Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation

Size: px
Start display at page:

Download "Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation"

Transcription

1 Metodološki zvezki, Vol. 13, No., 016, Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece iterval for the reciprocal of a ormal populatio mea with a kow coefficiet of variatio is proposed. This has applicatios i the area of uclear physics, agriculture ad ecoomic whe the researcher kows the coefficiet of variatio. The proposed cofidece iterval is based o the approximate expectatio ad variace of the estimator by Taylor series expasio. A Mote Carlo simulatio study was coducted to compare the performace of the proposed cofidece iterval with the existig cofidece iterval. Simulatio results show that the proposed cofidece iterval performs as well as the existig oe i terms of coverage probability. However, the approximate cofidece iterval is very easy to calculate compared with the exact cofidece iterval. 1 Itroductio The reciprocal of a ormal mea is applied i the area of uclear physics, agriculture ad ecoomic. For example, Lamaa et al. (1981) studied a charged particle mometum, reciprocal of a ormal mea is defied by p µ 1 where µ is the track curvature of a particle. The θ µ 1, where µ is the populatio mea. May researchers studied the reciprocal of a ormal mea. For istace, Zama (1981) discussed the estimators without momets i the case of the reciprocal of a ormal mea. The maximum likelihood estimate of the reciprocal of a ormal mea with a class of zero-oe loss fuctios was proposed by Zama (1985). Withers ad 1 Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techology, Thammasat Uiversity, Thailad; wararit@mathstat.sci.tu.ac.th

2 118 Wararit Paichkitkosolkul Nadarajah (013) preseted the theorem to costruct the poit estimators for the iverse powers of a ormal mea. Wogkhao et al. (013) proposed two cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio. Their cofidece itervals ca be applied whe the coefficiet of variatio of a cotrol group is kow. Oe of their cofidece itervals is developed based o a exact method i which this cofidece iterval is costructed from the pivotal statistics Z, where Z follows the stadard ormal distributio. The other cofidece iterval is costructed based o the geeralized cofidece iterval (Weerahadi, 1993). Simulatio results show that the coverage probabilities of the two cofidece itervals are ot sigificatly differet. However, the cofidece iterval based o the exact method is shorter tha the geeralized cofidece iterval. The exact method uses Taylor series expasio to fid the expectatio ad variace of the estimator of θ ad uses these results for costructig the cofidece iterval for θ. The lower ad upper limits of the cofidece iterval based o the exact method are difficult to compute sice they deped o a ifiite summatio. Therefore, our mai aim i this paper is to propose a approximate cofidece iterval for the reciprocal of a ormal mea with a kow coefficiet of variatio. The computatio of the ew proposed cofidece iterval is easier tha the exact cofidece iterval proposed by Wogkhao et al. (013). I additio, we also compare the estimated coverage probabilities of the ew proposed cofidece iterval ad existig cofidece iterval usig a Mote Carlo simulatio. The paper is orgaized as follows. I Sectio, the theoretical backgroud of the existig cofidece iterval for θ is discussed. We provide the theorem for costructig the approximate cofidece iterval for θ i Sectio 3. I Sectio 4, the performace of the cofidece itervals for θ is ivestigated through a Mote Carlo simulatio study. Coclusios are provided i the fial sectio. Existig Cofidece Iterval I this sectio, we review the theorem ad corollary proposed by Wogkhao et al. (013) ad use these to costruct the exact cofidece iterval for θ.

3 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 119 Theorem 1. (Wogkhao et al., 013) Let Y,..., 1 Y be a radom sample of size from a ormal distributio with mea µ ad variace σ. The estimator of θ is 1 1 where θˆ Y Y Y. The expectatio of ˆ θ ad j 1 j σ variatio, τ is kow, are respectively µ k ˆ ( k)! τ E( θ) θ 1+ k k 1 k! ˆ θ whe a coefficiet of (1) ad k ˆ (k + 1)! τ ( ). k k 1 k! E θ θ Proof of Theorem 1 See Wogkhao et al. (013) ˆ k From (1), lim E( ˆ θ ( k)! τ θ) θ ad E θ, w where w 1 +. k k 1 k! Thus, the ubiased estimator of θ is ˆ θ 1. w wy Corollary 1. From Theorem 1, θ ˆ var( θ) τ. Proof of Corollary 1 See Wogkhao et al. (013) Now we will use the fact that, from the cetral limit theorem, ˆ θ θ Z N(0,1). var( ˆ θ ) Based o Theorem 1 ad Corollary 1, we get Z ˆ θ θ w N(0,1). () θ τ Therefore, the 100(1 α)% exact cofidece iterval for θ based o Equatio () is ˆ θ ˆ θ CIexact ± z1 α / τ, w k ( k)! τ where w 1+ k k 1 k! ad z 1 α is the 100(1 α / ) percetile of the stadard / ormal distributio.

