Estimation of Gumbel Parameters under Ranked Set Sampling

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Estimation of Gumbel Parameters under Ranked Set Sampling"

Transcription

1 Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, Sameer A. Al-Subh Mutah Uiversity, Karak, Jorda, Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Yousef, Omar M. ad Al-Subh, Sameer A. (2014) "Estimatio of Gumbel Parameters uder Raked Set Samplig," Joural of Moder Applied Statistical Methods: Vol. 13 : Iss. 2, Article. DOI: /masm/ Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of

2 Joural of Moder Applied Statistical Methods November 2014, Vol. 13, No. 2, Copyright 2014 JMASM, Ic. ISSN Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa Applied Uiversity Zarqa, Jorda S. A. Al-Subh Mutah Uiversity Karak, Jorda Cosider the MLEs (maximum likelihood estimators) of the parameters of the Gumbel distributio usig SRS (simple radom sample) ad RSS (raked set sample) ad the MOMEs (method of momet estimators) ad REGs (regressio estimators) based o SRS. A compariso betwee these estimators usig bias ad MSE (mea square error) was performed usig simulatio. It appears that the MLE based o RSS ca be a robust competitor to the MLE based o SRS. Keywords: Raked set samplig; simple radom samplig, parameters, Gumbel distributio, maximum likelihood estimator, bias, mea square error, regressio estimator, method of momet estimator. Itroductio There are may areas of applicatio of the Gumbel distributio icludig evirometal scieces, system reliability, ad hydrology. I hydrology, for example, the Gumbel distributio may be used to represet the distributio of the miimum level of a river i a particular year based o miimum values for the past few years. It is useful for predictig the occurrece of extreme earthquakes, floods, ad other atural disasters. The potetial applicability of the Gumbel distributio to represet the distributio of miima relates to extreme value theory, which idicates that it is likely to be useful if the distributio of the uderlyig sample data is of the ormal or expoetial type. The problem of estimatio of the ukow parameters of the Gumbel distributio is cosidered by may authors uder simple radom samplig. Maciuas et al. (1979) cosidered the estimates of the parameters of the Gumbel distributio by the methods of probability weighted momets, momets, ad Omar M. Yousef is a lecturer i the Basic Scieces Departmet. him at Sameer A. Al-Subh is a assistat professor i the Departmet of Mathematics ad Statistics. him at 432

3 YOUSEF & AL-SUBH maximum likelihood. They used both idepedet ad serially correlated Gumbel umbers to derive the results from Mote Carlo experimets. They foud the method of probability weighted momets estimator is more efficiet tha the estimators. Leese (1973), derived the MLE (maximum likelihood estimator) of Gumbel distributio parameters i case of cesored samples ad he gave expressios for their large-sample stadard errors. Fioretio ad Gabriele (1984), give some modificatios of the MLE the Gumbel distributio parameters to reduce the bias of the estimators. Phie (1987) estimated the parameters of the Gumbel distributio by momets, MLE, maximum etropy ad probability weighted momets. He derived the asymptotic variace-covariace matrix of the MLEs ad used simulatio to compare betwee the various estimators. He foud that the MLE is best i terms of the root MSE (mea square error). Corsii et al. (1995), discussed the MLE ad Cramer-Rao (CR) bouds for the locatio ad scale parameters of the Gumbel distributio. Mousa et al. (2002), foud the Bayesia estimatio for the two parameters of the Gumbel distributio based o record values. RSS as itroduced by McItyre (1952) is a igeious samplig techique for selectig a sample which is more iformative tha a SRS to estimate the populatio mea. He used of RSS techique to estimate the mea pasture ad forage yields. RSS techique is very useful whe visual rakig of populatio uits is less expesive tha their actual quatificatios. Therefore, selectig a sample based o RSS ca reduce the cost ad icrease the efficiecy of estimatio. The basic idea behid selectig a sample uder RSS ca be described as follows: Select m radom samples each of size m, usig a visual ispectio or ay cheap method to rak the uits withi each sample with respect to the variable of iterest. The select, for actual measuremet, the i th smallest uit from the i th sample, i = 1,, m. I this way, a total of m measured uits are obtaied, oe from each sample. The procedure could be repeated r times util a sample of = mr measuremets are obtaied. These mr measuremets form RSS. Takahasi ad Wakimoto (1968) gave the theoretical backgroud for RSS. They showed that the mea of a RSS is a ubiased estimator of the populatio mea with variace smaller tha that of the mea of a SRS. Dell ad Clutter (1972) showed that the RSS mea remais ubiased ad more efficiet tha the SRS mea for estimatig the populatio eve if rakig is ot perfect. A comprehesive survey about developmets i RSS ca be foud i Che (2000) ad Muttlak ad Al-Saleh (2000). Because there are may attractive applicatios of Gumbel distributio, it is of iterest to coduct a statistical iferece for the Gumbel distributio. The statistical iferece icludes the study of some properties of Gumbel distributio, 433

