On the Distribution of General Quadratic Functions in Normal Vectors

A representation in terms of independent standard normal variables tor the general quadratic form in normal variables in the univariate case, obtained by DIK and DE GUNST (1985), is extended to the multivariate situation. A representation for the quadratic function in normal vectors X'AX , where X is a random matrix with normally distributed elements and A a real symmetric matrix, is given in terms of independent and identically distributed central normal vectors. The representation is valid only when the covariance structure of X is of a special form, but all known results, especially necessary and sufficient conditions for X'AX to have a Wishart distribution, can easily be derived from it.     

Analysis of innovation in SMEs using an innovative capability-based non-linear model: A study in the province of Seville (Spain)

This article analyzes organizational characteristics associated with innovative outcomes of SMEs in local economies with a low level of technological and R&D activities, where the indicators generally used to gauge technological innovation do not offer sufficient data. For the study of innovation in such economies, we propose a conceptual model based on innovative capability which also includes explanatory variables of environment and other contextual factors of the firms. Through a process of filtering spurious variables and incorporating quadratic components, we have varied the original linear structure to form a quadratic model with greater explanatory capacity and better results. The resulting quadratic model has undergone various contrasts of hypotheses, thus demonstrating the effectiveness of the model as a whole and of its basic components, as well as the importance of the main explanatory variables. Finally, the model has also proven to be valid in all sectors.     

Orthogonal expansions for the generalized Cramér-von Mises statistics

We provide a new proof for the representation of Cramér-von Mises statistics under (known) gamma and normal distributions. The new method uses orthogonal polynomials and provides an explicit form of the statistics from which the asymptotic distribution can be calculated.Acknowledgements This research was partially supported by FQM-331, FQM-270, BMF 2001-2378 and BMF2002-04525-C02-02. The authors are thankful to the referees for their suggestions and helpful comments.     

The expectation of products of quadratic forms in normal variables: the practice

Summary  A table is presented to simplify the computation of the expectation of a product of quadratic forms in normal variables. Some peculiarities of the table are discussed.     

Decomposition of the Koopmans-Beckmann Problem

Subsequent findings indicate that the Koopmans-Beckmann linear programming representation of the quadratic assignment problem may be more useful in solving for the best integer assignment and a set of sustaining prices than their initial results suggest. In response to the renewed interest in the quadratic assignment problem, this brief paper presents the Koopmans-Beckmann linear program in general matrix form, demostrates that the program is decomposable, and shows that the primal subprogram constitutes a simple linear assignment problem whose optimal solution set always contains at least one integer assignment of plants to locations.     

Realized volatility forecasting and market microstructure noise

We extend the analytical results for reduced form realized volatility based forecasting in ABM (2004) to allow for market microstructure frictions in the observed high-frequency returns. Our results build on the eigenfunction representation of the general stochastic volatility class of models developed byMeddahi (2001). In addition to traditional realized volatility measures and the role of the underlying sampling frequencies, we also explore the forecasting performance of several alternative volatility measures designed to mitigate the impact of the microstructure noise. Our analysis is facilitated by a simple unified quadratic form representation for all these estimators. Our results suggest that the detrimental impact of the noise on forecast accuracy can be substantial. Moreover, the linear forecasts based on a simple-to-implement ‘average’ (or ‘subsampled’) estimator obtained by averaging standard sparsely sampled realized volatility measures generally perform on par with the best alternative robust measures.     

On the evaluation of poly-t density functions

Poly-t densities are defined by the property that their kernel is a product, or a ratio of products, of multivariate t-density kernels. As discussed in Drèze (1977), these densities arise as Bayesian posterior densities for regression coefficients under a variety of specifications for the prior density and the data generating process. We have therefore developed methods and computer algorithms to evaluate integrating constants and other characteristics of poly-t densities with no more than a single quadratic form in the numerator (section 2). As a by-product of our analysis we have also derived an algorithm for the computation of moments of positive definite quadratic forms in Normal variables (section 3). In section 4 we discuss inference on the sampling variances associated with the models discussed in Drèze (1977).     

Design and investigation of EWMA and double EWMA with quadratic process model in R2R controllers

In the last few years, “Run-to-Run” (R2R) control techniques have been developed and used to control various processes in industries. These techniques combine control engineering and statistical process control techniques. The R2R controller is basically consists of a linear regression model that relates the input variables to the output variables using Exponentially Weighted Moving Average (EWMA) or double EWMA (dEWMA) scheme. In this paper, we have developed a quadratic R2R controller model as an extension of Box, Luceno and Del Castillo models. The validity and performance of the developed models were tested in the magnetic ceramic process. By this study we have found that the quadratic model improves capability of process and applying dEWMA procedure decreases variance of the response variables. The results of the study shows that the quadratic R2R control models outperform linear R2R controls in the certain manufacturing process.     

On decompositions of some random variables

The aim of this paper is to present several stochastic analogs of classical formulas for the gamma function. The obtained results provide representation of some random variables as finite or infinite products of independent random variables. Examples include generalized gamma, normal, beta and other distributions.     

A re‐interpretation of the linear quadratic model when inventories and sales are polynomially cointegrated

Estimation of the linear quadratic model, the workhorse of the inventory literature, traditionally takes inventories and sales to be first‐difference stationary series, and the ratio of the two variables to be stationary. However, these assumptions do not always match the properties of the data for the last two decades in the United States. We propose a model that allows for the non‐stationary characteristics of the data, using polynomial cointegration. We show that the closed‐form solution has other recent models as special cases. The resulting model performs well on aggregate and disaggregated data. Copyright © 2006 John Wiley & Sons, Ltd.     

