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

Size: px
Start display at page:

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

Transcription

1 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 Yarmou Uiversity, Irbid, Jorda, m_hassab@hotmail.com Follow this ad additioal wors at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Ebrahem, Mohammed Al-Ha (5) "Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests," Joural of Moder Applied Statistical Methods: Vol. 5 : Iss., Article. DOI:.37/masm/63546 Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods Copyright 6 JMASM, Ic. November, 6, Vol. 5, No., /5/$95. Bootstrap Itervals of the Parameters of Logormal Distributio Usig Power Rule Model ad Accelerated Life Tests Mohammed Al-Ha Ebrahem Departmet of Statistics Yarmou Uiversity Assumed that the distributio of the lifetime of ay uit follows a logormal distributio with parameters μ adσ. Also, assume that the relatioship betwee μ ad the stress level V is give by the power rule model. Several types of bootstrap itervals of the parameters were studied ad their performace was studied usig simulatios ad compared i term of attaimet of the omial cofidece level, symmetry of lower ad upper error rates ad the expected width. Coclusios ad recommedatios are give. Key words: Power rule model, logormal distributio, bootstrap itervals, accelerated life test. Itroductio The logormal distributio has may special features that allowed it to be used as a model i various real life applicatios. I particular, it is used i aalyzig biological data (Koch, 966), ad for aalyzig data i worplace exposure to cotamiats (Lyles & Kupper, 997). It is also of importace i modelig lifetimes of products ad idividuals (Lawless, 98). Various other motivatios ad applicatios of the logormal distributio may also be foud (see Johso et. al., 994, Scheider, 986). I a life testig experimet, the problem is that most uits have a very log life uder the ormal coditios. Therefore, by the time the experimet is completed ad a estimate of the reliability is obtaied, the results will be outdated. To overcome this delay, accelerated life testig was itroduced (Ma. et. al., 974). I a accelerated life testig experimet a certai umber of uits are subected to a stress that is higher tha the ormal stress. The Mohammed Al-Ha Ebrahem is Assistat Professor i the Departmet of Statistics at Yarmou Uiversity, Irbid-Jorda. His research iterest is i reliability, accelerated life test ad o-parametric regressio models. m_hassab@hotmail.com. experimet is repeated uder differet values of stress. I order to do so, some relatioship betwee the parameters of the time to failure distributio of the uit ad the correspodig stress level must be postulated. It is assumed that desity fuctio of the time to failure of a uit depeds o oe parameter sayθ, ad the eviromet depeds o oe stress V ad that the relatioship betwee C θ ad V is give by θ = where C ad P P V are positive costats. This relatioship is ow as the power rule model. Cosider the iterval estimatio for the parameters of the logormal distributio after reparametrizig the locatio parameter μ as a fuctio of the stress V usig power rule model. The performace of the bootstrap ad Jacife itervals (Efro & Tibshirai, 993) i term of attaimet of the omial cofidece level, symmetry of lower ad upper error rates ad the expected width of the itervals will be compared. The Model ad The Maximum lielihood Estimatio It is assumed that the lifetime (T) of ay uit follows a logormal distributio with locatio parameter μ ad scale parameter σ. The probability desity fuctio of T is give by (Lawless, 98): 38