4 ESTIMATION OF GUMBEL PARAMETERS emphasizig o estimatio of Gumbel parameters. The estimatio of the locatio ad scale parameters, deoted as α ad β respectively, of the Gumbel distributio uder SRS ad RSS is studied. The Gumbel parameters were estimated by usig several methods of estimatio i both cases of SRS ad RSS such as maximum likelihood, momets ad regressio. Furthermore, the performace of these estimators is ivestigated ad compared through simulatio. Bias, mea square error (MSE) ad efficiecy of these estimators were used for compariso. Parameter Estimatio Usig SRS The cdf ad pdf of the radom variable X which has a Gumbel distributio with parameters α ad β are give respectively by x Fx ( ;, ) exp exp, 1 x x f( x;, ) exp exp, (1) (2) where α is the locatio parameter ad β is the scale parameter, β > 0, x ad α (, ). Let X1, X2,, X be a radom sample from X. The MLEs, MOMEs (method of momet estimators) ad REGs (regressio estimators) will be examied i case of both parameters are ukow based o X1, X2,, X. MLEs Let X1, X2,, X be a radom sample from (2). The log-likelihood fuctio is give by xi xi l(, ) log exp. i1 i1 (3) 434

5 YOUSEF & AL-SUBH After takig the derivatives with respect to α ad β equatig to 0, the MLEs are obtaied as ˆ ad ˆ log. (4) ˆ x x w z MLE, S i i MLE, S MLE, S i1 where x i 1 zi zi exp, z zi ad wi. ˆ mle, S i1 z MOMEs The mea ad variace for Gumbel distributio are give by ad. (5) 6 The momet estimators of the two parameters are ˆMOME, S 6 s ad ˆ ˆ MOME, S x MOME, S (6) where s, x are the sample stadard deviatio ad mea, respectively, ad γ = is Euler s costat. REGs x x Let y Fx ( ;, ) exp exp l y= exp x x l y= exp t=l -l y = t a x b 1 where a ad b. 435

6 ESTIMATION OF GUMBEL PARAMETERS The regressio estimators of the two parameters are ˆ 1, ad ˆ ˆ,, ˆ REG S REG S REG S t ax (7) aˆ x x t t i i i1 where aˆ. 2 i1 x i x Parameter Estimatio Uder RSS MLEs Let X(i:m), i = 1,, m ad = 1,, r deote the i th order statistics from the i th set of size m of the th cycle be the RSS data for X with sample size = mr. Usig (1) ad (2), the pdf of X(i:m) is give by (Arold et al.,1992) i-1 m-i 1 fi : m( X ) cf( X ) 1- F( X ) f ( X ), where c B( i, m -i 1), 1 X X f( X ) exp exp ad F X by X ( ) exp exp. r l(, ) f ( X ) m i1 1 r m i1 1 i: m i r m i i1 1 The the likelihood fuctio is give -1 m-i = c F( X ) 1- F( X ) f ( X ) mr -1 m-i c F( X ) 1- F( X ) f ( X ). 436

