Location problems on a sphere

This paper reviews recent developments in facility location with geodesic distances. Geodesic distances are becoming more important as industrial, defence and energy networks become progressively more global. Discussed are the minisum or Weber problem, the minimax or Rawls problem, and the maximin or obnoxious facility problem. These problems are basic to location theory on the plane; they become more complex on the sphere than on the plane because of the topological nature of the surface of a sphere. Extensions and variations are discussed.     

Measuring scale efficiency from a parametric hyperbolic distance function

This paper shows how scale efficiency can be measured from an arbitrary parametric hyperbolic distance function with multiple outputs and multiple inputs. It extends the methods introduced by Ray (J Product Anal 11:183–194, 1998), and Balk (J Product Anal 15:159–183, 2001) and Ray (2003) that measure scale efficiency from a single-output multi-input distance function and from a multi-output and multi-input distance function, respectively. The method developed in the present paper is different from Ray’s and Balk’s in that it allows for simultaneous contraction of inputs and expansion of outputs. Theorems applicable to an arbitrary parametric hyperbolic distance function are introduced first, and then their uses in measuring scale efficiency are illustrated with the translog functional form.     

Undesirable outputs and a primal Divisia productivity index based on the directional output distance function

Despite their great popularity, all the conventional Divisia productivity indexes ignore undesirable outputs. The purpose of this study is to fill in this gap by proposing a primal Divisia-type productivity index that is valid in the presence of undesirable outputs. The new productivity index is derived by total differentiation of the directional output distance function with respect to a time trend and referred to as the Divisia–Luenberger productivity index. We also empirically compare the Divisia–Luenberger productivity index and a representative of the conventional Divisia productivity indexes–the radial-output-distance-function-based Feng and Serletis (2010) productivity index–using aggregate data on 15 OECD countries over the period 1981–2000. Our empirical results show that failure to take into account undesirable outputs not only leads to misleading rankings of countries both in terms of productivity growth and in terms of technological change, but also results in wrong conclusions concerning efficiency change.     

On an admissibility problem involving the moment generating function of the lognormal distribution

A new location invariant loss function is considered and the best invariant estimator of normal mean is obtained. This estimator is a function of the moment generating function of the lognormal distribution. The admissibility is studied of a class of linear estimators of the form cX+d, where X~N(/,C²), with / unknown and C² known. This yields the admissibility of the best invariant estimator of /.     

Transforms for finding probabilities and variate values of a distribution function in tables of a related distribution function

Het verband tussen enige bedkende verdelingen.
In dit artikel worden enkele schema's gegeven voor het verband tussen een aantal van de meest bekende verdelingen. Bovendien wordt een aantal andere relaties besproken. He doel van dit artikel is drieledig. Ten eerste kunnen statistische afdelingen, die in het bezit zijn van een uitgebreide tabellenverzamelign de gegeven transformaties gebruiken om bestaande tabellen uit andere tabellen te vervolledigen ten dienste van bepaalde praktische toepassingen. Ten tweede kunnen statistici, die niet in het bezit van volledige tabellenverzamelingen zijn, ontbreakende tabellen vervanden tussen deze verdelingen leiden tot meer inzicht in bepaalde toetsings-methoden. Dit laatste punt wordt niet uitvoerig uitgewerkt.
Par. 1. geeft een inleiding en overzicht; par. 2 vermeldt de gebruikte symbolen en verwijst naar bekende tabellenverzamelingen; par. 3 en 4 geven schematisch de transformaties weer; par. 5 geeft een schematisch overzicht van de relaties; par. 6 geeft een aantal numerieke voorbeelden; in de appendix worden de wiskundige formules van de verdelingen vermeld.
Afleidingen worden niet gegeven.     

The depth function of a population distribution

Tukey (1975) introduced the notion of halfspace depth in a data analytic context, as a multivariate analog of rank relative to a finite data set. Here we focus on the depth function of an arbitrary probability distribution on Â^p, and even of a non-probability measure. The halfspace depth of any point / in Â^p is the smallest measure of a closed halfspace that contains /. We review the properties of halfspace depth, enriched with some new results. For various measures, uniform as well as non-uniform, we derive an expression for the depth function. We also compute the Tukey median, which is the / in which the depth function attains its maximal value. Various interesting phenomena occur. For the uniform distribution on a triangle, a square or any regular polygon, the depth function has ridges that correspond to an 'inversion' of depth contours. And for a product of Cauchy distributions, the depth contours are squares. We also consider an application of the depth function to voting theory.     

Estimating a distribution function in the presence of auxiliary information

For estimating the distribution functionF of a population, the empirical or sample distribution functionF _n has been studied extensively. Qin and Lawless (1994) have proposed an alternative estimator for estimatingF in the presence of auxiliary information under a semiparametric model. They have also proved the point-wise asymptotic normality of . In this paper, we establish the weak convergence of to a Gaussian process and show that the asymptotic variance function of is uniformly smaller than that ofF _n . As an application of , we propose to employ the mean and varianceŜ _n ² of to estimate the population mean and variance in the presence of auxiliary information. A simulation study is presented to assess the finite sample performance of the proposed estimators , andŜ _n ² .     

Minimax invariant estimator of a continuous distribution function under a general loss function

The problem of invariant estimation of a continuous distribution function is considered under a general loss function. Minimaxity of the minimum risk invariant estimator of a continuous distribution function is proved for any sample size n ≥ 2.     

