Estimation of Parameters of Three Parameter Esscher Transformed Laplace Distribution
|
|
- Doris Johnson
- 5 years ago
- Views:
Transcription
1 Iteratioal Joural of Statistics ad Systems ISSN Volume 1, Number (017), pp Research Idia Publicatios Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio Dais George 1 ad Rimsha H 1 (Departmet of Statistics, Catholicate College/ MG Uiversity, Pathaamthitta, Kerala, Idia) Project Fellow (UGC Major Project, Catholicate College/ Research Scholar (Bharathiar Uiversity), Pathaamthitta, Kerala, Idia) Abstract I this article, we cosider the estimatio of parameters of three parameter Esscher trasformed Laplace distributio which is a alterative to various types of asymmetric Laplace distributios. The three parameter Esscher trasformed Laplace distributios is the locatio scale family of the oe parameter Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). Keywords: Asymmetric distributio, Esscher trasformed Laplace distributio, Heavy-tailed distributio, Maximum Likelihood Estimatio Method. I. INTRODUCTION I the last several decades various forms of asymmetric, heavy-tailed ad asymmetric heavy-tailed distributios have appeared i the literature, see McGill (196), Holla ad Bhattacharya (1968), Balakrisha ad Ambagaspitiya (199), Ghitay(005, 007), Kozumbowski ad Podg'orski (000) ad Poiraud-Casaova ad Thomas Agam (000), Jayakumar ad Kuttykrisha (006), ] VaidaBartkute ad LeoidasSakalauskas (007).
2 670 Dais George ad Rimsha H I this paper, we cosider the estimatio of three parameter Esscher trasformed Laplace distributio, which is the locatio scale family of Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). It is a alterative to the existig heavy-tailed ad asymmetric distributios. Esscher trasformed Laplace distributio satisfies may importat statistical properties like ifiite divisibility, geometric iifiite divisibility, stability with respect to geometric summatio, maximum etropy, fiiteess of momets, simplicity etc. ad subclasses of geometric stable distributios. This model provides more flexibility, allowig for asymmetry, peakedess ad tail heaviess tha the ormal model, which are commo features of may data sets, see Sebastia George ad Dais George (01). The probability desity fuctio of ad the distributio fuctio of oe parameter Esscher traformed Laplace distributio, deoted by ETL( ) is give by ad f(x; ) ={ (1 θ ) (1 θ ) e [x(1+θ)], x < 0, θ < 1 e [ x(1 θ)], x 0, θ < 1, (1) F(x) = { (1 ) (1 ) e [x(1+θ)], x < 0, θ < 1 + (1+θ) (1 e [ x(1 θ)] ), x 0, θ < 1. () Here θ is the parameter of the distributio. Graphs of the p.d.f of ETL( ) for various values of are give i Figure 1. Figure1. Desities of Esscher trasformed Laplace distributio for (a) (-1, 0), (b) = 0 (classical Laplace) ad (c) (0, 1).
3 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 671. THREE PARAMETER ESSCHER TRANSFORMED LAPLACE DISTRIBUTION Itroducig the locatio parameter(µ) ad scale parameter(σ) i the ETL( ) distributio, the p.d.f ad d.f of the ETL(,µ,σ) distributio are as follows: ad f(x;,µ,σ) ={ F(x;,µ,σ) ={ (1 θ ) σ (1 θ ) (1 ) σ 1 (1+ ) e[(x μ σ )(1+θ)], x < μ e [(μ x σ )(1 θ)], x μ, e [(x μ σ )(1+θ)], x < μ e [(μ x σ )(1 θ)], x μ, θ < 1, σ > 0, (3) θ < 1, σ > 0, () Graphs of the desity fuctio of ETL(,µ,σ) for µ = 5 ad for various values Of ad σ i Figure. Figure : Desities of Esscher Trasformed Laplace Distributio for (a) σ = 0. ad (-1; 0), (b) σ = 0. ad (0; 1) (c) = -0:6 ad Various Values of σ ad (d) = 0:6 ad Various Values of σ.