7 YOUSEF & AL-SUBH The log-likelihood fuctio is give by r L(, ) mr log c ( i 1)log F( X ) m 1 i1 r m r m ( m i)log(1 F( X )) log f ( X ). 1 i1 1 i1 (8) Takig the derivatives of (8) with respect to α ad β respectively, ad equatig the resultig quatities to zero. Because there is o explicit solutio for (8), the equatios eed to be solved umerically to fid ˆ ˆ MLE, R ad MLE, R. Ad-hoc Estimators These are the same as the estimators i (6) ad (7) with SRS replaced by RSS Estimator Compariso A compariso betwee all above estimators for both parameters of the Gumbel distributio was carried out uder SRS ad RSS usig simulatio. The package R has bee used to coduct the simulatio. The followig values of the parameters ad sample sizes have bee cosidered: α = 0.5, β = 1; α = 1, β = 0.5; α = 1, β = 1; α = 1, β = 2; α = 2, β = 1, = ad =. For each, a set (m;r) is decided such that = mr. The bias ad the MSE are computed for all the estimators uder cosideratio. The efficiecy betwee all estimators with respect to the MLE based o SRS are calculated where the efficiecy of the estimator is defied as where If ˆ ˆ 2 1 ˆ ˆ ˆ MSE( 1) eff ( 2, 1) MSE( ˆ ) 10,000 ˆ 1 2 ˆ 2t 2 MSE. 10, 000 eff, 1 the ˆ 2 is better tha ˆ 1. The bias of the estimators is reported i Tables 1 ad 3 ad the efficiecies of the estimators is reported i Tables 2 ad 4. t

8 ESTIMATION OF GUMBEL PARAMETERS Table 1. The bias ad MSE of estimators of α Bias MSE (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

9 YOUSEF & AL-SUBH Table 2. The efficiecy of estimators of α (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

10 ESTIMATION OF GUMBEL PARAMETERS Table 3. The bias ad MSE of estimators of β Bias MSE (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r=

11 YOUSEF & AL-SUBH Table 4. The efficiecy of estimators of β (α, b) =mr ˆ mle,s ˆ moe,s ˆ reg,s ˆ mle,r ˆ moe,r (1,1) (1,2) (2,1) (0.5,1) (1,0.5) m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= m=3, r= m=4, r= m=2, r= m=3, r= m=4, r= From Tables 1 to 4, the followig coclusios are put forth i) I geeral, the bias is large for all estimators. Therefore, all the estimators are cosidered as biased estimators for α. ii) From Table 1, it ca be oticed that the REG uder SRS has the smallest bias as compared to the other estimators cosidered i most cases. I geeral, for all estimators of α uder RSS, the bias is less tha the case uder SRS. iii) For fixed α, the MSE of ˆ decreases as the sample size icreases. iv) It is oticed that from Table 2 that MLE uder RSS is the most efficiet tha the MLE based o SRS. 441

12 ESTIMATION OF GUMBEL PARAMETERS v) The efficiecy of the other estimators (MOMEs ad REGs based o SRS ad RSS) are ot cosistet, sometimes less oe ad other times greater tha 1. Similar remarks ca be oticed for the case of β. Coclusio Based o this study, it may be cocluded that all estimators are biased. Because the MLEs uder RSS are more efficiet tha the MLE uder SRS, RSS is recommeded i case orderig ca be doe visually or by a iexpesive method. The other estimators are ot recommeded because they are ot cosistet. Refereces Arold, B. C., Balakrisha, N., & Nagaraa, H. N. (1992). A First Course i Order Statistics. New York: Joh Wiley ad Sos. Che, Z. (2000). O raked-set sample quatiles ad their applicatios. Joural of Statistical Plaig ad Iferece, 83, Corsii, G., Gii, F., Greco, M. V., & Verrazzai, L. (1995). Cramer-Rao bouds ad estimatio of the parameters of the Gumbel distributio. Aerospace ad Electroic Systems, 31(3), Dell, D. R., & Clutter, J. L. (1972). Raked set samplig theory with order statistics backgroud. Biometrics, 28, Fioretio, M., & Gabriele, S. (1984). A correctio for the bias of maximum-likelihood estimators of Gumbel parameters. Joural of Hydrology, 73(1-2), Gumbel, E. J. (1958). Statistics of Extremes. New York: Columbia Uiversity Press. Leese, M. N. (1973). Use of cesored data i the estimatio of Gumbel distributio parameters for aual maximum flood series. Water Resources Research, 9(6), doi: /WR009i006p01534 Maciuas, J., Matalas, N. C., & Wallis, J. R. (1979). Probability weighted momets compared with some traditioal techiques i estimatig Gumbel Parameters ad quatiles. Water Resources Research, 15(5), doi: /WR015i005p

