Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values
|
|
- Gregory Neal Welch
- 5 years ago
- Views:
Transcription
1 Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp Witer 2014 Joural homepage: Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values M. Abdi * Received: 31 July 2013 ; Accepted: 4 December 2013 Abstract I this paper we study the estimatio problems for the two-parameter expoetiated Gumbel distributio based o lower record values. A exact cofidece iterval ad a exact joit cofidece regio for the parameters are costructed. A simulatio study is coducted to study the performace of the proposed cofidece iterval ad regio. Fially a umerical example with real data set is give to illustrate the proposed procedures. Keywords Cofidece Iterval Expoetiated Gumbel Distributio Exact Joit Cofidece Regio Record Values. 1 Itroductio The Gumbel distributio is perhaps the most widely applied statistical distributio for climate modelig. Some of its applicatio areas i climate modelig iclude: global warmig problems flood frequecy aalysis raifall modelig ad wid speed modelig. A recet book by Kotz ad Nadarajah [1] which describes this distributio lists over 50 applicatios ragig from accelerated life testig through to earthquakes floods horse racig raifall sea currets wid speeds ad track race records (to metio just a few). I literature expoetiated family of distributios defied i two ways. If G( x; ) is cumulative distributio fuctio (cdf) of a baselie distributio the by addig oe more parameter (say ) the cdf of expoetiated baselie distributio F( x; ) is give by a) F x; G x; 0 x R b) F x; 1 1 G x; 0 x R where is parameter space. Gupta et al. [2] itroduced the expoetiated expoetial (EE) distributio as a geeralizatio of the expoetial distributio. The two parameters EE distributio associated with defiitio (a) have bee studied i detail by Gupta ad Kudu [3] which is a sub-model of the expoetiated Weibull distributio itroduced by Mudholkar ad Shrivastava [4]. Nadarajah [5] itroduced expoetiated Gumbel distributio usig (b). Some of its applicatio areas i climate modelig iclude global warmig problem flood frequecy aalysis offshore modelig raifall modelig ad wid speed modelig. Persso ad Ryde [6] discussed * Correspodig Author. () me.abdi@bam.ac.ir. (M. Abdi) M. Abdi M.Sc Departmet of Mathematics ad Computig Higher Educatio Complex of Bam Bam Ira.
2 62 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) estimatio of T-year retur values for sigificat wave height i a case study ad compare poit estimates ad their ucertaities to the results give by alterative approaches usig Gumbel or Geeralized Extreme Value distributios. A radom variable X is said to have Gumbel distributio if its cdf is x ; e G x e x 0 0 By itroducig a shape parameter θ 0 ad usig defiitio (a) the cdf of the expoetiated Gumbel distributio is x e F x; G x; e x (1) th which is simply the θ power of cdf of the Gumbel distributio. The probability desity fuctio (pdf) correspodig to (1) is x x e f x; e x (2) We shall write X EG( ) to deote a absolutely cotiuous radom variable X havig the two-parameter expoetiated Gumbel distributio with shape ad scale parameters ad σ respectively whose pdf is give by (2). The purpose of this paper is to costruct the iterval estimatio for the two-parameter expoetiated Gumbel distributio based o lower record values. The rest of this paper is orgaized as follows. Sectio 2 provides some prelimiaries. I Sectio 3 we preset a exact cofidece iterval for scale parameter ad a exact joit cofidece regio for the parameters ( ) based o lower record values. I Sectio 4 a Mote Carlo simulatio is coducted to study the performace of the proposed cofidece iterval ad regio. Fially i sectio 5 a umerical example with real data set is preseted to illustrate the proposed methods. 2 Prelimiaries Let X1 X 2 be a sequece of idepedet ad idetically distributed (iid) cotiuous radom variables with cdf F( x) ad pdf f ( x ). A observatio X j is called a upper (lower) record value of this sequece its value exceeds (is lower tha) that of all previous observatios. Geerally let us defie T1 1 U1 X1 ad for 2 T mi j T : X X U X. -1 j T -1 T The the sequece { U }({ T }) is kow as upper record statistics (upper record times). Similarly the lower record times S ad the lower record values L are defied as follows:
3 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 63 S1 1 X1 ad for 2 S = -1 mi j S : X X L X. For more details o j S -1 S records ad its applicatios see Nevzorov[7] Ahsaullah[8] ad Arold et. al. [9]. The followig lemmas are useful i this paper. Lemma 2.1. Let L2 be the first m observed lower record values from a populatio with cdf F (.). Defie U l[ F( L )] i 1 2 m. i i The U1 U2 Um are the first m upper record values from a stadard expoetial distributio. Proof. From the joit pdf of L2 ad usig a simple Jacobia argumet we ca easily obtai the joit pdf of U1 U2 Um as f u u u e 0 u u... u -um U1 U2... Um 1 2 m 1 2 m which is the joit pdf of the first m upper record values from a stadard expoetial distributio (see Arold et al. [9]). The proof is thus obtaied. Lemma 2.2. If U1 U2 Um are the first m upper record values from a stadard expoetial distributio. The the spacigs U1 U2 U1 Um Um 1 are iid radom variables from a stadard expoetial distributio. Proof. The proof ca be foud i Arold et al. [9]. 3 Mai Result Let L2 be the first m observed lower record values from the expoetiated Gumbel distributio. I this sectio a 100(1 )% cofidece iterval for scale parameter ad a 100(1 )% joit cofidece regio for ( ) are costructed based o the observed lower records L2. Li Let us defie Yi exp i 1 2 m. The by Lemma 2.1 Y1 Y2 Ym are the first m upper record values from a stadard expoetial distributio. Moreover by Lemma 2.2 we ca observe that Z1 Y1 Z2 Y2 Y1 Z Y Y m m m 1 (3)
4 64 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) are iid radom variables from a stadard expoetial distributio. Hece V 2Z 2 Y (4) 1 1 has a chi-square distributio with 2j degrees of freedom ad m U 2 Z 2 Y Y i m 1 (5) i2 has a chi-square distributio with 2( m 1) degrees of freedom. We ca also fid that U ad V are idepedet radom variables. Let T U / 2( m 1) U 1 Y Y (6) m 1 V / 2 ( m 1) V m 1 Y1 Ad S U V 2 Y m (7) It is easy to show that T has a F distributio with 2( m 1) ad 2 degrees of freedom ad S has a chi-square distributio with 2m degrees of freedom. Furthermore T ad S are idepedet see Johso et al. ([10] P. 350). Let F ( 1 2 ) be the percetile of F distributio with right-tail probability ad 1 ad 2 degrees of freedom. Next theorem gives a exact cofidece iterval for the scale parameter base o lower record values. Theorem 3.1. Suppose that L2 be the first m observed lower record values from EG distributio i (1). The for ay 0 1 l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 1 is a 100(1 )% cofidece iterval for. Proof. From (6) we kow that the pivot T 1 e e 1 e 1 L 1 m 1 m 1 e has a F distributio with 2( m 1) ad 2 degrees of freedom. We ote that T ( ) is strictly
5 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 65 decreasig fuctio of σ. Hece for 0 1 we obtai F 2 m 1 2 T F 2 m is equivalet to the evet l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 1 this completes the proof. It should be metioed here that we ca also use T ( ) to test ull hypothesis H0 : 0. Let 2 ( ) deote the percetile of 2 distributio with right-tail probability ad degrees of freedom. Next theorem gives a exact joit cofidece regio for the parameters ad. Theorem 3.2. Suppose that L2 be the first m observed lower record values from EG distributio. The the followig iequalities determie 100(1 )% joit cofidece regio for ad : l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] (2 m) (2 m) Proof. From (7) we kow that L m S 2 e 2 has a distributio with 2m degrees of freedom ad it is idepedet of T. Hece for 0 1 we have P [ F (2( m 1) 2) T F (2( m 1) 2)] ad P[ (2 m) S (2 m)] From these relatioships we coclude that
