ENSURING POSITIVENESS OF THE SCALED DIFFERENCE CHISQUARE TEST STATISTIC ALBERT SATORRA UNIVERSITAT POMPEU FABRA UNIVERSITY OF CALIFORNIA


 Hope Heath
 2 years ago
 Views:
Transcription
1 PSYCHOMETRIKA VOL. 75, NO. 2, JUNE 200 DOI: 0.007/S Y ENSURING POSITIVENESS OF THE SCALED DIFFERENCE CHISQUARE TEST STATISTIC ALBERT SATORRA UNIVERSITAT POMPEU FABRA PETER M. BENTLER UNIVERSITY OF CALIFORNIA A scale ifference test statistic T that can be compute from stanar software of structural equation moels (SEM) by han calculations was propose in Satorra an Bentler (Psychometrika 66:507 54, 200). The statistic T is asymptotically equivalent to the scale ifference test statistic T introuce in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neuecker, pp , 2000), which requires more involve computations beyon stanar output of SEM software. The test statistic T has been wiely use in practice, but in some applications it is negative ue to negativity of its associate scaling correction. Using the implicit function theorem, this note evelops an improve scaling correction leaing to a new scale ifference statistic T that avois negative chisquare values. Key wors: momentstructures, goonessoffit test, chisquare ifference test statistic, chisquare istribution, nonnormality.. Introuction Moment structure analysis is wiely use in behavioral, social, an economic stuies to analyze structural relations between variables, some of which may be latent; see, e.g., Bollen an Curran (2006), Grace (2006), Yuan an Bentler (2007), an references therein. In such analyses, it frequently happens that two neste moels M 0 an M, are compare using estimation methos that are nonoptimal (asymptotically) given the istribution of the ata; e.g., maximum likelihoo (ML) estimation is use when the ata are not multivariate normal. In those circumstances, the usual chisquare ifference test T = T 0 T, base on the separate moels gooness of fit test statistics, is not χ 2 istribute. A correction to T by a scaling factor was propose by Satorra (2000) an Satorra an Bentler (200) to improve the chisquare approximation. The latter is the focus of this paper. Satorra an Bentler s (200) correction provie a simple proceure to obtain an approximate scale chisquare statistic using han calculations on regular output of structural equation moeling (SEM) analysis; it has the rawback, however, that a positive value for the scaling correction is not assure. The present paper evelops a simple proceure by which a researcher can compute the exact scaling correction of ifference test statistic base only on output from stanar SEM programs. Research supporte by grants SEJ an PR from the Spanish Ministry of Science an Technology an by USPHS grants DA0007 an DA0070. Requests for reprints shoul be sent to Albert Satorra, Department of Economics an Business, Universitat Pompeu Fabra, Ramon Trias Fargas 2527, Barcelona, Spain The Psychometric Society 243
2 244 PSYCHOMETRIKA 2. Setup an Notation Throughout we ahere to the notation an results of Satorra an Bentler (200). Let σ an s be pimensional vectors of population an sample moments, respectively, where s tens in probability to σ as sample size n +.Let n(s σ)be asymptotically normally istribute with a finite asymptotic variance matrix Γ (p p). Consier the moel M 0 : σ = σ(θ) for the moment vector σ, where σ( ) is a twicecontinuously ifferentiable vectorvalue function of θ, a qimensional parameter vector. Consier a iscrepancy function F = F(s,σ) in the sense of Browne (984), an the estimator ˆθ base on F or on an (asymptotically equivalent) weighte least squares (WLS) analysis with weight matrix V = 2 2 F(s,σ)/ σ σ evaluate at σ = s. Let M 0 : σ = σ (δ), a(δ) = 0, an M : σ = σ (δ) be two neste moels for σ. Here, δ is a (q + m)imensional vector of parameters, an σ (.) an a(.) (an mvalue function) are twicecontinuously ifferentiable vectorvalue functions of δ, a compact subset of R q+m.our interest is in the test of a null hypothesis H 0 : a(δ) = 0 against the alternative H : a(δ) 0. For the evelopments that follow, we require the Jacobian matrices Π(p (q + m)) := σ (δ)/ δ an A(m (q + m)) := a(δ)/ δ, which we assume to be regular at the true value of δ, sayδ 0. We also assume that A is of full row rank. By using the implicit function theorem, associate to M 0 (more specifically, to the restrictions a(δ) = 0), there exists (locally in a neighborhoo of δ 0 ) a onetoone function δ = δ(θ) efine in an open an compact subset SofR q, an a θ 0 in the interior of S such that δ(θ 0 ) = δ 0 an σ(δ(θ)) satisfies the moel M 0.LetH = δ(θ)/ θ ((q + m) q) be the corresponing Jacobian matrix evaluate at θ 0. Hence, by the chain rule of ifferentiation, Δ = σ/ θ = ( σ (δ)/ δ )( δ(θ)/ θ ) = ΠH. Since a(δ(θ)) = 0, it hols that with A evaluate at δ 0, AH = 0 with r(a) + r(h) = q + m, an r(.) enoting the rank of a matrix. Thus, H is an orthogonal complement of A.Let P ((q + m) (q + m)) := Π VΠ. Associate to M, the less restricte moel σ = σ (δ), the goonessoffit test statistic is T = nf (s, σ), where σ is the fitte moment vector in moel M with associate egrees of freeom r = r 0 m an scaling factor c given by c := r tr{u Γ }= r tr{vγ} r tr { P Π VΓVΠ }, () where U := V VΠP Π V. (2) We refer to Satorra an Bentler (200) for further etails. 