13 YOUSEF & AL-SUBH McItyre, G. A. (1952). A method of ubiased selective samplig usig raked sets. Australia Joural of Agricultural Research, 3, Mousa, M. A. M., Jahee, Z. F., & Ahmad, A. A. (2002). Bayesia Estimatio, Predictio ad Characterizatio for the Gumbel Model Based o Records. A Joural of Theoretical ad Applied Statistics, 36(1), Muttlak, H. A., & Al-Saleh, M. F Recet developmets i raked set samplig, Pakista Joural of Statistics, 16, Phie, H. N. (1987). A review of methods of parameter estimatio for the extreme value type-1 distributio. Joural of Hydrology, 90(3 4), Takahasi, K., & Wakitmoto, K. (1968). O ubiased estimates of the populatio mea based o the sample stratified by meas of orderig. Aals of the Istitute of Statistical Mathematics, 20,

New Entropy Estimators with Smaller Root Mean Squared Error

New Entropy Estimators with Smaller Root Mean Squared Error Joural of Moder Applied Statistical Methods Volume 4 Issue 2 Article 0 --205 New Etropy Estimators with Smaller Root Mea Squared Error Amer Ibrahim Al-Omari Al al-bayt Uiversity, Mafraq, Jorda, alomari_amer@yahoo.com

More information

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. 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 information

The Sampling Distribution of the Maximum. Likelihood Estimators for the Parameters of. Beta-Binomial Distribution

The 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 information

MOMENT-METHOD ESTIMATION BASED ON CENSORED SAMPLE

MOMENT-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 information

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION

ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION STATISTICA, ao LXXIII,. 4, 013 ESTIMATION AND PREDICTION BASED ON K-RECORD VALUES FROM NORMAL DISTRIBUTION Maoj Chacko Departmet of Statistics, Uiversity of Kerala, Trivadrum- 695581, Kerala, Idia M. Shy

More information

Estimation of the Population Mean in Presence of Non-Response

Estimation 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 information

Linear Regression Models

Linear Regression Models Liear Regressio Models Dr. Joh Mellor-Crummey 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 information

A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s)

A 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 information

R. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State

R. 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 information

Element sampling: Part 2

Element 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 information

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses

More information

Varanasi , India. Corresponding author

Varanasi , India. Corresponding author 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

More information

Parameter, Statistic and Random Samples

Parameter, 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

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis

Modified 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 information

The 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.

The 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 information

It should be unbiased, or approximately unbiased. Variance of the variance estimator should be small. That is, the variance estimator is stable.

It 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 information

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 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 information

Modified Lilliefors Test

Modified Lilliefors Test Joural of Moder Applied Statistical Methods Volume 14 Issue 1 Article 9 5-1-2015 Modified Lilliefors Test Achut Adhikari Uiversity of Norther Colorado, adhi2939@gmail.com Jay Schaffer Uiversity of Norther

More information

Section 14. Simple linear regression.

