## 1 Introduction

Production theory and efficiency analysis examine how firms or production units in a particular sector of activity transform their inputs (e.g. labour, energy and capital) into quantities of outputs, that is, the goods or services that are produced by the firms. The analysis is not limited to business firms such as manufacturing concerns, electricity plants, banks and for-profit hospitals; it is used to examine schools, universities, provision of public goods and services, non-profit organisations including hospitals and credit unions and so on. The efficient production frontier is defined in the relevant input–output space as the locus of the maximal attainable level of outputs corresponding to given levels of inputs. Alternatively, if prices of inputs are available, one can consider a cost frontier defined by the minimal cost of producing various levels of outputs. Intermediate cases are also possible; for example, one might consider maximisation of quantities of a subset of outputs while minimising quantities of some (perhaps all) inputs, holding other quantities fixed. In all cases, the problem amounts to estimating a boundary surface in the relevant space (e.g. the input–output space in the case of a production frontier or the output–cost space in terms of a cost frontier) under shape constraints induced by some economic assumptions (e.g. monotonicity or concavity). The technical efficiency of a particular production plan (characterised geometrically by a point in the input–output space) is then determined by an appropriate measure of distance between this point and the optimal frontier. The background of the economic theory behind this analysis is due to *Koopmans* (1951) and *Debreu*(1951); see *Shephard* (1970) for a comprehensive presentation of the underlying economic theory.

In empirical studies, the attainable set in the input–output space is unobserved, and hence, the efficiency of a given firm is also unknown. These quantities must be estimated from a sample of observed combinations of input and output quantities obtained from existing production units operating in the activity sector being studied. Many different approaches have been investigated in the literature, including statistical models of varying degrees of sophistication and ranging from fully parametric to fully non-parametric approaches. This literature has developed in a variety of academic fields, including economics, management and management science, operations research, econometrics and statistics; in each case field, papers ranging from ‘very theoretical’ to ‘very applied’ can be found.

This ‘guided tour’ focuses on statistical results obtained in the non-parametric branch of the literature, while stressing the inherent difficulty of the problem and solutions that have been developed. The tour begins in Section 2 by defining the basics of an economic model for production theory. The most popular non-parametric estimators, based on envelopment techniques, are then presented in Section 3. Statistical properties of the estimators and practical aspects of inference making (mostly by bootstrap methods) are discussed in Section 4. Section 5 presents various extensions that have been proposed in the literature to address some of the inherent drawbacks of envelopment estimators (e.g. sensitivity to extreme data points and outliers). Section 6 shows how environmental factors that may influence the production process can be included in the analysis, allowing for heterogeneity. Finally, Section 7 briefly describes additional interesting issues and challenges that remain open questions, including (i) how to use non-parametric methods to improve some parametric estimators, (ii) introduction of noise in the observational process; (iii) testing issues; and (iv) non-parametric frontier models for panel data. The existing first solutions to these problems are described, but more work is needed on these issues.