A primal Divisia technical change index based on the output distance function

We derive a primal Divisia technical change index based on the output distance function and further show the validity of this index from both economic and axiomatic points of view. In particular, we derive the primal Divisia technical change index by total differentiation of the output distance function with respect to a time trend. We then show that this index is dual to the Jorgenson and Griliches (1967) dual Divisia total factor productivity growth (TFPG) index when both the output and input markets are competitive; dual to the Diewert and Fox (2008) markup-adjusted revenue-share-based dual Divisia technical change index when market power is limited to output markets; dual to the Denny et al. (1981) and Fuss (1994) cost-elasticity-share-based dual Divisia TFPG index when market power is limited to output markets and constant returns to scale is present; and also dual to a markup-and-markdown-adjusted Divisia technical change index when market power is present in both output and input markets. Finally, we show that the primal Divisia technical change index satisfies the properties of identity, commensurability, monotonicity, and time reversal. It also satisfies the property of proportionality in the presence of path independence, which in turn requires separability between inputs and outputs and homogeneity of subaggregator functions.     

Deregulation and dairy production systems: a Bayesian distance function approach

Milk quota trading rules differ across EU member countries. In Denmark a biannual milk quota exchange was set up in 1997 to promote a more efficient reallocation of milk quotas as well as to reduce transaction costs related to the searching and matching of sellers and buyers. Using two comprehensive unbalanced panel data sets on organic and conventional milk farms this study attempts to disentangle the effects of the introduction of quota transferability on the production structure of those farms as well as the probability of market entry/exit. Bayesian estimation techniques are used to estimate an input oriented generalized Leontief distance function as well as a curvature constrained specification. The results suggest that the deregulation in the quota allocation mechanism led to an increased allocative efficiency of organic as well as conventional milk production as well as a relative shift of the PPF in favor of the production of organic milk.     

Identifying bank outputs and inputs with a directional technology distance function

The bank efficiency literature lacks an agreed definition of bank outputs and inputs. This is problematic given the long-standing controversy concerning the status of deposits, but also because bank efficiency estimates are known to be affected by the inclusion of additional outputs such as non-traditional (fee-based) activities or risk measures. This paper proposes a data-driven identification of bank outputs and inputs using the directional technology distance function. While previous applications of this tool used symmetric expansion or contraction directions, we focus on a set of orthogonal directions, each corresponding to an assumption on the input/output status of an individual variable. These directions correspond to a set of different specifications, whose estimated coefficients can be used to determine the input or output status of all variables except the regressand. Our empirical analysis revealed a very consistent pattern across the alternative specifications estimated. There is strong evidence that customer deposits are an input, and that non-performing loans are an important undesirable output. Finally, the orthogonal expansions/contractions we consider avoid the simultaneity problem raised by the “convenient normalization” commonly used to impose linear homogeneity in stochastic frontier estimation.     

Kernel estimation of a smooth distribution function based on censored data

The problem of estimating a smooth distribution functionF at a pointτ based on randomly right censored data is treated under certain smoothness conditions onF. The asymptotic performance of a certain class of kernel estimators is compared to that of the Kaplan-Meier estimator ofF(τ). It is shown that the relative deficiency of the Kaplan-Meier estimator ofF(τ) with respect to the appropriately chosen kernel type estimator tends to infinity as the sample sizen increases to infinity. Strong uniform consistency and the weak convergence of the normalized process are also proved. Research Surported in part by NIH grant 1R01GM28405.     

On a generalised stirling distribution

In this note we consider a generalisation of theStirling noumber of the second kind. The distribution of the sum ofn independent zero-truncated Poisson variables, which can be expressed in terms of such a gneralised number, may be called horizontally generalised Stirling distribution of the second kind. A recurrence relation for the probability function of this distribution, which will be useful for tabulation purposes, is given. The distribution function is obtained in terms of a linear combination of incomplete gamma functions.     

Revisiting the decomposition of cost efficiency for non-homothetic technologies: a directional distance function approach

In the early 1980’s Kopp and Diewert proposed a popular method to decompose cost efficiency into allocative and technical efficiency for parametric functional forms based on the radial approach initiated by Farrell. We show that, relying on recently proposed homogeneity and duality results, their approach is unnecessary for self-dual homothetic production functions, while it is inconsistent in the non-homothetic case. By stressing that for homothetic technologies the radial distance function can be correctly interpreted as a technical efficiency measure, since allocative efficiency is independent of the output level and radial input reductions leave it unchanged, we contend that for non-homothetic technologies this is not the case because optimal input demands depend on the output targeted by the firm, as does the inequality between marginal rates of substitution and market prices—allocative inefficiency. We demonstrate that a correct definition of technical efficiency corresponds to the directional distance function because its flexibility ensures that allocative efficiency is kept unchanged through movements in the input production possibility set when solving technical inefficiency, and therefore the associated cost reductions can be solely—and rightly—ascribed to technical-engineering-improvements. The new methodology allowing for a consistent decomposition of cost inefficiency is illustrated resorting to simple examples of non-homothetic production functions.     

On a recent characterization of the bivariate normal distribution

Summary A recent characterization of the bivariate normal distribution is recalled and a few remarks regarding this characterization and its extension to the multivariate case are given.     

Profit efficiency in the European Union banking industry: a directional technology distance function approach

We employ the directional technology distance function approach and present estimates of profit efficiency in the 25 European Union (EU) member states over the period 1998–2008. This method decomposes profit efficiency into its technical and allocative components. We investigate potential efficiency differences across the old EU region and the new EU member states, across countries and across banks of different size. Our results indicate a significant level of profit inefficiency for the EU region, which is predominantly attributed to allocative inefficiency. Our findings also suggest that banks operating in the old EU region are, on average, more profit efficient than credit institutions in the new EU member states. Overall, we observe considerable variation of efficiency scores across countries and different patterns in efficiency change over time, as well as a negative relationship between bank size and efficiency.