4 67 Dais George ad Rimsha H The mea, variace, momet geeratig fuctio, characteristic fuctio ad th momet of the Esscher trasformed Laplace distributio are give by Mea = µ+ σ (1 ), Variace = σ (1+ ) (1 ), Momet Geeratig Fuctio = e tµ 1 t σ (1 ) + t σ (1 ), ad Characteristic Fuctio = e itµ 1 it σ (1 ) + t σ (1 ) th momet of y about μ =! σ [ (1 θ)+1 +(1+θ) +1 (1 θ ) ]. 3. PARAMETER ESTIMATION I this sectio, we study the problem of estimatig the parameters of three parameter Esscher trasformed Laplace distributio. Note that our distributios are essetially covolutios of expoetial radom variables of differet sigs, ad commo estimatio procedures for mixtures of positive expoetial distributios are ot applicable i this case. We shall focus o the method of maximum likelihood method of estimatio. 3.1 Maximum Likelihood Estimatio Let X1,X,,X be a i.i.d radom sample from a ETL(,µ,σ) distributio with desity g,µ,σ give by (3) ad let x1, x, x be their particular realizatio. Our goal is to fid the MLEs of the parameters. The likelihood fuctio ca be as where L(X,,µ,σ) = (1 θ ) (x i μ) + = { x i μ, x i μ 0, x i < μ The the log-likelihood fuctio is e[ (1+θ σ ) i=1 (x i μ) ( 1 θ σ ) i=1 (x i μ) + ] (σ) ad, (5) (x i μ) = { μ x i, x i μ 0, x i < μ. logl(x,,µ,σ ) = log(1 - θ ) - log logσ- D σ where D = D(,µ) = (1+ ) i=1(x i μ) + (1 ) i=1 (x i μ) +
5 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 673 The estimates are give i the followig Table 1. TABLE 1 Cases Parameters Estimates Asympt.variace 1. μ is ukow (σ,θ kow). σ is ukow (μ,θ kow) σ μ = Xj():. Where j() = (1 θ) + 1 [[x]]deote the itegral part of x = 1 [(1 θ) (x i μ) + + (1 i=1 + θ) (x i μ) ] i=1 1 θ σ σ 1 θ 3. θ is ukow (μ,σ kow) θ is uique solutio g(y,, β, )= Log(1-y )-(1-y)α +(1+y) = 0 1 θ (1 θ) + 1 α(μ ) = = 1 [ (x i=1 i μ) + ] (μ )= 1 [ i=1 ] (x i μ). μ,σ ukow θ is kow σ μ = Xj():. = 1 [(1 θ) (x i μ) + + (1 i=1 + θ) (x i μ) ] = [ σ (1 θ ) 0 0 ] σ (1 θ ) i=1 5. θ,σ ukow μ is kow σ = α(μ) = β(μ) θ β(μ) α(μ) [ α(μ) + β(μ) ] = σ (1 + (1 8 θ) + 1) a c [ b ] where a = 1 σ, b= (1+θ)( θ ), (1 θ) c= (1+θ) 1 (1 θ) σ θ θ 6. θ,μ ukow σ is kow α(μ) = 1 σ 1 (x i μ) + i=1 ad = σ (1 θ) [a c b ] Where
6 67 Dais George ad Rimsha H (μ) = 1 σ 1 (x i=1 i μ) R J 1, R J, R J (0, 1 1 ]; J=[-1, ) (μ 1,θ 1 ) (μ,θ ) (μ,θ ) b= a= θ 1 θ σ (1+θ) c= 1 σ 7. μ, θ,σ is kow Fid 1< r such that h(x r:) h(x j:) for j= 1,, where h(μ) = log( α(μ) + (μ) + α(μ) β(μ) μ = X r: = β(μ) θ σ = α(μ) Where α(μ) β(μ) [ α(μ) + β(μ) ] α(μ ) = = 1 [ (x i=1 i μ) + ] (μ )= 1 [ i=1 ] (x i μ) a b c = σ [ d f] g e= Where a = b = 1 σ (1+θ) c = θ ( θ) d= σ (1+θ) σ(1+θ) 1 θ. CONCLUSION I this paper, we studied the estimatio problem of three parameter Esscher trasformed Laplace distributio, which is the locatio scale family of the oe parameter Esscher trasformed Laplace distributio itroduced by Sebastia George ad Dais George (01). Beig heavy-tailed ad asymmetric, this distributio act as a alterative to various asymmetric distributios ad ca be used for modelig real data sets of that ature which are commoly occurrig from diverse fields such as busiess, telecommuicatios, physical ad biological scieces ad hece the estimatio of parameters holds its importace.