6 66 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) P F m T F m m S m or equivaletly l[1 m 1 F (2( m 1) 2)] l[1 m 1 F (2( m 1) 2)] 4 Simulatio study (2 m) (2 m) I this sectio a Mote Carlo simulatio is coducted to study the performace of the proposed cofidece iterval ad joit cofidece regio. I this simulatio we radomly geerated lower record sample L2... L m from the Gumbel distributio with the value of parameters ad the computed 95% cofidece itervals ad regios usig the results preseted i Sectio 3. We the replicated the process 5000 times. We preseted i Table 1 the simulated average cofidece legth for parameter cofidece area for the parameters ( ) ad the 95% coverage probabilities of the proposed cofidece itervals ad regios. From Table 1 we observe whe m icreases the average cofidece legth for ad the average cofidece area for ( ) are decreased. The simulatio results shows that the coverage probabilities of the exact cofidece itervals for parameter ad joit cofidece regios for parameters ( ) are close to the desired level of 0.95 for differet sample sizes. Hece our proposed methods for costructig exact cofidece itervals ad joit cofidece regios ca becused reliably.. Table 1 The simulated average cofidece legth (CL) cofidece area (CA) ad 95% coverage probabilities (CP) for the parameters. m CL ( ) CA( ) CP( ) CP( )
7 Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values 67 5 Numerical example I this sectio real example with climate record data are give to illustrate the proposed cofidece itervals ad joit cofidece regios. We preset a data aalysis ad illustrate applicatio of the results i Sectio 3 to the seasoal (July 1-Jue 30) raifall i iches recorded at Los Ageles Civic Ceter from 1962 to 2012 (see the website of Los Ageles Almaac: weather/we13.htm). The data are as follows: Here we checked the validity of the expoetiated Gumbel Model based o the parameters ˆ ˆ usig the Kolmogorov Smirov (K-S) test. It is observed that the K-S distace is K S with a correspodig P Value So the expoetiated Gumbel model provides a good fit to the above data. If oly the lower record values of the seasoal raifall have bee observed these are To fid a 95% cofidece iterval for ad a joit cofidece regio for ad we eed the followig percetiles: F F F (10 2) F (10 2) ad (12) By Theorem 3.1 the 95% CI for is ( ) with cofidece legth By Theorem 3.2 the 95% JCR for ad is determied by the followig iequalities: with area Figure 1 shows the above joit cofidece regio for the parameters.
8 68 M. Abdi / IJAOR Vol. 4 No Witer 2014 (Serial #11) Fig. 1 Joit cofidece regio for parameters ad. Ackowledgmets The authors would like to thak the Editor ad the referees for their useful commets which improved the paper. Refereces 1. Kotz S. Nadarajah S. (2000). Extreme Value Distributios: Theory ad Applicatios. Lodo: Imperial College Pres:. 2. Gupta R. C. Gupta P. L. Gupta R. D. (1998). Modelig failure time data by Lehma alteratives. Commuicatios i Statistics-Theory ad Methods 27: Mudholkar G. S. Srivastava D. (1993). The expoetiated Weibull family for aalyzig bathtub failure-rate data. IEEE Tras. Reliability. 42: Nadarajah S. (2005). The expoetiated Gumbel distributio with climate applicatio. Evirometrics. 17 (1) Persso K. Ryde J. (2010). Expoetiated Gumbel Distributio for Estimatio of Retur Levels of Sigificat Wave Height Joural of Evirometal Statistics. 1(3) Nevzorov V. B. (1988). Records. Theory of Probability ad its Applicatio Ahsaullah M. (1995). Record Statistics. New York : Nova Sciece Publishers Ic. Commack. 8. Arold B. C. Balakrisha N. Nagaraja H. N. (1998). Records. New York: Joh Wiley & Sos. 9. Gupta R.D. Kudu D. (2001). Expoetiated expoetial family: a alterative to gamma ad Weibull distributios. Biometrical Joural Johso N. L. Kotz S. Balakrisha N. (1994). Cotiuous Uivariate Distributios. New York: Joh Wiley & Sos.
