Akaike Information Criterion and Fourth-Order Kernel Method for Line Transect Sampling (LTS)
|
|
- Dominick Hubbard
- 5 years ago
- Views:
Transcription
1 Appl. Math. If. Sci. 10, No. 1, ( Applied Mathematics & Iformatio Scieces A Iteratioal Joural Akaike Iformatio Criterio ad Fourth-Order Kerel Method for Lie Trasect Samplig (LTS Ali Algari 1, ad Ahmad Almutlg 2 1 Statistics Departmet, Faculty of Sciece, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia 2 Mathematics Departmet, Faculty of Sciece, Qassim Uiversity, Qassim, Saudi Arabia Received: 7 Mar. 2015, Revised: 20 May 2015, Accepted: 12 Aug Published olie: 1 Ja Abstract: Parametric ad oparametric approaches were used to fit lie trasect data. Differet parametric detectio fuctios are suggested to compute the smoothig parameter of the oparametric fourth-order kerel estimator. Amog the differet cadidate parametric detectio fuctios, the researcher suggests to use Akaike Iformatio Criterio (AIC to select the most appropriate oe of them to fit lie trasect data. More specifically, four differet parametric models are cosidered i this research. Where as two models were take to satisfy the shoulder coditio assumptio, the other two do ot. Oce the appropriate model is determied, it ca be used to select the smoothig parameter of the oparametric fourth-order kerel estimator. As the researcher expected, this techique leads to improve the performaces of the fourth-order kerel estimator. For a wide rage of target desities, a simulatio study is performed to study the properties of the proposed estimators which show the superiority of the resultig proposed fourth-order kerel estimator over the classical kerel estimator i most cosidered cases. Keywords: Akaike iformatio criterio, Maximum likelihood estimatio, Fourth-order kerel estimator, Expoetial model, Halformal model, Weighted Expoetial model, Reversed Logistic model. 1 Itroductio The smoothig parameter h of the fourth-order kerel estimator plays a vital role i its performace. Its performace becomes acceptable compared to the classical kerel estimator whe the method of determiig the smoothig parameter is chose correctly. I other words, there is a scope to improve the performaces of the fourth-order kerel estimator by cosiderig a suitable parametric method to determie h. mor survey give i Che [1, Clie ad Hart [2, Eidous [, Eidous ad bal-masri [4, Cowlig ad Hall [5, Eidous [6, Eidous ad Alshakhatreh [7, Gasser ad Muller [8, Gasser et al. [9, Eidous [10, Mack ad Quag [11, Wad ad Joes [12, Zhag ad Karuamui [1, Eidous [14. The four parametric models will be agai cosidered i this paper. However, choosig a suitable parametric model for the data is ot depedet o guesswork, istead the (AIC will be used for this purpose i this paper. To ivestigate the statistical properties of the resultat estimator, a simulatio study is performed lately i research Akaike Iformatio Criterio: Burham ad Aderso [16 illustrated that the (AIC is a quatitative method to select a suitable model for data. Iformatio theory ad a extesio of the maximum likelihood priciple usig AIC give i Akaike [17. They defied the AIC as: AIC= 2logL+2q, (1 where log L is the atural logarithm of the likelihood fuctio evaluated at the MLE of the model parameters ad q is the umber of parameters i the model. I this paper, the formulas for AIC will be computed for four parametric models. These parametric models are: Expoetial model f(x= 1 θ exp ( x θ, x>0, (2 Half-ormal model ( 1 x 2 f(x= exp 2Πσ 2 2σ 2, x>0, ( Correspodig author ahalgari@kau.edu.sa Natural Scieces Publishig Cor.