Section 14. Simple linear regression. Sectio 14 Simple liear regressio. Let us look at the cigarette dataset from [1] (available to dowload from joural s website) ad []. The cigarette dataset cotais measuremets of tar, icotie, weight ad carbo

More information

Review 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

Review 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 information

STATISTICAL INFERENCE

STATISTICAL INFERENCE STATISTICAL INFERENCE POPULATION AND SAMPLE Populatio = all elemets of iterest Characterized by a distributio F with some parameter θ Sample = the data X 1,..., X, selected subset of the populatio = sample

More information

The standard deviation of the mean

The 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 information

Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation

Some 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 information

Monte Carlo Integration

Monte 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 information

EDGEWORTH SIZE CORRECTED W, LR AND LM TESTS IN THE FORMATION OF THE PRELIMINARY TEST ESTIMATOR

EDGEWORTH 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 information

New Ratio Estimators Using Correlation Coefficient

New 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. e-mails : kadilar@hacettepe.edu.tr ; hcigi@hacettepe.edu.tr

More information

Probability and statistics: basic terms

Probability and statistics: basic terms Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample

More information

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 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 information

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS

A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS J. Japa Statist. Soc. Vol. 41 No. 1 2011 67 73 A RANK STATISTIC FOR NON-PARAMETRIC K-SAMPLE AND CHANGE POINT PROBLEMS Yoichi Nishiyama* We cosider k-sample ad chage poit problems for idepedet data i a

More information

The (P-A-L) Generalized Exponential Distribution: Properties and Estimation

The (P-A-L) Generalized Exponential Distribution: Properties and Estimation Iteratioal Mathematical Forum, Vol. 12, 2017, o. 1, 27-37 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.610140 The (P-A-L) Geeralized Expoetial Distributio: Properties ad Estimatio M.R.

More information

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.

Econ 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1. Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio

More information

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,

More information

o <Xln <X2n <... <X n < o (1.1)

o <Xln <X2n <... <X n < o (1.1) Metrika, Volume 28, 1981, page 257-262. 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 information

KLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions

KLMED8004 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 information

Applying least absolute deviation regression to regressiontype estimation of the index of a stable distribution using the characteristic function

Applying least absolute deviation regression to regressiontype estimation of the index of a stable distribution using the characteristic function Applyig least absolute deviatio regressio to regressiotype estimatio of the idex of a stable distributio usig the characteristic fuctio J. MARTIN VAN ZYL Departmet of Mathematical Statistics ad Actuarial

More information

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

A Risk Comparison of Ordinary Least Squares vs Ridge Regression Joural of Machie Learig Research 14 (2013) 1505-1511 Submitted 5/12; Revised 3/13; Published 6/13 A Risk Compariso of Ordiary Least Squares vs Ridge Regressio Paramveer S. Dhillo Departmet of Computer

More information

Statistical inference: example 1. Inferential Statistics

Statistical 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 information

A LARGER SAMPLE SIZE IS NOT ALWAYS BETTER!!!

A 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

3/3/2014. CDS M Phil Econometrics. Types of Relationships. Types of Relationships. Types of Relationships. Vijayamohanan Pillai N.

3/3/2014. 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 information

The Sample Variance Formula: A Detailed Study of an Old Controversy

The 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 information

Chapter 6 Principles of Data Reduction

Chapter 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 information

On The Gamma-Half Normal Distribution and Its Applications

On The Gamma-Half Normal Distribution and Its Applications Joural of Moder Applied Statistical Methods Volume Issue Article 5 5--3 O The Gamma-Half Normal Distributio ad Its Applicatios Ayma Alzaatreh Austi Peay State Uiversity, Clarksville, TN Kriste Kight Austi

More information

Statistical Theory MT 2009 Problems 1: Solution sketches

Statistical Theory MT 2009 Problems 1: Solution sketches Statistical Theory MT 009 Problems : Solutio sketches. Which of the followig desities are withi a expoetial family? Explai your reasoig. (a) Let 0 < θ < ad put f(x, θ) = ( θ)θ x ; x = 0,,,... (b) (c) where

More information

Improved exponential estimator for population variance using two auxiliary variables

