Varanasi , India. Corresponding author


 Silas Parker
 1 years ago
 Views:
Transcription
1 A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet of Statistics, Baaras Hidu Uiversity Varaasi005, Idia Departmet of Statistics, Rajastha Uiversity, Jaipur, Idia Correspodig author Abstract This paper proposes a geeral family of estimators for estimatig the populatio mea i systematic samplig i the presece of orespose adaptig the family of estimators proposed by Khoshevisa et al. (007). I this paper we have discussed the geeral properties of the proposed family icludig optimum property. The results have bee illustrated umerically by takig a empirical populatio cosidered i the literature. Keywords: Family of estimators, Auxiliary iformatio, Mea square error, Norespose, Systematic samplig.. Itroductio The method of systematic samplig, first studied by Madow ad Madow (944), is used widely i surveys of fiite populatios. Whe properly applied, the methods pocks up ay obvious or hidde stratificatio i the populatio ad thus ca be more precise tha radom samplig. I additio, systematic samplig is implemeted easily, thus reducig costs. I this variat of radom samplig, oly the first uit of the sample is selected at radom from the populatio. The subsequet uits are the selected by followig some defiite rule. Systematic samplig has bee cosidered i detail by Cochra (946) ad Lahiri (954). Reviews of the work doe i the field have bee give by ates (948) ad
2 Bucklad (95). The applicatio of systematic samplig to forest surveys has bee illustrated by Hasel (94), Fiey (948) ad Nair ad Bhargava (95). Use of systematic samplig i estimatig catch of fish has bee demostrated by Sukhatme et al. (958). The use of auxiliary iformatio has bee permeated the importat role to improve the efficiecy of the estimator. Kushwaha ad Sigh (989) suggested a class of almost ubiased ratio ad product type estimators for estimatig the populatio mea usig jackkife techique iitiated by Queouille (956). Afterward Baarasi et al. (993) ad Sigh ad Sigh (998) have proposed the estimators of populatio mea usig auxiliary iformatio i systematic samplig. Khoshevisa et al. (007) suggested a geeral family of estimators for estimatig the populatios mea usig kow values of some populatio parameters i simple radom samplig, give by a + b t = y (.) α(ax + b) + ( α)(a + b) where y ad x are the sample meas of study ad auxiliary variables respectively. is the populatio mea of auxiliary variable. a 0 ad b are either real umbers or fuctios of kow parameters of auxiliary variable. α ad g are the real umbers which are to be determied. Here we would like to metio that the choice of the estimator depeds o the availability ad values of the various parameter(s) used (for choice of the parameters a ad b refer to Sigh et al. (008) ad Sigh ad Kumar(0)). I this paper we have proposed a geeral family of estimators for estimatig the populatio mea i systematic samplig usig auxiliary iformatio i the presece of orespose followig Khoshevisa et al. (007). We have also derived the expressios for miimum mea square errors (MSE) of the family with respect to α. A comparative study is also carried out to compare the optimum estimators of the family with respect to usual mea estimator with the help of umerical data. g. Proposed Family of Estimators Let us suppose that a populatio cosists of N uits umbered from to N i some order ad a sample of size is to be draw such that N = k ( k is a iteger). Thus
3 there will be k samples each of uits ad we select oe sample from the set of k samples. Let ad be the study ad auxiliary variable with respective meas ad. Let us cosider yij(xij) be the th j observatio i the th i systematic sample uder study (auxiliary) variable ( i =...k : j =... ). Wwe assume that the orespose is observed oly o study variable ad auxiliary variable is free from orespose. Usig HaseHurwitz (946) techique of subsamplig of orespodets, the estimator of populatio mea, ca be defied as where y ad y y yh = (.) + y h are, respectively the meas based o respodet uits from the systematic sample of uits ad subsample of h uits selected from o respodet uits i the systematic sample. The estimator of populatio mea of auxiliary variable based o the systematic sample of size uits, is give by x ij j= x = ( i =... k ) (.) Obviously, y ad x are ubiased estimators. The variace expressio for the estimators ad ( x) where y ad x are, respectively, give by N V y = L { + ρ} S + WS V = { + ( ) ρ } S (.3) (.4) ρ ad ρ are the correlatio coefficiets betwee a pair of uits withi the systematic sample for the study ad auxiliary variables respectively. S ad respectively the mea squares of the etire group for study ad auxiliary variable. S are be the mea square of orespose group uder study variable, W is the orespose rate i the populatio ad L =. h S
