No abstract

In frontier analysis, most of the nonparametric approaches (FDH,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). 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.

Spatial interaction or gravity models have been used to model flows that take many forms, for example population migration, commodity flows, traffic flows, all of which reflect movements between origin and destination regions. We focus on how to interpret estimates from spatial autoregressive extensions to the conventional regression‐based gravity models that relax the assumption of independence between flows. These models proposed by LeSage and Pace (, ) define spatial dependence involving flows between regions. We show how to calculate partial derivative expressions for these models that can be used to quantify these various types of effect that arise from changes in the characteristics/explanatory variables of the model.

Constrained smoothing splines are discussed under order restrictions on the shape of the function m. We consider shape constraints of the type m ( r) > 0, i.e. positivity, monotonicity, convexity, F F F. (Here for an integer r > 0, m ( r) denotes the rth derivative of m.) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a ®rst step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is``projected'' onto the constrained set. The projection is calculated with respect to a Sobolev-type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the in®nite number of constraints can be replaced by a ®nite number with only a small loss of accuracy, this is discussed for estimation of a convex function.

When the aim is to model market-shares as a function of explanatory variables, 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 widely used in econometrics for discrete choice modeling. But aggregated multinomial logit models (MNL) and the so-called market-share models or generalized multiplicative competitive interaction models (GMCI) present some limitations: in their simpler version they do not specify brand-specific and cross-effect parameters. Introducing all possible cross effects is not possible in the MNL and would imply a very large number of parameters in the case of the GMCI. In this paper, we consider alternative models which are the Dirichlet covariate 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 last 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 main properties of the models in order to explain why CODA and DIR models can outperform traditional market-share models. The benefits of highlighting these relationships is on one hand to propose new models to the marketing literature and on the other hand to improve the interpretation of the CODA and DIR models using the elasticities of the econometrics literature. Finally, an application to the automobile market is presented where we model brands market-shares as a function of media investments, controlling for the brands average price and a scrapping incentive dummy variable. We compare the goodness-of-fit of the various models in terms of quality measures adapted to shares.