COMPARISON OF RATIO ESTIMATORS WITH TWO AUXILIARY VARIABLES K. RANGA RAO. College of Dairy Technology, SPVNR TSU VAFS, Kamareddy, Telangana, India
|
|
- Maria Malone
- 6 years ago
- Views:
Transcription
1 COMPARISON OF RATIO ESTIMATORS WITH TWO AUXILIARY VARIABLES K. RANGA RAO College of Dairy Technology, SPVNR TSU VAFS, Kamareddy, Telangana, India Abstract: Many estimators of the population parameters are constructed using known auxiliary variables. The classical well known ratio estimator is one of them. Ratio estimators are frequently employed in sample surveys when estimating the population mean Y of a variate Y with the help of the known population means of a correlated auxiliary variables. Various improvements of this ratio estimator have been considered by many authors. Some estimators use two auxiliary variables. Some other use groups of estimators are composite. In case two auxiliary variables are known, the ratio-cum-product estimator may be used. It is well known that when the auxiliary information is to be used at the estimation stage, the ratio, product and regression estimators are widely utilied in many situations. There are no analytical procedures to compare the ratio estimators, since their mean square errors are approximated up to some extent. Theoretically, it is hard to compare the bias, mean square error, skewness and kurtosis of the estimators over the other estimators. However, we can compute these of the estimators using Monte Carlo simulation. This method is suitable when the theoretical comparisons fail. In this paper, some ratio estimators with two auxiliary variables available in literature are reviewed and their efficiencies are compared by simulation for different distributions with known correlation coefficients. The results show that the simulation method is more appropriate when there is no closed expression for the bias and mean squared error of the estimators. Keywords: Ratio estimator, Auxiliary variable and Simulation. InterStat 015 Page 1
2 1. Introduction In sample surveys it is usual to make use of information on auxiliary variables to obtain improved designs and more efficient estimators. It is well known that when the auxiliary information is to be used at the estimation stage, the ratio, product and regression estimators are widely utilied in many situations. Theoretically, it has been established that, in general, the regression estimator is more efficient than the ratio and product estimators except when the regression line of the character under study on the auxiliary character passes through the neighbourhood of the origin. In this case the efficiency of the estimators is almost equal. However, due to the stronger intuitive appeal, statisticians are more inclined towards the use of ratio and product estimators. Perhaps that is why an extensive work has been done in the direction of improving the performance of these estimators. In sampling literature, when a single auxiliary variable is available, many estimators have been proposed which, under some realistic conditions, are more efficient than the sample mean, the ratio and product estimators and as efficient as the regression estimator in the optimum case. Suppose the population mean of x, vi. X is known. Suppose further that the random sample of, say, sie n taken from the population is enumerated for both x and y, if x and y are the sample means of x and y, then as an estimator of the population mean, Y, of y, we may consider y R, rather than y itself, where y R y = X = RX ˆ, (1.1) x where y R ˆ = r 0 =, (1.) x InterStat 015 Page
3 being the estimator of the population ratio Y X. Although this estimator y R is biased, one can see that it is almost unbiased for large n and that the variance may, indeed, be smaller than the variance of y in some situations. This paper is concerned with the problem of estimating the population ratio using two auxiliary variables. When two or more auxiliary variables are available many estimators may be defined linking together different estimators, such as ratio, product or regression, each one utiliing a single variable. Olkin (1958) suggested the use of information on more than one supplementary character, positively correlated with the variable under study, using a linear combination of ratio estimators based on each auxiliary variable separately. Singh (1967) gave a multivariate expression of product estimator, while Raj (1965) suggested a method of using multi-auxiliary variables considering a linear combination of single difference estimators. Many other contributions are present in sampling literature when two auxiliary variables are available some estimators obtained from the ratio cum product estimators developed by Singh (1965, 1967). estimators Estimation of R can also be made in a different way by employing alternative Λ Y and Λ X respectively in terms of an auxiliary variable, having high degree of associations with y and x. For example, in a labour force survey, the survey statistician may be interested in to estimate the proportion of female workers employed in industry, number of female workers and female population for a sampling unit as y, x and respectively. In this context, possibility of creating estimators for R has been indicated by Tripathi (1980) who defined an estimator by considering difference estimators for Y and X in terms of the auxiliary variable. In their text, Sarndal, Swenson and Wretman (1991) also pointed out that if the presence of such an auxiliary variable improves the accuracy of one or both means, efficiency can also gained in the estimation of R. The authors encouraged the use of InterStat 015 Page 3