4 Let us assume that the populatio mea is kow. Thus the usual ratio ad product estimators of the populatio mea uder orespose based o a systematic sample of size, ca be respectively defied as ad y y R = (.5) x y P = y x (.6) To obtai the biases ad mea square errors, we use large sample approximatio. y = ( + ) e 0 x = ( + ) e e such that E ( e 0 ) = ( ) ( ) E e 0 = ( ) e V y V( x) E = ad E ( e 0 e ) = where respectively. E = 0, ad L S = { + ρ } C + W, = { + } C, Cov y, x ρ = { + ρ } { + ρ} ρcc C ad C are the coefficiets of variatio of study ad auxiliary variables Expressig the equatios (.5) ad (.6) i terms of i expectatios the bias expressios of the estimators of by ad y R B = + y P { ρ}( Kρ ) C B = { + ρ} Kρ C e s ( 0,) i = ad takig y R ad y P, are respectively give (.7) (.8)
5 where, ρ = { + ρ} { + ρ } C ad K = ρ. C The mea square errors (MSE s) of y R N MSE = + ad P y MSE = + y R ad y P, are respectively, give by + ρ L C K C + W S { } ( ) ρ ρ N { } ( ) ρ ρ C + + Kρ C + L W S (.9) (.0) Motivated by Khoshevisa et al. (007), we ow defie a family of estimators of populatio mea based o a systematic sample of size i the presece of orespose as t g a + b = y (.) α( ax + b) + ( α)( a + b) This family ca geerate the orespose versios of a umber of estimators of populatio mea icludig the usual ratio ad product estimators o differet choices of a, b, α ad g.. Properties of Expressig t t a where λ =. a + b t i terms of e i s, we get ( + e )( + αλe ) g = y 0 (.) We assume that λ e < so that the right had side of the equatio (.) is expadable i terms of power series. Expadig the right had side of the equatio (.) ad eglectig the terms i e i s havig power greater tha two, we have
6 g(g + ) t = e0 gαλe + α λ e gαλe0e (.3) Takig expectatio both the sides of equatio (.3), we get the bias of t up to the first order of approximatio, as ( t ) B = { + ρ } ( g + ) N g C α λ gαλkρ (.4) Squarig both the sides of the equatio (.3) ad the takig the expectatio, we obtai the MSE of t up to the first order of approximatio, as ( t ) N MSE = +. Optimum Choice of α { } ( ) ρ ρ C + g α λ gαλρ K C I order to obtai the miimum MSE of respect to α ad equatig the derivative to zero, we get { + ρ }[ αg λ gλρ K] C L + ( ) t, we differetiate the MSE of The equatio (.6) provides the optimum values of α as W S (.5) t with = 0 (.6) ρ K α = gλ (.7) Puttig the optimum value of α from equatio (.7) ito the equatio (.5), we get the miimum MSE of t, as ( t ) mi MSE = + { ρ }[ C K C ] ρ L + ( ) W S (.8)
7 The miimum MSE of t, is same as the mea square error of the usual regressio estimator i systematic samplig uder orespose. 3. Empirical Study For umerical illustratio, we have cosidered the data give i Murthy (967, p. 33). The data are based o legth () ad timber volume () for 76 forest strips. Murthy (967, p.49) ad Kushwaha ad Sigh (989) reported the values of itraclass correlatio coefficiets ρ ad ρ approximately equal for the systematic sample of size 6 by eumeratig all possible systematic samples after arragig the data i ascedig order of strip legth. The details of populatio parameters are : N = 76, = 6, = 8.636, = , S = , S = , ρ = 0.870, 3 S = S 4 = Table shows the percetage relative efficiecy (PRE) of t (optimum) with respect to y for the differet choices of W ad L. Table : PRE of t (optimum) with respect to y W L PRE
8 Coclusio I this paper, we have proposed a geeral family of estimators of populatio mea i systematic samplig usig a auxiliary variable i the presece of orespose. The optimum property of the family has bee discussed. The study cocludes that the suggested family coverges to the usual regressio estimator of populatio mea i systematic samplig uder orespose if the parameter α attais its optimum value. From Table, it ca easily be see that the estimator t (optimum) performs always better tha the usual estimator y. It is also observed that the percetage relative efficiecy (PRE) of t (optimum) with respect to y decreases with icrease i orespose rate W as well as L. Refereces. Baarasi, Kushwaha, S.N.S. ad Kushwaha, K.S. (993): A class of ratio, product ad differece (RPD) estimators i systematic samplig, Microelectro. Reliab., 33, 4,