On the independence of integer and fractional parts

We use the independence of the integer and fractional parts of exponentially distributed random variables to obtain expressions for the order statistics from a geometric distribution. As our main result we show that a strong form of this independence characterizes the exponential distribution.     

Erwartungswerte,Varianzen und Kovarianzen mehrerer Maxima der Abstände unabhängiger rechteckverteilter Zufallsvariabler

Zusammenfassung Unter Benutzung derRényi'schen Darstellung der geordneten Stichprobe unabhängiger exponentiell-verteilter Zufallsvariabler und der Darstellung der geordneten Stichprobe unabhängiger rechteckverteilter Variabler werden die gesuchten Größen sehr bequem aus einer Darstellung gewonnen, die auch für weitere Ergebnisse nützlich ist.

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Summary By means ofRényi's representation of the ordered sample of exponentially distributed variables and the representation of the ordered sample of uniformly distributed variables a useful representation for the maximum, the second-maximum etc. of the distances of uniformly distributed variables is given which delivers expectations and variances-covariances immediately.

     

Estimating demand with distance functions: Parameterization in the primal and dual

Our purpose is to investigate the ability of different parametric forms to ‘correctly’ estimate consumer demands based on distance functions using Monte Carlo methods. Our approach combines economic theory, econometrics and quadratic approximation. We begin by deriving parameterizations for transformed quadratic functions which are linear in parameters and characterized by either homogeneity or which satisfy the translation property. Homogeneity is typical of Shephard distance functions and expenditure functions, whereas translation is characteristic of benefit/shortage or directional distance functions. The functional forms which satisfy these conditions and include both first- and second-order terms are the translog and quadratic forms, respectively. We then derive a primal characterization which is homogeneous and parameterized as translog and a dual model which satisfies the translation property and is specified as quadratic. We assess functional form performance by focusing on empirical violations of the regularity conditions. Our analysis corroborates results from earlier Monte Carlo studies on the production side suggesting that the quadratic form more closely approximates the ‘true’ technology or in our context consumer preferences than the translog.     

Computing the covariance of two Brownian area integrals

We compute the expected product of two correlated Brownian area integrals, a problem that arises in the analysis of a popular sorting algorithm. Along the way we find three different formulas for the expectation of the product of the absolute values of two standard normal random variables with correlation θ . These two formulas are found: (a) via conditioning and the non-central chi-square distribution; (b) via Mehler's formula; (c) by representing the correlated normal random variables in terms of independent normal's and integration using polar coordinates.     

A note on the distribution of the product of zero‐mean correlated normal random variables

The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate.     

Afschatting van verdelingen van kwadratische vormen met toepassingen op aanpassingstoetsen voor normaliteit en exponentialiteit

Summary In this paper a simple approximation is given for the distribution of the quadratic form being a weighted sum of squares of independent, identically distributed standardized normal variates. Using the formulae, the Monte Carlo results concerning some goodness of fit tests for normality and exponentiality are verified and extended.     

Interview design for revealing preferences of policy makers

A policy maker is asked a few simple questions about his preference. Then the model represents it by a quadratic utility function, which can be made monotonic and quasi-concave (= to provide the convexity of the preference). The design of the interview with a policy maker is aimed at attaining the following goals: (a) no ambiguous output (= degeneration of the model), (b) ordinal approach to preferences (= asking questions about ordinal preferences and providing the uniqueness of the ordinal preference at the model output, regardless of its representation by a quadratic utility function), (c) stability of the model (= the model's input–output transformation is continuous). We also describe briefly the implementation of our model in a user-friendly interface to a corresponding computer program.     

Controlling a linear dynamic system according to asymmetric preferences

In this paper an attempt is made to introduce an asymmetric loss function in the context of a dynamic decision problem, where the target variables and instruments are related linearly, and where uncertainty is introduced by means of additive disturbances. Because of the particular form of the loss function an optimal linear feedback rule and an analytical expression for the minimal expected loss can only be formulated in the case of as many instruments as target variables. In the case of more targets than instruments an open-loop policy has to be adopted. Numerical experiments with a small artificial model, in which this open-loop policy is compared with the closed-loop policies resulting from two quadratic approximations are presented. Both in the case of a deterministic as well as a stochastic model the open-loop policy turns out to be better than or at least as well as the other approaches.     

On classes of estimators of the variance function of a linear estimator

Summary The variance function of a linear estimator can be expressed into a quadratic form. The present paper presents classes of estimators of this quadratic form along the lines implicitly suggested byHorvitz andThompson [1952] while formulating the classes of linear estimators. Accordingly it is noted that there exist nine principal classes of estimators out of which one principal class is examined in detail. Furthermore to illustrate the theory an example is considered where the expression for a unique estimator variance of the best estimator in theT ₁ class is derived.     

On estimating nonnegative definite quadratic forms

It is often required to estimate a quadratic form in survey sampling, especially when one has to estimate the mean squared error of a linear estimator of the population total. In this note we consider the problem of obtaining uniformly nonnegative quadratic unbiased estimators for nonnegative definite quadratic forms. The estimators considered here are necessarily quadratic. Received January 1997