4 regression estimators for this purpose. However, this approach encourages us to estimate R by a generalied estimator of the form Rˆ = Y ˆ X ˆ which is able to produce many estimators.. Some Ratio Estimators with Two Auxiliary Variables Let P = {1,,...N} be a finite population of N elements, Y be the study variable and X and Z two auxiliary variables. Assume that the mean,y, of the variable of interest is unknown. Let a sample of sie n be drawn from the population according to the simple random sampling without replacement (srswor) and let y, x and be the sample means of the variables Y, X and Z. Supposing that the population mean Z of is known, let us consider Λ Y = y Λ x [ 1+ θ ( c c )]Z and X = 1+ ( c c ) [ θ x ]Z as Tin s (1965) type AUR estimators for Y and X respectively, where 1 1 θ =, c = s /, cx = s x / x and n N c = s /.Then Λ R reduces to an estimator defined by r 1 = r 0 [ 1+ θ ( c c )] [ 1+ θ ( c c )] x (.1) another estimator [ 1+ θ c ] r = r0, for R. (.) [ 1+ θ c ] x InterStat 015 Page 4
5 In many surveys, information on auxiliary variates which are highly correlated with the variable under study is readily available and can be used for improving sampling design. Ratio estimators and linear regression estimators make use of auxiliary information for increasing precision. It was seen that the ratio estimator provides a precise estimate of the population mean if regression is linear and the line passes through the origin. When regression is linear and line does not go through the origin, it is better to use estimators based on linear regression. In other words, if the study variate is approximately a constant and a multiple of the auxiliary variate, it is more precise to estimate the population mean or total by fitting a linear regression. Such an estimator is called regression estimator. We will discuss the applicability of regressions estimators in ratio estimation. Regression estimators are used to estimate the population means individually and by taking the ratio of these estimators, one can get the following ratio estimator. Employing regression estimators ( Z ) and x b ( Z ) y b for Y and X respectively explained in terms of, the estimator of R is then x ( Z ) ( Z ) y b r3 =. (.3) x b x It may be mentioned here that this estimator was considered by Tripathi (1980). Chand (1975) and Sukhatme and Chand (1977) proposed a technique of changing the available information on auxiliary characteristics with the main characteristic. This technique does not involve unknown weights like Olkin(1958), Raj (1965) and Rao and Mudholkar (1967) estimators and at the most assume knowledge of population mean of auxiliary character least correlated with the main character. Kiregyera (1980, 1984) also proposed some chain type ratio and regression estimators based on two auxiliary variables. InterStat 015 Page 5
6 Many estimators of the population parameters are constructed using known auxiliary variables. The classical well known ratio estimator is one of them. Various improvements of this ratio estimator have been considered by many authors. Some estimators use two auxiliary variables. Some other use groups of estimators are composite. In case two auxiliary variables are known, the ratio-cum-product estimator may be used (Singh 1965). This estimator behaves similarly as simple ratio estimator. Singh s (1965, 1967) ratio cum product estimators given by r 4 = r0 Z / (.4) and r 5 = r0 / Z (.5) The product method of estimation is generally used when the study variable Y is negatively correlated with an auxiliary characteristics X whose population mean is assumed to be known. In order to improve the efficiency of product estimation, sometimes producttype estimators are used which are developed by mixing product estimator with usual mean estimator. Kadilar and Cingi (004) and Sahoo and Sahoo (1993) used a second auxiliary variable Z closely related to X and suggested different improved competitive estimators, assuming the availability of information on for all units of the population. Several researchers have established many classes of ratio and product-type estimators in the past that reduce the bias and the mean square error by improving the auxiliary variables. The above review clearly motivates the need for a comparative study of these estimators on the basis of their design based properties like biasedness, efficiency, and InterStat 015 Page 6
7 approach to normality (asymmetry) etc., before we decide to use any one of them in practice. It is however difficult to investigate analytically the behaviour of the estimators. Because the results derived through Taylor s lineariation method [e.g., Tin (1965)] are not only in asymptotical forms but also very much complicated to lead to make a good choice among different estimators. 3. Empirical Study First we generate the uniform random numbers and then normal random numbers with specified parameters is carried out to define the following populations of sie 000. The generation of the populations involves the following two steps based on the algorithm proposed by Rao (009). Population I: Step 1: Generate three independent random variables Z, Z 1 and Z from N( 3,4) distribution using Box-Muller method. Step : Using steps, 3, and 4 of algorithm given by Rao (009), generate the population-i of the triplet ( Y X, Z ),. Population II: Step 1: Generate three independent random variables Z, Z 1 and Z using Box-Muller method with Z ~ N (3,4), Z ~ (4,9) and Z ~ (5,16) 1 N N distributions. Step : Using steps, 3, and 4 of algorithm given by Rao (009), generate the population- II of the triplet ( Y X, Z ),. InterStat 015 Page 7