9 . Bucklad, W. R. (95): A review of the literature of systematic samplig, JRSS, (B), 3, Cochra, W. G. (946): Relative accuracy of systematic ad stratified radom samples for a certai class of populatio, AMS, 7, Fiey, D.J. (948): Radom ad systematic samplig i timber surveys, Forestry,, Hase, M. H. ad Hurwitz, W. N. (946) : The problem of orespose i sample surveys, Jour. of The Amer. Stat. Assoc., 4, Hasel, A. A. (94): Estimatio of volume i timber stads by strip samplig, AMS, 3, Khoshevisa, M., Sigh, R., Chauha, P., Sawa, N. ad Smaradache, F. (007): A geeral family of estimators for estimatig populatio mea usig kow value of some populatio parameter(s). Far East J. Theor. Statist.,, Kushwaha, K. S. ad Sigh, H.P. (989): Class of almost ubiased ratio ad product estimators i systematic samplig, Jour. Id. Soc. Ag. Statistics, 4,, Lahiri, D. B. (954): O the questio of bias of systematic samplig, Proceedigs of World Populatio Coferece, 6, Madow, W. G. ad Madow, L.H. (944): O the theory of systematic samplig, I. A. Math. Statist., 5, 4.. Murthy, M.N. (967): Samplig Theory ad Methods. Statistical Publishig Society, Calcutta.. Nair, K. R. ad Bhargava, R. P. (95): Statistical samplig i timber surveys i Idia, Forest Research Istitute, Dehradu, Idia forest leaflet, Queouille, M. H. (956): Notes o bias i estimatio, Biometrika, 43, Sigh, R ad Sigh, H. P. (998): Almost ubiased ratio ad product type estimators i systematic samplig, Questiio,,3, Sigh, R., Kumar, M. ad Smaradache, F. (008): Almost Ubiased Estimator for Estimatig Populatio Mea Usig Kow Value of Some Populatio Parameter(s). Pak. J. Stat. Oper. Res., 4() pp6376.
10 6. Sigh, R. ad Kumar, M. (0): A ote o trasformatios o auxiliary variable i survey samplig. MASA, 6:, Sukhatme, P. V., Paes, V. G. ad Sastry, K. V. R. (958): Samplig techiques for estimatig the catch of sea fish i Idia, Biometrics, 4, ates, F. (948): Systematic samplig, Philosophical Trasactios of Royal Society, (A), 4,
Some Exponential RatioProduct Type Estimators using information on Auxiliary Attributes under Second Order Approximation
; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 09770)]; ISSN 0975556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial RatioProduct
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 informationImproved exponential estimator for population variance using two auxiliary variables
OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 66767 ISSN 5657, ISBN 979735550, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables
More informationEstimation of the Population Mean in Presence of NonResponse
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 NoRespose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,
More informationModified Ratio Estimators Using Known Median and CoEfficent of Kurtosis
America Joural of Mathematics ad Statistics 01, (4): 95100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad CoEfficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry
More informationA Family of Efficient Estimator in Circular Systematic Sampling
olumbia Iteratioal Publishig Joural of dvaced omputig (0) Vol. o. pp. 668 doi:0.776/jac.0.00 Research rticle Famil of Efficiet Estimator i ircular Sstematic Samplig Hemat K. Verma ad Rajesh Sigh * Received
More informationA Generalized Class of Unbiased Estimators for Population Mean Using Auxiliary Information on an Attribute and an Auxiliary Variable
Iteratioal Joural of Computatioal ad Applied Mathematics. ISSN 894966 Volume, Number 07, pp. 8 Research Idia ublicatios http://www.ripublicatio.com A Geeralized Class of Ubiased Estimators for opulatio
More informationOn ratio and product methods with certain known population parameters of auxiliary variable in sample surveys
Statistics & Operatios Research Trasactios SORT 34 JulyDecember 010, 157180 ISSN: 169681 www.idescat.cat/sort/ Statistics & Operatios Research c Istitut d Estadística de Cataluya Trasactios sort@idescat.cat
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 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. Crosssectioal data. 2. Time series data.