3 38 PARAMETERS OF LOGNORMAL DISTRIBUTION f () t = exp tσ π ( lt μ) σ, < t <. () The locatio parameter μ was reparameterized as a fuctio of the stress V usig the power rule C model μ =, therefore c ad σ are the ew V P parameters of the model. The uow parameters c ad σ were estimated usig complete samples. The -th sample is obtaied by usig uits ad the value V for the stress, =,,.,. The lielihood fuctio of the complete samples is give by: ad σˆ Cˆ = i= = = l t i / v = / v p p cˆ l ti p = i= v = (4) (5) L( μ, σ ) = e = σ σ = i= (π ) (l t = i μ ) Π Π t = i= i () C Usig the power rule model μ = =,, P V.,, the lielihood fuctio is give by: L( C, σ ) = e = σ σ = i= = (π ) c l ti p v o Π Πt = i= i (3) It is easy to show that the Maximum lielihood estimators of C ad σ are give by: It is obvious that Ĉ is a ubiased estimator of C while σˆ is a biased estimator of σ. The Percetile Iterval The methods of derivig cofidece itervals preseted i this sectio ad sectio 4 are based o the parametric bootstrap approach (Efro & Tibshirai, 993); they are costructed by resamplig from the estimated parametric distributio. To costruct the percetile iterval, a simulatio of the bootstrap distributio of Ĉ ad σˆ is doe by resamplig from the parametric model of the origial data. That is, a B bootstrap sample is geerated ad for each sample Ĉ ad σˆ are calculated usig equatio (4) ad (5) respectively. The calculated values * * are deoted by Ĉ ad ˆ σ. Let Ĝ deotes the cumulative * distributio of Ĉ, the ( α ) % percetile iterval of C is ˆ α ˆ α G, G, similarly let Ĝ deotes the cumulative * distributio of ˆ σ, the ( α ) % percetile iterval of σ is ˆ α ˆ α G, G.

4 MOHAMMED AL-HAJ EBARAHEM 383 The Bias Corrected ad Accelerated Iterval (BCa Iterval) The bias corrected ad accelerated iterval is costructed by calculatig two umbers â ad ẑ called the accelerated ad the bias correctio factor respectively, they are calculated usig the followig formulas ( Cˆ(.) Cˆ( i) ) i= a ˆ = (6) 3 / 6 ( Cˆ(.) Cˆ( i) ) i= where C ˆ( i ) is the maximum lielihood estimator of C usig the origial data excludig the i-th Cˆ( i) i= observatio ad C ˆ (.) =, = = The value of ẑ is give by 3 Cˆ * #( < Cˆ) zˆ = Φ (7) B where Φ(.) is the stadard ormal cumulative distributio fuctio. The ( α ) % BCa ˆ ˆ G α, G α where ( ) iterval of C is ( ) ( ) ad α = Φ z ˆ zˆ + zα / + aˆ( zˆ + z α / ) zˆ + z α / α = Φ + z ˆ (8) aˆ( zˆ + z α / ) where z α is the α quatile of the stadard ormal distributio. I the same way, the ( α ) % BCa iterval of σ ca be costructed. Jacife Iterval A ( α ) % Jacife iterval of C ( Efro ad Tibshirai, 993) is costructed as follows:. Cˆ(.) ± Z ˆ, where (α / ) S Jac ( Cˆ(.) Cˆ( i) ) Sˆ Jac =, Ĉ (.), C ˆ( i ) i= ad were defied i sectio 4. Similarly, the ( α ) % Jacife iterval of σ by replacig C by σ i the above iterval. Simulatio Study A simulatio study is coducted to ivestigate the performace of the itervals discussed i sectios 3, 4 ad 5 above. The idices of the simulatio study are: : The umber of logormal populatios, i this study =. : Sample size from the first logormal populatio, i this study = 5,, 3. : Sample size from the secod logormal populatio, i this study = 5,, 3. C : Parameter of the power rule model, i this study C = 3. P : I this study P =.3. V : The value of stress for the first logormal populatio, i this study V =. V : The value of stress for the secod logormal populatio, i this study V =. σ : I this study σ =. B: The umber of bootstrap samples, i this study B =. For each combiatio of ad samples are geerated ad a ( α ) % Percetile iterval is costructed, BCa iterval ad Jacife iterval for C ad σ. Two values are cosidered for α,.5 ad.. The followig were obtaied for each iterval: - The expected width (IW): the average of widths of the itervals. - Lower error rate (LER): the fractio of itervals that fall etirely above the true parameter.