8 From the above two populations, 1000 simple random samples without replacement of sie n=10, 30 and 50 are drawn and for each of the sample, the 6 ratio estimators are computed. For each ratio estimator, the estimate i Rˆ, standard error ( ) SE ˆ, relative bias RB( R ˆ i ), skewness β 1 and kurtosis β are computed using the simulation method and presented against the sample sie n= 10, 30, 50 and the correlations between the variables in the following tables. Results are reported up to two decimals in the tables, but the conclusions are drawn from the original values. R i Table 1. Comparison of ratio estimators using two auxiliary variables from the Population-I Correlation Coefficient r yx =0.7 r x =0.80 r =0.90 Sample Sie (n) Ratio Estimator Rˆi S. E. ( ˆ ) ( R i ) R i RB ˆ 1 β β r r r r r r r r r r r r r r r r r r InterStat 015 Page 8
9 Table. Comparison of ratio estimators using two auxiliary variables from the Population-II Correlation Coefficient r yx =0.48 r x =0.66 r =0.7 Sample Sie (n) ( ) ( ) Ratio Estimator Rˆi S E. Rˆ i RB R i β1 β r r r r r r r r r r r r r r r r r r Conclusion It is observed from the above tables, from the population-i, the estimators r 1 & r are more efficient than the other estimators when the sample sie is small, whereas the estimator r 3 is more efficient estimator than the other estimators for large samples. The estimator r 3 is relatively, less biased as compared with other estimators for the small samples, whereas the estimators r 1 & r are less biased as compared with other estimators for the large samples under population-i. InterStat 015 Page 9
10 Similarly, in the second population, relatively best estimators are r 1 & r for small samples and r 3 is for large samples. The estimators r 1 and r are less biased under the second population. From the above study, it is observed that the Tin s (1965) type AUR estimators with two auxiliary variables (r 1 & r ) are more efficient for small samples, whereas the Tripathi (1980) estimator (r 3 ) is more efficient for large samples. It is also observed that the variation of the auxiliary variables resulting in the performance of the estimators when the small samples are selected. From the coefficients of skewness and kurtosis, it is observed that the estimators slightly deviating from the normal distributions whenever the variances of the auxiliary variables are different (Population-II), whereas in population-i, empirically observed that the considered estimators follows asymptotic normal distribution. References: 1. Chand, L. (1975), Some ratio-type estimators based on two or more auxiliary variables, Unpublished Ph. D. Dissertation, Iowa State University, Ames, Iowa.. Kadilar, C. and Cingi, H. (004), Estimator of a population mean using two auxiliary variables in simple random sampling, Int. J. Math., 4, Kiregyera, B. (1980), A chain ratio-type estimator in finite population double sampling using two auxiliary variables, Metrika, 7, Kiregyera, B. (1984), Regression type estimators using two auxiliary variables and the model of double sampling from finite populations, Metrika,31, Olkin, I. (1958), Multivariate ratio estimation for finite population, Biometrika, 45, InterStat 015 Page 10
11 6. Raj, D. (1965), On a method of using multi-auxiliary information in sample surveys, Jour. Amer. Stat. Assoc., 60, Rao, P.S.R.S. and Mudholkar, G.S. (1967), Generalied multivariate estimations for the mean of finite populations, Jour. Amer. Stat. Assoc.,6, Rao, K. R. (009), Comparison of ratio estimators using Monte Carlo Simulation, Unpublished Ph.D. Thesis, Department of Statistics, Osmania University, Hyderabad. 9. Sahoo, L.N. (1987a), On a class of almost unbiased estimator for population ratio, Statistics, 18, Sarndal, C.E., Swensson, B. and Wretman, J. (1991), Model assisted survey sampling, Springer Verlag, New York. 11. Singh, M.P. (1965), On the estimation of ratio and product of the population parameters, Sankhya,7, Singh, M.P. (1967), Ratio cum product method of estimation, Metrika,1, Sukhatme, B.V. and Chand, L. (1977), Multivariate ratio-type estimators, Proceedings of American Statistical Association, Social Statistics Section, Tin, M. (1965), Comparison of some ratio estimators, Jour. Amer. Stat. Assoc.,60, Tripathi, T.P. (1969), A regression type estimator in sampling with PPS and with replacement, Aust. Jour. Stat.,11, Tripathi, T.P. (1980), A general class of estimators for population ratio, Sankhya,C 4, InterStat 015 Page 11
Journal of Modern Applied Statistical Methods
Journal of Modern Applied Statistical Methods Volume 3 Issue Article 3 5--04 Two Parameter Modified Ratio Estimators with Two Auxiliar Variables for Estimation of Finite Population Mean with Known Skewness,
More informationGeneralized Modified Ratio Type Estimator for Estimation of Population Variance
Sri Lankan Journal of Applied Statistics, Vol (16-1) Generalized Modified Ratio Type Estimator for Estimation of Population Variance J. Subramani* Department of Statistics, Pondicherry University, Puducherry,
More informationA CLASS OF PRODUCT-TYPE EXPONENTIAL ESTIMATORS OF THE POPULATION MEAN IN SIMPLE RANDOM SAMPLING SCHEME
STATISTICS IN TRANSITION-new series, Summer 03 89 STATISTICS IN TRANSITION-new series, Summer 03 Vol. 4, No., pp. 89 00 A CLASS OF PRODUCT-TYPE EXPONENTIAL ESTIMATORS OF THE POPULATION MEAN IN SIMPLE RANDOM
More informationModified ratio estimators of population mean using linear combination of co-efficient of skewness and quartile deviation
CSIRO PUBLISHING The South Pacific Journal of Natural and Applied Sciences, 31, 39-44, 2013 www.publish.csiro.au/journals/spjnas 10.1071/SP13003 Modified ratio estimators of population mean using linear
More informationAvailable online at (Elixir International Journal) Statistics. Elixir Statistics 44 (2012)
7411 A class of almost unbiased modified ratio estimators population mean with known population parameters J.Subramani and G.Kumarapandiyan Department of Statistics, Ramanujan School of Mathematical Sciences
More informationEFFICIENT ESTIMATORS FOR THE POPULATION MEAN
Hacettepe Journal of Mathematics and Statistics Volume 38) 009), 17 5 EFFICIENT ESTIMATORS FOR THE POPULATION MEAN Nursel Koyuncu and Cem Kadılar Received 31:11 :008 : Accepted 19 :03 :009 Abstract M.