7 Estimatio of Parameters of Three Parameter Esscher Trasformed Laplace Distributio 675 REFERENCES [1] N. Balakrisha, ad R. S. Ambagaspitiya. O skewed Laplace distributios. Report, McMaster Uiversity, Hamilto, Otario, Caada., (199). [] Ghitay, M. E., Al-Hussaii, E. K., Al-Jarallah, R. A. (005) MarshallOlki exteded Weibull distributio ad its applicatio to cesored data,j.appl. Stat.,3, [3] Ghitay, M. E., Al-Awadhi F. A., AlkhalfaL, A. (007) Marshall-Olki exteded lomax distributio ad its applicatio to cesored data, Com mu. Statist. Theory Methods, 36, [] F.Esscher. O the probability fuctio m the collective theory of rtsk. Scadmaoza Actuarial Joural, 15, (193), [5] M. S Holla ad S. K.,. Bhattacharya. O a compoud Gaussia distributio. A. Ist. Statist. Math., 0, (1968), [6] Jayakumar, K. ad Kuttykrisha, A.P. A ew asymmetric Laplace autoregressive process. Joural of Statistical Theory ad Applicatios 7, (006),: [7] S. Kotz, T.J. Kozubowski ad Podg\'{o}rski K. The Laplace distributio ad Geeralizatios: A Revisit with Applicatios to Commuicatios, Ecoomics,Egieerig ad Fiace. Birkhauser. Bosto., Techical, (000). [8] T.J Kozubowski ad K. Podg\'{o}rski, Asymmetric Laplace distributio, Math.Sci., 5, (000), [9] W. J. McGill, Radom uctuatios of respose rate. Psychometrica, 7, (196), [10] S.Poiraud-Casaova, ad C. Thomas-Aga,. About mootoe regressio quatiles. Statist. Probab. Lett., 8, (000). [11] Sebastia George ad Dais George. Esscher Trasformed Laplace Distributio ad Its Applicatios. Joural of Probability ad Statistical Sciece, 10(), (01), [1] VaidaBartkute ad Leoidas Sakalauskas. Three Parameter Estimatio of the Weibull distributio by Order Statistics /publicatio/887810, (015),DOI: 10.11/ _001.
8 676 Dais George ad Rimsha H
Bayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution
Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume, Number 4 (07, pp. 7-73 Research Idia Publicatios http://www.ripublicatio.com Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet
More informationExam 1 Spring 2015 Statistics for Applications 3/5/2015
8.443 Exam Sprig 05 Statistics for Applicatios 3/5/05. Log Normal Distributio: A radom variable X follows a Logormal(θ, σ ) distributio if l(x) follows a Normal(θ, σ ) distributio. For the ormal radom
More informationIntroduction to Probability and Statistics Chapter 7
Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based
More information. (The calculated sample mean is symbolized by x.)
Stat 40, sectio 5.4 The Cetral Limit Theorem otes by Tim Pilachowski If you have t doe it yet, go to the Stat 40 page ad dowload the hadout 5.4 supplemet Cetral Limit Theorem. The homework (both practice
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpeCourseWare http://ocwmitedu 430 Itroductio to Statistical Methods i Ecoomics Sprig 009 For iformatio about citig these materials or our Terms of Use, visit: http://ocwmitedu/terms 430 Itroductio
More informationAn Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions
A Empirical Study of the Behaviour of the Sample Kurtosis i Samples from Symmetric Stable Distributios J. Marti va Zyl Departmet of Actuarial Sciece ad Mathematical Statistics, Uiversity of the Free State,
More informationNOTES ON ESTIMATION AND CONFIDENCE INTERVALS. 1. Estimation
NOTES ON ESTIMATION AND CONFIDENCE INTERVALS MICHAEL N. KATEHAKIS 1. Estimatio Estimatio is a brach of statistics that deals with estimatig the values of parameters of a uderlyig distributio based o observed/empirical
More information5. Best Unbiased Estimators
Best Ubiased Estimators http://www.math.uah.edu/stat/poit/ubiased.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 5. Best Ubiased Estimators Basic Theory Cosider agai
More informationx satisfying all regularity conditions. Then
AMS570.01 Practice Midterm Exam Sprig, 018 Name: ID: Sigature: Istructio: This is a close book exam. You are allowed oe-page 8x11 formula sheet (-sided). No cellphoe or calculator or computer is allowed.
More information5 Statistical Inference
5 Statistical Iferece 5.1 Trasitio from Probability Theory to Statistical Iferece 1. We have ow more or less fiished the probability sectio of the course - we ow tur attetio to statistical iferece. I statistical
More informationThese characteristics are expressed in terms of statistical properties which are estimated from the sample data.