More informationEstimation of Population Mean in Presence of NonResponse in Double Sampling
J. Stat. Appl. Pro. 6, No. 2, 345353 (2017) 345 Joural of Statistics Applicatios & Probability A Iteratioal Joural http://dx.doi.org/10.18576/jsap/060209 Estimatio of Populatio Mea i Presece of NoRespose
More informationNew Ratio Estimators Using Correlation Coefficient
New atio Estimators Usig Correlatio Coefficiet Cem Kadilar ad Hula Cigi Hacettepe Uiversit, Departmet of tatistics, Betepe, 06800, Akara, Turke. emails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr
More informationEstimation of Gumbel Parameters under Ranked Set Sampling
Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 112014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com
More informationEstimation of Population Ratio in PostStratified Sampling Using Variable Transformation
Ope Joural o Statistics, 05, 5, 9 Published Olie Februar 05 i SciRes. http://www.scirp.org/joural/ojs http://dx.doi.org/0.436/ojs.05.500 Estimatio o Populatio Ratio i PostStratiied Samplig Usig Variable
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 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 informationUnbiased Estimation. February 712, 2008
Ubiased Estimatio February 72, 2008 We begi with a sample X = (X,..., X ) of radom variables chose accordig to oe of a family of probabilities P θ where θ is elemet from the parameter space Θ. For radom
More informationThe Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. BetaBinomial Distribution
Iteratioal Mathematical Forum, Vol. 8, 2013, o. 26, 12631277 HIKARI Ltd, www.mhikari.com http://d.doi.org/10.12988/imf.2013.3475 The Samplig Distributio of the Maimum Likelihood Estimators for the Parameters
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 5766 ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationActivity 3: Length Measurements with the FourSided Meter Stick
Activity 3: Legth Measuremets with the FourSided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a foursided meter
More informationIt should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.
Chapter 10 Variace Estimatio 10.1 Itroductio Variace estimatio is a importat practical problem i survey samplig. Variace estimates are used i two purposes. Oe is the aalytic purpose such as costructig
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 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 informationGeneralized Exponential Type Estimator for Population Variance in Survey Sampling
Revista Colombiaa de Estadística Juio 2014, volume 37, o. 1, pp. 211 a 222 Geeralized Expoetial Type Estimator for Populatio Variace i Survey Samplig Estimadores tipo expoecial geeralizado para la variaza
More informationOn an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, HigashiOsaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol DiscreteEvent System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol DiscreteEvet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationo <Xln <X2n <... <X n < o (1.1)
Metrika, Volume 28, 1981, page 257262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3337 HIKARI Ltd, www.mhikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationGoodnessOfFit For The Generalized Exponential Distribution. Abstract
GoodessOfFit 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 information71. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
71 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7 Sectio 1. Samplig Distributio 73 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture  9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
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 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 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 informationSampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals
Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationSTATISTICAL method is one branch of mathematical
40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati
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 informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 00900 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Twoparameter 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 informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 705041010,
More informationA LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!
A LARGER SAMLE SIZE IS NOT ALWAYS BETTER!!! Nagaraj K. Neerchal Departmet of Mathematics ad Statistics Uiversity of Marylad Baltimore Couty, Baltimore, MD 2250 Herbert Lacayo ad Barry D. Nussbaum Uited
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More informationResearch Article Health Monitoring for a Structure Using Its Nonstationary Vibration
Hidawi Publishig Corporatio Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu
More informationARIMA Models. Dan Saunders. y t = φy t 1 + ɛ t
ARIMA Models Da Sauders I will discuss models with a depedet variable y t, a potetially edogeous error term ɛ t, ad a exogeous error term η t, each with a subscript t deotig time. With just these three
More informationLinear Regression Models
Liear Regressio Models Dr. Joh MellorCrummey Departmet of Computer Sciece Rice Uiversity johmc@cs.rice.edu COMP 528 Lecture 9 15 February 2005 Goals for Today Uderstad how to Use scatter diagrams to ispect
More information3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.
3/3/04 CDS M Phil Old Least Squares (OLS) Vijayamohaa Pillai N CDS M Phil Vijayamoha CDS M Phil Vijayamoha Types of Relatioships Oly oe idepedet variable, Relatioship betwee ad is Liear relatioships Curviliear
More informationChapter 1 (Definitions)
FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TINspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform largesample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationSummary: CORRELATION & LINEAR REGRESSION. GC. Students are advised to refer to lecture notes for the GC operations to obtain scatter diagram.
