Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function
|
|
- Elvin Golden
- 6 years ago
- Views:
Transcription
1 Iteratioal Joural of Statistics ad Systems ISSN Volume 12, Number 4 (217), pp Research Idia Publicatios Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh Distributio- A Bayesia Approach uder Liex Loss Fuctio Isha Gupta Departmet of Statistics; Uiversity of Jammu, Jammu, Idia. Abstract I this paper, Bayesia ad E Bayesia method of estimatio are proposed for estimatig the parameter of Rayleigh distributio. The Bayes estimate of the parameter is derived uder the assumptio that the prior distributio is iformative i.e. gamma prior usig Liex loss fuctio. Further, compariso betwee the E-Bayes estimators with the associated Bayes estimators have bee carried out through simulatio study usig MATLAB software. Keywords: Rayleigh distributio, Liex Loss fuctio, Bayes ad E-Bayes estimators, Gamma prior. 1. INTRODUCTION The Rayleigh distributio is a cotiuous probability distributio servig as a special case of the well-kow Weibull distributio. This distributio has log bee cosidered to have sigificat applicatios i fields such as survival aalysis, reliability theory ad especially commuicatio egieerig. The Rayleigh distributio provides a populatio model which is useful i several areas of statistics (Feradez, 2). I the literature, may researcher studied properties of the Rayleigh distributio, particularly i life testig ad reliability. Whe cosiderig the complete Rayleigh model, the probability desity fuctio is give by f (x, θ) = 2 θ x e θx2, x, θ > ; (1) usig the parameterizatio of the distributio as proposed by Bhattacharya ad Tyagi
2 792 Isha Gupta (199), ad is deoted by X Rayleigh(θ). The parameter θ is a scale parameter, ad characterizes the lifetime of the object uder cosideratio i applicatio. May authors have studied Rayleigh distributio e.g. Ferreira et al. (216) have proposed Bayes estimators icludig shrikage estimators of the ukow parameter of the cesored Rayleigh distributio usig Al-Bayyati loss fuctio cosiderig differet objective prior distributio. Dey (212) has obtaied Bayes estimator of parameter ad reliability fuctio of iverse Rayleigh distributio uder two loss fuctios ad also obtaied associated risk fuctios of the Bayes estimator. Saat et al. (216) have proposed Bayes estimate of parameter of Rayleigh distributio usig Quasi prior uder differet loss fuctios. Ahmed et al. (213) have obtaied Bayes estimate of parameter of Rayleigh distributio usig Jeffrey s ad extesio of Jeffrey s prior uder Squared error ad Al-Bayyati s loss fuctio. The mai objective of this paper is to itroduce a statistical compariso betwee the Bayesia ad Expected- Bayesia procedures for estimatig the parameter of Rayleigh distributio. The resultig estimators are obtaied by usig Liex loss fuctio. The layout of the paper is as follow. I Sectio 2, Bayes estimate of parameter have bee obtaied usig cojugate prior uder Liex loss fuctio. I Sectio 3, E- Bayes estimate have also bee obtaied usig three differet prior distributios. Fially, compariso betwee Bayes ad E-Bayes estimates have bee made usig simulatio study i Sectio 4. Some cocludig remarks have bee give i Sectio BAYESIAN ESTIMATION. I this sectio, Bayes estimate of parameter of Rayleigh distributio is obtaied by usig Liex loss fuctio. Let X 1, X 2, X 3..be a sequece of radom variables from Rayleigh distributio, whose desity fuctio is give by (1), the the likelihood fuctio is give by L( x, θ ) = (2θ) i=1 x i e θ i=1 (x i )2 (2) We use the gamma cojugate prior desity for the parameter θ ad the pdf of gamma prior desity with scale parameter r is give by g( θ / r) = e θ θ r 1 ; r >, θ > (3) Г(r) O combiig (2) ad (3), ad usig Bayes theorem, the posterior desity of θ give x is give by P ( θ x) α L ( x, θ) g( θ / r) = (2θ) i=1 x i e θ i=1 (x i )2 e θ θ r 1 Г(r) ; r >, θ >, x >