4 10 Wararit Paichkitkosolkul 3 Proposed Cofidece Iterval To fid a simple approximate expressio for the expectatio of ˆ, θ we use a Taylor series expasio of 1 y aroud µ : 1 y ( ) ( ) ( ) y µ + y µ + O y µ y y y y y y y y µ µ µ (3) Theorem. Let with mea µ ad variace Y,..., 1 Y be a radom sample of size from a ormal distributio σ. The estimator of θ is θˆ Y 1 where 1 Y Yi i 1 The approximate expectatio ad variace of ˆ θ whe a coefficiet of variatio, σ τ is kow, are respectively µ ˆ 1 τ E( θ ) 1+ (3) µ ad var( ˆ θ θ) τ. (4). Proof of Theorem. Cosider radom variable Y where Y has support (0, ). Let ˆ 1. θ Y Fid approximatios for E( ˆ θ ) ad var( ˆ θ ) usig Taylor series expasio of ˆ θ aroud µ as i Equatio (3). The mea of ˆ θ ca be foud by applyig the expectatio operator to the idividual terms (igorig all terms higher tha two), E( ˆ θ ) E 1 Y E E ( Y EY ( )) E ( Y EY ( )) O ( ) Y Y Y Y Y 3 µ µ µ var( Y ) µ ( EY ( )) 1 var( Y + ) µ µ 1 σ 1 + µ µ 1 τ 1 +. µ 3 (4)

5 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 11 A approximatio of the variace of ˆ θ is obtaied by usig the first-order terms of the Taylor series expasio: var( ˆ θ ) var 1 Y 1 1 E E Y Y 1 1 E Y µ E + ( Y EY ( )) µ Y Y µ 1 Y Y var( Y ) 4 µ σ 4 µ var( Y ) µ µ θ τ. (5) It is clear from Equatio (4) that ˆ θ is asymptotically ubiased ( lim E( ˆ θ) θ) ˆ θ τ ad E θ, v where v 1 +. Therefore, the ubiased estimator of θ is From Equatio (5), ˆ θ is cosistet ˆ ( θ ) lim var( ) 0. ˆ θ 1. v vy We the apply the cetral limit theorem ad Theorem, Z ˆ θ θ v N(0,1). θ τ Therefore, it is easily see that the (1 α)100% approximate cofidece iterval for θ is CIapprox ˆ θ ± z v 1 α / ˆ θ τ,

6 1 Wararit Paichkitkosolkul τ where v 1+ ad z1 α / is the 100(1 α / ) percetile of the stadard ormal distributio. 4 Simulatio Study A Mote Carlo simulatio was coducted usig the R statistical software [16] versio 3..1 to compare the estimated coverage probabilities of the ew proposed cofidece iterval ad the exact cofidece iterval. Source code is available i Appedix. The estimated coverage probability (based o M replicates) are give by 1 α #( L θ U) / M, where #( L θ U) deotes the umber of simulatio rus for which the true reciprocal of a ormal mea θ lies withi the cofidece iterval. From two previous sectios, we foud that the legths of both cofidece itervals are equal to z ˆ 1 α / θτ / which the expected legths are ot cosidered i simulatio study. The sets of ormal data were geerated with θ 0.1, 0., 0.5, 1, 5 ad 10, ad the coefficiet of variatio τ 0.05, 0.1, 0., 0.33, 0.5 ad The sample sizes were set at 10, 0, 30, 50, 100 ad 500. The umber of simulatio rus was 10,000 ad the omial cofidece level 1 α was fixed at The results are demostrated i Figure 1 ad Tables 1-4. Both cofidece itervals have estimated coverage probabilities close to the omial cofidece level for almost situatios. However, the estimated coverage probabilities of the exact cofidece iterval are very poor whe the coefficiet of variatio τ is close to 1 ad small sample sizes. Additioally, the estimated coverage probabilities of the cofidece itervals do ot icrease or decrease accordig to the values of τ ad. The estimated coverage probabilities of the proposed cofidece iterval are ot sigificatly differet from these of the exact cofidece iterval i ay situatio. However, the approximate cofidece iterval is very easy to calculate compared with the exact cofidece iterval because the exact cofidece iterval is based o a ifiite summatio.