5 384 PARAMETERS OF LOGNORMAL DISTRIBUTION 3- Upper error rate (UER): the fractio of itervals that fall etirely below the true parameter. 4- Total error rate (TER): the fractio of itervals that did ot cotai the true parameter value. Results ad Coclusios The results are give i tables. Table has simulatio results of the percetile iterval of the parameter C usig α =. 5. Table has simulatio results of the BCa iterval of the parameter C usig α =. 5. Table 3 has simulatio results of the Jacife iterval of the parameter C usig α =. 5. Table 4 has simulatio results of the percetile iterval of the parameter C usig α =.. Table 5 has simulatio results of the BCa iterval of the parameter C usigα =.. Table 6 has simulatio results of the Jacife iterval of the parameter C usig α =.. Table 7 has simulatio results of the percetile iterval of the parameter σ usig α =. 5. Table 8 has simulatio results of the BCa iterval of the parameter σ usig α =.5. Table 9 has simulatio results of the Jacife iterval of the parameter σ usig α =.5. Table has simulatio results of the percetile iterval of the parameter σ usig α =.. Table has simulatio results of the BCa iterval of the parameter σ usig α =.. Table has simulatio results of the Jacife iterval of the parameter σ usig α =.. From these results the followig ca be cocluded: For the parameter C, the three itervals have almost the same expected width, ad the expected width decreases as the sample sizes icreases. I term of attaimet of coverage probability ad symmetry of lower ad upper rates, the three itervals behave i the same way. It is recommeded that the Jacife iterval be used because its calculatio is simpler tha the BCa ad the percetile itervals. For the parameter σ, the expected width for the percetile iterval is early smaller tha the other two itervals. O the other had, i term of attaimet of coverage probability ad symmetry of lower ad upper rates, the BCa iterval behaves the best. It is therefore recommeded that the BCa iterval be used i this case.

6 MOHAMMED AL-HAJ EBARAHEM 385 Table. Percetile Iterval of the parameter C with α =. 5 IW LER UER TER Table. BCa Iterval of the parameter C with α =. 5 IW LER UER TER Table 3. Jacife Iterval of the parameter C with α =. 5 IW LER UER TER

7 386 PARAMETERS OF LOGNORMAL DISTRIBUTION Table 4. Percetile Iterval of the parameter C with α =. IW LER UER TER Table 5. BCa Iterval of the parameter C with α =. IW LER UER TER Table 6. Jacife Iterval of the parameter C with α =. IW LER UER TER

8 MOHAMMED AL-HAJ EBARAHEM 387 Table 7. Percetile Iterval of the parameter σ with α =. 5 IW LER UER TER Table 8. BCa Iterval of the parameter σ with α =. 5 IW LER UER TER Table 9. Jacife Iterval of the parameter σ with α =. 5 IW LER UER TER

9 388 PARAMETERS OF LOGNORMAL DISTRIBUTION Table. Percetile Iterval of the parameter σ with α =. IW LER UER TER Table. BCa Iterval of the parameter σ with α =. IW LER UER TER Table. Jacife Iterval of the parameter σ with α =. IW LER UER TER

10 MOHAMMED AL-HAJ EBARAHEM 389 Refereces Efro, B. & Tibshirai, R. (993). A itroductio to the bootstrap. New Yor: Chapma ad Hall. Johso, N. L, Kotz, S., & Balarisha. (994). Cotiuous uivariate distributios: vol. New Yor: Wiley Koch, A.L. (966). The logarithm i biology. Joural of Theoretical Biology,, Lawless, J.F. (98). Statistical models ad methods for lifetime data. New Yor: Wiley. Lyles, R. H., Kupper, L. L., & Rappaport, S. M. (997). Assessig regulatory compliace of occupatioal exposures via the balaced oe-way radom effects ANOVA model. Joural of Agricultural, Biological, ad Evirometal Statististics,, Ma, N. R., Schafer, R. E., & Sigpurwalla, N. D. (974). Methods for statistical aalysis of reliability ad life data. New Yor: Joh Wiley ad Sos Ic. Scheider, H. (986). Trucated ad cesored samples from ormal populatios. New Yor: Marcel Deer. Efro, B. & Tibshirai, R. (993). A itroductio to the bootstrap. New Yor, N.Y.: Chapma ad Hall. Jeigs, D. (987). How do we udge cofidece itervals adequacy? The America Statisticia, 4(4), Kulldorff, G. (96). Estimatio from grouped ad partially grouped samples. New Yor, N.Y.: Wiley. Meeer, Jr, W. (986). Plaig Life tests i which uits are ispected for failure. IEEE Tras. o Reliability R-35, Pettitt, A. N., & Stephes, M. A. (977). The Kolmogrov-Smirov goodess-of-fit statistic with discrete ad grouped data. Techometrics, 9, 5.