3 Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh 793 = k θ +r 1 e θ [ i=1 (x i )2 + 1 ] ; where k is idepedet of θ Thus, Posterior desity is give by P ( θ x) = e θ [ (x i ) 2 i=1 + 1 ] θ +r 1 [ i=1 (x i ) 2 + 1] +r Г (+r) Bayesia estimatio of θ uder liex loss fuctio. Zeller represet Liex ( i.e. liear expoetial ) loss fuctio as L (θ, θ ) = a {exp[b(θ θ) b(θ θ) 1]} ; ; x >, θ >, r > (4) where a >, b ; a is scale of loss fuctio ad b determies its shape. Without loss of geerality, we assume a = 1 ad obtai Bayes estimate of θ. Here, E[L (θ, θ )] = L (θ, θ ) θ [ e i=1 (x i )2 + 1 ] θ +r 1 [ i=1 (x i ) 2 + 1] +r Г (+r) P ( θ x) dθ = [ e [b( θ θ)] b(θ θ) 1] = e bθ [ dθ i=1 (x i ) b + (x i ) 2 +r ] bθ + i=1 + 1 b ( + r ) Thus Bayes estimator of θ uder Liex loss fuctio is give by θ BL = +r + 1 i= (x i ) 2 log b [b + i=1 (x i )2 ] (5) i=1 (x i ) EXPECTED - BAYESIAN ESTIMATE OF PARAMETER USING LINEX LOSS FUNCTION Ha (27) itroduced a ew method, amed E- Bayesia method to estimate failure probability. The E Bayes estimate of θ i.e. expectatio of the Bayes estimate of θ is give by θ EB = E ( θ x) = θ BE π(θ, r) dr ;. Q where Q is the domai of r for which the prior desity is decreasig i θ. θ BE is the Bayes estimate of θ uder the Liex loss fuctio. I order to obtai E-Bayes estimates of θ, we have to choose prior distributio of hyper parameter r. These distributios are used to study the impact of differet prior distributios o E-Bayes estimatio of θ. The followig distributios of r are give by π 1 (θ, r) = 2 (c r) c 2 ; < r < c, (6)
4 794 Isha Gupta π 2 (θ, r) = 1 ; < r < c, (7) c π 3 (θ, r) = 2 r c2 ; < r < c, (8) E-Bayesia Estimatio of θ uder Liex loss fuctio : E-Bayesia estimate of θ relative to Liex loss fuctio based o π 1 (θ, r), is deoted by θ EBL1 ad is obtaied by usig (5) ad (6). c θ EBL1 = θ BE π 1 (θ, r) dr c = { + r b log [b + i=1 (x i) i=1 (x i ) ]} 2 (c r) c 2 dr = [ 3+c ] log 3b [b + i=1 (x i )2 + 1 ] (9) i=1 (x i ) E-Bayesia estimates of θ based o π 2 (θ, r), is deoted by θ EBL2 ad is obtaied by usig (5) ad (7). θ EBL2 = [ 2 + c 2b ] log [ b + i=1 (x i )2 + 1 ] (1) i=1 + 1 E-Bayesia estimates of θ based o π 3 (θ, r), is deoted by θ EBL3 ad is obtaied by usig (5) ad (8). θ EBL3 = [ 3 +2 c 3b (x i ) 2 ] log [ b + i=1 (x i )2 + 1 ] (11) i=1 + 1 (x i ) 2 4. SIMULATION STUDY I order to compare the performace of Bayes ad E-Bayes techiques of estimatio, a simulatio study was coducted usig Matlab software for differet sample sizes ad for differet values of loss parameter. The followig steps were coducted: a. For give value of the prior parameter c, we geerate samples to fid value of r from uiform priors (6-8) respectively. b. For give value of r, we geerate θ from the gamma prior desity (3), c. For kow values of θ, we geerate sample from Rayleigh distributio with pdf (1), ad the Bayes ad Expected Bayes estimates usig Liex loss fuctio are computed from (5), (9), (1) ad (11) respectively. d. The above steps are repeated 5 times ad the mea square error of the Bayes ad E-Bayes estimates are computed ad the results are show i table 1.