2 268 A. Algari, A. Almutlg: Akaike iformatio criterio... Weighted Expoetial model f(x= 2θ exp( θx(2 exp( θx, x>0, (4 log Reversed Logistic model 2θexp( θx, x>0, (5 log(1+2exp( θx Also, we derived the smoothig parameter of the fourthorder kerel estimator based o each model separately. If X 1, X 2,..., X is a radom sample of size represetig the perpedicular distaces ad followig oe of the above pdf models f(x The correspodig smoothig parameters for each model are, h exp = (θ1 ( 1, where( ˆθ 1 =x (6 h half = (σ( 1 9,where ˆσ = x2 i h weight = ( 1 θ ( 1 7,where ˆθ = 7 6x h Rever 1 =.5400( (,where 1 ˆθ 4 = (9 θ 4 x I the followig sectio, we derived the AIC that correspods to each model. 2 Akaike Iformatio Criterio for Some Parametric Models For local likelihood desity estimatio i lie trasect samplig see Barabesi [18 ad Barabesi [19 Let X 1, X 2,..., X be a radom sample of size perpedicular distaces followig the expoetial model. The likelihood fuctio is ( 1 1 L(θ 1 = exp( θ 1 θ 1 x i By takig the atural logarithm of both sides, we obtai logl(θ 1 = 1 θ 1 (7 (8 (10 x i logθ 1 (11 Multiply both sides of the last equatio by ( -2 we obtai 2logL(θ 1 = 2 θ 1 x i + 2logθ 1 (12 Sice there is oe parameter that eeds to be estimated, q=1. Therefore, the AIC for the expoetial model is Note that xis the ML estimator for θ 1. I the same way we ca derive the AIC for the other three models, which are give below: * For the Half-ormal model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are ˆσ L ( σ 2 ( ( 1 = exp 2Πσ 2 1 2σ 2 logl ( σ 2 = log 2Π logσ 1 2σ 2 This gives the AIC by x 2 i AIC half = 2log2+log [ 2π [ ˆσ = 2log2+log 2π x2 i (14 x 2 i (15 (16 where x2 i is the ML estimator of σ 2. * For the weighted expoetial model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are exp( L(θ = ( 2θ θ i (2 exp( θ x, x logl(θ =log( 2θ θ x i +log[ (2 exp( θ x, The AIC is AIC weight = 2log( 2 ˆθ + 2 ˆθ 2 log( 2 exp ( ˆθ x + 2. (17 (18 (19 where ˆθ is the ML estimator of θ, which does ot exist i closed form ad a umerical method such as the Newto- Raphso method is eeded to obtai the value of θ. * Fially, for the reversed logistic model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are L(θ 4 = ( 2θ4 log exp( θ 4 x i logl(θ 4 =log( 2θ4 log = log[ The AIC is (1+2exp( θ 4 x θ 4 x i (1+2exp( θ 4 x. (20 (21 AIC exp = 2ˆθ 1 x i+ 2log ˆθ = 2(1+logx+2. (1 AIC Revr = 2log ( 2 ˆθ 4 + 2log(+ 2 ˆθ 4 x 2 log( 1+2exp ( ˆθ 4 x + 2. (22 Natural Scieces Publishig Cor.
3 Appl. Math. If. Sci. 10, No. 1, (2016 / where ˆθ 4 is the ML estimator of θ 4. Agai, a closed form for the value of ˆθ 4 does ot exist i closed form ad a umerical method is eeded to obtai it. Simulatio Study Desig we performed a simulatio study to ivestigate the properties of the proposed estimator. This estimator is the fourth-order kerel whose parameter h ca be computed by cosiderig ay reasoable parametric model. Oe of the four parametric models metioed i Sectio (.2 of this chapter is ow used to compute h. The AIC is a criterio for choosig the most appropriate model based o the data. The resultig estimator is deoted by ˆf(0. The proposed estimator ˆf(0 is computed as follows: 1.Simulate the perpedicular distaces X 1, X 2,..., X from oe of the target models metioed i Sectio ( For the simulated perpedicular distaces, compute the value of h that correspods to the expoetial, half-ormal, weighted expoetial, ad reversed logistic models. The values of these smoothig parameters are deoted by h exp, h hal f, h weight, ad h Rever, respectively..compute the AIC for the expoetial, half-ormal, weighted expoetial, ad reversed logistic models. 4.Select the model with the smallest AIC as a referece model to compute the value of h based o the selected model. The formula for computig h for each model are give i Sectio (.2. 