0. Key Statistical Measures of Data Four pricipal features which characterize a set of observatios o a radom variable are: (i) the cetral tedecy or the value aroud which all other values are buched, (ii)
More informationCombining imperfect data, and an introduction to data assimilation Ross Bannister, NCEO, September 2010
Combiig imperfect data, ad a itroductio to data assimilatio Ross Baister, NCEO, September 00 rbaister@readigacuk The probability desity fuctio (PDF prob that x lies betwee x ad x + dx p (x restrictio o
More informationSTAT 135 Solutions to Homework 3: 30 points
STAT 35 Solutios to Homework 3: 30 poits Sprig 205 The objective of this Problem Set is to study the Stei Pheomeo 955. Suppose that θ θ, θ 2,..., θ cosists of ukow parameters, with 3. We wish to estimate
More information18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013
18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/2013 1. Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric
More informationLecture 4: Parameter Estimation and Confidence Intervals. GENOME 560 Doug Fowler, GS
Lecture 4: Parameter Estimatio ad Cofidece Itervals GENOME 560 Doug Fowler, GS (dfowler@uw.edu) 1 Review: Probability Distributios Discrete: Biomial distributio Hypergeometric distributio Poisso distributio
More informationParametric Density Estimation: Maximum Likelihood Estimation
Parametric Desity stimatio: Maimum Likelihood stimatio C6 Today Itroductio to desity estimatio Maimum Likelihood stimatio Itroducto Bayesia Decisio Theory i previous lectures tells us how to desig a optimal
More informationFINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices?
FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural
More informationTopic 14: Maximum Likelihood Estimation
Toic 4: November, 009 As before, we begi with a samle X = (X,, X of radom variables chose accordig to oe of a family of robabilities P θ I additio, f(x θ, x = (x,, x will be used to deote the desity fuctio
More informationECON 5350 Class Notes Maximum Likelihood Estimation
ECON 5350 Class Notes Maximum Likelihood Estimatio 1 Maximum Likelihood Estimatio Example #1. Cosider the radom sample {X 1 = 0.5, X 2 = 2.0, X 3 = 10.0, X 4 = 1.5, X 5 = 7.0} geerated from a expoetial
More informationMaximum Empirical Likelihood Estimation (MELE)
Maximum Empirical Likelihood Estimatio (MELE Natha Smooha Abstract Estimatio of Stadard Liear Model - Maximum Empirical Likelihood Estimator: Combiatio of the idea of imum likelihood method of momets,
More informationResearch Article The Probability That a Measurement Falls within a Range of n Standard Deviations from an Estimate of the Mean
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 70806, 8 pages doi:0.540/0/70806 Research Article The Probability That a Measuremet Falls withi a Rage of Stadard Deviatios
More informationDepartment of Mathematics, S.R.K.R. Engineering College, Bhimavaram, A.P., India 2
Skewess Corrected Cotrol charts for two Iverted Models R. Subba Rao* 1, Pushpa Latha Mamidi 2, M.S. Ravi Kumar 3 1 Departmet of Mathematics, S.R.K.R. Egieerig College, Bhimavaram, A.P., Idia 2 Departmet
More informationChapter 8: Estimation of Mean & Proportion. Introduction
Chapter 8: Estimatio of Mea & Proportio 8.1 Estimatio, Poit Estimate, ad Iterval Estimate 8.2 Estimatio of a Populatio Mea: σ Kow 8.3 Estimatio of a Populatio Mea: σ Not Kow 8.4 Estimatio of a Populatio
More informationChapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1
Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for
More informationEstimation of Population Variance Utilizing Auxiliary Information
Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume 1, Number (017), pp. 303-309 Research Idia Publicatios http://www.ripublicatio.com Estimatio of Populatio Variace Utilizig Auxiliary Iformatio
More informationUnbiased estimators Estimators
19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.
More informationChapter 10 - Lecture 2 The independent two sample t-test and. confidence interval
Assumptios Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Upooled case Idepedet Samples - ukow σ 1, σ - 30 or m 30 - Pooled case Idepedet samples - Pooled variace - Large samples Chapter 10 - Lecture The
More informationA Bayesian perspective on estimating mean, variance, and standard-deviation from data
Brigham Youg Uiversity BYU ScholarsArchive All Faculty Publicatios 006--05 A Bayesia perspective o estimatig mea, variace, ad stadard-deviatio from data Travis E. Oliphat Follow this ad additioal works
More informationExam 2. Instructor: Cynthia Rudin TA: Dimitrios Bisias. October 25, 2011
15.075 Exam 2 Istructor: Cythia Rudi TA: Dimitrios Bisias October 25, 2011 Gradig is based o demostratio of coceptual uderstadig, so you eed to show all of your work. Problem 1 You are i charge of a study
More informationStatistics for Economics & Business
Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie
More informationSummary. Recap. Last Lecture. .1 If you know MLE of θ, can you also know MLE of τ(θ) for any function τ?