Confidence Intervals For P(X less than Y) In The Exponential Case With Common Location Parameter

Confidence Intervals For P(X less than Y) In The Exponential Case With Common Location Parameter Joural of Moder Applied Statistical Methods Volume Issue Article 7 --3 Cofidece Itervals For P(X less tha Y I he Expoetial Case With Commo Locatio Parameter Ayma Baklizi Yarmouk Uiversity, Irbid, Jorda,

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

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

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

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

Control Charts for Mean for Non-Normally Correlated Data

Control Charts for Mean for Non-Normally Correlated Data Joural of Moder Applied Statistical Methods Volume 16 Issue 1 Article 5 5-1-017 Cotrol Charts for Mea for No-Normally Correlated Data J. R. Sigh Vikram Uiversity, Ujjai, Idia Ab Latif Dar School of Studies

More 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

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

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

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

Approximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece

More 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

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

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

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

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

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

CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION

CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION Petros Maravelakis, Joh Paaretos ad Stelios Psarakis Departmet of Statistics Athes Uiversity of Ecoomics ad Busiess 76 Patisio St., 4 34, Athes, GREECE. Itroductio

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

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

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

Comparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes

Comparison of Minimum Initial Capital with Investment and Non-investment Discrete Time Surplus Processes The 22 d Aual Meetig i Mathematics (AMM 207) Departmet of Mathematics, Faculty of Sciece Chiag Mai Uiversity, Chiag Mai, Thailad Compariso of Miimum Iitial Capital with Ivestmet ad -ivestmet Discrete Time

More 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

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

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

Resampling modifications for the Bagai test

Resampling modifications for the Bagai test Joural of the Korea Data & Iformatio Sciece Society 2018, 29(2), 485 499 http://dx.doi.org/10.7465/jkdi.2018.29.2.485 한국데이터정보과학회지 Resamplig modificatios for the Bagai test Youg Mi Kim 1 Hyug-Tae Ha 2 1

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

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 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

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY

ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY ANOTHER WEIGHTED WEIBULL DISTRIBUTION FROM AZZALINI S FAMILY Sulema Nasiru, MSc. Departmet of Statistics, Faculty of Mathematical Scieces, Uiversity for Developmet Studies, Navrogo, Upper East Regio, Ghaa,

More 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

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

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

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

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

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

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

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

A new distribution-free quantile estimator

A new distribution-free quantile estimator Biometrika (1982), 69, 3, pp. 635-40 Prited i Great Britai 635 A ew distributio-free quatile estimator BY FRANK E. HARRELL Cliical Biostatistics, Duke Uiversity Medical Ceter, Durham, North Carolia, U.S.A.