More informationCalibration Approach Separate Ratio Estimator for Population Mean in Stratified Sampling
Article International Journal of Modern Mathematical Sciences, 015, 13(4): 377-384 International Journal of Modern Mathematical Sciences Journal homepage: www.modernscientificpress.com/journals/ijmms.aspx
More informationBias Reduction Using the Bootstrap
Bias Reduction Using the Bootstrap Find f t (i.e., t) so that or E(f t (P, P n ) P) = 0 E(T(P n ) θ(p) + t P) = 0. Change the problem to the sample: whose solution is so the bias-reduced estimate is E(T(P
More informationA RIDGE REGRESSION ESTIMATION APPROACH WHEN MULTICOLLINEARITY IS PRESENT
Fundamental Journal of Applied Sciences Vol. 1, Issue 1, 016, Pages 19-3 This paper is available online at http://www.frdint.com/ Published online February 18, 016 A RIDGE REGRESSION ESTIMATION APPROACH
More informationCalibration approach estimators in stratified sampling
Statistics & Probability Letters 77 (2007) 99 103 www.elsevier.com/locate/stapro Calibration approach estimators in stratified sampling Jong-Min Kim a,, Engin A. Sungur a, Tae-Young Heo b a Division of
More informationPower of t-test for Simple Linear Regression Model with Non-normal Error Distribution: A Quantile Function Distribution Approach
Available Online Publications J. Sci. Res. 4 (3), 609-622 (2012) JOURNAL OF SCIENTIFIC RESEARCH www.banglajol.info/index.php/jsr of t-test for Simple Linear Regression Model with Non-normal Error Distribution:
More informationGENERATION OF STANDARD NORMAL RANDOM NUMBERS. Naveen Kumar Boiroju and M. Krishna Reddy
GENERATION OF STANDARD NORMAL RANDOM NUMBERS Naveen Kumar Boiroju and M. Krishna Reddy Department of Statistics, Osmania University, Hyderabad- 500 007, INDIA Email: nanibyrozu@gmail.com, reddymk54@gmail.com
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationEffects of skewness and kurtosis on model selection criteria
Economics Letters 59 (1998) 17 Effects of skewness and kurtosis on model selection criteria * Sıdıka Başçı, Asad Zaman Department of Economics, Bilkent University, 06533, Bilkent, Ankara, Turkey Received
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationA New Test for Correlation on Bivariate Nonnormal Distributions
Journal of Modern Applied Statistical Methods Volume 5 Issue Article 8 --06 A New Test for Correlation on Bivariate Nonnormal Distributions Ping Wang Great Basin College, ping.wang@gbcnv.edu Ping Sa University
More informationCalibration Estimation under Non-response and Missing Values in Auxiliary Information
WORKING PAPER 2/2015 Calibration Estimation under Non-response and Missing Values in Auxiliary Information Thomas Laitila and Lisha Wang Statistics ISSN 1403-0586 http://www.oru.se/institutioner/handelshogskolan-vid-orebro-universitet/forskning/publikationer/working-papers/
More informationPARAMETRIC AND NON-PARAMETRIC BOOTSTRAP: A SIMULATION STUDY FOR A LINEAR REGRESSION WITH RESIDUALS FROM A MIXTURE OF LAPLACE DISTRIBUTIONS
PARAMETRIC AND NON-PARAMETRIC BOOTSTRAP: A SIMULATION STUDY FOR A LINEAR REGRESSION WITH RESIDUALS FROM A MIXTURE OF LAPLACE DISTRIBUTIONS Melfi Alrasheedi School of Business, King Faisal University, Saudi
More informationSTAT 509: Statistics for Engineers Dr. Dewei Wang. Copyright 2014 John Wiley & Sons, Inc. All rights reserved.
STAT 509: Statistics for Engineers Dr. Dewei Wang Applied Statistics and Probability for Engineers Sixth Edition Douglas C. Montgomery George C. Runger 7 Point CHAPTER OUTLINE 7-1 Point Estimation 7-2
More informationThe histogram should resemble the uniform density, the mean should be close to 0.5, and the standard deviation should be close to 1/ 12 =
Chapter 19 Monte Carlo Valuation Question 19.1 The histogram should resemble the uniform density, the mean should be close to.5, and the standard deviation should be close to 1/ 1 =.887. Question 19. The
More informationOmitted Variables Bias in Regime-Switching Models with Slope-Constrained Estimators: Evidence from Monte Carlo Simulations
Journal of Statistical and Econometric Methods, vol. 2, no.3, 2013, 49-55 ISSN: 2051-5057 (print version), 2051-5065(online) Scienpress Ltd, 2013 Omitted Variables Bias in Regime-Switching Models with
More informationSampling and sampling distribution
Sampling and sampling distribution September 12, 2017 STAT 101 Class 5 Slide 1 Outline of Topics 1 Sampling 2 Sampling distribution of a mean 3 Sampling distribution of a proportion STAT 101 Class 5 Slide
More informationRobust Critical Values for the Jarque-bera Test for Normality
Robust Critical Values for the Jarque-bera Test for Normality PANAGIOTIS MANTALOS Jönköping International Business School Jönköping University JIBS Working Papers No. 00-8 ROBUST CRITICAL VALUES FOR THE