Last Lecture Biostatistics 60 - Statistical Iferece Lecture Cramer-Rao Theorem Hyu Mi Kag February 9th, 03 If you kow MLE of, ca you also kow MLE of τ() for ay fuctio τ? What are plausible ways to compare
More informationpoint estimator a random variable (like P or X) whose values are used to estimate a population parameter
Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity
More informationEstimating the Parameters of the Three-Parameter Lognormal Distribution
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 3-30-0 Estimatig the Parameters of the Three-Parameter Logormal Distributio Rodrigo J. Aristizabal
More informationA New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions
A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhu-qi Lu, Ph.D. Abstract A -valued two-variable truth fuctio is called fuctioally complete,
More informationSampling Distributions and Estimation
Samplig Distributios ad Estimatio T O P I C # Populatio Proportios, π π the proportio of the populatio havig some characteristic Sample proportio ( p ) provides a estimate of π : x p umber of successes
More informationInferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,
Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike 4.0
More informationii. Interval estimation:
1 Types of estimatio: i. Poit estimatio: Example (1) Cosider the sample observatios 17,3,5,1,18,6,16,10 X 8 X i i1 8 17 3 5 118 6 16 10 8 116 8 14.5 14.5 is a poit estimate for usig the estimator X ad
More informationConfidence Intervals based on Absolute Deviation for Population Mean of a Positively Skewed Distribution
Iteratioal Joural of Computatioal ad Theoretical Statistics ISSN (220-59) It. J. Comp. Theo. Stat. 5, No. (May-208) http://dx.doi.org/0.2785/ijcts/0500 Cofidece Itervals based o Absolute Deviatio for Populatio
More informationLecture 5 Point Es/mator and Sampling Distribu/on
Lecture 5 Poit Es/mator ad Samplig Distribu/o Fall 03 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Road map Poit Es/ma/o Cofidece Iterval
More informationControl Charts for Mean under Shrinkage Technique
Helderma Verlag Ecoomic Quality Cotrol ISSN 0940-5151 Vol 24 (2009), No. 2, 255 261 Cotrol Charts for Mea uder Shrikage Techique J. R. Sigh ad Mujahida Sayyed Abstract: I this paper a attempt is made to
More informationSampling Distributions and Estimation
Cotets 40 Samplig Distributios ad Estimatio 40.1 Samplig Distributios 40. Iterval Estimatio for the Variace 13 Learig outcomes You will lear about the distributios which are created whe a populatio is
More informationLecture 4: Probability (continued)
Lecture 4: Probability (cotiued) Desity Curves We ve defied probabilities for discrete variables (such as coi tossig). Probabilities for cotiuous or measuremet variables also are evaluated usig relative
More informationA point estimate is the value of a statistic that estimates the value of a parameter.
Chapter 9 Estimatig the Value of a Parameter Chapter 9.1 Estimatig a Populatio Proportio Objective A : Poit Estimate A poit estimate is the value of a statistic that estimates the value of a parameter.
More informationFOUNDATION ACTED COURSE (FAC)
FOUNDATION ACTED COURSE (FAC) What is the Foudatio ActEd Course (FAC)? FAC is desiged to help studets improve their mathematical skills i preparatio for the Core Techical subjects. It is a referece documet
More information4.5 Generalized likelihood ratio test
4.5 Geeralized likelihood ratio test A assumptio that is used i the Athlete Biological Passport is that haemoglobi varies equally i all athletes. We wish to test this assumptio o a sample of k athletes.
More informationKernel Density Estimation. Let X be a random variable with continuous distribution F (x) and density f(x) = d
Kerel Desity Estimatio Let X be a radom variable wit cotiuous distributio F (x) ad desity f(x) = d dx F (x). Te goal is to estimate f(x). Wile F (x) ca be estimated by te EDF ˆF (x), we caot set ˆf(x)
More informationA New Approach to Obtain an Optimal Solution for the Assignment Problem
Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 231-7064 Idex Copericus Value (2013): 6.14 Impact Factor (2015): 6.31 A New Approach to Obtai a Optimal Solutio for the Assigmet Problem A. Seethalakshmy
More informationA STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS *
Page345 ISBN: 978 0 9943656 75; ISSN: 05-6033 Year: 017, Volume: 3, Issue: 1 A STOCHASTIC GROWTH PRICE MODEL USING A BIRTH AND DEATH DIFFUSION GROWTH RATE PROCESS WITH EXTERNAL JUMP PROCESS * Basel M.