5 Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh 795 Table 1: Averaged values of MSE for Bayes ad E-Bayes estimates of the parameter θ. (sample size) c = 2, r =.5, b =.75 c = 2, r =.5, b = -.75 θ BL θ EBL1 θ EBL2 θ EBL3 θ BL θ EBL1 θ EBL2 θ EBL CONCLUSION I this paper, Bayes ad Expected-Bayes methods are used for estimatio of parameter of Rayleigh distributio usig Liex loss fuctio. It has bee oticed from the results of simulatio study, that the E-Bayes estimates have smaller Mea Square Error as compared with the associated Bayes estimate. It has also bee observed that E-Bayes estimate, i most cases, ted to be more efficiet tha Bayes estimate except (for = 35, b=.75 ad = 3, 4,b= -.75) where Bayes ad E-Bayes are equally efficiet. REFERENCES [1] Ahmed, A., Ahmad S. P. ad Reshi, J. A. (213). Bayesia aalysis of Rayleigh distributio. Iteratioal Joural of Scietific ad Research Publicatios, Vol. 3, 1-9. [2] Bhattacharya ad Tyagi (199). Bayesia survival aalysis based o the Rayleigh model. Trabajos de Estadistica, 5(1), [3] Dey, S. (212). Bayesia estimatio of the parameter ad reliability fuctio of a Iverse Rayleigh distributio. Malaysia Joural of Mathematical Scieces, 6(1), [4] Feradez, A. J. (2). Bayesia iferece from type II cesored Rayleigh data. Statistics ad Probability Letters, 48, [5] Ferreira, J. T., Bekker, A. ad Arashi M. (216). Objective Bayesia estimators for the right cesored Rayleigh distributio evaluatig the Al-
6 796 Isha Gupta Bayyati loss fuctio. Revstat- Statistical Joural, Vol. 14, No. 4, [6] Ha, M., (27). E-Bayesia estimatio of failure probability ad its applicatio. Mathematical ad Computer Modellig, 45, [7] Saat. P ad Srivastava, R. S. (216). Bayesia aalysis of Rayleigh distributio uder quasi prior for differet loss fuctios. Iteratioal Joural of Sciece ad Research, Vol. 5, [8] Zeller, A. (1986). Bayesia Estimatio ad Predictio Usig Asymmetric Loss Fuctios, Jour. Amer. Statist. Assoc., 81,
Bayesian 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 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 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 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 informationConfidence 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 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 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 informationComparison of Methods for Estimation of Sample Sizes under the Weibull Distribution
Iteratioal Joural of Applied Egieerig Research ISSN 0973-4562 Volume 12, Number 24 (2017) pp. 14273-14278 Research Idia Publicatios. http://www.ripublicatio.com Compariso of Methods for Estimatio of Sample
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 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 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 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 informationAl- Mustansiriyah J. Sci. Vol. 24, No 5, 2013
Al- Mustasiriyah J. Sci. Vol. 24, No 5, 23 Usig Etropy Loss Fuctio to Estimate the Scale Parameter for Laplace Distributio Huda A. Rasheed, Akbal J. Sulta ad Nadia J. Fazah Departmet of Mathematics, college
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 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 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 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 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 a Mixture of Two Weibull Distributions under Generalized Order Statistics
IOSR Joural of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X Volume 12, Issue 2 Ver V (Mar - Apr 2016), PP 25-35 wwwiosrjouralsorg Estimatio of a Mixture of Two Weibull Distributios uder Geeralized
More informationModeling and Estimation of a Bivariate Pareto Distribution using the Principle of Maximum Entropy
Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral
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 informationA Method of Proposing New Distribution and its Application to Bladder Cancer Patients Data
J. Stat. Appl. Pro. Lett. 2, No. 3, 235-245 (2015) 235 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020306 A Method of Proposig New Distributio
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
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 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 informationOn Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers
Pujab Uiversity Joural of Mathematics ISSN 1016-2526) Vol. 502)2018) pp. 11-19 O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers P. Nasiri Departmet of Statistics, Uiversity
More informationECE 901 Lecture 12: Complexity Regularization and the Squared Loss
ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality
More informationTHE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR
Iteratioal Joural of Iovative Maagemet, Iformatio & Productio ISME Iteratioal c2013 ISSN 2185-5439 Volume 4, Number 1, Jue 2013 PP. 17-24 THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR
More informationEstimating the Population Mean using Stratified Double Ranked Set Sample
Estimatig te Populatio Mea usig Stratified Double Raked Set Sample Mamoud Syam * Kamarulzama Ibraim Amer Ibraim Al-Omari Qatar Uiversity Foudatio Program Departmet of Mat ad Computer P.O.Box (7) Doa State
More informationNon-Bayes, Bayes and Empirical Bayes Estimators for the Shape Parameter of Lomax Distribution