More informationدرس هفتم یادگیري ماشین. (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی
یادگیري ماشین توزیع هاي نمونه و تخمین نقطه اي پارامترها Sampling Distributions and Point Estimation of Parameter (Machine Learning) دانشگاه فردوسی مشهد دانشکده مهندسی رضا منصفی درس هفتم 1 Outline Introduction
More informationAn Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process
Computational Statistics 17 (March 2002), 17 28. An Improved Saddlepoint Approximation Based on the Negative Binomial Distribution for the General Birth Process Gordon K. Smyth and Heather M. Podlich Department
More informationELEMENTS OF MONTE CARLO SIMULATION
APPENDIX B ELEMENTS OF MONTE CARLO SIMULATION B. GENERAL CONCEPT The basic idea of Monte Carlo simulation is to create a series of experimental samples using a random number sequence. According to the
More informationUsing Halton Sequences. in Random Parameters Logit Models
Journal of Statistical and Econometric Methods, vol.5, no.1, 2016, 59-86 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2016 Using Halton Sequences in Random Parameters Logit Models Tong Zeng
More informationEcon 300: Quantitative Methods in Economics. 11th Class 10/19/09
Econ 300: Quantitative Methods in Economics 11th Class 10/19/09 Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write. --H.G. Wells discuss test [do
More informationPoint Estimation. Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage
6 Point Estimation Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Point Estimation Statistical inference: directed toward conclusions about one or more parameters. We will use the generic
More informationSmall area estimation for poverty indicators
Small area estimation for poverty indicators Risto Lehtonen (University of Helsinki) Ari Veijanen (Statistics Finland) Mikko Myrskylä (Max Planck Institute for Demographic Research) Maria Valaste (Social
More informationMEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL
MEASURING PORTFOLIO RISKS USING CONDITIONAL COPULA-AR-GARCH MODEL Isariya Suttakulpiboon MSc in Risk Management and Insurance Georgia State University, 30303 Atlanta, Georgia Email: suttakul.i@gmail.com,
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
More informationMuch of what appears here comes from ideas presented in the book:
Chapter 11 Robust statistical methods Much of what appears here comes from ideas presented in the book: Huber, Peter J. (1981), Robust statistics, John Wiley & Sons (New York; Chichester). There are many
More informationSMALL AREA ESTIMATES OF INCOME: MEANS, MEDIANS
SMALL AREA ESTIMATES OF INCOME: MEANS, MEDIANS AND PERCENTILES Alison Whitworth (alison.whitworth@ons.gsi.gov.uk) (1), Kieran Martin (2), Cruddas, Christine Sexton, Alan Taylor Nikos Tzavidis (3), Marie
More informationPoint Estimation. Copyright Cengage Learning. All rights reserved.
6 Point Estimation Copyright Cengage Learning. All rights reserved. 6.2 Methods of Point Estimation Copyright Cengage Learning. All rights reserved. Methods of Point Estimation The definition of unbiasedness
More informationEstimation of Population Variance Utilizing Auxiliary Information
Iteratioal Joural of Statistics ad Systems ISSN 0973-675 Volume 1, Number (017), pp. 303-309 Research Idia Publicatios http://www.ripublicatio.com Estimatio of Populatio Variace Utilizig Auxiliary Iformatio
More informationImproved Inference for Signal Discovery Under Exceptionally Low False Positive Error Rates
Improved Inference for Signal Discovery Under Exceptionally Low False Positive Error Rates (to appear in Journal of Instrumentation) Igor Volobouev & Alex Trindade Dept. of Physics & Astronomy, Texas Tech
More informationIntroduction to Algorithmic Trading Strategies Lecture 8
Introduction to Algorithmic Trading Strategies Lecture 8 Risk Management Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Value at Risk (VaR) Extreme Value Theory (EVT) References
More informationSample Size for Assessing Agreement between Two Methods of Measurement by Bland Altman Method
Meng-Jie Lu 1 / Wei-Hua Zhong 1 / Yu-Xiu Liu 1 / Hua-Zhang Miao 1 / Yong-Chang Li 1 / Mu-Huo Ji 2 Sample Size for Assessing Agreement between Two Methods of Measurement by Bland Altman Method Abstract:
More informationAnomalies under Jackknife Variance Estimation Incorporating Rao-Shao Adjustment in the Medical Expenditure Panel Survey - Insurance Component 1
Anomalies under Jackknife Variance Estimation Incorporating Rao-Shao Adjustment in the Medical Expenditure Panel Survey - Insurance Component 1 Robert M. Baskin 1, Matthew S. Thompson 2 1 Agency for Healthcare
