The new class of Kummer beta generalized distributions
|
|
- Claude Ethan Singleton
- 5 years ago
- Views:
Transcription
1 The ew class of Kummer beta geeralized distributios Rodrigo Rossetto Pescim 12 Clarice Garcia Borges Demétrio 1 Gauss Moutiho Cordeiro 3 Saralees Nadarajah 4 Edwi Moisés Marcos Ortega 1 1 Itroductio Geeralized beta distributios have bee widely studied i statistics ad umerous authors have developed various classes of these distributios. I the last decade, [2] proposed a geeral class of distributios for a radom variable defied from the logit of the beta radom variable by employig two parameters whose role is to itroduce skewess ad to vary tail weight. However, the beta geeralized distributios do ot offer more flexibility to extremes (right ad left) of the curves of the desity fuctios ad therefore they are ot suitable for aalyzig data sets with high degree of asymmetry. The Kummer beta (KB) distributio o the uit iterval (,1) was proposed by [4] with cumulative distributio fuctio (cdf) ad probability desity fuctio (pdf) give by ad x F KB (x) = K t a 1 (1 t) b 1 exp( ct)dt, f KB (x) = K x a 1 (1 x) b 1 exp( cx), < x < 1, (1) where a >, b > ad < c <. Here, ad 1F 1 (a;a + b; c) = 1 LCE - ESALQ/USP. rrpescim@usp.br 2 Agradecimeto ao CNPq pelo apoio fiaceiro. 3 DE - UFPE. 4 Uiversity of Machester - UK K 1 = Γ(a)Γ(b) Γ(a + b) 1 F 1 (a;a + b; c) (2) Γ(a + b) 1 t a 1 (1 t) b 1 (a) k ( c) exp( ct)dt = Γ(a)Γ(b) k k= (a + b) k k! 1
2 is the cofluet hypergeometric fuctio, Γ( ) is the gamma fuctio ad (d) k = d(d +1)...(d + k 1) deotes the ascedig factorial. This distributio is a extesio of the beta distributio, ad for a < 1 (ad certai values of the parameter c) yields bimodal distributios o fiite rage. Cosider startig from a paret cotiuous distributio fuctio G(x). A atural way of geeratig families of distributios o some other support from a simple startig paret distributio with desity fuctio g(x) = dg(x)/dx is to apply the quatile fuctio to a family of distributios o the iterval (,1). We ow use the same methodology of [2] ad [1] to costruct a ew class of Kummer beta geeralized (KBG) distributios. From a arbitrary paret cumulative distributio G(x), the cdf F(x) of the KBG family of distributios is defied by G(x) F(x) = K t a 1 (1 t) b 1 exp( ct)dt, (3) where a > ad b > are shape parameters which itroduce skewess, ad thereby promote weight variatio of the tails, whereas the parameter < c < squeezes the desity fuctio to the left or right, i.e., it leads the tail weights of the desity to extreme values. The desity fuctio correspodig to (3) ca be writte as f (x) = K g(x)g(x) a 1 {1 G(x)} b 1 exp{ c G(x)}, (4) where K is defied i (2). For each cotiuous G distributio (here ad heceforth G deotes the baselie distributio), we ca associate the KBG-G distributio with three extra parameters a, b ad c defied by the desity fuctio (4). 2 Special KBG Geeralized Distributios The KBG desity fuctio (4) allows for greater flexibility of its tails ad promotes the variatio of the tail weights to the extremes of the distributio. It ca be widely applied i may areas of egieerig ad biological scieces. The desity fuctio (4) will be most tractable whe the cdf G(x) ad the pdf g(x) have simple aalytic expressios. We ow defie some of the may distributios which ca arise as special sub-models withi the KBG class of distributios. 2.1 KBG-Normal The KBGN desity fuctio is obtaied from (4) by takig G( ) ad g( ) to be the cdf ad pdf of the ormal distributio, N(µ, 2 ), so that f (x) = K ( ){ x µ ϕ Φ )} a 1 { 1 Φ )} b 1 exp{ c Φ )}, 2