3. Scaling Correction for the Difference Test When both moels M 0 an M are fitte, for example, by ML, then we can test the restriction a(δ) = 0, assuming M hols, using the chisquare ifference test statistic T := T 0 T. Uner the null hypothesis, we woul like T to have a χ 2 istribution with egrees of freeom m = r 0 r. This is the restricte test of M 0 within M. For general istributions of the ata, the asymptotic chisquare approximation may not hol. To improve on the chisquare approximation, Satorra (2000) gave explicit formulae that exten the scaling corrections propose by Satorra an Bentler (994) to the case of ifference, Wal, an score types of test statistics. General expressions for those corrections were also put forwar in Satorra (989, p. 46). Specifically, for the test statistic T we are consiering, Satorra (2000, p. 24) propose the following scale test statistic: T := T /ĉ, where c := m tr{u Γ } (3)
3 ALBERT SATORRA AND PETER M. BENTLER 245 with U = VΠP A ( AP A ) AP Π V. (4) Here, ĉ enotes c after substituting consistent estimates of V an Γ, an evaluating the Jacobians A an Π at the estimate ˆδ 0 when fitting M 0 (or M ). Since tr{u Γ } can be expresse as the trace of the prouct of two positive efinite matrices, tr{u Γ } > 0, an thus c > 0; the same for ĉ > 0. Consequently, T is ensure to be a nonnegative number. A practical problem with the statistic T is that it requires computations that are outsie the stanar output of current SEM programs. Furthermore, ifference tests are usually han compute from ifferent moeling runs. Satorra an Bentler (200) propose a proceure to combine the estimates of the scaling corrections c 0 an c associate to the chisquare goonessoffit test for the two fitte moels M 0 an M in orer to compute a consistent estimate of the scaling correction c for the ifference test statistic. A moifie (easy to compute) scale test statistic T with the same asymptotic istribution as T was propose. Both statistics were shown to be asymptotically equivalent uner a sequence of local alternatives (so they have the same asymptotic local power). Their proceure to compute T is as follows (see Satorra an Bentler, 200, p. 5).. Obtain the unscale an scale goonessoffit tests when fitting M 0 an M, respectively; that is, T 0 an T 0 when fitting M 0, an T an T when fitting M ; 2. Compute the scaling corrections ĉ 0 = T 0 / T 0, ĉ = T / T, an the unscale chisquare ifference T = T 0 T an its egrees of freeom m = r 0 r ; 3. Compute the scale ifference test statistic as T := T / c with c = (r 0 ĉ 0 r ĉ )/m. Here, r 0 an r are the respective egrees of freeom of the moels M 0 an M. The basis for computing the scaling correction for the ifference test statistic is the following alternative expression for U of (4) (see Satorra an Bentler, 200, p. 50) U = U 0 U, (5) where U is given in (2) an U 0 := V VΠH(H Π VΠH) H Π V. Since (5) implies mc = tr{u Γ }=tr { (U 0 U )Γ } = r 0 c 0 r c, (6) it follows that c = (r 0 c 0 r c )/m. This is the theoretical basis for Satorra an Bentler s (200) proposal as given in steps 3 above. 4. Problem with the Current Scale Difference Test ForanarbitrarymatrixV>0, (5) an (6) are exact equalities when the matrices U, U 0 an U are evaluate at a common value δ (as, e.g., the fitte value uner moel M 0 or uner M ); however, they are just asymptotic equalities when U, U 0 an U are evaluate at ifferent estimates that converge to the same true value uner the null hypothesis. Uner Satorra an Bentler s (200) proposal, c evaluates U 0 an U at the estimates ˆδ 0 an ˆδ, respectively. Since ˆδ will in general not satisfy the null moel M 0 (i.e., it will not be of the form δ = δ(θ) for the function implie by the implicit function theorem), when it eviates highly from M 0,the estimate ifference c = (r 0 ĉ 0 r ĉ )/m may turn out to be negative. This may happen in small samples, or when M 0 is highly incorrect; a result can be an improper value for T. Satorra an Bentler (200, p. 5) warne on the possibility that an improper value of T coul arise.