More informationVARIANCE ESTIMATION FROM CALIBRATED SAMPLES
VARIANCE ESTIMATION FROM CALIBRATED SAMPLES Douglas Willson, Paul Kirnos, Jim Gallagher, Anka Wagner National Analysts Inc. 1835 Market Street, Philadelphia, PA, 19103 Key Words: Calibration; Raking; Variance
More informationStrategies for Improving the Efficiency of Monte-Carlo Methods
Strategies for Improving the Efficiency of Monte-Carlo Methods Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu Introduction The Monte-Carlo method is a useful
More informationNew SAS Procedures for Analysis of Sample Survey Data
New SAS Procedures for Analysis of Sample Survey Data Anthony An and Donna Watts, SAS Institute Inc, Cary, NC Abstract Researchers use sample surveys to obtain information on a wide variety of issues Many
More informationBEHAVIOUR OF PASSAGE TIME FOR A QUEUEING NETWORK MODEL WITH FEEDBACK: A SIMULATION STUDY
IJMMS 24:24, 1267 1278 PII. S1611712426287 http://ijmms.hindawi.com Hindawi Publishing Corp. BEHAVIOUR OF PASSAGE TIME FOR A QUEUEING NETWORK MODEL WITH FEEDBACK: A SIMULATION STUDY BIDYUT K. MEDYA Received
More informationSTRESS-STRENGTH RELIABILITY ESTIMATION
CHAPTER 5 STRESS-STRENGTH RELIABILITY ESTIMATION 5. Introduction There are appliances (every physical component possess an inherent strength) which survive due to their strength. These appliances receive
More informationON JARQUE-BERA TESTS FOR ASSESSING MULTIVARIATE NORMALITY
Journal of Statistics: Advances in Theory and Alications Volume, umber, 009, Pages 07-0 O JARQUE-BERA TESTS FOR ASSESSIG MULTIVARIATE ORMALITY KAZUYUKI KOIZUMI, AOYA OKAMOTO and TAKASHI SEO Deartment of
More informationA NEW NOTION OF TRANSITIVE RELATIVE RETURN RATE AND ITS APPLICATIONS USING STOCHASTIC DIFFERENTIAL EQUATIONS. Burhaneddin İZGİ
A NEW NOTION OF TRANSITIVE RELATIVE RETURN RATE AND ITS APPLICATIONS USING STOCHASTIC DIFFERENTIAL EQUATIONS Burhaneddin İZGİ Department of Mathematics, Istanbul Technical University, Istanbul, Turkey
More informationConsistent estimators for multilevel generalised linear models using an iterated bootstrap
Multilevel Models Project Working Paper December, 98 Consistent estimators for multilevel generalised linear models using an iterated bootstrap by Harvey Goldstein hgoldstn@ioe.ac.uk Introduction Several
More informationSection 2.4. Properties of point estimators 135
Section 2.4. Properties of point estimators 135 The fact that S 2 is an estimator of σ 2 for any population distribution is one of the most compelling reasons to use the n 1 in the denominator of the definition
More informationTwo-term Edgeworth expansions of the distributions of fit indexes under fixed alternatives in covariance structure models
Economic Review (Otaru University of Commerce), Vo.59, No.4, 4-48, March, 009 Two-term Edgeworth expansions of the distributions of fit indexes under fixed alternatives in covariance structure models Haruhiko
More informationOn the Distribution of Kurtosis Test for Multivariate Normality
On the Distribution of Kurtosis Test for Multivariate Normality Takashi Seo and Mayumi Ariga Department of Mathematical Information Science Tokyo University of Science 1-3, Kagurazaka, Shinjuku-ku, Tokyo,
More informationAn Improved Skewness Measure
An Improved Skewness Measure Richard A. Groeneveld Professor Emeritus, Department of Statistics Iowa State University ragroeneveld@valley.net Glen Meeden School of Statistics University of Minnesota Minneapolis,
More informationLinear Regression with One Regressor
Linear Regression with One Regressor Michael Ash Lecture 9 Linear Regression with One Regressor Review of Last Time 1. The Linear Regression Model The relationship between independent X and dependent Y
More informationAccelerated Option Pricing Multiple Scenarios
Accelerated Option Pricing in Multiple Scenarios 04.07.2008 Stefan Dirnstorfer (stefan@thetaris.com) Andreas J. Grau (grau@thetaris.com) 1 Abstract This paper covers a massive acceleration of Monte-Carlo
More informationImportance Sampling for Fair Policy Selection
Importance Sampling for Fair Policy Selection Shayan Doroudi Carnegie Mellon University Pittsburgh, PA 15213 shayand@cs.cmu.edu Philip S. Thomas Carnegie Mellon University Pittsburgh, PA 15213 philipt@cs.cmu.edu
More informationRalph S. Woodruff, Bureau of the Census
130 THE USE OF ROTATING SAMPTRS IN THE CENSUS BUREAU'S MONTHLY SURVEYS By: Ralph S. Woodruff, Bureau of the Census Rotating panels are used on several of the monthly surveys of the Bureau of the Census.
More informationIntroduction Dickey-Fuller Test Option Pricing Bootstrapping. Simulation Methods. Chapter 13 of Chris Brook s Book.