3 where x R, µ R is a locatio parameter, > is a scale parameter, a ad b are positive shape parameters, c R, ad ϕ( ) ad Φ( ) are the pdf ad cdf of the stadard ormal distributio, respectively. A radom variable with the above desity fuctio is deoted by X KBGN(a,b,c,µ, 2 ). For µ = ad = 1, we have the stadard KBGN distributio. 2.2 KBG-Weibull The cdf of the Weibull distributio with parameters β > ad α > is G(x) = 1 exp{ (βx) α } for x >. Correspodigly, the KBG-Weibull desity fuctio, say KBGW(a, b, c, α, β), reduces to f (x) = K α β α x α 1 [1 exp{ (βx) α }] a 1 exp{ c[1 exp{ (βx) α }] b(βx) α }, where x,a,b,β > ad c R. For α = 1, we obtai the KBG-expoetial (KBGE) distributio. The KBGW(1, b,, 1, β) distributio correspods to the expoetial distributio with parameter β = bβ. 2.3 KBG-Gamma Let Y be a gamma radom variable with cdf G(y) = Γ βy (α)/γ(α) for y,α,β >, where Γ( ) is the gamma fuctio ad Γ z (α) = z t α 1 e t dt is the icomplete gamma fuctio. The desity fuctio of a radom variable X havig the KBGGa distributio, say X KBGGa(a,b,c,β,α), ca be expressed as f (x) = K βα x α 1 e βx Γ(α) a+b 1 { exp c Γ } βx(α) Γ Γ(α) βx (α) a 1 { Γ(α) Γ βx (α) } b 1. For α = 1 ad c =, we obtai the KBGE distributio. The KBGGa(1,b,,β,1) distributio reduces to the expoetial distributio with parameter β = bβ. 3 Iferece Let γ be the p-dimesioal parameter vector of the baselie distributio i equatios (3) ad (4). We cosider idepedet radom variables X 1,...,X, each X i followig a KBG-G distributio with parameter vector θ = (a, b, c, γ). The log-likelihood fuctio l = l(θ) for the model parameters obtaied from (4) is l(θ) = log(k) + + (a 1) logg(x i ;γ) c G(x i ;γ) log{g(x i ;γ)} + (b 1) log{1 G(x i ;γ)}. 3
4 We ca ote that the elemets of score vector deped o the specified baselie distributio. Numerical maximizatio of the log-likelihood above is accomplished by usig the RS method which is available i the gamlss package i statistical software R. 4 Applicatio - Voltage data Here, we compare the results of the fits of the KBGW, beta Weibul (BW), expoetiated Weibull (EW) ad Weibull distributios to the data set studied by [3]. The data represet the times of failure ad ruig times for a sample of devices from a field-trackig study of a larger system. At a certai poit i time, 3 uits were istalled i ormal service coditios. Two causes of failure were observed for each uit that failed: the failure caused by a accumulatio of radomly occurrig damage from power-lie voltages pikes durig electric storms ad failure caused by ormal product wear. Table 1 lists the MLEs of the model parameters ad the values of the followig statistics for some models: Akaike Iformatio Criterio (AIC) ad Bayesia Iformatio Criterio (BIC). The computatios were doe usig statistical software R. These Tabela 1: MLEs ad iformatio criteria for voltage data. Model d β a b c AIC BIC KBGW BW EW Weibull results idicate that the KBGW model has the smallest values for the AIC ad BIC statistics amog the fitted models, ad therefore it could be chose as the best model. More iformatio ca be provided by a visual compariso of the histograms of the data with the fitted desities. I order to assess if the model is appropriate, we provide i Figure 1a the histogram of the data ad the fitted KBGW, BW, EW ad Weibull desity fuctios. Further, i Figure 1b, we plot the empirical ad estimated survival fuctios of the KBGW, BW, EW ad Weibull distributios. We coclude that the KBGW distributio yields a good fit for these data. 5 Coclusios Followig the idea of the class of beta geeralized distributios ad the distributio by [4], we defie a ew family of Kummer beta geeralized (KBG) distributios to exted several widely kow distributios such as the ormal, Weibull ad Gumbel distributios. For each cotiuous G distributio, we ca defie the correspodig KBG-G distributio usig simple formulae. We discuss maximum likelihood estimatio ad iferece o the parameters. A 4