More informationTopic-7. Large Sample Estimation
Topic-7 Large Sample Estimatio TYPES OF INFERENCE Ò Estimatio: É Estimatig or predictig the value of the parameter É What is (are) the most likely values of m or p? Ò Hypothesis Testig: É Decidig about
More informationEXERCISE - BINOMIAL THEOREM
BINOMIAL THOEREM / EXERCISE - BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9
More informationBOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS. January 2008
BOUNDS FOR TAIL PROBABILITIES OF MARTINGALES USING SKEWNESS AND KURTOSIS V. Betkus 1,2 ad T. Juškevičius 1 Jauary 2008 Abstract. Let M = X 1 + + X be a sum of idepedet radom variables such that X k 1,
More informationStatistics for Business and Economics
Statistics for Busiess ad Ecoomics Chapter 8 Estimatio: Additioal Topics Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1 8. Differece Betwee Two Meas: Idepedet Samples Populatio meas,
More informationDiscriminating Between The Log-normal and Gamma Distributions
Discrimiatig Betwee The Log-ormal ad Gamma Distributios Debasis Kudu & Aubhav Maglick Abstract For a give data set the problem of selectig either log-ormal or gamma distributio with ukow shape ad scale
More informationSolution to Tutorial 6
Solutio to Tutorial 6 2012/2013 Semester I MA4264 Game Theory Tutor: Xiag Su October 12, 2012 1 Review Static game of icomplete iformatio The ormal-form represetatio of a -player static Bayesia game: {A
More informationRafa l Kulik and Marc Raimondo. University of Ottawa and University of Sydney. Supplementary material
Statistica Siica 009: Supplemet 1 L p -WAVELET REGRESSION WITH CORRELATED ERRORS AND INVERSE PROBLEMS Rafa l Kulik ad Marc Raimodo Uiversity of Ottawa ad Uiversity of Sydey Supplemetary material This ote
More informationGenerative Models, Maximum Likelihood, Soft Clustering, and Expectation Maximization
Geerative Models Maximum Likelihood Soft Clusterig ad Expectatio Maximizatio Aris Aagostopoulos We will see why we desig models for data how to lear their parameters ad how by cosiderig a mixture model
More informationSampling Distributions & Estimators
API-209 TF Sessio 2 Teddy Svoroos September 18, 2015 Samplig Distributios & Estimators I. Estimators The Importace of Samplig Radomly Three Properties of Estimators 1. Ubiased 2. Cosistet 3. Efficiet I
More information1 Basic Growth Models
UCLA Aderso MGMT37B: Fudametals i Fiace Fall 015) Week #1 rofessor Eduardo Schwartz November 9, 015 Hadout writte by Sheje Hshieh 1 Basic Growth Models 1.1 Cotiuous Compoudig roof: lim 1 + i m = expi)
More informationApplication of Esscher Transformed Laplace Distribution in Microarray Gene Expression Data
Journal of Modern Applied Statistical Methods Volume 15 Issue 1 Article 31 5-1-016 Application of Esscher Transformed Laplace Distribution in Microarray Gene Expression Data Shanmugasundaram Devika Christian
More informationPoint Estimation by MLE Lesson 5
Poit Estimatio b MLE Lesso 5 Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative
More informationAn Improved Estimator of Population Variance using known Coefficient of Variation
J. Stat. Appl. Pro. Lett. 4, No. 1, 11-16 (017) 11 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/04010 A Improved Estimator of Populatio Variace
More informationA random variable is a variable whose value is a numerical outcome of a random phenomenon.
The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss
More informationProceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 17-19, 2005 (pp )
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 7-9, 005 (pp488-49 Realized volatility estimatio: ew simulatio approach ad empirical study results JULIA
More informationMonetary Economics: Problem Set #5 Solutions
Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.
