propose and discuss two classes of estimators which are smooth with respect to the response variable as well as to the covariate. Some simulations demonstrate that the new methods have better mean square error performances than the generalized Kaplan-Meier estimator introduced by Beran (1981) and considered in the literature by Dabrowska (1989, 1992) and Gonzalez-Manteiga and Cadarso-Suarez (1994).

Spatial point process models for location-allocation problems

The problem of finding an optimal location frequently occurs in geomarketing, economics and other fields: positioning a new branch of a bank, a supermarket, a fire station, a plant, designing a traffic network, etc. The optimal location of the source facility is the argument-minimum of an optimization problem parametrized by some characteristics of the clients. The random nature of some of these characteristics has already been recognized, but few stochastic models for location-allocation problems address the issue of uncertainty of the locations of the clients, and even then they do it with very naive tools. It is proposed to recognize uncertainty in the spatial positions of the clients, and possible spatial autocorrelation as well, by considering the random inputs of the optimization as one realization of a spatial marked point process. The method, called SPP location-allocation, involves fitting a point process model, simulating from the adjusted process, and solving a family of optimization problems for each simulated set of observations. The advantage of this approach over the deterministic one is twofold: it gives an indication of the spatial variability of the optimal solution, and it allows one to solve larger problems. Finally an application to the optimal positioning of a new fire station in the Toulouse area (France) is presented with some heuristic algorithms.

About predictions in spatial autoregressive models: optimal and almost optimal strategies

ABSTRACT About predictions in spatial autoregressive models: optimal and almost optimal strategies. Spatial Economic Analysis. This paper addresses the problem of prediction in the spatial autoregressive (SAR) model for areal data, which is classically used in spatial econometrics. With kriging theory, prediction using the best linear unbiased predictors (BLUPs) is at the heart of the geostatistical literature. From a methodological point of view, we explore the limits of the extension of BLUP formulas in the context of SAR models for out-of-sample prediction simultaneously at several sites. We propose a more tractable ‘almost best’ alternative and clarify the relationship between the BLUP and a proper expectation–maximization (EM) algorithm predictor. From an empirical perspective, we present data-based simulations to compare the efficiency of classical formulas with the best and almost best predictions.

Measuring and testing spatial mass concentration with micro-geographic data

Abstract We address the question of measuring and testing industrial spatial concentration based on micro-geographic data with distance-based methods. We discuss the basic requirements for such measures and we propose four additional requirements. We also discuss the null assumptions classically used for testing aggregation of a particular sector and propose an alternative point of view. Our general index measure involves a cumulative and a non-cumulative version. This allows us to propose an alternative version of the Duranton–Overman index with a proper baseline as well as a cumulative version of this same index. We present simulations to evaluate the respective powers of this new approach and the classical ones.

Using compositional and Dirichlet models for market share regression

ABSTRACT When the aim is to model market shares, the marketing literature proposes some regression models which can be qualified as attraction models. They are generally derived from an aggregated version of the multinomial logit model. But aggregated multinomial logit models (MNL) and the so-called generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross effect parameters. In this paper, we consider alternative models: the Dirichlet model (DIR) and the compositional model (CODA). DIR allows to introduce brand-specific parameters and CODA allows additionally to consider cross effect parameters. We show that these two models can be written in a similar fashion, called attraction form, as the MNL and the GMCI models. As market share models are usually interpreted in terms of elasticities, we also use this notion to interpret the DIR and CODA models. We compare the properties of the models in order to explain why CODA and DIR models can outperform traditional market share models. An application to the automobile market is presented where we model brands market shares as a function of media investments, controlling for the brands price and scrapping incentive. We compare the quality of the models using measures adapted to shares.

Efficiency Measurement: A Nonparametric Approach

The aim of the paper is to present a statistical methodology allowing a meaningful comparison of the production performance of firms without resorting to the usual concept of production frontier. We introduce an efficiency measure based on a nonstandard conditional distribution and propose a two-stage estimation procedure with a smoothing step followed by an isotonization step. We illustrate the approach through a simulated example and an analysis of the performance of Spanish electricity distributors.

A Collection of Examples

New reproducing kernels with interesting applications continually appear in the literature. In Section 4 of the present chapter we list major examples for which the kernel and the associated norm and space are explicitly described. They can be used to illustrate aspects of the theory or to practically implement some of the tools presented in the book. In Section 2 and 3 we give examples of effective constructions of kernels. In Section 2 we apply the general characterization theorem proved in Chapter 1. In Section 3 we consider the class of factorizable kernels to which belong markovian kernels defined in Chapter 2, and show how to construct them from functions of one single variable. The present chapter does not end with exercises as others but we would like to encourage the reader to use it as a basis to construct new kernels, norms and spaces with amazing features and applications!