7 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 13 Coverage Probabilities Exact Approx 0.1 Coverage Probabilities Exact Approx Coverage Probabilities Exact Approx 0.5 Coverage Probabilities Exact Approx Coverage Probabilities Exact Approx 5 Coverage Probabilities Exact Approx Figure 1: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe 30 (solid lie) ad 100 (dash lie)

8 14 Wararit Paichkitkosolkul Table 1: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 0.1 ad 0.. τ θ 0.1 θ 0. Exact Approx. Exact Approx

9 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 15 Table : Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 0.5 ad 1. τ θ 0.5 θ 1 Exact Approx. Exact Approx

10 16 Wararit Paichkitkosolkul Table 3: Estimated coverage probabilities of cofidece itervals for the reciprocal of a ormal mea with a kow coefficiet of variatio whe θ 5 ad 10. τ θ 5 θ 10 Exact Approx. Exact Approx A Illustrative Example To illustrate a example of two cofidece iterval for the reciprocal of a ormal mea proposed i the previous sectio, we used the weights (i kilograms) of 61 oe-moth old ifats listed as follows:

11 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea The data were take from the study by Ziegler et al. (007) (cited i Ledolter ad Hogg, 010, p.87). From past experiece, we assume that the coefficiet of variatio of the weights of 61 oe-moth old ifats is about The histogram, desity plot, Box-ad-Whisker plot ad ormal quatile-quatile plot are displayed i Figure. Algorithm 1 shows the result of the Shapiro-Wilk ormality test. (a) (b) Frequecy Desity weight weight (c) Sample Quatiles (d) Theoretical Quatiles Figure : (a) Histogram, (b) desity plot, (c) Box-ad-Whisker plot ad (d) ormal quatile-quatile plot of the weight of a oe-moth old ifat

12 18 Wararit Paichkitkosolkul Shapiro-Wilk ormality test data: weight W 0.978, p-value Algorithm 1: Shapiro-Wilk test for ormality of the weight of a oe-moth old ifat The 95% exact ad approximate cofidece itervals for the reciprocal of a ormal mea are calculated ad reported i Table 4. The lower ad upper limits of the both cofidece itervals are ot differet. Table 4: The 95% cofidece itervals for the reciprocal of a ormal mea of the weight of a oe-moth old ifat. Methods Cofidece Itervals Lower Limit Upper Limit Legths Exact Approximate Coclusios I this paper, we proposed a approximate cofidece iterval for the reciprocal of a ormal populatio mea with a kow coefficiet of variatio. Normally, this arises whe the coefficiet of variatio of the cotrol group is kow. The approximate cofidece iterval proposed uses the approximatio of the expectatio ad variace of the estimator. The proposed ew cofidece iterval is compared with the exact cofidece iterval costructed by Wogkhao et al. (013) through a Mote Carlo simulatio study. The approximate cofidece iterval performs as efficietly as the exact cofidece iterval i terms of coverage probability. Moreover, approximate cofidece iterval also is easy to compute compared with the exact cofidece iterval. Appedix: Source R code for all cofidece itervals ci.exact <- fuctio(y,tao,alpha) { <- legth(y) ybar <- mea(y) zeta.hat <- 1/ybar w <- cal.w(tao,)

