In this paper, segmented concave least squares (SCES) is introduced. SCLS is a nonparametric piecewise linear regression problem in which the estimated function is (monotonic) concave and the number of linear segments (k) is pre-specified. Ordinary least squares (k = 1) and concave least squares (k = n, the number of observations) are two extreme cases of this problem. An application of SCLS is to estimate a hedonic function. Using this method, observations are categorized into k groups and a piecewise linear hedonic function is estimated such that there is one linear segment for every group. The estimated hedonic function holds the principle of diminishing marginal utility. In this paper, SCLS is used to categorize hotels in Finland into three groups. A trade-off between the number of groups and the goodness of fit measure is used to determine the number of groups. Based on the similarities of the pricing methods, hotels in the sample are endogenously classified and the shadow prices for each group are calculated. The results reveal that the hotels do not value hotel attributes similarly and there are significant differences among groups. Hedonic pricing model via SCLS provides a novel categorization of hotels that cannot be obtained by using ordinary least squares. (C) 2017 Elsevier B.V. All rights reserved.

Controlling the number of active assets (cardinality of the portfolio) in a mean-variance portfolio problem is practically important but computationally demanding. Such task is ordinarily a mixed integer quadratic programming (MIQP) problem. We propose a novel approach to reformulate the problem as a mixed integer linear programming (MILP) problem for which computer codes are readily available. For numerical tests, we find cardinality constrained minimum variance portfolios of stocks in S&P500. A significant gain in robustness and computational effort by our MILP approach relative to MIQP is reported. Similarly, our MILP approach also competes favorably against cardinality constrained portfolio optimization with risk measures CVaR and MASD. For illustrations, we depict portfolios in a portfolio map where cardinality provides a third criterion in addition to risk and return. Fast solution allows an interactive search for a desired portfolio.

We propose a penalized method for the least squares estimator of a multivariate concave regression function. This estimator is formulated as a quadratic programing (QP) problem with O(n(2)) constraints, where n is the number of observations. Computing such an estimator is a very time-consuming task, and the computational burden rises dramatically as the number of observations increases. By introducing a quadratic penalty function, we reformulate the concave least squares estimator as a QP with only non-negativity constraints. This reformulation can be adapted for estimating variants of shape restricted least squares, i.e. the monotonic-concave/convex least squares. The experimental results and an empirical study show that the reformulated problem and its dual are solved significantly faster than the original problem. The Matlab and R codes for implementing the penalized problems are provided in the paper. (C) 2016 Elsevier B.V. All rights reserved.

Standard axioms of free disposability, convexity and constant returns to scale employed in Data Envelopment Analysis (DEA) implicitly assume continuous, real-valued inputs and outputs. However, the implicit assumption of continuous data will never hold with exact precision in real world data. To address the discrete nature of data explicitly, various formulations of Integer DEA (IDEA) have been suggested. Unfortunately, the axiomatic foundations and the correct mathematical formulation of IDEA technology has caused considerable confusion in the literature. This chapter has three objectives. First, we re-examine the axiomatic foundations of IDEA, demonstrating that some IDEA formulations proposed in the literature fail to satisfy the axioms of free disposability of continuous inputs and outputs, and natural disposability of discrete inputs and outputs. Second, we critically examine alternative efficiency metrics available for IDEA. We complement the IDEA formulations for the radial input measure with the radial output measure and the directional distance function. We then critically discuss the additive efficiency metrics, demonstrating that the optimal slacks are not necessarily unique. Third, we consider estimation of the IDEA technology under stochastic noise, modeling inefficiency and noise as Poisson distributed random variables.