4 246 PSYCHOMETRIKA In orer to be sure to avoi a negative value for c, an hence T, currently one woul nee to resort to computing T using (3). Unfortunately, this is impractical or impossible for most applie researchers who only have access to stanar SEM software. Fortunately, as we show next, the exact value of T can also be obtaine from the stanar output of SEM software, using a new han computation. 5. A New Scale Test Statistic T Denote by M 0 the fit of moel M to a moel setup with starting values taken as the final estimates obtaine from moel M 0, an with number of iterations set to 0. Consier ĉ (0) := T (0) / T (0), where T (0) an T (0) are the stanar unscale an scale test statistic of this aitional run. Note that the estimate ĉ (0) uses moel M but the matrices Π an A are now evaluate at ˆδ 0 := δ(ˆθ), where ˆθ is the estimate uner M 0. Since now all the matrices involve in (5) are evaluate at ˆδ 0, the equality (6) hols exactly, an not only asymptotically, as when U 0 is evaluate at ˆδ 0 an U at ˆδ. The scaling correction that is now compute is ĉ (0) := ( r 0 ĉ 0 r ĉ (0) ) /m, (7) which, in view of (6), is the scaling correction of (3) when U is evaluate at the estimate ˆδ 0 when fitting M 0, an thus it is a positive number. The new scale ifference statistic is thus efine as T (0) := (T 0 T )/ĉ (0). (8) Clearly, T (0) = T ; that is, T (0) coincies numerically with the scale statistic (3) propose in Satorra (2000). 6. An Illustration We use these ata just for illustrative purposes, an because they provie an example where the stanar scaling correction fails to be positive. We use a latent variable moel iscusse for ata by Bentler, Satorra, an Yuan (2009). The Bonett Woowar Ranall (2002) test shows that these ata have significant excess kurtosis inicative of nonnormality at a onetail 0.05 level, so test statistics erive from ML estimation may not be appropriate an we o the scaling corrections. The moel consiere is a structure means moel, with the mean cigarette sales inirectly affecting the mean rates of the various cancers. The specifie moel is V j = λ j F + E j, j = 2,...,5, F = βv + D, V = μ + E, where the V j s enote observe variables; F, D, an E j are the common, resiual common, an unique factors, respectively; λ j enotes a factor loaing parameter, β is the effect of cigarette smoking on the cancer factor, an μ is the mean parameter for rates of smoking. The units of measurement for the factor were tie to V 2, with λ 2 =. The following values for the ML an Satorra Bentler (994) (SB) scale chisquare statistics are obtaine T = , T = , r = 9, ĉ =.6434, along with the egrees of freeom r an the scaling correction ĉ. The moel oes not fit, though for the sake of the illustration we are aiming for, this is not of concern to us.