13 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea 19 } z <- qorm(1-alpha/) T1 <- (tao^)/(*(ybar^)) lower <- (zeta.hat/w)-z*sqrt(t1) upper <- (zeta.hat/w)+z*sqrt(t1) out <- cbid(lower,upper) retur(out) ci.approx <- fuctio(y,tao,alpha) { <- legth(y) ybar <- mea(y) zeta.hat <- 1/ybar v <- 1+(tao^)/ z <- qorm(1-alpha/) T1 <- ((zeta.hat^)*(tao^))/ lower <- (zeta.hat/v)-z*sqrt(t1) upper <- (zeta.hat/v)+z*sqrt(t1) out <- cbid(lower,upper) retur(out) } cal.w <- fuctio(tao,) { temp <- rep(0,50) for (k i 1:50) { temp[k] <- (factorial(*k)/((^k)*factorial(k)))*(((tao^)/)^k) } w <- 1+sum(temp) retur(w) } Ackowledgemets The author is grateful to two aoymous referees for their valuable commets ad commets, which have sigificatly ehaced the quality ad presetatio of this paper. The author is also thakful for the support i the form of the research fuds awarded by Thammasat Uiversity. Refereces [1] Ihaka, R. ad Getlema, R. (1996): R: A laguage for data aalysis ad graphics. Joural of Computatioal ad Graphical Statistics, 5,

14 130 Wararit Paichkitkosolkul [] Lamaa, E., Romao, G. ad Sgrbi, C. (1981): Curvature measuremets i uclear emulsios. Nuclear Istrumets ad Methods, 187, [3] Ledolter, L., Hogg, R.V. (010): Applied Statistics for Egieers ad Physical Scietists, Pearso, New Jersey. [4] Weerahadi, S. (1993): Geeralized cofidece itervals, Joural of the America Statistical Associatio, 88, [5] Withers, C.S. ad Nadarajah, S. (013): Estimators for the iverse powers of a ormal mea, Joural of Statistical Plaig ad Iferece, 143, [6] Wogkhao, A., Niwitpog, S. ad Niwitpog, S. (013): Cofidece iterval for the iverse of a ormal mea with a kow coefficiet of variatio. Iteratioal Joural of Mathematical, Computatioal, Statistical, Natural ad Physical Egieer, 7, [7] Zama, A. (1981): Estimators without momets: the case of the reciprocal of a ormal mea. Joural of Ecoometrics, 15, [8] Zama, A. (1985): Admissibility of the maximum likelihood estimate of the reciprocal of a ormal mea with a class of zero-oe loss fuctios. Sakhyā: The Idia Joural of Statistics, Series A, 47, [9] Ziegler, E., Nelso, S.E., Jeter, J.M. (007): Early iro supplemetatio of breastfed ifats, Departmet of Pediatrics, Uiversity of Iowa, Iowa City, USA.

Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation

Confidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology

More information

Research Article Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean

Research Article Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean Probability ad Statistics Volume 2013, Article ID 324940, 11 pages http://dx.doi.org/10.1155/2013/324940 Research Article Cofidece Itervals for the Coefficiet of Variatio i a Normal Distributio with a

More information

Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values

Confidence interval for the two-parameter exponentiated Gumbel distribution based on record values Iteratioal Joural of Applied Operatioal Research Vol. 4 No. 1 pp. 61-68 Witer 2014 Joural homepage: www.ijorlu.ir Cofidece iterval for the two-parameter expoetiated Gumbel distributio based o record values

More information

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n. Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

More information

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.

Statistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample. Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized

More 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

Access to the published version may require journal subscription. Published with permission from: Elsevier.

Access to the published version may require journal subscription. Published with permission from: Elsevier. This is a author produced versio of a paper published i Statistics ad Probability Letters. This paper has bee peer-reviewed, it does ot iclude the joural pagiatio. Citatio for the published paper: Forkma,

More information

Department of Mathematics

Department of Mathematics Departmet of Mathematics Ma 3/103 KC Border Itroductio to Probability ad Statistics Witer 2017 Lecture 19: Estimatio II Relevat textbook passages: Larse Marx [1]: Sectios 5.2 5.7 19.1 The method of momets

More information

Maximum likelihood estimation from record-breaking data for the generalized Pareto distribution

Maximum likelihood estimation from record-breaking data for the generalized Pareto distribution METRON - Iteratioal Joural of Statistics 004, vol. LXII,. 3, pp. 377-389 NAGI S. ABD-EL-HAKIM KHALAF S. SULTAN Maximum likelihood estimatio from record-breakig data for the geeralized Pareto distributio

More information

This is a author produced versio of a paper published i Commuicatios i Statistics - Theory ad Methods. This paper has bee peer-reviewed ad is proof-corrected, but does ot iclude the joural pagiatio. Citatio

More information

Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution

Double Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.