More informationCHANGE POINT TREND ANALYSIS OF GNI PER CAPITA IN SELECTED EUROPEAN COUNTRIES AND ISRAEL
The 9 th Iteratioal Days of Statistics ad Ecoomics, Prague, September 0-, 05 CHANGE POINT TREND ANALYSIS OF GNI PER CAPITA IN SELECTED EUROPEAN COUNTRIES AND ISRAEL Lia Alatawa Yossi Yacu Gregory Gurevich
More informationThe Valuation of the Catastrophe Equity Puts with Jump Risks
The Valuatio of the Catastrophe Equity Puts with Jump Risks Shih-Kuei Li Natioal Uiversity of Kaohsiug Joit work with Chia-Chie Chag Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk
More informationOutline. Populations. Defs: A (finite) population is a (finite) set P of elements e. A variable is a function v : P IR. Population and Characteristics
Outlie Populatio Characteristics Types of Samples Sample Characterstics Sample Aalogue Estimatio Populatios Defs: A (fiite) populatio is a (fiite) set P of elemets e. A variable is a fuctio v : P IR. Examples
More informationAY Term 2 Mock Examination
AY 206-7 Term 2 Mock Examiatio Date / Start Time Course Group Istructor 24 March 207 / 2 PM to 3:00 PM QF302 Ivestmet ad Fiacial Data Aalysis G Christopher Tig INSTRUCTIONS TO STUDENTS. This mock examiatio
More information1 Estimating sensitivities
Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter
More informationMODIFICATION OF HOLT S MODEL EXEMPLIFIED BY THE TRANSPORT OF GOODS BY INLAND WATERWAYS TRANSPORT
The publicatio appeared i Szoste R.: Modificatio of Holt s model exemplified by the trasport of goods by ilad waterways trasport, Publishig House of Rzeszow Uiversity of Techology No. 85, Maagemet ad Maretig
More informationAsymmetric Type II Compound Laplace Distributions and its Properties
CHAPTER 4 Asymmetric Type II Compound Laplace Distributions and its Properties 4. Introduction Recently there is a growing trend in the literature on parametric families of asymmetric distributions which
More informationBASIC STATISTICS ECOE 1323
BASIC STATISTICS ECOE 33 SPRING 007 FINAL EXAM NAME: ID NUMBER: INSTRUCTIONS:. Write your ame ad studet ID.. You have hours 3. This eam must be your ow work etirely. You caot talk to or share iformatio
More informationHeavy-tailed modeling of CROBEX
Heavy-tailed modelig of CROBEX DANIJEL GRAHOVAC, PhD* NENAD ŠUVAK, PhD* Article** JEL: C15, C, C51 doi: 10.336/fitp.39.4.4 * The authors would like to thak two aoymous referees for their helpful commets
More informationPoint Estimation by MLE Lesson 5
Poit Estimatio b MLE Lesso 5 Review Defied Likelihood Maximum Likelihood Estimatio Step : Costruct Likelihood Step : Maximize fuctio Take Log of likelihood fuctio Take derivative of fuctio Set derivative
More informationLecture 5: Sampling Distribution
Lecture 5: Samplig Distributio Readigs: Sectios 5.5, 5.6 Itroductio Parameter: describes populatio Statistic: describes the sample; samplig variability Samplig distributio of a statistic: A probability
More informationSubject CT1 Financial Mathematics Core Technical Syllabus
Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig
More informationChapter 10 Statistical Inference About Means and Proportions With Two Populations. Learning objectives
Chater 0 Statistical Iferece About Meas ad Proortios With Two Poulatios Slide Learig objectives. Uderstad ifereces About the Differece Betwee Two Poulatio Meas: σ ad σ Kow. Uderstad Ifereces About the
More informationLecture 9: The law of large numbers and central limit theorem
Lecture 9: The law of large umbers ad cetral limit theorem Theorem.4 Let X,X 2,... be idepedet radom variables with fiite expectatios. (i) (The SLLN). If there is a costat p [,2] such that E X i p i i=
More informationBIAS-CORRECTED MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF THE WEIGHTED LINDLEY DISTRIBUTION
Michiga Techological Uiversity Digital Commos @ Michiga Tech Dissertatios, Master's Theses ad Master's Reports - Ope Dissertatios, Master's Theses ad Master's Reports 2015 BIAS-CORRECTED MAXIMUM LIKELIHOOD
More information1. Suppose X is a variable that follows the normal distribution with known standard deviation σ = 0.3 but unknown mean µ.