Key Cocepts: 1) Sketchig of scatter diagram The scatter diagram of bivariate (i.e. cotaiig two variables) data ca be easily obtaied usig GC. Studets are advised to refer to lecture otes for the GC operatios
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 9732675 Volume 12, Number 4 (217), pp. 791796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E Bayesia Method of Estimatio of Parameter
More informationSession 5. (1) Principal component analysis and KarhunenLoève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad KarhueLoève trasformatio Topic 2 of this course explais the image
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
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 informationSolutions to Odd Numbered End of Chapter Exercises: Chapter 4
Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd Numbered Ed of Chapter Exercises: Chapter 4 (This versio July 2, 24) Stock/Watso  Itroductio to Ecoometrics
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 I1 Part I  203A A radom variable X is distributed with the margial desity: >
More informationThe Sample Variance Formula: A Detailed Study of an Old Controversy
The Sample Variace Formula: A Detailed Study of a Old Cotroversy Ky M. Vu PhD. AuLac Techologies Ic. c 00 Email: kymvu@aulactechologies.com Abstract The two biased ad ubiased formulae for the sample variace
More informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationEstimation of the ratio, product and mean using multi auxiliary variables in the presence of nonresponse
Chilea Joural of Statistics Vol. 5, No. 1, April 014, 49 7 Samplig Theory Research Paper Estimatio of the ratio, prouct a mea usig multi auxiliary variables i the presece of orespose Suil Kumar Alliace
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationTHE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS
R775 Philips Res. Repts 26,414423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS  DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
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 informationCURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:
CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION
More informationSubject: Differential Equations & Mathematical ModelingIII
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical ModeligIII Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationMOMENTMETHOD ESTIMATION BASED ON CENSORED SAMPLE
Vol. 8 o. Joural of Systems Sciece ad Complexity Apr., 5 MOMETMETHOD ESTIMATIO BASED O CESORED SAMPLE I Zhogxi Departmet of Mathematics, East Chia Uiversity of Sciece ad Techology, Shaghai 37, Chia. Email:
More informationStability Analysis of the Euler Discretization for SIR Epidemic Model
Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we
More informationOn Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers
Pujab Uiversity Joural of Mathematics ISSN 10162526) Vol. 502)2018) pp. 1119 O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers P. Nasiri Departmet of Statistics, Uiversity
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. 4355 Bagladesh ISSN 256422 X EDGEORTH SIZE CORRECTED, AND TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR Zahirul Hoque Departmet of Statistics
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationDimensionfree PACBayesian bounds for the estimation of the mean of a random vector
Dimesiofree PACBayesia bouds for the estimatio of the mea of a radom vector Olivier Catoi CREST CNRS UMR 9194 Uiversité Paris Saclay olivier.catoi@esae.fr Ilaria Giulii Laboratoire de Probabilités et
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 11
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract We will itroduce the otio of reproducig kerels ad associated Reproducig Kerel Hilbert Spaces (RKHS). We will cosider couple
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More information(all terms are scalars).the minimization is clearer in sum notation:
7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oedimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationREGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
More informationHow to Maximize a Function without Really Trying
How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet
More informationSymmetric Division Deg Energy of a Graph
Turkish Joural of Aalysis ad Number Theory, 7, Vol, No 6, 9 Available olie at http://pubssciepubcom/tat//6/ Sciece ad Educatio Publishig DOI:69/tat6 Symmetric Divisio Deg Eergy of a Graph K N Prakasha,
More informationMonte Carlo Integration
Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce
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 informationSome New Iterative Methods for Solving Nonlinear Equations
World Applied Scieces Joural 0 (6): 870874, 01 ISSN 1818495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida
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 AlHa Ebrahem
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationSEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
More informationMechanical Efficiency of Planetary Gear Trains: An Estimate
Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia Email: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept.
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationBernoulli numbers and the EulerMaclaurin summation formula
Physics 6A Witer 006 Beroulli umbers ad the EulerMaclauri summatio formula I this ote, I shall motivate the origi of the EulerMaclauri summatio formula. I will also explai why the coefficiets o the right
More information