No-Bayes, Bayes ad Empirical Bayes Estimators for the Shape Parameter of Lomax Distributio Dr. Nadia H. Al-Noor* ad Shahad Saad Alwa *Dept. of Mathematics / College of Sciece / AL- Mustasiriya Uiversity
More informationEstimation of the Population Mean in Presence of Non-Response
Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationSome Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product
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 (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 informationReliability Measures of a Series System with Weibull Failure Laws
Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume, Number 2 (26), pp. 73-86 Research Idia Publicatios http://www.ripublicatio.com Reliability Measures of a Series System with Weibull Failure
More informationTrimmed Mean as an Adaptive Robust Estimator of a Location Parameter for Weibull Distribution
World Academy of Sciece Egieerig ad echology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol: No:6 008 rimmed Mea as a Adaptive Robust Estimator of a Locatio Parameter for Weibull Distributio
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 informationA General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)
Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig
More informationInference of Bivariate Generalized Exponential. Distribution Based on Copula Functions
Applied Mathematical Scieces, Vol. 11, 2017, o. 24, 1155-1186 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7398 Iferece of Bivariate Geeralized Expoetial Distributio Based o Copula Fuctios
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 informationBayesian Estimation of the Parameters of Two- Component Mixture of Rayleigh Distribution under Doubly Censoring
Joural of Moder Applied Statistical Methods Volume 3 Issue Article 4-04 Bayesia Estimatio of the Parameters of Two- Compoet Mixture of Rayleigh Distributio uder Doubly Cesorig Tahassum N. Sidhu Quaid-i-Azam
More informationParameter Estimation In Weighted Rayleigh Distribution
Joural of Moder Applied Statistical Methods Volume 6 Issue Article 4 December 07 Parameter Estimatio I Weighted Rayleigh Distributio M. Ajami Vali-e-Asr Uiversity of Rafsaja Rafsaja Ira m.ajami@vru.ac.ir
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 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 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 informationOn an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
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 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 informationLecture 11 and 12: Basic estimation theory
Lecture ad 2: Basic estimatio theory Sprig 202 - EE 94 Networked estimatio ad cotrol Prof. Kha March 2 202 I. MAXIMUM-LIKELIHOOD ESTIMATORS The maximum likelihood priciple is deceptively simple. Louis
More informationUniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations
Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie
More informationA Family of Unbiased Estimators of Population Mean Using an Auxiliary Variable
Advaces i Computatioal Scieces ad Techolog ISSN 0973-6107 Volume 10, Number 1 (017 pp. 19-137 Research Idia Publicatios http://www.ripublicatio.com A Famil of Ubiased Estimators of Populatio Mea Usig a
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 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 informationPARAMETER ESTIMATION BASED ON CUMU- LATIVE KULLBACK-LEIBLER DIVERGENCE
PARAMETER ESTIMATION BASED ON CUMU- LATIVE KULLBACK-LEIBLER DIVERGENCE Authors: Yaser Mehrali Departmet of Statistics, Uiversity of Isfaha, 81744 Isfaha, Ira (y.mehrali@sci.ui.ac.ir Majid Asadi Departmet
More informationNew Entropy Estimators with Smaller Root Mean Squared Error
Joural of Moder Applied Statistical Methods Volume 4 Issue 2 Article 0 --205 New Etropy Estimators with Smaller Root Mea Squared Error Amer Ibrahim Al-Omari Al al-bayt Uiversity, Mafraq, Jorda, alomari_amer@yahoo.com
More informationThe New Probability Distribution: An Aspect to a Life Time Distribution
Math. Sci. Lett. 6, No. 1, 35-4 017) 35 Mathematical Sciece Letters A Iteratioal Joural http://dx.doi.org/10.18576/msl/060106 The New Probability Distributio: A Aspect to a Life Time Distributio Diesh
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 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 informationDiscriminating between Generalized Exponential and Gamma Distributions
Joural of Probability ad Statistical Sciece 4, 4-47, Aug 6 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios Orawa Supapueg Kamo Budsaba Adrei I Volodi Praee Nilkor Thammasat Uiversity Uiversity
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 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 informationA Generalized Gamma-Weibull Distribution: Model, Properties and Applications
Marquette Uiversity e-publicatios@marquette Mathematics, Statistics ad Computer Sciece Faculty Research ad Publicatios Mathematics, Statistics ad Computer Sciece, Departmet of --06 A Geeralized Gamma-Weibull
More informationBayes Estimators for the Shape Parameter of Pareto Type I. Distribution under Generalized Square Error Loss Function
Mathematical Theory ad Modelig ISSN 2224-584 (aper) ISSN 2225-522 (Olie) Vol.4, No.11, 214 Bayes Estimators for the Shape arameter of areto Type I Distributio uder Geeralized Square Error Loss Fuctio Dr.