5 ALBERT SATORRA AND PETER M. BENTLER 247 Restricte Moel M 0 The same moel is now fitte with the ae restriction that the error variances of the kiney an leukemia cancers, E 4 an E 5, are equal. This moel gives the following statistics T 0 = , T 0 = , r 0 = 0, ĉ 0 = Difference Test Our main interest lies in testing the ifference between M 0 an M, which we o with the chisquare ifference test. The ML ifference statistic is T = = , which, with egree of freeom (m), rejects the null hypothesis that the error variances for E4 an E5 are equal. Since the ata are not normal, we compute the SB (200) T scale ifference statistic. This requires computing the scaling factor c = (r 0 ĉ 0 r ĉ )/m given by [ 0(.4322) 9(.6434) ] / = = The scaling factor c is negative, so the SB ifference test cannot be carrie out; or if carrie out, it results in an improper negative chisquare value. New Scale Difference Test As escribe above, to compute the scale statistic T, we implement (7) an (8). The output that is missing in the prior runs is T (0) the value of the SB statistic obtaine at the final parameter estimates for moel M 0 when moel M is evaluate. This can be obtaine by creating a moel setup M 0 that contains the parameterization of M with start values taken from the output of moel M 0. Moel M 0 is run with zero iterations, so that the parameter values o not change before output incluing test statistics is prouce. The new result gives T (0) = , T (0) = , r = 9, ĉ (0) =.469, where as expecte, T (0) = T 0 as reporte above (i.e., the ML statistics are ientical), an the value ĉ 0 is han compute. As a result, we can compute ĉ (0) = ( r 0 ĉ 0 r ĉ (0) ) [ ] /m = (0)(.4322) (9)(.469) =.0, which, in contrast to the SB (200) c computations, is positive. Finally, we can compute the propose new SB correcte chisquare statistic as T = T (0) = (T 0 T )/ĉ (0) = ( )/.0 = 29.79, which can be referre to a χ 2 variate for evaluation. 7. Discussion The implicit function theorem was use to provie a theoretical basis for the evelopment of a practical version of the computationally more ifficult scale ifference statistic propose by For this particular example, the Appenix of Satorra an Bentler (2008) illustrates this proceure with EQS (Bentler, 2008). In the same reference, there is a secon illustration with a larger egree of freeom.
6 248 PSYCHOMETRIKA Satorra (2000). 2 The propose metho is only marginally more ifficult to compute than that of Satorra an Bentler (200) an solves the problem of an uninterpretable negative χ 2 ifference test that applie researchers have complaine about for some time. Like the metho it is replacing, the propose proceure applies to a general moeling setting. The vector of parameters σ to be moele may contain various types of moments: means, prouctmoments, frequencies (proportions), an so forth. Thus, this scale ifference test applies to methos such as factor analysis, simultaneous equations for continuous variables, loglinear multinomial parametric moels, etc. It can easily be seen that the proceure applies also in the case where the matrix Γ is singular, when the ata is compose of various samples, as in multisample analysis, an to other estimation methos. It applies also to the case where the estimate of Γ reflects the fact that we have intraclass correlation among observations, as in complex samples. Hence, this new statistic shoul be useful in a variety of applie moeling contexts. Simulation work will be neee to unerstan its virtues an limitations, relative to other alternatives, in such contexts. References Bentler, P.M. (2008). EQS 6 structural equations program manual. Encino: Multivariate Software. Bentler, P.M., Satorra, A., & Yuan, K.H. (2009). Smoking an cancers: caserobust analysis of a classic ata set. Structural Equation Moeling, 6, Bollen, K.A., & Curran, P.J. (2006). Latent curve moels: a structural equation perspective. New York: Wiley. Bonett, D.G., Woowar, J.A., & Ranall, R.L. (2002). Estimating pvalues for Maria s coefficients of multivariate skewness an kurtosis. Computational Statistics, 7, Browne, M.W. (984). Asymptotically istributionfree methos for the analysis of covariance structures. British Journal of Mathematical an Statistical Psychology, 37, Grace, J.B. (2006). Structural equation moeling an natural systems. New York: Cambrige University Press. Satorra, A. (989). Alternative test criteria in covariance structure analysis: a unifie approach. Psychometrika, 54, 3 5. Satorra, A. (2000). Scale an ajuste restricte tests in multisample analysis of moment structures. In D.D.H. Heijmans & D.S.G. Pollock, A. Satorra (Es.), Innovations in multivariate statistical analysis: a festschrift for Heinz Neuecker (pp ). Dorrecht: Kluwer Acaemic. Satorra, A., & Bentler, P.M. (994). Corrections to test statistics an stanar errors in covariance structure analysis. In A. von Eye & C.C. Clogg (Es.) Latent variables analysis: applications for evelopmental research (pp ). Thousan Oaks: Sage. Satorra, A., & Bentler, P.M. (200). A scale ifference chisquare test statistic for moment structure analysis. Psychometrika, 66, Satorra, A., & Bentler, P.M. (2008). Ensuring positiveness of the scale ifference chisquare test statistic. Department of Statistics, UCLA Department of Statistics Preprint. Yuan, K.H., & Bentler, P.M. (2007). Structural equation moeling. In C.R. Rao & S. Sinharay (Es.) Hanbook of statistics 26: Psychometrics (pp ). Amsteram: NorthHollan. Manuscript Receive: 4 JUN 2008 Final Version Receive: 25 MAY 2009 Publishe Online Date: 20 JUN Satorra (2000) provies also Monte Carlo evience on a specific moel context an various sample sizes of the superiority of the scaling correction over other alternatives such as the ajuste (mean an variance correcte) statistic.