5 (a) (b) f(x) KBGW BW EW Weibull S(x) Kapla Meier KBGW BW EW Weibull x x Figura 1: (a) Estimated KBGW, BW, EW ad Weibull desity fuctios for voltage data. (b) Estimated survival fuctios ad the empirical survival for voltage data. applicatio of the ew family to real data set is give to show the feasibility of the proposed class of models. We hope this geeralizatio may attract wider applicatios i statistics. Referêcias [1] CORDEIRO, G.M.; de CASTRO, M. A ew family of geeralized distributios. Joural of Statistical Computatio ad Simulatio. Elselvier. v. 81, p , 211. [2] EUGENE, N.; LEE, C.; FAMOYE, F. Beta-ormal distributio ad its applicatios. Commuicatio i Statistics - Theory ad Methods. Elselvier. v. 31, p , 22. [3] MEEKER, W.Q.; ESCOBAR, L.A. Statistical Methods for Reliability Data. Joh Wiley, New York p. [4] NG, K.W.; KOTZ, S. Kummer-Gamma ad Kummer-Beta uivariate ad multivariate distributios. Research Report. v. 84, Departmet of Statistics, The Uiversity of Hog Kog, Hog Kog,
Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values
Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values
More informationANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY
ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,
More informationMaximum likelihood estimation from record-breaking data for the generalized Pareto distribution
METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio
More informationBayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationThe Inverse Weibull-Geometric Distribution
Article Iteratioal Joural of Moder Mathematical Scieces, 016, 14(: 134-146 Iteratioal Joural of Moder Mathematical Scieces Joural homepage: www.moderscietificpress.com/jourals/ijmms.aspx The Iverse Weibull-Geometric
More informationMathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical
More informationBayesian inference for Parameter and Reliability function of Inverse Rayleigh Distribution Under Modified Squared Error Loss Function
Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:99-878 EISSN: 239-844 Joural home page: www.ajbasweb.com Bayesia iferece
More informationExponential Families and Bayesian Inference
Computer Visio Expoetial Families ad Bayesia Iferece Lecture Expoetial Families A expoetial family of distributios is a d-parameter family f(x; havig the followig form: f(x; = h(xe g(t T (x B(, (. where
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationMOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMET-METHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationBayesian Control Charts for the Two-parameter Exponential Distribution
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com 2 Uiversity of the Free State Abstract By usig data that are the mileages
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More information1.010 Uncertainty in Engineering Fall 2008
MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval
More informationA Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution
A Note o Box-Cox Quatile Regressio Estimatio of the Parameters of the Geeralized Pareto Distributio JM va Zyl Abstract: Makig use of the quatile equatio, Box-Cox regressio ad Laplace distributed disturbaces,
More informationA New Distribution Using Sine Function- Its Application To Bladder Cancer Patients Data
J. Stat. Appl. Pro. 4, No. 3, 417-47 015 417 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.1785/jsap/040309 A New Distributio Usig Sie Fuctio- Its Applicatio To
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 1263-1277 HIKARI Ltd, www.m-hikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More informationDepartment of Mathematics
Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets
More informationAccess to the published version may require journal subscription. Published with permission from: Elsevier.
This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,
More informationThe (P-A-L) Generalized Exponential Distribution: Properties and Estimation
Iteratioal Mathematical Forum, Vol. 12, 2017, o. 1, 27-37 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610140 The (P-A-L) Geeralized Expoetial Distributio: Properties ad Estimatio M.R.