Maximum 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
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 informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
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 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 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 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 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 informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
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 (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 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 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 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 informationChapter 13: Tests of Hypothesis Section 13.1 Introduction
Chapter 13: Tests of Hypothesis Sectio 13.1 Itroductio RECAP: Chapter 1 discussed the Likelihood Ratio Method as a geeral approach to fid good test procedures. Testig for the Normal Mea Example, discussed
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 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 informationPower Comparison of Some Goodness-of-fit Tests
Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 7-6-2016 Power Compariso of Some Goodess-of-fit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750
More informationLecture 6 Simple alternatives and the Neyman-Pearson lemma
STATS 00: Itroductio to Statistical Iferece Autum 06 Lecture 6 Simple alteratives ad the Neyma-Pearso lemma Last lecture, we discussed a umber of ways to costruct test statistics for testig a simple ull
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 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 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 information7.1 Convergence of sequences of random variables
Chapter 7 Limit theorems Throughout this sectio we will assume a probability space (Ω, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationThe new class of Kummer beta generalized distributions
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
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 information32 estimating the cumulative distribution function
32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio
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 informationTesting Statistical Hypotheses for Compare. Means with Vague Data
Iteratioal Mathematical Forum 5 o. 3 65-6 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach
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 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 informationSince X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain
Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the
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 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 informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationA Note on the Kolmogorov-Feller Weak Law of Large Numbers
Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI:10.3770/j.iss:095-651.015.0.013 Http://jmre.dlut.edu.c A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu
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 information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationLecture 5. Materials Covered: Chapter 6 Suggested Exercises: 6.7, 6.9, 6.17, 6.20, 6.21, 6.41, 6.49, 6.52, 6.53, 6.62, 6.63.
STT 315, Summer 006 Lecture 5 Materials Covered: Chapter 6 Suggested Exercises: 67, 69, 617, 60, 61, 641, 649, 65, 653, 66, 663 1 Defiitios Cofidece Iterval: A cofidece iterval is a iterval believed to
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 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 informationOverview. p 2. Chapter 9. Pooled Estimate of. q = 1 p. Notation for Two Proportions. Inferences about Two Proportions. Assumptions
Chapter 9 Slide Ifereces from Two Samples 9- Overview 9- Ifereces about Two Proportios 9- Ifereces about Two Meas: Idepedet Samples 9-4 Ifereces about Matched Pairs 9-5 Comparig Variatio i Two Samples
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationExam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.
Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #10 19 1 Exam II Topics **Will emphasize material
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationBIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov
Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios
More informationProblem Set 4 Due Oct, 12
EE226: Radom Processes i Systems Lecturer: Jea C. Walrad Problem Set 4 Due Oct, 12 Fall 06 GSI: Assae Gueye This problem set essetially reviews detectio theory ad hypothesis testig ad some basic otios
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
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 informationThis section is optional.
4 Momet Geeratig Fuctios* This sectio is optioal. The momet geeratig fuctio g : R R of a radom variable X is defied as g(t) = E[e tx ]. Propositio 1. We have g () (0) = E[X ] for = 1, 2,... Proof. Therefore
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 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 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 informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationSTAT Homework 1 - Solutions
STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
More informationTesting Statistical Hypotheses with Fuzzy Data
Iteratioal Joural of Statistics ad Systems ISS 973-675 Volume 6, umber 4 (), pp. 44-449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui
More informationSection 14. Simple linear regression.
Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo
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 informationOutput Analysis and Run-Length Control
IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%
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 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 informationBayesian Methods: Introduction to Multi-parameter Models
Bayesia Methods: Itroductio to Multi-parameter Models Parameter: θ = ( θ, θ) Give Likelihood p(y θ) ad prior p(θ ), the posterior p proportioal to p(y θ) x p(θ ) Margial posterior ( θ, θ y) is Iterested
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 informationA Question. Output Analysis. Example. What Are We Doing Wrong? Result from throwing a die. Let X be the random variable
A Questio Output Aalysis Let X be the radom variable Result from throwig a die 5.. Questio: What is E (X? Would you throw just oce ad take the result as your aswer? Itroductio to Simulatio WS/ - L 7 /
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 information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
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 informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationEDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR
Joural of Statistical Research 26, Vol. 37, No. 2, pp. 43-55 Bagladesh ISSN 256-422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics
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 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 informationThere is no straightforward approach for choosing the warmup period l.
B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.
More informationInvestigating the Significance of a Correlation Coefficient using Jackknife Estimates
Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
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 informationThe variance of a sum of independent variables is the sum of their variances, since covariances are zero. Therefore. V (xi )= n n 2 σ2 = σ2.
SAMPLE STATISTICS A radom sample x 1,x,,x from a distributio f(x) is a set of idepedetly ad idetically variables with x i f(x) for all i Their joit pdf is f(x 1,x,,x )=f(x 1 )f(x ) f(x )= f(x i ) The sample
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 information