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms
More informationMeasures of distance between samples: Euclidean
4 Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More information10.2 Systems of Linear Equations: Matrices
SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix
More informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationCrossOver Analysis Using TTests
Chapter 35 CrossOver Analysis Using ests Introuction his proceure analyzes ata from a twotreatment, twoperio (x) crossover esign. he response is assume to be a continuous ranom variable that follows
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/
More informationDetecting Possibly Fraudulent or ErrorProne Survey Data Using Benford s Law
Detecting Possibly Frauulent or ErrorProne Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationJitter effects on Analog to Digital and Digital to Analog Converters
Jitter effects on Analog to Digital an Digital to Analog Converters Jitter effects copyright 1999, 2000 Troisi Design Limite Jitter One of the significant problems in igital auio is clock jitter an its
More informationOptimal Control Policy of a Production and Inventory System for multiproduct in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:15927415 Optimal Control Policy of a Prouction an Inventory System for multiprouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More information(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i
An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of
More informationModelling and Resolving Software Dependencies
June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software
More informationState of Louisiana Office of Information Technology. Change Management Plan
State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change
More informationREINVENTING GUTTMAN SCALING AS A STATISTICAL MODEL: ABSOLUTE SIMPLEX THEORY
REINVENTING GUTTMAN SCALING AS A STATISTICAL MODEL: ABSOLUTE SIMPLEX THEORY Peter M. Bentler* University of California, Los Angeles *With thanks to Prof. KeHai Yuan, University of Notre Dame 1 Topics
More informationPrinciples and Practice of Scaled Difference ChiSquare Testing. Fred B. Bryant. Albert Satorra
Principles and practice 1 Running Head: SCALED DIFFERENCE CHISQUARE TESTING Structural Equation Modeling (in press) Principles and Practice of Scaled Difference ChiSquare Testing Fred B. Bryant LOYOLA
More informationOption Pricing for Inventory Management and Control
Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories
More informationData Center Power System Reliability Beyond the 9 s: A Practical Approach
Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of missioncritical
More informationSimple Second Order ChiSquare Correction
Simple Second Order ChiSquare Correction Tihomir Asparouhov and Bengt Muthén May 3, 2010 1 1 Introduction In this note we describe the second order correction for the chisquare statistic implemented
More informationA New Evaluation Measure for Information Retrieval Systems
A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ailabor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ailabor.e
More informationCalibration of the broad band UV Radiometer
Calibration of the broa ban UV Raiometer Marian Morys an Daniel Berger Solar Light Co., Philaelphia, PA 19126 ABSTRACT Mounting concern about the ozone layer epletion an the potential ultraviolet exposure
More informationSensitivity Analysis of Nonlinear Performance with Probability Distortion
Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 2429, 214 Sensitivity Analysis of Nonlinear Performance with Probability Distortion
More informationf(x) = a x, h(5) = ( 1) 5 1 = 2 2 1
Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying
More informationThe Inefficiency of Marginal cost pricing on roads
The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that
More informationThe oneyear nonlife insurance risk
The oneyear nonlife insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on nonlife insurance reserve risk has been evote to the ultimo risk, the risk in the
More informationarxiv:1309.1857v3 [grqc] 7 Mar 2014
Generalize holographic equipartition for FriemannRobertsonWalker universes WenYuan Ai, Hua Chen, XianRu Hu, an JianBo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.