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 informationControl Charts for Mean for Non-Normally Correlated Data
Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies
More informationOn Constructing Super-Models to Enhance Failure. Forecasting: A Comparative Study of Four Real. Data Sets
Applied Mathematical Scieces, Vol. 12, 2018, o. 31, 1571-1600 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811167 O Costructig Super-Models to Ehace Failure Forecastig: A Comparative
More informationSolutions: Homework 3
Solutios: Homework 3 Suppose that the radom variables Y,...,Y satisfy Y i = x i + " i : i =,..., IID where x,...,x R are fixed values ad ",...," Normal(0, )with R + kow. Fid ˆ = MLE( ). IND Solutio: Observe
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 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 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 informationEstimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches
Iteratioal Joural of Mathematical Aalysis Vol. 8, 2014, o. 48, 2375-2383 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49287 Estimatig Cofidece Iterval of Mea Usig Classical, Bayesia,
More informationSurveying the Variance Reduction Methods
Iteratioal Research Joural of Applied ad Basic Scieces 2013 Available olie at www.irjabs.com ISSN 2251-838X / Vol, 7 (7): 427-432 Sciece Explorer Publicatios Surveyig the Variace Reductio Methods Arash
More informationProvläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE
TECHNICAL REPORT CISPR 16-4-3 2004 AMENDMENT 1 2006-10 INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE Amedmet 1 Specificatio for radio disturbace ad immuity measurig apparatus ad methods Part 4-3:
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 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 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 informationADVANCED SOFTWARE ENGINEERING
ADVANCED SOFTWARE ENGINEERING COMP 3705 Exercise Usage-based Testig ad Reliability Versio 1.0-040406 Departmet of Computer Ssciece Sada Narayaappa, Aeliese Adrews Versio 1.1-050405 Departmet of Commuicatio
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 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 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 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 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 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 informationControl chart for number of customers in the system of M [X] / M / 1 Queueing system
Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.
More information4.5 Multiple Imputation
45 ultiple Imputatio Itroductio Assume a parametric model: y fy x; θ We are iterested i makig iferece about θ I Bayesia approach, we wat to make iferece about θ from fθ x, y = πθfy x, θ πθfy x, θdθ where
More informationOverdispersion study of poisson and zero-inflated poisson regression for some characteristics of the data on lamda, n, p
Iteratioal Joural of Advaces i Itelliget Iformatics ISSN: 2442-6571 140 Overdispersio study of poisso ad zero-iflated poisso regressio for some characteristics of the data o lamda,, p Lili Puspita Rahayu
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 informationAAEC/ECON 5126 FINAL EXAM: SOLUTIONS
AAEC/ECON 5126 FINAL EXAM: SOLUTIONS SPRING 2015 / INSTRUCTOR: KLAUS MOELTNER This exam is ope-book, ope-otes, but please work strictly o your ow. Please make sure your ame is o every sheet you re hadig
More informationPREDICTION INTERVALS FOR FUTURE SAMPLE MEAN FROM INVERSE GAUSSIAN DISTRIBUTION
Qatar Uiv. Sci. J. (1991), 11: 19-26 PREDICTION INTERVALS FOR FUTURE SAMPLE MEAN FROM INVERSE GAUSSIAN DISTRIBUTION By MUHAMMAD S. ABU-SALIH ad RAFIQ K. AL-BAITAT Departmet of Statistics, Yarmouk Uiversity,
More informationLECTURE NOTES 9. 1 Point Estimation. 1.1 The Method of Moments
LECTURE NOTES 9 Poit Estimatio Uder the hypothesis that the sample was geerated from some parametric statistical model, a atural way to uderstad the uderlyig populatio is by estimatig the parameters of
More informationImproved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA
More informationProbability and MLE.
10-701 Probability ad MLE http://www.cs.cmu.edu/~pradeepr/701 (brief) itro to probability Basic otatios Radom variable - referrig to a elemet / evet whose status is ukow: A = it will rai tomorrow Domai
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 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 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 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 informationOn Posterior Analysis of Mixture of Two Components of Gumbel Type II Distribution
Iteratioal Joural of Probability ad Statistics 0, (4: 9-3 DOI: 0.593/j.ijps.0004.05 O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Navid Feroze,*, Muhammad Aslam Departmet
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 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 informationFirst Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise
First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >
More information