Improved exponential estimator for population variance using two auxiliary variables OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 667-67 ISSN -5657, ISBN 97-973-55-5-0, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables

More information

Estimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches

Estimating Confidence Interval of Mean Using. Classical, Bayesian, and Bootstrap Approaches Iteratioal Joural of Mathematical Aalysis Vol. 8, 2014, o. 48, 2375-2383 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49287 Estimatig Cofidece Iterval of Mea Usig Classical, Bayesia,

More information

Statisticians use the word population to refer the total number of (potential) observations under consideration

Statisticians 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 information

Solutions to Odd Numbered End of Chapter Exercises: Chapter 4

Solutions 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 information

Final Examination Solutions 17/6/2010

Final 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 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

Regression with an Evaporating Logarithmic Trend

Regression with an Evaporating Logarithmic Trend Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5,

More information

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

Some 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 70504-1010,

More information

Statistical Inferences for Lomax Distribution Based on Record Values (Bayesian and Classical)

Statistical Inferences for Lomax Distribution Based on Record Values (Bayesian and Classical) Joural of Moder Applied Statistical Methods Volume Issue Article --0 Statistical Ifereces for Lomax Distriutio Based o Record Values (Bayesia ad Classical Parviz Nasiri Uiversity of Payame Noor, Tehra,

More information

Testing Statistical Hypotheses for Compare. Means with Vague Data

Testing Statistical Hypotheses for Compare. Means with Vague Data Iteratioal Mathematical Forum 5 o. 3 65-6 Testig Statistical Hypotheses for Compare Meas with Vague Data E. Baloui Jamkhaeh ad A. adi Ghara Departmet of Statistics Islamic Azad iversity Ghaemshahr Brach

More information

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation

Chapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.

More information

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1

PH 425 Quantum Measurement and Spin Winter SPINS Lab 1 PH 425 Quatum Measuremet ad Spi Witer 23 SPIS Lab Measure the spi projectio S z alog the z-axis This is the experimet that is ready to go whe you start the program, as show below Each atom is measured

More information

The Distribution of the Concentration Ratio for Samples from a Uniform Population

The Distribution of the Concentration Ratio for Samples from a Uniform Population Applied Mathematics, 05, 6, 57-70 Published Olie Jauary 05 i SciRes. http://www.scirp.or/joural/am http://dx.doi.or/0.436/am.05.6007 The Distributio of the Cocetratio Ratio for Samples from a Uiform Populatio

More information

(all terms are scalars).the minimization is clearer in sum notation:

(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 information

Lecture 9: September 19

Lecture 9: September 19 36-700: Probability ad Mathematical Statistics I Fall 206 Lecturer: Siva Balakrisha Lecture 9: September 9 9. Review ad Outlie Last class we discussed: Statistical estimatio broadly Pot estimatio Bias-Variace

More information

Sampling Distributions, Z-Tests, Power

Sampling Distributions, Z-Tests, Power Samplig Distributios, Z-Tests, Power We draw ifereces about populatio parameters from sample statistics Sample proportio approximates populatio proportio Sample mea approximates populatio mea Sample variace

More information

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise

First Year Quantitative Comp Exam Spring, Part I - 203A. f X (x) = 0 otherwise First Year Quatitative Comp Exam Sprig, 2012 Istructio: There are three parts. Aswer every questio i every part. Questio I-1 Part I - 203A A radom variable X is distributed with the margial desity: >

More information

MASSACHUSETTS 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

MASSACHUSETTS 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 information

Time series models 2007

Time series models 2007 Norwegia Uiversity of Sciece ad Techology Departmet of Mathematical Scieces Solutios to problem sheet 1, 2007 Exercise 1.1 a Let Sc = E[Y c 2 ]. The This gives Sc = EY 2 2cEY + c 2 ds dc = 2EY + 2c = 0