More informationDigital barrier option contract with exponential random time
IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationCh 10. Arithmetic Average Options and Asian Opitons
Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options
More informationOn Adaboost and Optimal Betting Strategies
On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore
More informationFactoring Dickson polynomials over finite fields
Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University
More informationOn the Multivariate t Distribution
Technical report from Automatic Control at Linköpings universitet On the Multivariate t Distribution Michael Roth Division of Automatic Control Email: roth@isy.liu.se 7th April 03 Report no.: LiTHISYR3059
More informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUSMALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,
More informationAn intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations
This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi
More informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationDEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS
DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS Paolo Presciani*, Monica Malvezzi #, Giuseppe Luigi Bonacci +, Monica Balli + * FS Trenitalia Unità Tecnologie Materiale Rotabile Direzione
More informationEconomics 241B Hypothesis Testing: Large Sample Inference
Economics 241B Hyothesis Testing: Large Samle Inference Statistical inference in largesamle theory is base on test statistics whose istributions are nown uner the truth of the null hyothesis. Derivation
More informationMath 230.01, Fall 2012: HW 1 Solutions
Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The
More informationWitt#5e: Generalizing integrality theorems for ghostwitt vectors [not completed, not proofread]
Witt vectors. Part 1 Michiel Hazewinkel Sienotes by Darij Grinberg Witt#5e: Generalizing integrality theorems for ghostwitt vectors [not complete, not proofrea In this note, we will generalize most of
More informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
More informationA Monte Carlo Simulation of Multivariate General
A Monte Carlo Simulation of Multivariate General Pareto Distribution an its Application 1 1 1 1 1 1 1 1 0 1 Luo Yao 1, Sui Danan, Wang Dongxiao,*, Zhou Zhenwei, He Weihong 1, Shi Hui 1 South China Sea
More informationImproved division by invariant integers
1 Improve ivision by invariant integers Niels Möller an Torbjörn Granlun Abstract This paper consiers the problem of iviing a twowor integer by a singlewor integer, together with a few extensions an
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationA Data Placement Strategy in Scientific Cloud Workflows
A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,
More informationS 4 Flavor Symmetry Embedded into SU(3) and Lepton Masses and Mixing
S 4 Flavor Symmetry Embee into SU() an Lepton Masses an Mixing Yoshio Koie Institute for Higher Eucation esearch an Practice, Osaka University, 11 Machikaneyama, Toyonaka, Osaka 50004, Japan Email aress:
More informationThe most common model to support workforce management of telephone call centers is
Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering
More informationMODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND
art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 23 ransport an elecommunication Institute omonosova iga V9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN
More informationRisk Adjustment for Poker Players
Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk
More informationLowComplexity and Distributed Energy Minimization in Multihop Wireless Networks
LowComplexity an Distribute Energy inimization in ultihop Wireless Networks Longbi Lin, Xiaojun Lin, an Ness B. Shroff Center for Wireless Systems an Applications (CWSA) School of Electrical an Computer
More informationWeb Appendices to Selling to Overcon dent Consumers
Web Appenices to Selling to Overcon ent Consumers Michael D. Grubb MIT Sloan School of Management Cambrige, MA 02142 mgrubbmit.eu www.mit.eu/~mgrubb May 2, 2008 B Option Pricing Intuition This appenix
More informationA Theory of Exchange Rates and the Term Structure of Interest Rates
Review of Development Economics, 17(1), 74 87, 013 DOI:10.1111/roe.1016 A Theory of Exchange Rates an the Term Structure of Interest Rates HyoungSeok Lim an Masao Ogaki* Abstract This paper efines the
More informationStock Market Value Prediction Using Neural Networks
Stock Market Value Preiction Using Neural Networks Mahi Pakaman Naeini IT & Computer Engineering Department Islamic Aza University Paran Branch email: m.pakaman@ece.ut.ac.ir Hamireza Taremian Engineering
More informationProduct Differentiation for SoftwareasaService Providers
University of Augsburg Prof. Dr. Hans Ulrich Buhl Research Center Finance & Information Management Department of Information Systems Engineering & Financial Management Discussion Paper WI99 Prouct Differentiation
More informationMathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition
Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an PierreEmmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and