Simulation Methods Chapter 13 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 26, 2017 Christopher
More informationAn Empirical Examination of Traditional Equity Valuation Models: The case of the Athens Stock Exchange
European Research Studies, Volume 7, Issue (1-) 004 An Empirical Examination of Traditional Equity Valuation Models: The case of the Athens Stock Exchange By G. A. Karathanassis*, S. N. Spilioti** Abstract
More informationHeterogeneity in Returns to Wealth and the Measurement of Wealth Inequality 1
Heterogeneity in Returns to Wealth and the Measurement of Wealth Inequality 1 Andreas Fagereng (Statistics Norway) Luigi Guiso (EIEF) Davide Malacrino (Stanford University) Luigi Pistaferri (Stanford University
More informationChapter 7 - Lecture 1 General concepts and criteria
Chapter 7 - Lecture 1 General concepts and criteria January 29th, 2010 Best estimator Mean Square error Unbiased estimators Example Unbiased estimators not unique Special case MVUE Bootstrap General Question
More informationAn Efficient Class of Exponential Estimator of Finite Population Mean Under Double Sampling Scheme in Presence of Non-Response
Global Journal of Pure an Applie Mathematics. ISSN 0973-768 Volume 3, Number 9 (07), pp. 599-533 Research Inia Publications http://www.ripublication.com An Efficient Class of Eponential Estimator of Finite
More informationChapter 5 Univariate time-series analysis. () Chapter 5 Univariate time-series analysis 1 / 29
Chapter 5 Univariate time-series analysis () Chapter 5 Univariate time-series analysis 1 / 29 Time-Series Time-series is a sequence fx 1, x 2,..., x T g or fx t g, t = 1,..., T, where t is an index denoting
More informationTechnical Note: An Improved Range Chart for Normal and Long-Tailed Symmetrical Distributions
Technical Note: An Improved Range Chart for Normal and Long-Tailed Symmetrical Distributions Pandu Tadikamalla, 1 Mihai Banciu, 1 Dana Popescu 2 1 Joseph M. Katz Graduate School of Business, University
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUniversity of California Berkeley
University of California Berkeley Improving the Asmussen-Kroese Type Simulation Estimators Samim Ghamami and Sheldon M. Ross May 25, 2012 Abstract Asmussen-Kroese [1] Monte Carlo estimators of P (S n >
More informationSmall Sample Performance of Instrumental Variables Probit Estimators: A Monte Carlo Investigation
Small Sample Performance of Instrumental Variables Probit : A Monte Carlo Investigation July 31, 2008 LIML Newey Small Sample Performance? Goals Equations Regressors and Errors Parameters Reduced Form
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationStock Price Sensitivity
CHAPTER 3 Stock Price Sensitivity 3.1 Introduction Estimating the expected return on investments to be made in the stock market is a challenging job before an ordinary investor. Different market models
More informationEstimation of dynamic term structure models
Estimation of dynamic term structure models Greg Duffee Haas School of Business, UC-Berkeley Joint with Richard Stanton, Haas School Presentation at IMA Workshop, May 2004 (full paper at http://faculty.haas.berkeley.edu/duffee)
More informationA New Multivariate Kurtosis and Its Asymptotic Distribution
A ew Multivariate Kurtosis and Its Asymptotic Distribution Chiaki Miyagawa 1 and Takashi Seo 1 Department of Mathematical Information Science, Graduate School of Science, Tokyo University of Science, Tokyo,
More informationKERNEL PROBABILITY DENSITY ESTIMATION METHODS
5.- KERNEL PROBABILITY DENSITY ESTIMATION METHODS S. Towers State University of New York at Stony Brook Abstract Kernel Probability Density Estimation techniques are fast growing in popularity in the particle
More informationDetermination of the Optimal Stratum Boundaries in the Monthly Retail Trade Survey in the Croatian Bureau of Statistics
Determination of the Optimal Stratum Boundaries in the Monthly Retail Trade Survey in the Croatian Bureau of Statistics Ivana JURINA (jurinai@dzs.hr) Croatian Bureau of Statistics Lidija GLIGOROVA (gligoroval@dzs.hr)
More informationAnalysis of The Efficacy of Black-scholes Model - An Empirical Evidence from Call Options on Nifty-50 Index
Analysis of The Efficacy of Black-scholes Model - An Empirical Evidence from Call Options on Nifty-50 Index Prof. A. Sudhakar Professor Dr. B.R. Ambedkar Open University, Hyderabad CMA Potharla Srikanth
More informationBasics. STAT:5400 Computing in Statistics Simulation studies in statistics Lecture 9 September 21, 2016
STAT:5400 Computing in Statistics Simulation studies in statistics Lecture 9 September 21, 2016 Based on a lecture by Marie Davidian for ST 810A - Spring 2005 Preparation for Statistical Research North
More informationChapter 5: Statistical Inference (in General)
Chapter 5: Statistical Inference (in General) Shiwen Shen University of South Carolina 2016 Fall Section 003 1 / 17 Motivation In chapter 3, we learn the discrete probability distributions, including Bernoulli,
More informationBias in Reduced-Form Estimates of Pass-through
Bias in Reduced-Form Estimates of Pass-through Alexander MacKay University of Chicago Marc Remer Department of Justice Nathan H. Miller Georgetown University Gloria Sheu Department of Justice February
More informationList of tables List of boxes List of screenshots Preface to the third edition Acknowledgements
Table of List of figures List of tables List of boxes List of screenshots Preface to the third edition Acknowledgements page xii xv xvii xix xxi xxv 1 Introduction 1 1.1 What is econometrics? 2 1.2 Is
More informationA general approach to calculating VaR without volatilities and correlations