More information5. Likelihood Ratio Tests
1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More information4. Partial Sums and the Central Limit Theorem
1 of 10 7/16/2009 6:05 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 4. Partial Sums ad the Cetral Limit Theorem The cetral limit theorem ad the law of large umbers are the two fudametal theorems
More informationA proposed discrete distribution for the statistical modeling of
It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical
More informationA New Lifetime Distribution For Series System: Model, Properties and Application
Joural of Moder Applied Statistical Methods Volume 7 Issue Article 3 08 A New Lifetime Distributio For Series System: Model, Properties ad Applicatio Adil Rashid Uiversity of Kashmir, Sriagar, Idia, adilstat@gmail.com
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationBootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests
Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem
More information[ ] ( ) ( ) [ ] ( ) 1 [ ] [ ] Sums of Random Variables Y = a 1 X 1 + a 2 X 2 + +a n X n The expected value of Y is:
PROBABILITY FUNCTIONS A radom variable X has a probabilit associated with each of its possible values. The probabilit is termed a discrete probabilit if X ca assume ol discrete values, or X = x, x, x 3,,
More informationOptimal Design of Accelerated Life Tests with Multiple Stresses
Optimal Desig of Accelerated Life Tests with Multiple Stresses Y. Zhu ad E. A. Elsayed Departmet of Idustrial ad Systems Egieerig Rutgers Uiversity 2009 Quality & Productivity Research Coferece IBM T.
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationExpectation and Variance of a random variable
Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio
More informationApproximations and more PMFs and PDFs
Approximatios ad more PMFs ad PDFs Saad Meimeh 1 Approximatio of biomial with Poisso Cosider the biomial distributio ( b(k,,p = p k (1 p k, k λ: k Assume that is large, ad p is small, but p λ at the limit.
More informationOn Marshall-Olkin Extended Weibull Distribution
Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 O Marshall-Olki Exteded Weibull Distributio Haa Haj Ahmad Departmet of Mathematics, Uiversity of Hail Hail, KSA haaahm@yahoo.com Omar
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationIE 230 Seat # Name < KEY > Please read these directions. Closed book and notes. 60 minutes.
IE 230 Seat # Name < KEY > Please read these directios. Closed book ad otes. 60 miutes. Covers through the ormal distributio, Sectio 4.7 of Motgomery ad Ruger, fourth editio. Cover page ad four pages of
More informationComparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes
The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time
More informationIntroducing a Novel Bivariate Generalized Skew-Symmetric Normal Distribution
Joural of mathematics ad computer Sciece 7 (03) 66-7 Article history: Received April 03 Accepted May 03 Available olie Jue 03 Itroducig a Novel Bivariate Geeralized Skew-Symmetric Normal Distributio Behrouz
More informationOn The Gamma-Half Normal Distribution and Its Applications
Joural of Moder Applied Statistical Methods Volume Issue Article 5 5--3 O The Gamma-Half Normal Distributio ad Its Applicatios Ayma Alzaatreh Austi Peay State Uiversity, Clarksville, TN Kriste Kight Austi
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More information6. Sufficient, Complete, and Ancillary Statistics
Sufficiet, Complete ad Acillary Statistics http://www.math.uah.edu/stat/poit/sufficiet.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 6. Sufficiet, Complete, ad Acillary
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationIE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.
Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (
More informationPROPERTIES OF THE FOUR-PARAMETER WEIBULL DISTRIBUTION AND ITS APPLICATIONS
Pak. J. Statist. 207 Vol. 33(6), 449-466 PROPERTIES OF THE FOUR-PARAMETER WEIBULL DISTRIBUTION AND ITS APPLICATIONS T.H.M. Abouelmagd &4, Saeed Al-mualim &2, M. Elgarhy 3, Ahmed Z. Afify 4 ad Muir Ahmad
More informationStat 319 Theory of Statistics (2) Exercises
Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationPOWER AKASH DISTRIBUTION AND ITS APPLICATION
POWER AKASH DISTRIBUTION AND ITS APPLICATION Rama SHANKER PhD, Uiversity Professor, Departmet of Statistics, College of Sciece, Eritrea Istitute of Techology, Asmara, Eritrea E-mail: shakerrama009@gmail.com
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationDirection: This test is worth 150 points. You are required to complete this test within 55 minutes.
Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem
More informationStatistical Theory MT 2008 Problems 1: Solution sketches
Statistical Theory MT 008 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. a) Let 0 < θ < ad put fx, θ) = θ)θ x ; x = 0,,,... b) c) where α
More informationGeneralized Semi- Markov Processes (GSMP)
Geeralized Semi- Markov Processes (GSMP) Summary Some Defiitios Markov ad Semi-Markov Processes The Poisso Process Properties of the Poisso Process Iterarrival times Memoryless property ad the residual
More informationESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION
STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationJournal of Multivariate Analysis. Superefficient estimation of the marginals by exploiting knowledge on the copula
Joural of Multivariate Aalysis 102 (2011) 1315 1319 Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: www.elsevier.com/locate/jmva Superefficiet estimatio of the margials
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationLecture 7: Properties of Random Samples
Lecture 7: Properties of Radom Samples 1 Cotiued From Last Class Theorem 1.1. Let X 1, X,...X be a radom sample from a populatio with mea µ ad variace σ
More informationInternational Journal of Mathematical Archive-5(7), 2014, Available online through ISSN
Iteratioal Joural of Mathematical Archive-5(7), 214, 11-117 Available olie through www.ijma.ifo ISSN 2229 546 USING SQUARED-LOG ERROR LOSS FUNCTION TO ESTIMATE THE SHAPE PARAMETER AND THE RELIABILITY FUNCTION
More informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 016 MODULE : Statistical Iferece Time allowed: Three hours Cadidates should aswer FIVE questios. All questios carry equal marks. The umber
More informationQuantile regression with multilayer perceptrons.
Quatile regressio with multilayer perceptros. S.-F. Dimby ad J. Rykiewicz Uiversite Paris 1 - SAMM 90 Rue de Tolbiac, 75013 Paris - Frace Abstract. We cosider oliear quatile regressio ivolvig multilayer
More informationStatistical Theory MT 2009 Problems 1: Solution sketches
Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where
More informationPROBABILITY DISTRIBUTION RELATIONSHIPS. Y.H. Abdelkader, Z.A. Al-Marzouq 1. INTRODUCTION
STATISTICA, ao LXX,., 00 PROBABILITY DISTRIBUTION RELATIONSHIPS. INTRODUCTION I spite of the variety of the probability distributios, may of them are related to each other by differet kids of relatioship.
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationIntroduction to Extreme Value Theory Laurens de Haan, ISM Japan, Erasmus University Rotterdam, NL University of Lisbon, PT
Itroductio to Extreme Value Theory Laures de Haa, ISM Japa, 202 Itroductio to Extreme Value Theory Laures de Haa Erasmus Uiversity Rotterdam, NL Uiversity of Lisbo, PT Itroductio to Extreme Value Theory
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationA NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos
.- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September,
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More informationMinimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions
America Joural of heoretical ad Applied Statistics 6; 5(4): -7 http://www.sciecepublishiggroup.com/j/ajtas doi:.648/j.ajtas.654.6 ISSN: 6-8999 (Prit); ISSN: 6-96 (Olie) Miimax Estimatio of the Parameter
More informationStatistical Inference Based on Extremum Estimators
T. Rotheberg Fall, 2007 Statistical Iferece Based o Extremum Estimators Itroductio Suppose 0, the true value of a p-dimesioal parameter, is kow to lie i some subset S R p : Ofte we choose to estimate 0
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationRAINFALL PREDICTION BY WAVELET DECOMPOSITION
RAIFALL PREDICTIO BY WAVELET DECOMPOSITIO A. W. JAYAWARDEA Departmet of Civil Egieerig, The Uiversit of Hog Kog, Hog Kog, Chia P. C. XU Academ of Mathematics ad Sstem Scieces, Chiese Academ of Scieces,
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationChapter 2 Descriptive Statistics
Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationDepartment of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution
Departmet of Civil Egieerig-I.I.T. Delhi CEL 899: Evirometal Risk Assessmet HW5 Solutio Note: Assume missig data (if ay) ad metio the same. Q. Suppose X has a ormal distributio defied as N (mea=5, variace=
More informationClosed book and notes. No calculators. 60 minutes, but essentially unlimited time.