More informationA SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION
REVSTAT Statistical Journal Volume 9, Number 2, June 2011, 155 180 A SPATIAL UNIT LEVEL MODEL FOR SMALL AREA ESTIMATION Authors: Pero S. Coelho ISEGI Universiae Nova e Lisboa, Portugal Faculty of Economics,
More informationThroughputScheduler: Learning to Schedule on Heterogeneous Hadoop Clusters
ThroughputScheuler: Learning to Scheule on Heterogeneous Haoop Clusters Shehar Gupta, Christian Fritz, Bob Price, Roger Hoover, an Johan e Kleer Palo Alto Research Center, Palo Alto, CA, USA {sgupta, cfritz,
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More informationPerformance And Analysis Of Risk Assessment Methodologies In Information Security
International Journal of Computer Trens an Technology (IJCTT) volume 4 Issue 10 October 2013 Performance An Analysis Of Risk Assessment ologies In Information Security K.V.D.Kiran #1, Saikrishna Mukkamala
More informationSensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm
Sensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,
More informationProfessional Level Options Module, Paper P4(SGP)
Answers Professional Level Options Moule, Paper P4(SGP) Avance Financial Management (Singapore) December 2007 Answers Tutorial note: These moel answers are consierably longer an more etaile than woul be
More informationExponential Functions: Differentiation and Integration. The Natural Exponential Function
46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential
More informationDETERMINING OPTIMAL STOCK LEVEL IN MULTIECHELON SUPPLY CHAINS
HUNGARIAN JOURNA OF INDUSTRIA CHEMISTRY VESZPRÉM Vol. 39(1) pp. 107112 (2011) DETERMINING OPTIMA STOCK EVE IN MUTIECHEON SUPPY CHAINS A. KIRÁY 1, G. BEVÁRDI 2, J. ABONYI 1 1 University of Pannonia, Department
More informationFirewall Design: Consistency, Completeness, and Compactness
C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an XiangYang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 787121188,
More informationOptimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs
Applie Mathematics ENotes, 8(2008), 194202 c ISSN 16072510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items
More informationPROBLEMS. A.1 Implement the COINCIDENCE function in sumofproducts form, where COINCIDENCE = XOR.
724 APPENDIX A LOGIC CIRCUITS (Corrispone al cap. 2  Elementi i logica) PROBLEMS A. Implement the COINCIDENCE function in sumofproucts form, where COINCIDENCE = XOR. A.2 Prove the following ientities
More informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility BlackScholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
More informationA Comparison of Performance Measures for Online Algorithms
A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55,
More informationMathematics Review for Economists
Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school
More informationMultilevel Modeling of Complex Survey Data
Multilevel Modeling of Complex Survey Data Tihomir Asparouhov 1, Bengt Muthen 2 Muthen & Muthen 1 University of California, Los Angeles 2 Abstract We describe a multivariate, multilevel, pseudo maximum
More information! # % & ( ) +,,),. / 0 1 2 % ( 345 6, & 7 8 4 8 & & &&3 6
! # % & ( ) +,,),. / 0 1 2 % ( 345 6, & 7 8 4 8 & & &&3 6 9 Quality signposting : the role of online information prescription in proviing patient information Liz Brewster & Barbara Sen Information School,
More information1 HighDimensional Space
Contents HighDimensional Space. Properties of HighDimensional Space..................... 4. The HighDimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More informationDIFFRACTION AND INTERFERENCE
DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe
More informationOptimized Capacity Bounds for the HalfDuplex Gaussian MIMO Relay Channel
Optimize Capacity Bouns for the HalfDuplex Gaussian MIMO elay Channel Lennart Geres an Wolfgang Utschick International ITG Workshop on mart Antennas (WA) February 211 c 211 IEEE. Personal use of this
More informationRules for Finding Derivatives
3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to
More informationUnsteady Flow Visualization by Animating EvenlySpaced Streamlines
EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating EvenlySpace Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,
More informationOptimal Energy Commitments with Storage and Intermittent Supply
Submitte to Operations Research manuscript OPRE200909406 Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton,
More informationTowards a Framework for Enterprise Architecture Frameworks Comparison and Selection
Towars a Framework for Enterprise Frameworks Comparison an Selection Saber Aballah Faculty of Computers an Information, Cairo University Saber_aballah@hotmail.com Abstract A number of Enterprise Frameworks
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationThe Quick Calculus Tutorial
The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,
More informationMannheim curves in the threedimensional sphere
Mannheim curves in the threeimensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.