More information

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4

MATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4 MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.

More 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

Random Variables, Sampling and Estimation

Random Variables, Sampling and Estimation Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos

A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES. Dennis D. Boos .- A NEW METHOD FOR CONSTRUCTING APPROXIMATE CONFIDENCE INTERVALS FOR M-ESTU1ATES by Deis D. Boos Departmet of Statistics North Carolia State Uiversity Istitute of Statistics Mimeo Series #1198 September,

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

MATH/STAT 352: Lecture 15

MATH/STAT 352: Lecture 15 MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet

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

Stat 421-SP2012 Interval Estimation Section

Stat 421-SP2012 Interval Estimation Section Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible

More information

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors

ECONOMETRIC THEORY. MODULE XIII Lecture - 34 Asymptotic Theory and Stochastic Regressors ECONOMETRIC THEORY MODULE XIII Lecture - 34 Asymptotic Theory ad Stochastic Regressors Dr. Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Asymptotic theory The asymptotic

More information

1 Inferential Methods for Correlation and Regression Analysis

1 Inferential Methods for Correlation and Regression Analysis 1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet

More information

Bootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests

Bootstrap Intervals of the Parameters of Lognormal Distribution Using Power Rule Model and Accelerated Life Tests Joural of Moder Applied Statistical Methods Volume 5 Issue Article --5 Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem

More 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

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.

Inferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process. Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike

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

Output Analysis and Run-Length Control

Output Analysis and Run-Length Control IEOR E4703: Mote Carlo Simulatio Columbia Uiversity c 2017 by Marti Haugh Output Aalysis ad Ru-Legth Cotrol I these otes we describe how the Cetral Limit Theorem ca be used to costruct approximate (1 α%

More information

Simulation. Two Rule For Inverting A Distribution Function

Simulation. Two Rule For Inverting A Distribution Function Simulatio Two Rule For Ivertig A Distributio Fuctio Rule 1. If F(x) = u is costat o a iterval [x 1, x 2 ), the the uiform value u is mapped oto x 2 through the iversio process. Rule 2. If there is a jump

More information

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700 Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2

More information

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15

17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15 17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

More 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

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More information

Lecture 33: Bootstrap

Lecture 33: Bootstrap Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece

More information

BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov

BIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios

More information

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain

Since X n /n P p, we know that X n (n. Xn (n X n ) Using the asymptotic result above to obtain an approximation for fixed n, we obtain Assigmet 9 Exercise 5.5 Let X biomial, p, where p 0, 1 is ukow. Obtai cofidece itervals for p i two differet ways: a Sice X / p d N0, p1 p], the variace of the limitig distributio depeds oly o p. Use the

More information

Properties and Hypothesis Testing

Properties and Hypothesis Testing Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.

More information

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

More information

32 estimating the cumulative distribution function

32 estimating the cumulative distribution function 32 estimatig the cumulative distributio fuctio 4.6 types of cofidece itervals/bads Let F be a class of distributio fuctios F ad let θ be some quatity of iterest, such as the mea of F or the whole fuctio

More information

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

Introduction to Econometrics (3 rd Updated Edition) Solutions to Odd- Numbered End- of- Chapter Exercises: Chapter 3 Itroductio to Ecoometrics (3 rd Updated Editio) by James H. Stock ad Mark W. Watso Solutios to Odd- Numbered Ed- of- Chapter Exercises: Chapter 3 (This versio August 17, 014) 015 Pearso Educatio, Ic. Stock/Watso

More information

Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be

More 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

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9

BIOS 4110: Introduction to Biostatistics. Breheny. Lab #9 BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous

More information

Stochastic Simulation

Stochastic Simulation Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso

More information

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes.

Direction: This test is worth 250 points. You are required to complete this test within 50 minutes. Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely

More information

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M.