More information

DISCRIMINATING BETWEEN NORMAL AND GUMBEL DISTRIBUTIONS

DISCRIMINATING BETWEEN NORMAL AND GUMBEL DISTRIBUTIONS DISCRIMINATING BETWEEN NORMAL AND GUMBEL DISTRIBUTIONS Authors: Abdelaziz Qaffou Departmet of Applied Mathematics, Faculty of Scieces ad Techiques, Sulta Moulay Slimae Uiversity, Bei Mellal. Morocco aziz.qaffou@gmail.com)

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 4

Introduction to Econometrics (3 rd Updated Edition) Solutions 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 August 7, 204) 205 Pearso Educatio, Ic. Stock/Watso

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

Sampling, Sampling Distribution and Normality

Sampling, Sampling Distribution and Normality 4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig

More information

Introducing Sample Proportions

Introducing Sample Proportions Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,

More information

DS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10

DS 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 information

Chapter 1 (Definitions)

Chapter 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 information

The DOA Estimation of Multiple Signals based on Weighting MUSIC Algorithm

The DOA Estimation of Multiple Signals based on Weighting MUSIC Algorithm , pp.10-106 http://dx.doi.org/10.1457/astl.016.137.19 The DOA Estimatio of ultiple Sigals based o Weightig USIC Algorithm Chagga Shu a, Yumi Liu State Key Laboratory of IPOC, Beijig Uiversity of Posts

More information

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { } UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig

More information

Topics Machine learning: lecture 2. Review: the learning problem. Hypotheses and estimation. Estimation criterion cont d. Estimation criterion

Topics Machine learning: lecture 2. Review: the learning problem. Hypotheses and estimation. Estimation criterion cont d. Estimation criterion .87 Machie learig: lecture Tommi S. Jaakkola MIT CSAIL tommi@csail.mit.edu Topics The learig problem hypothesis class, estimatio algorithm loss ad estimatio criterio samplig, empirical ad epected losses

More information

Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector

Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector Dimesio-free PAC-Bayesia 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 information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (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 information

STATISTICAL method is one branch of mathematical

STATISTICAL 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 information

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:

STA 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 large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT

WHAT 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 information

On Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers

On Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers Pujab Uiversity Joural of Mathematics ISSN 1016-2526) Vol. 502)2018) pp. 11-19 O Bayesia Shrikage Estimator of Parameter of Expoetial Distributio with Outliers P. Nasiri Departmet of Statistics, Uiversity

More information

Asymptotic Results for the Linear Regression Model

Asymptotic Results for the Linear Regression Model Asymptotic Results for the Liear Regressio Model C. Fli November 29, 2000 1. Asymptotic Results uder Classical Assumptios The followig results apply to the liear regressio model y = Xβ + ε, where X is

More information

Intermittent demand forecasting by using Neural Network with simulated data

Intermittent demand forecasting by using Neural Network with simulated data Proceedigs of the 011 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Jauary 4, 011 Itermittet demad forecastig by usig Neural Network with simulated data Nguye Khoa

More information

BOOTSTRAP BIAS CORRECTION IN SEMIPARAMETRIC ESTIMATION METHODS FOR ARFIMA MODELS

BOOTSTRAP BIAS CORRECTION IN SEMIPARAMETRIC ESTIMATION METHODS FOR ARFIMA MODELS A pesquisa Operacioal e os Recursos Reováveis 4 a 7 de ovembro de 2003, Natal-RN BOOTSTRAP BIAS CORRECTION IN SEMIPARAMETRIC ESTIMATION METHODS FOR ARFIMA MODELS Glaura C. Fraco Depto. Estatística UFMG

More information

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?

Instructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters? CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter

More information

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes.