More 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

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

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

Bayesian Control Charts for the Two-parameter Exponential Distribution

Bayesian Control Charts for the Two-parameter Exponential Distribution Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com 2 Uiversity of the Free State Abstract By usig data that are the mileages

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

MidtermII Review. Sta Fall Office Hours Wednesday 12:30-2:30pm Watch linear regression videos before lab on Thursday

MidtermII Review. Sta Fall Office Hours Wednesday 12:30-2:30pm Watch linear regression videos before lab on Thursday Aoucemets MidtermII Review Sta 101 - Fall 2016 Duke Uiversity, Departmet of Statistical Sciece Office Hours Wedesday 12:30-2:30pm Watch liear regressio videos before lab o Thursday Dr. Abrahamse Slides

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

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

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES*

POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* Kobe Uiversity Ecoomic Review 50(2004) 3 POWER COMPARISON OF EMPIRICAL LIKELIHOOD RATIO TESTS: SMALL SAMPLE PROPERTIES THROUGH MONTE CARLO STUDIES* By HISASHI TANIZAKI There are various kids of oparametric

More 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

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

Stat 319 Theory of Statistics (2) Exercises

Stat 319 Theory of Statistics (2) Exercises Kig Saud Uiversity College of Sciece Statistics ad Operatios Research Departmet Stat 39 Theory of Statistics () Exercises Refereces:. Itroductio to Mathematical Statistics, Sixth Editio, by R. Hogg, J.

More information

Parameter Estimation In Weighted Rayleigh Distribution

Parameter Estimation In Weighted Rayleigh Distribution Joural of Moder Applied Statistical Methods Volume 6 Issue Article 4 December 07 Parameter Estimatio I Weighted Rayleigh Distributio M. Ajami Vali-e-Asr Uiversity of Rafsaja Rafsaja Ira m.ajami@vru.ac.ir

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

Pattern Classification

Pattern Classification Patter Classificatio All materials i these slides were tae from Patter Classificatio (d ed) by R. O. Duda, P. E. Hart ad D. G. Stor, Joh Wiley & Sos, 000 with the permissio of the authors ad the publisher

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

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

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

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

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

Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function

Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume 12, Number 4 (217), pp. 791-796 Research Idia Publicatios http://www.ripublicatio.com Bayesia ad E- Bayesia Method of Estimatio of Parameter

More 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

Estimation for Complete Data

Estimation for Complete Data Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of

More 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

5. Likelihood Ratio Tests

5. Likelihood Ratio Tests 1 of 5 7/29/2009 3:16 PM Virtual Laboratories > 9. Hy pothesis Testig > 1 2 3 4 5 6 7 5. Likelihood Ratio Tests Prelimiaries As usual, our startig poit is a radom experimet with a uderlyig sample space,

More information

THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR

THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR OF VAR Iteratioal Joural of Iovative Maagemet, Iformatio & Productio ISME Iteratioal c2013 ISSN 2185-5439 Volume 4, Number 1, Jue 2013 PP. 17-24 THE DATA-BASED CHOICE OF BANDWIDTH FOR KERNEL QUANTILE ESTIMATOR

More information

There is no straightforward approach for choosing the warmup period l.

There is no straightforward approach for choosing the warmup period l. B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.

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

3 Resampling Methods: The Jackknife

3 Resampling Methods: The Jackknife 3 Resamplig Methods: The Jackkife 3.1 Itroductio I this sectio, much of the cotet is a summary of material from Efro ad Tibshirai (1993) ad Maly (2007). Here are several useful referece texts o resamplig

More information

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution

Department of Civil Engineering-I.I.T. Delhi CEL 899: Environmental Risk Assessment HW5 Solution Departmet of Civil Egieerig-I.I.T. Delhi CEL 899: Evirometal Risk Assessmet HW5 Solutio Note: Assume missig data (if ay) ad metio the same. Q. Suppose X has a ormal distributio defied as N (mea=5, variace=

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

4.1 Non-parametric computational estimation

4.1 Non-parametric computational estimation Chapter 4 Resamplig Methods 4.1 No-parametric computatioal estimatio Let x 1,...,x be a realizatio of the i.i.d. r.vs X 1,...,X with a c.d.f. F. We are iterested i the precisio of estimatio of a populatio

More information

Estimating the Change Point of Bivariate Binomial Processes Experiencing Step Changes in Their Mean