Confidence intervals summary Conservative and approximate confidence intervals for a binomial p Examples. MATH1005 Statistics. Lecture 24. M. MATH1005 Statistics Lecture 24 M. Stewart School of Mathematics ad Statistics Uiversity of Sydey Outlie Cofidece itervals summary Coservative ad approximate cofidece itervals for a biomial p The aïve iterval

More information

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract

Goodness-Of-Fit For The Generalized Exponential Distribution. Abstract Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated

More information

A Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution

A Note on Box-Cox Quantile Regression Estimation of the Parameters of the Generalized Pareto Distribution A Note o Box-Cox Quatile Regressio Estimatio of the Parameters of the Geeralized Pareto Distributio JM va Zyl Abstract: Makig use of the quatile equatio, Box-Cox regressio ad Laplace distributed disturbaces,

More information

Lecture 19: Convergence

Lecture 19: Convergence Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may

More information

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS

THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS R775 Philips Res. Repts 26,414-423, 1971' THE SYSTEMATIC AND THE RANDOM. ERRORS - DUE TO ELEMENT TOLERANCES OF ELECTRICAL NETWORKS by H. W. HANNEMAN Abstract Usig the law of propagatio of errors, approximated

More information

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical

More information

Estimation of the Mean and the ACVF

Estimation of the Mean and the ACVF Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators

More information

Minimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions

Minimax Estimation of the Parameter of Maxwell Distribution Under Different Loss Functions America Joural of heoretical ad Applied Statistics 6; 5(4): -7 http://www.sciecepublishiggroup.com/j/ajtas doi:.648/j.ajtas.654.6 ISSN: 6-8999 (Prit); ISSN: 6-96 (Olie) Miimax Estimatio of the Parameter

More information

Discriminating between Generalized Exponential and Gamma Distributions

Discriminating between Generalized Exponential and Gamma Distributions Joural of Probability ad Statistical Sciece 4, 4-47, Aug 6 Discrimiatig betwee Geeralized Expoetial ad Gamma Distributios Orawa Supapueg Kamo Budsaba Adrei I Volodi Praee Nilkor Thammasat Uiversity Uiversity

More information

Estimation of Gumbel Parameters under Ranked Set Sampling

Estimation of Gumbel Parameters under Ranked Set Sampling Joural of Moder Applied Statistical Methods Volume 13 Issue 2 Article 11-2014 Estimatio of Gumbel Parameters uder Raked Set Samplig Omar M. Yousef Al Balqa' Applied Uiversity, Zarqa, Jorda, abuyaza_o@yahoo.com

More information

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons

Statistical Analysis on Uncertainty for Autocorrelated Measurements and its Applications to Key Comparisons Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA Outlies. Itroductio.

More information

Sample Size Determination (Two or More Samples)

Sample Size Determination (Two or More Samples) Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie

More information

Modied moment estimation for the two-parameter Birnbaum Saunders distribution

Modied moment estimation for the two-parameter Birnbaum Saunders distribution Computatioal Statistics & Data Aalysis 43 (23) 283 298 www.elsevier.com/locate/csda Modied momet estimatio for the two-parameter Birbaum Sauders distributio H.K.T. Ng a, D. Kudu b, N. Balakrisha a; a Departmet

More information

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01

ENGI 4421 Confidence Intervals (Two Samples) Page 12-01 ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly

More information

Central limit theorem and almost sure central limit theorem for the product of some partial sums

Central limit theorem and almost sure central limit theorem for the product of some partial sums Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp. 289 294. Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

More 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

Statistical Intervals for a Single Sample

Statistical Intervals for a Single Sample 3/5/06 Applied Statistics ad Probability for Egieers Sixth Editio Douglas C. Motgomery George C. Ruger Chapter 8 Statistical Itervals for a Sigle Sample 8 CHAPTER OUTLINE 8- Cofidece Iterval o the Mea

More information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

Discrete Mathematics for CS Spring 2008 David Wagner Note 22

Discrete Mathematics for CS Spring 2008 David Wagner Note 22 CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

Extreme Value Charts and Analysis of Means (ANOM) Based on the Log Logistic Distribution

Extreme Value Charts and Analysis of Means (ANOM) Based on the Log Logistic Distribution Joural of Moder Applied Statistical Methods Volume 11 Issue Article 0 11-1-01 Extreme Value Charts ad Aalysis of Meas (ANOM) Based o the Log Logistic istributio B. Sriivasa Rao R.V.R & J.C. College of