5.Compute the value of the fourth-order kerel estimator based o the selected h of step (4 ad based o the perpedicular distaces X 1, X 2,..., X. The data are simulated from the 16 models that were give i Sectio (2.6 with the same values of, β, ad ϖ. For a simple compariso, the results of a classical kerel estimator ˆf k (0 are also preseted. The relative bias (RB, relative mea error (RMX, ad the efficiecy (EFF of f k (0 with respect to ˆf k (0 are give i Tables ( Note that, EFF= MSE( ˆf k (0 MSE( f k (0 (2 as illustrated i Sectio (2. 4 Results Depedig o the simulatio results of Tables (.1-.4, several coclusios ca be draw by ispectig the results with regard to,, ad EFF. 1.It is obvious that the RBs that are associated with the proposed estimators fk (0 are smaller tha (i their magitude the correspodig RBs that are associated with the classical kerel estimator ˆf k (0. 2.The RMEs for the two estimators fk (0 ad ˆf k (0 decrease whe the sample size icreases. This is a strog idicatio of the cosistecy of these estimators..compared to fk (0, the performace of the classical kerel estimator seems to be reasoable for the EP model with β =2 at which the smoothig parameter of ˆf k (0 is computed uder this model (i.e., uder the half- ormal model. For the two cases, the HR model with (β,ω=(2.5,10 ad for the BE model with large value of β, the classical kerel estimator performs better tha fk (0. However, the efficiecies i these cases are aroud 1, which idicates that the performaces of the two estimators are similar. 4.With the exceptio of the three cases metioed i (, the performace of fk (0 is better tha that of ˆf k (0. I fact, out of the 16 target models, there are 12 cases i which fk (0 beats ˆf k (0. 5.I geeral, the performace of the fourth-order kerel estimator fk (0 is very good for almost all cosidered target models. I additio, for the cases i which ˆf k (0 seems to be better tha fk (0, the gai of efficiecy is ot sigificat compared with the large efficiecies of fk (0 over ˆf k (0 i certai cases. For example, the efficiecy of fk (0 for the model HR with (β,ω,=(1,5,200 is 4.086, which meas that the performace of ˆf k (0 becomes the same as that of fk (0 for a sample size of approximately = Fially, we ca say that Tables (.1-.4 demostrate clearly that there is a sigificat improvemet whe applyig the estimator fk (0 istead of the classical kerel estimator ˆf k (0. Refereces [1 Che, S. X. (1996. A kerel estimate for the desity of a biological populatio by usig lie trasect samplig. Applied Statistics, 45, [2 Clie, D. B. H., & Hart, J. D. (1991. Kerel estimatio of desities of discotiuous derivatives. Statistics, 22, [ Eidous, O. M., (2009. Kerel method startig with half-ormal detectio fuctio for lie trasect desity estimatio. Commu. i Statist.-Theory ad Methods. 8, [4 Eidous, O. M.,& Al-Masri, A. (2010. Fourth-order kerel method usig lie trasect samplig. Advaces ad Applicatios i Statistics, 19, [5 Cowlig, A., & Hall, P. (1996. O pseudodata method for removig boudary effects i kerel desity estimatio. Joural of the Royal Statistical Society, Ser. B, 58, [6 Eidous, O. M., (2012. A ew kerel estimator for abudace usig lie trasect samplig without the shoulder coditio. Joural of the Koria Statistical Society, 41, [7 Eidous, O. M., & Alshakhatreh, M. K. (2011. Asymptotic ubiased kerel estimator for lie trasect samplig. Natural Scieces Publishig Cor.
4 270 A. Algari, A. Almutlg: Akaike iformatio criterio... Table.1: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from expoetial power (EP detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = Table.2: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from hazard-rate (HR detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = Table.: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from the beta (BE family detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = ˆf k (0 β = fk (0 ω = Table.4: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from the beta (BE family detectio fuctio. Parameter Estimator =50 =100 =200 β = 0.6 ˆf k (0 b= fk (0 ω = ˆf k (0 b= fk (0 ω = ˆf k (0 b= fk (0 ω = ˆf k (0 b= fk (0 ω = Natural Scieces Publishig Cor.