Isotonic nonparametric least squares (INLS) is a regression method for estimating a monotonic function by fitting a step function to data. In the literature of frontier estimation, the free disposal hull (FDH) method is similarly based on the minimal assumption of monotonicity. In this paper, we link these two separately developed nonparametric methods by showing that FDH is a sign-constrained variant of INLS. We also discuss the connections to related methods such as data envelopment analysis (DEA) and convex nonparametric least squares (CNLS). Further, we examine alternative ways of applying isotonic regression to frontier estimation, analogous to corrected and modified ordinary least squares (COLS/MOLS) methods known in the parametric stream of frontier literature. We find that INLS is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings. In the absence of noise, the corrected INLS (CINLS) has a higher discriminating power than FDH. In the case of noisy data, we propose to apply the method of non-convex stochastic envelopment of data (non-convex StoNED), which disentangles inefficiency from noise based on the skewness of the INLS residuals. The proposed methods are illustrated by means of simulated examples. (C) 2013 Elsevier B.V. All rights reserved.

In this paper, we consider non-singular linear transformation of the inputand outputvariables in the Data Envelopment Analysis (DEA). The transformation is useful in selecting variables and dealing, for instance, with interval scale variables. We will develop general theory and show that the results are invariant due to non-singular linear transformation provided the concept of “dominance” is defined accordingly. The invariance property is valid only for non-singular linear transformation. Finally, we briefly discuss in singular linear transformation and illustrate some pitfalls, which may lead to wrong results.

This paper explores the changes in value added (VA) of a sample of schools for cohorts of students finishing secondary education between 2005 and 2008. VA estimates are based on distance measures obtained from DEA models. These measures are computed for each pupil in each school, and evaluate the distance between the school frontier in a given year and a pooled frontier comprising all schools analysed. The school VA is then computed by aggregating the VA scores for the cohort of pupils attending that school in a given year. The ratio between VA estimates for two consecutive cohorts, that attended the school in different years, is taken as the index of VA change. However, the evolution of school performance over time should consider not only the movements of the school frontier, but should also take into account other effects, such as the proximity of the students to the best-practices, represented by the school frontier, observed over time. For that purpose we developed an enhanced Malmquist index to evaluate the evolution of school performance over time. One of the components of the Malmquist index proposed measures VA change, and the other measures the ability of all school students to move closer to their own school best practices over time. The approach developed is applied to a sample of Portuguese secondary schools.

This paper presents a model for ranking efficient units by a new approach. In the proposed method, the idea of excluding the unit being scored from the production possibility set is changed to the idea of weakening the unit being scored. We propose a model for ranking efficient DMUs that is more efficient and less problematic than the models based on excluding the under evaluation unit. # $ &' ( )*' + ! , : ! !# . $! # % & '( ! #) *'+ $,-)/ )0 0 1 + . $! % (*2 ,3 ! + # ! ) 4, % * . * 5 % 67 8 9 : ; ? @A B + # C ;+ D * E ? ! + + F 3 G! ) C . ( $! 67 ? > H C I?) 8 %: 8J $,-)/ )0 0 K@A # # & 7 5 ) / ? .

This paper presents a model for ranking efficient units by a new approach. In the proposed method, the idea of excluding the unit being scored from the production possibility set is changed to the idea of weakening the unit being scored. We propose a model for ranking efficient DMUs that is more efficient and less problematic than the models based on excluding the under evaluation unit. # $ &' ( )*' + ! , : ! !# . $! # % & '( ! #) *'+ $,-)/ )0 0 1 + . $! % (*2 ,3 ! + # ! ) 4, % * . * 5 % 67 8 9 : ; ? @A B + # C ;+ D * E ? ! + + F 3 G! ) C . ( $! 67 ? > H C I?) 8 %: 8J $,-)/ )0 0 K@A # # & 7 5 ) / ? .

Production Possibility Set (PPS) based on Free Disposal Hull assumption describes the minimum PPS for evaluating efficiency of DMUs and presents one reference for each unit. Tulkens (Journal of Productivity Analysis, 4(1), 183–210) proposed a mathematical program and a procedure for solving FDH model that can be used for only VRS technology. In this paper, we extend the method for solving all four standard technologies (VRS, CRS, NDRS and NIRS) by a numeration algorithm without using LP or MILP regular solving methods.