IE 230 Probability & Statistics in Engineering I. Closed book and notes. No calculators. 120 minutes. Closed book ad otes. No calculators. 120 miutes. Cover page, five pages of exam, ad tables for discrete ad cotiuous distributios. Score X i =1 X i / S X 2 i =1 (X i X ) 2 / ( 1) = [i =1 X i 2 X 2 ] / (

More information

STA 4032 Final Exam Formula Sheet

STA 4032 Final Exam Formula Sheet Chapter 2. Probability STA 4032 Fial Eam Formula Sheet Some Baic Probability Formula: (1) P (A B) = P (A) + P (B) P (A B). (2) P (A ) = 1 P (A) ( A i the complemet of A). (3) If S i a fiite ample pace

More information

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN HARDMEKO 004 Hardess Measuremets Theory ad Applicatio i Laboratories ad Idustries - November, 004, Washigto, D.C., USA TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN Koichiro HATTORI, Satoshi

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON 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 information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

Stat 200 -Testing Summary Page 1

Stat 200 -Testing Summary Page 1 Stat 00 -Testig Summary Page 1 Mathematicias are like Frechme; whatever you say to them, they traslate it ito their ow laguage ad forthwith it is somethig etirely differet Goethe 1 Large Sample Cofidece

More information

Confidence Intervals for the Coefficients of Variation with Bounded Parameters

Confidence Intervals for the Coefficients of Variation with Bounded Parameters Vol:7, No:9, 03 Cofidece Itervals for the Coefficiets of Variatio with Bouded Parameters Jeerapa Sappakitkamjor, Sa-aat Niwitpog Iteratioal Sciece Idex, Mathematical ad Computatioal Scieces Vol:7, No:9,

More information

arxiv: v1 [math.pr] 13 Oct 2011

arxiv: v1 [math.pr] 13 Oct 2011 A tail iequality for quadratic forms of subgaussia radom vectors Daiel Hsu, Sham M. Kakade,, ad Tog Zhag 3 arxiv:0.84v math.pr] 3 Oct 0 Microsoft Research New Eglad Departmet of Statistics, Wharto School,

More information

1 Covariance Estimation

1 Covariance Estimation Eco 75 Lecture 5 Covariace Estimatio ad Optimal Weightig Matrices I this lecture, we cosider estimatio of the asymptotic covariace matrix B B of the extremum estimator b : Covariace Estimatio Lemma 4.

More information

Monte Carlo Methods: Lecture 3 : Importance Sampling

Monte Carlo Methods: Lecture 3 : Importance Sampling Mote Carlo Methods: Lecture 3 : Importace Samplig Nick Whiteley 16.10.2008 Course material origially by Adam Johase ad Ludger Evers 2007 Overview of this lecture What we have see... Rejectio samplig. This

More information

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.

MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR 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 information

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang d Iteratioal Coferece o Machiery, Materials Egieerig, Chemical Egieerig ad Biotechology (MMECEB 015) Research o real time compesatio of thermal errors of CNC lathe based o liear regressio theory Qiu Yogliag

More information

On Marshall-Olkin Extended Weibull Distribution

On Marshall-Olkin Extended Weibull Distribution Joural of Statistical Theory ad Applicatios, Vol. 6, No. March 27) 7 O Marshall-Olki Exteded Weibull Distributio Haa Haj Ahmad Departmet of Mathematics, Uiversity of Hail Hail, KSA haaahm@yahoo.com Omar

More information

True Nature of Potential Energy of a Hydrogen Atom

True Nature of Potential Energy of a Hydrogen Atom True Nature of Potetial Eergy of a Hydroge Atom Koshu Suto Key words: Bohr Radius, Potetial Eergy, Rest Mass Eergy, Classical Electro Radius PACS codes: 365Sq, 365-w, 33+p Abstract I cosiderig the potetial

More information

An Extreme Value Theory Approach for Analyzing the Extreme Risk of the Gold Prices

An Extreme Value Theory Approach for Analyzing the Extreme Risk of the Gold Prices 2007 6 97-109 A Extreme Value Theory Approach for Aalyzig the Extreme Risk of the Gold Prices Jiah-Bag Jag Abstract Fiacial markets frequetly experiece extreme movemets i the egative side. Accurate computatio

More information

Estimation of Population Ratio in Post-Stratified Sampling Using Variable Transformation

Estimation of Population Ratio in Post-Stratified 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 Post-Stratiied Samplig Usig Variable

More information