5 Appl. Math. If. Sci. 10, No. 1, (2016 / Commu. i Statist.-Theory ad Methods, 40(24, [8 Gasser, T., & Muller, H. G. (1979. Kerel estimatio of regressio fuctios. I: Gasser, T., & Roseblatt, M. (Eds, Smoothig techiques for curve estimatio. Lecture otes i mathematics, Vol. 757, Sprig, Heidelberg, [9 Gasser, T., Muller, H. G., & Mammitzsch, V. (1985. Kerels for oparametric curve estimatio. Joural of the Royal Statistical Society, Ser. B, 47, [10 Eidous, O. M. (2005a. O improvig kerel estimator usig lie trasect samplig. Commu. I statistic.- Theory ad Methods, 4, [11 Mack, Y. P. & Quag, P. X. (1998. Kerel methods i lie ad poit trasect samplig. Biometrics, 54, [12 Wad, M. P. & Joes, M. C. (1995. Kerel smoothig. Lodo: Chapma & Hall. [1 Zhag, S., & Karuamui, R. J. (1998. O kerel desity estimatio ear edpoits. Joural of Statistical Plaig ad Iferece, 70, [14 Eidous, O. M., (2011a. Variable locatio kerel method usig lie trasect samplig. Eviromets, 22, [15 Ababeh, F. ad Eidous, O. M. (2012. A weighted expoetial detectio fuctio model for lie trasect data. Joural of Moder Applied Statistical Methods, 11(1, [16 Burham, K. P., & Aderso, D. R. (1976. Mathematical models for oparametric ifereces from lie trasect data. Biometrics, 2, [17 Akaike, H. (197. Iformatio theory ad a extesio of the maximum likelihood priciple. A iteratioal Symposium of Iformatio Theory, 2ded, Budapest, Hugary, [18 Barabesi, L. (2000. Local likelihood desity estimatio i lie trasect samplig. Evirometrics, 11, [19 Barabesi, L. (2001. Local parametric desity estimatio method i lie trasect samplig. Metro, LIX, Ahmad Almutlg Takig his Diploma i Risk Maagemet from Europea Busiess (Learig Academy i the UK [October December 2010.I Dec.2014 takig the master i Statistics from Kig Abdulaziz Uiversity, Jeddah. Thesis Topic Akaike Iformatio Criterio ad Fourth-Order Kerel Method for Lie Trasect Samplig. Ali Algari has obtaied his PhD degree at 201 from School of Mathematics ad Applied Statistics, Wollogog Uiversity, Australia. He is a Ivestigator ad Data Aalyst at ceter of survey methodology ad data aalysis,wollogog. Australia. He is a member, Saudi Associatio of Mathematics ad Applied Statistics. He is presetly employed as assistat professor i Statistics Departmet, Faculty of Sciece, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia. Natural Scieces Publishig Cor.
Double 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 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 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 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 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 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 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 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 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 informationReview Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn
Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom
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 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 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 informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
More informationAsymptotic distribution of products of sums of independent random variables
Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege
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 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 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 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 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 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 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 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 informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 5
CS434a/54a: Patter Recogitio Prof. Olga Veksler Lecture 5 Today Itroductio to parameter estimatio Two methods for parameter estimatio Maimum Likelihood Estimatio Bayesia Estimatio Itroducto Bayesia Decisio
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 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 informationECE 901 Lecture 14: Maximum Likelihood Estimation and Complexity Regularization
ECE 90 Lecture 4: Maximum Likelihood Estimatio ad Complexity Regularizatio R Nowak 5/7/009 Review : Maximum Likelihood Estimatio We have iid observatios draw from a ukow distributio Y i iid p θ, i,, where
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 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 informationECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors
ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic
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 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 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 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 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 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 informationBecause it tests for differences between multiple pairs of means in one test, it is called an omnibus test.
Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal
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 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 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 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 informationModified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
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 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 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 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 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 informationECE 901 Lecture 13: Maximum Likelihood Estimation
ECE 90 Lecture 3: Maximum Likelihood Estimatio R. Nowak 5/7/009 The focus of this lecture is to cosider aother approach to learig based o maximum likelihood estimatio. Ulike earlier approaches cosidered
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 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 informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationOn stratified randomized response sampling
Model Assisted Statistics ad Applicatios 1 (005,006) 31 36 31 IOS ress O stratified radomized respose samplig Jea-Bok Ryu a,, Jog-Mi Kim b, Tae-Youg Heo c ad Chu Gu ark d a Statistics, Divisio of Life
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
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 informationImproved Ratio Estimators of Population Mean In Adaptive Cluster Sampling
J. Stat. Appl. Pro. Lett. 3, o. 1, 1-6 (016) 1 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.18576/jsapl/030101 Improved Ratio Estimators of Populatio
More informationLecture 13: Maximum Likelihood Estimation
ECE90 Sprig 007 Statistical Learig Theory Istructor: R. Nowak Lecture 3: Maximum Likelihood Estimatio Summary of Lecture I the last lecture we derived a risk (MSE) boud for regressio problems; i.e., select
More informationSimulation. Two Rule For Inverting A Distribution Function
Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump
More information10. Comparative Tests among Spatial Regression Models. Here we revisit the example in Section 8.1 of estimating the mean of a normal random
Part III. Areal Data Aalysis 0. Comparative Tests amog Spatial Regressio Models While the otio of relative likelihood values for differet models is somewhat difficult to iterpret directly (as metioed above),
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 informationAClassofRegressionEstimatorwithCumDualProductEstimatorAsIntercept
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 15 Issue 3 Versio 1.0 Year 2015 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic.
More informationCastiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
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 informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationLast Lecture. Wald Test
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 22 Hyu Mi Kag April 9th, 2013 Is the exact distributio of LRT statistic typically easy to obtai? How about its asymptotic distributio? For testig
More informationEECS564 Estimation, Filtering, and Detection Hwk 2 Solns. Winter p θ (z) = (2θz + 1 θ), 0 z 1
EECS564 Estimatio, Filterig, ad Detectio Hwk 2 Sols. Witer 25 4. Let Z be a sigle observatio havig desity fuctio where. p (z) = (2z + ), z (a) Assumig that is a oradom parameter, fid ad plot the maximum
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationThis exam contains 19 pages (including this cover page) and 10 questions. A Formulae sheet is provided with the exam.
Probability ad Statistics FS 07 Secod Sessio Exam 09.0.08 Time Limit: 80 Miutes Name: Studet ID: This exam cotais 9 pages (icludig this cover page) ad 0 questios. A Formulae sheet is provided with the
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 informationMaximum Likelihood Estimation and Complexity Regularization
ECE90 Sprig 004 Statistical Regularizatio ad Learig Theory Lecture: 4 Maximum Likelihood Estimatio ad Complexity Regularizatio Lecturer: Rob Nowak Scribe: Pam Limpiti Review : Maximum Likelihood Estimatio
More informationPOWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES*
Kobe Uiversity Ecoomic Review 50(2004) 3 POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* By HISASHI TANIZAKI There are various kids of oparametric
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
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 informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationCentral limit theorem and almost sure central limit theorem for the product of some partial sums
Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics
More informationElement sampling: Part 2
Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig
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 informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationStopping oscillations of a simple harmonic oscillator using an impulse force
It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 (ISSN: 2347-2529) IJAAMM Joural homepage: www.ijaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Stoppig oscillatios of a simple harmoic
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 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 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 informationEfficient GMM LECTURE 12 GMM II
DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet
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 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 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 informationCHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION
CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION 4. Itroductio Numerous bivariate discrete distributios have bee defied ad studied (see Mardia, 97 ad Kocherlakota ad Kocherlakota, 99) based o various methods
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 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 informationA new distribution-free quantile estimator
Biometrika (1982), 69, 3, pp. 635-40 Prited i Great Britai 635 A ew distributio-free quatile estimator BY FRANK E. HARRELL Cliical Biostatistics, Duke Uiversity Medical Ceter, Durham, North Carolia, U.S.A.
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 informationKernel density estimator
Jauary, 07 NONPARAMETRIC ERNEL DENSITY ESTIMATION I this lecture, we discuss kerel estimatio of probability desity fuctios PDF Noparametric desity estimatio is oe of the cetral problems i statistics I
More informationAbstract. Ranked set sampling, auxiliary variable, variance.
Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we
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 informationUniformly Consistency of the Cauchy-Transformation Kernel Density Estimation Underlying Strong Mixing
Appl. Mat. If. Sci. 7, No. L, 5-9 (203) 5 Applied Matematics & Iformatio Scieces A Iteratioal Joural c 203 NSP Natural Scieces Publisig Cor. Uiformly Cosistecy of te Caucy-Trasformatio Kerel Desity Estimatio
More informationIntroductory statistics
CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key
More information