More informationSupporting Adaptive Workflows in Advanced Application Environments
Supporting aptive Workflows in vance pplication Environments Manfre Reichert, lemens Hensinger, Peter Daam Department Databases an Information Systems University of Ulm, D89069 Ulm, Germany Email: {reichert,
More informationRUNESTONE, an International Student Collaboration Project
RUNESTONE, an International Stuent Collaboration Project Mats Daniels 1, Marian Petre 2, Vicki Almstrum 3, Lars Asplun 1, Christina Björkman 1, Carl Erickson 4, Bruce Klein 4, an Mary Last 4 1 Department
More informationIndices of Model Fit STRUCTURAL EQUATION MODELING 2013
Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit A recommended minimal set of fit indices that should be reported and interpreted when reporting the results of SEM analyses:
More informationCOPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono
COPULA INFERENCE FOR MULTIPLE LIVES ANALYSIS PRELIMINARIES * Yogo Purwono Department of Management Faculty of Economics, University of Inonesia Depok, INDONESIA Abstract. Copula moels are becoming increasingly
More informationNonparametric Estimation of StatePrice Densities Implicit in Financial Asset Prices
THE JOURNAL OF FINANCE VOL LIII, NO. 2 APRIL 1998 Nonparametric Estimation of StatePrice Densities Implicit in Financial Asset Prices YACINE AÏTSAHALIA an ANDREW W. LO* ABSTRACT Implicit in the prices
More informationOptimizing Multiple Stock Trading Rules using Genetic Algorithms
Optimizing Multiple Stock Traing Rules using Genetic Algorithms Ariano Simões, Rui Neves, Nuno Horta Instituto as Telecomunicações, Instituto Superior Técnico Av. Rovisco Pais, 04000 Lisboa, Portugal.
More informationFAST JOINING AND REPAIRING OF SANDWICH MATERIALS WITH DETACHABLE MECHANICAL CONNECTION TECHNOLOGY
FAST JOINING AND REPAIRING OF SANDWICH MATERIALS WITH DETACHABLE MECHANICAL CONNECTION TECHNOLOGY Jörg Felhusen an Sivakumara K. Krishnamoorthy RWTH Aachen University, Chair an Insitute for Engineering
More informationForecasting and Staffing Call Centers with Multiple Interdependent Uncertain Arrival Streams
Forecasting an Staffing Call Centers with Multiple Interepenent Uncertain Arrival Streams Han Ye Department of Statistics an Operations Research, University of North Carolina, Chapel Hill, NC 27599, hanye@email.unc.eu
More informationHeuristics for allocation of reconfigurable resources in a serial line with reliability considerations
IIE Transactions ISSN: 0740817X (Print) 15458830 (Online) Journal homepage: http://www.tanfonline.com/loi/uiie20 Heuristics for allocation of reconfigurable resources in a serial line with reliability
More informationA New Pricing Model for Competitive Telecommunications Services Using Congestion Discounts
A New Pricing Moel for Competitive Telecommunications Services Using Congestion Discounts N. Keon an G. Ananalingam Department of Systems Engineering University of Pennsylvania Philaelphia, PA 191046315
More informationGeTec Ingenieurgesellschaft für Informations und Planungstechnologie mbh. www.getecac.de. Presented by
The Design of vibro replacement Dipl.Ing. Heinz J. Priebe Presente by GeTec Ingenieurgesellschaft für Informations un Planungstechnologie mbh RheinMain Office +49 69 800 6624 Fax +49 69 800 4977 Aachen
More informationWeb Appendices of Selling to Overcon dent Consumers
Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the
More informationA modelbased approach to goodnessoffit evaluation in item response theory
A MODELBASED APPROACH TO GOF EVALUATION IN IRT 1 A modelbased approach to goodnessoffit evaluation in item response theory DL Oberski JK Vermunt Dept of Methodology and Statistics, Tilburg University,
More informationStudy on the Price Elasticity of Demand of Beijing Subway
Journal of Traffic an Logistics Engineering, Vol, 1, No. 1 June 2013 Stuy on the Price Elasticity of Deman of Beijing Subway Yanan Miao an Liang Gao MOE Key Laboratory for Urban Transportation Complex
More information11 CHAPTER 11: FOOTINGS
CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationDynamic Network Security Deployment Under Partial Information
Dynamic Network Security Deployment Uner Partial nformation nvite Paper) George Theoorakopoulos EPFL Lausanne, Switzerlan Email: george.theoorakopoulos @ epfl.ch John S. Baras University of Marylan College
More information