IE 230 Seat # Closed book ad otes. No calculators. 60 miutes, but essetially ulimited time. Cover page, four pages of exam, ad Pages 8 ad 12 of the Cocise Notes. This test covers through Sectio 4.7 of
More informationJanuary 25, 2017 INTRODUCTION TO MATHEMATICAL STATISTICS
Jauary 25, 207 INTRODUCTION TO MATHEMATICAL STATISTICS Abstract. A basic itroductio to statistics assumig kowledge of probability theory.. Probability I a typical udergraduate problem i probability, we
More informationAkaike Information Criterion and Fourth-Order Kernel Method for Line Transect Sampling (LTS)
Appl. Math. If. Sci. 10, No. 1, 267-271 (2016 267 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://dx.doi.org/10.18576/amis/100127 Akaike Iformatio Criterio ad Fourth-Order Kerel Method
More informationRecord Values from T-X Family of. Pareto-Exponential Distribution with. Properties and Simulations
Applied Mathematical Scieces, Vol. 3, 209, o., 33-44 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.209.879 Record Values from T-X Family of Pareto-Epoetial Distributio with Properties ad Simulatios
More information( θ. sup θ Θ f X (x θ) = L. sup Pr (Λ (X) < c) = α. x : Λ (x) = sup θ H 0. sup θ Θ f X (x θ) = ) < c. NH : θ 1 = θ 2 against AH : θ 1 θ 2
82 CHAPTER 4. MAXIMUM IKEIHOOD ESTIMATION Defiitio: et X be a radom sample with joit p.m/d.f. f X x θ. The geeralised likelihood ratio test g.l.r.t. of the NH : θ H 0 agaist the alterative AH : θ H 1,
More informationQuick Review of Probability
Quick Review of Probability Berli Che Departmet of Computer Sciece & Iformatio Egieerig Natioal Taiwa Normal Uiversity Refereces: 1. W. Navidi. Statistics for Egieerig ad Scietists. Chapter & Teachig Material.
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationCONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION
CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION Petros Maravelakis, Joh Paaretos ad Stelios Psarakis Departmet of Statistics Athes Uiversity of Ecoomics ad Busiess 76 Patisio St., 4 34, Athes, GREECE. Itroductio
More informationSome illustrations of possibilistic correlation
Some illustratios of possibilistic correlatio Robert Fullér IAMSR, Åbo Akademi Uiversity, Joukahaisekatu -5 A, FIN-252 Turku e-mail: rfuller@abofi József Mezei Turku Cetre for Computer Sciece, Joukahaisekatu
More informationx = Pr ( X (n) βx ) =
Exercise 93 / page 45 The desity of a variable X i i 1 is fx α α a For α kow let say equal to α α > fx α α x α Pr X i x < x < Usig a Pivotal Quatity: x α 1 < x < α > x α 1 ad We solve i a similar way as
More informationA Comparative Study of Traditional Estimation Methods and Maximum Product Spacings Method in Generalized Inverted Exponential Distribution
J. Stat. Appl. Pro. 3, No. 2, 153-169 (2014) 153 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.12785/jsap/030206 A Comparative Study of Traditioal Estimatio Methods
More informationHOMEWORK I: PREREQUISITES FROM MATH 727
HOMEWORK I: PREREQUISITES FROM MATH 727 Questio. Let X, X 2,... be idepedet expoetial radom variables with mea µ. (a) Show that for Z +, we have EX µ!. (b) Show that almost surely, X + + X (c) Fid the
More information