More information

Power Comparison of Some Goodness-of-fit Tests

Power Comparison of Some Goodness-of-fit Tests Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School 7-6-2016 Power Compariso of Some Goodess-of-fit Tests Tiayi Liu tliu019@fiu.edu DOI: 10.25148/etd.FIDC000750

More 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

Introductory statistics

Introductory statistics CM9S: Machie Learig for Bioiformatics Lecture - 03/3/06 Itroductory statistics Lecturer: Sriram Sakararama Scribe: Sriram Sakararama We will provide a overview of statistical iferece focussig o the key

More information

Chapter 8: Estimating with Confidence

Chapter 8: Estimating with Confidence Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig

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

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

Abstract. Ranked set sampling, auxiliary variable, variance.

Abstract. Ranked set sampling, auxiliary variable, variance. Hacettepe Joural of Mathematics ad Statistics Volume (), 1 A class of Hartley-Ross type Ubiased estimators for Populatio Mea usig Raked Set Samplig Lakhkar Kha ad Javid Shabbir Abstract I this paper, we

More 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

Investigating the Significance of a Correlation Coefficient using Jackknife Estimates

Investigating the Significance of a Correlation Coefficient using Jackknife Estimates Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN 2307-4531 (Prit & Olie) http://gssrr.org/idex.php?joural=jouralofbasicadapplied ---------------------------------------------------------------------------------------------------------------------------

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

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

Comparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading

Comparing Two Populations. Topic 15 - Two Sample Inference I. Comparing Two Means. Comparing Two Pop Means. Background Reading Topic 15 - Two Sample Iferece I STAT 511 Professor Bruce Craig Comparig Two Populatios Research ofte ivolves the compariso of two or more samples from differet populatios Graphical summaries provide visual

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

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

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

More information

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2

2 1. The r.s., of size n2, from population 2 will be. 2 and 2. 2) The two populations are independent. This implies that all of the n1 n2 Chapter 8 Comparig Two Treatmets Iferece about Two Populatio Meas We wat to compare the meas of two populatios to see whether they differ. There are two situatios to cosider, as show i the followig examples:

More information

This is an introductory course in Analysis of Variance and Design of Experiments.

This is an introductory course in Analysis of Variance and Design of Experiments. 1 Notes for M 384E, Wedesday, Jauary 21, 2009 (Please ote: I will ot pass out hard-copy class otes i future classes. If there are writte class otes, they will be posted o the web by the ight before class

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

TAMS24: Notations and Formulas

TAMS24: Notations and Formulas TAMS4: Notatios ad Formulas Basic otatios ad defiitios X: radom variable stokastiska variabel Mea Vätevärde: µ = X = by Xiagfeg Yag kpx k, if X is discrete, xf Xxdx, if X is cotiuous Variace Varias: =

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

The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples

The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples The performace of uivariate goodess-of-fit tests for ormality based o the empirical characteristic fuctio i large samples By J. M. VAN ZYL Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity

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

(7 One- and Two-Sample Estimation Problem )

(7 One- and Two-Sample Estimation Problem ) 34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:

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

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals

More information

1.010 Uncertainty in Engineering Fall 2008

1.010 Uncertainty in Engineering Fall 2008 MIT OpeCourseWare http://ocw.mit.edu.00 Ucertaity i Egieerig Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu.terms. .00 - Brief Notes # 9 Poit ad Iterval

More information

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight)

Tests of Hypotheses Based on a Single Sample (Devore Chapter Eight) Tests of Hypotheses Based o a Sigle Sample Devore Chapter Eight MATH-252-01: Probability ad Statistics II Sprig 2018 Cotets 1 Hypothesis Tests illustrated with z-tests 1 1.1 Overview of Hypothesis Testig..........

More information

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes.

Direction: This test is worth 150 points. You are required to complete this test within 55 minutes. Term Test 3 (Part A) November 1, 004 Name Math 6 Studet Number Directio: This test is worth 10 poits. You are required to complete this test withi miutes. I order to receive full credit, aswer each problem

More information

Approximations to the Distribution of the Sample Correlation Coefficient

Approximations to the Distribution of the Sample Correlation Coefficient World Academy of Sciece Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:5 No:4 0 Approximatios to the Distributio of the Sample Correlatio Coefficiet Joh N Haddad ad

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

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

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