Chapter 9 Exercises Suppose X is a variable that follows the ormal distributio with kow stadard deviatio σ = 03 but ukow mea µ (a) Costruct a 95% cofidece iterval for µ if a radom sample of = 6 observatios
More informationAsymptotics: Consistency and Delta Method
ad Delta Method MIT 18.655 Dr. Kempthore Sprig 2016 1 MIT 18.655 ad Delta Method Outlie Asymptotics 1 Asymptotics 2 MIT 18.655 ad Delta Method Cosistecy Asymptotics Statistical Estimatio Problem X 1,...,
More informationEcient estimation of log-normal means with application to pharmacokinetic data
STATISTICS IN MEDICINE Statist. Med. 006; 5:303 3038 Published olie 3 December 005 i Wiley IterSciece (www.itersciece.wiley.com. DOI: 0.00/sim.456 Eciet estimatio of log-ormal meas with applicatio to pharmacokietic
More informationSolutions to Problem Sheet 1
Solutios to Problem Sheet ) Use Theorem.4 to prove that p log for all real x 3. This is a versio of Theorem.4 with the iteger N replaced by the real x. Hit Give x 3 let N = [x], the largest iteger x. The,
More informationGame Theory. Lecture Notes By Y. Narahari. Department of Computer Science and Automation Indian Institute of Science Bangalore, India July 2012
Game Theory Lecture Notes By Y. Narahari Departmet of Computer Sciece ad Automatio Idia Istitute of Sciece Bagalore, Idia July 01 Chapter 4: Domiat Strategy Equilibria Note: This is a oly a draft versio,
More informationGranularity Adjustment in a General Factor Model
Graularity Adjustmet i a Geeral Factor Model Has Rau-Bredow Uiversity of Cologe, Uiversity of Wuerzburg E-mail: has.rau-bredow@mail.ui-wuerzburg.de May 30, 2005 Abstract The graularity adjustmet techique
More informationChapter 8 Interval Estimation. Estimation Concepts. General Form of a Confidence Interval
Chapter 8 Iterval Estimatio Estimatio Cocepts Usually ca't take a cesus, so we must make decisios based o sample data It imperative that we take the risk of samplig error ito accout whe we iterpret sample
More informationNeighboring Optimal Solution for Fuzzy Travelling Salesman Problem
Iteratioal Joural of Egieerig Research ad Geeral Sciece Volume 2, Issue 4, Jue-July, 2014 Neighborig Optimal Solutio for Fuzzy Travellig Salesma Problem D. Stephe Digar 1, K. Thiripura Sudari 2 1 Research
More information0.1 Valuation Formula:
0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()
More informationBasic formula for confidence intervals. Formulas for estimating population variance Normal Uniform Proportion
Basic formula for the Chi-square test (Observed - Expected ) Expected Basic formula for cofidece itervals sˆ x ± Z ' Sample size adjustmet for fiite populatio (N * ) (N + - 1) Formulas for estimatig populatio
More informationACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. 2 INTEREST, AMORTIZATION AND SIMPLICITY. by Thomas M. Zavist, A.S.A.
ACTUARIAL RESEARCH CLEARING HOUSE 1990 VOL. INTEREST, AMORTIZATION AND SIMPLICITY by Thomas M. Zavist, A.S.A. 37 Iterest m Amortizatio ad Simplicity Cosider simple iterest for a momet. Suppose you have
More information(Hypothetical) Negative Probabilities Can Speed Up Uncertainty Propagation Algorithms
Uiversity of Texas at El Paso DigitalCommos@UTEP Departmetal Techical Reports (CS) Departmet of Computer Sciece 2-2017 (Hypothetical) Negative Probabilities Ca Speed Up Ucertaity Propagatio Algorithms
More informationRandom Sequences Using the Divisor Pairs Function
Radom Sequeces Usig the Divisor Pairs Fuctio Subhash Kak Abstract. This paper ivestigates the radomess properties of a fuctio of the divisor pairs of a atural umber. This fuctio, the atecedets of which
More informationEstimating Proportions with Confidence
Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter
More informationREVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS. Guangwu Liu L. Jeff Hong
Proceedigs of the 2008 Witer Simulatio Coferece S. J. Maso, R. R. Hill, L. Möch, O. Rose, T. Jefferso, J. W. Fowler eds. REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS Guagwu Liu L. Jeff
More informationEstimation of generalized Pareto distribution
Estimatio of geeralized Pareto distributio Joa Del Castillo, Jalila Daoudi To cite this versio: Joa Del Castillo, Jalila Daoudi. Estimatio of geeralized Pareto distributio. Statistics ad Probability Letters,
More informationr i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i
The iformatio required by the mea-variace approach is substatial whe the umber of assets is large; there are mea values, variaces, ad )/2 covariaces - a total of 2 + )/2 parameters. Sigle-factor model:
More informationToday: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3)
Today: Fiish Chapter 9 (Sectios 9.6 to 9.8 ad 9.9 Lesso 3) ANNOUNCEMENTS: Quiz #7 begis after class today, eds Moday at 3pm. Quiz #8 will begi ext Friday ad ed at 10am Moday (day of fial). There will be
More informationConfidence Intervals Introduction
Cofidece Itervals Itroductio A poit estimate provides o iformatio about the precisio ad reliability of estimatio. For example, the sample mea X is a poit estimate of the populatio mea μ but because of
More informationThe material in this chapter is motivated by Experiment 9.
Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi
More informationQuantitative Analysis
EduPristie FRM I \ Quatitative Aalysis EduPristie www.edupristie.com Momets distributio Samplig Testig Correlatio & Regressio Estimatio Simulatio Modellig EduPristie FRM I \ Quatitative Aalysis 2 Momets
More information