page 19 A general approach to calculating VaR without volatilities and correlations Peter Benson * Peter Zangari Morgan Guaranty rust Company Risk Management Research (1-212) 648-8641 zangari_peter@jpmorgan.com
More informationConsistent weighting of the LFS - monthly, quarterly, annual and longitdinal data
Memorandum Consistent weighting of the LFS - monthly, quarterly, annual and longitdinal data Martijn Souren summary This paper describes the challenges that come with pursuing internal consistency for
More informationOn the Use of Stock Index Returns from Economic Scenario Generators in ERM Modeling
On the Use of Stock Index Returns from Economic Scenario Generators in ERM Modeling Michael G. Wacek, FCAS, CERA, MAAA Abstract The modeling of insurance company enterprise risks requires correlated forecasts
More informationPoint Estimators. STATISTICS Lecture no. 10. Department of Econometrics FEM UO Brno office 69a, tel
STATISTICS Lecture no. 10 Department of Econometrics FEM UO Brno office 69a, tel. 973 442029 email:jiri.neubauer@unob.cz 8. 12. 2009 Introduction Suppose that we manufacture lightbulbs and we want to state
More informationBrooks, Introductory Econometrics for Finance, 3rd Edition
P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Chris Brooks,
More informationPRE CONFERENCE WORKSHOP 3
PRE CONFERENCE WORKSHOP 3 Stress testing operational risk for capital planning and capital adequacy PART 2: Monday, March 18th, 2013, New York Presenter: Alexander Cavallo, NORTHERN TRUST 1 Disclaimer
More informationThe misleading nature of correlations
The misleading nature of correlations In this note we explain certain subtle features of calculating correlations between time-series. Correlation is a measure of linear co-movement, to be contrasted with
More informationOn the Distribution and Its Properties of the Sum of a Normal and a Doubly Truncated Normal
The Korean Communications in Statistics Vol. 13 No. 2, 2006, pp. 255-266 On the Distribution and Its Properties of the Sum of a Normal and a Doubly Truncated Normal Hea-Jung Kim 1) Abstract This paper
More informationThe Simple Regression Model
Chapter 2 Wooldridge: Introductory Econometrics: A Modern Approach, 5e Definition of the simple linear regression model Explains variable in terms of variable Intercept Slope parameter Dependent variable,
More information2 Control variates. λe λti λe e λt i where R(t) = t Y 1 Y N(t) is the time from the last event to t. L t = e λr(t) e e λt(t) Exercises
96 ChapterVI. Variance Reduction Methods stochastic volatility ISExSoren5.9 Example.5 (compound poisson processes) Let X(t) = Y + + Y N(t) where {N(t)},Y, Y,... are independent, {N(t)} is Poisson(λ) with
More informationGamma. The finite-difference formula for gamma is
Gamma The finite-difference formula for gamma is [ P (S + ɛ) 2 P (S) + P (S ɛ) e rτ E ɛ 2 ]. For a correlation option with multiple underlying assets, the finite-difference formula for the cross gammas
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationMarket Risk Analysis Volume IV. Value-at-Risk Models
Market Risk Analysis Volume IV Value-at-Risk Models Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume IV xiii xvi xxi xxv xxix IV.l Value
More informationAssicurazioni Generali: An Option Pricing Case with NAGARCH
Assicurazioni Generali: An Option Pricing Case with NAGARCH Assicurazioni Generali: Business Snapshot Find our latest analyses and trade ideas on bsic.it Assicurazioni Generali SpA is an Italy-based insurance
More informationFinal Exam Suggested Solutions
University of Washington Fall 003 Department of Economics Eric Zivot Economics 483 Final Exam Suggested Solutions This is a closed book and closed note exam. However, you are allowed one page of handwritten
More informationINLEDNING. Promemorior från P/STM / Statistiska centralbyrån. Stockholm : Statistiska centralbyrån, Nr 1-24.
INLEDNING TILL Promemorior från P/STM / Statistiska centralbyrån. Stockholm : Statistiska centralbyrån, 1978-1986. Nr 1-24. Efterföljare: Promemorior från U/STM / Statistiska centralbyrån. Stockholm :
More informationWeek 7 Quantitative Analysis of Financial Markets Simulation Methods
Week 7 Quantitative Analysis of Financial Markets Simulation Methods Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 November
More informationManagement of cash in Public sector Enterprises - A case study of ECIL, Hyderabad
IOSR Journal of Business and Management (IOSR-JBM) e-issn: 2278-487X, p-issn: 2319-7668 PP 50-55 www.iosrjournals.org Management of cash in Public sector Enterprises - A case study of ECIL, Hyderabad Dr.N.Jyothi
More informationOn the Distribution of Multivariate Sample Skewness for Assessing Multivariate Normality
On the Distribution of Multivariate Sample Skewness for Assessing Multivariate Normality Naoya Okamoto and Takashi Seo Department of Mathematical Information Science, Faculty of Science, Tokyo University
More informationAlternative VaR Models
Alternative VaR Models Neil Roeth, Senior Risk Developer, TFG Financial Systems. 15 th July 2015 Abstract We describe a variety of VaR models in terms of their key attributes and differences, e.g., parametric
More informationManagement Science Letters
Management Science Letters 3 (2013) 73 80 Contents lists available at GrowingScience Management Science Letters homepage: www.growingscience.com/msl Investigating different influential factors on capital
More informationModule 4: Point Estimation Statistics (OA3102)
Module 4: Point Estimation Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 8.1-8.4 Revision: 1-12 1 Goals for this Module Define
More informationTime Observations Time Period, t
Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard Time Series and Forecasting.S1 Time Series Models An example of a time series for 25 periods is plotted in Fig. 1 from the numerical
More information