Estimating the Change Point of Bivariate Binomial Processes Experiencing Step Changes in Their Mean Proceedigs of the 202 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet Istabul, Turey, July 3 6, 202 Estimatig the Chage Poit of Bivariate Biomial Processes Experiecig Step Chages i Their

More information

A Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors

A Generalized Class of Estimators for Finite Population Variance in Presence of Measurement Errors Joural of Moder Applied Statistical Methods Volume Issue Article 3 --03 A Geeralized Class of Estimators for Fiite Populatio Variace i Presece of Measuremet Errors Praas Sharma Baaras Hidu Uiversit, Varaasi,

More information

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable

Estimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya

More information

The new class of Kummer beta generalized distributions

The new class of Kummer beta generalized distributions The ew class of Kummer beta geeralized distributios Rodrigo Rossetto Pescim 12 Clarice Garcia Borges Demétrio 1 Gauss Moutiho Cordeiro 3 Saralees Nadarajah 4 Edwi Moisés Marcos Ortega 1 1 Itroductio Geeralized

More information

Lecture 2: Monte Carlo Simulation

Lecture 2: Monte Carlo Simulation STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?

More information

PROBABILITY DISTRIBUTION RELATIONSHIPS. Y.H. Abdelkader, Z.A. Al-Marzouq 1. INTRODUCTION

PROBABILITY DISTRIBUTION RELATIONSHIPS. Y.H. Abdelkader, Z.A. Al-Marzouq 1. INTRODUCTION STATISTICA, ao LXX,., 00 PROBABILITY DISTRIBUTION RELATIONSHIPS. INTRODUCTION I spite of the variety of the probability distributios, may of them are related to each other by differet kids of relatioship.

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

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

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

Improved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling

Improved Class of Ratio -Cum- Product Estimators of Finite Population Mean in two Phase Sampling Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 4 Issue 2 Versio.0 Year 204 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. (USA

More information

A proposed discrete distribution for the statistical modeling of

A proposed discrete distribution for the statistical modeling of It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical

More information

Estimating the Population Mean using Stratified Double Ranked Set Sample

Estimating the Population Mean using Stratified Double Ranked Set Sample Estimatig te Populatio Mea usig Stratified Double Raked Set Sample Mamoud Syam * Kamarulzama Ibraim Amer Ibraim Al-Omari Qatar Uiversity Foudatio Program Departmet of Mat ad Computer P.O.Box (7) Doa State

More information

ON THE RESAMPLING METHOD IN SAMPLE MEDIAN ESTIMATION

ON THE RESAMPLING METHOD IN SAMPLE MEDIAN ESTIMATION ACTA UNIVERSITATIS LODZIENSIS FOLIA OECONOMICA 3(302), 2014 * ON THE RESAMPLING METHOD IN SAMPLE MEDIAN ESTIMATION Abstract. Bootstrap is oe of the resamplig statistical methods. This method was proposed

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

GUIDELINES ON REPRESENTATIVE SAMPLING

GUIDELINES ON REPRESENTATIVE SAMPLING DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue

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

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

Pattern Classification

Pattern Classification Patter Classificatio All materials i these slides were tae from Patter Classificatio (d ed) by R. O. Duda, P. E. Hart ad D. G. Stor, Joh Wiley & Sos, 000 with the permissio of the authors ad the publisher

More information

ADVANCED SOFTWARE ENGINEERING

ADVANCED SOFTWARE ENGINEERING ADVANCED SOFTWARE ENGINEERING COMP 3705 Exercise Usage-based Testig ad Reliability Versio 1.0-040406 Departmet of Computer Ssciece Sada Narayaappa, Aeliese Adrews Versio 1.1-050405 Departmet of Commuicatio

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

CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION

CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION 4. Itroductio Numerous bivariate discrete distributios have bee defied ad studied (see Mardia, 97 ad Kocherlakota ad Kocherlakota, 99) based o various methods

More 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

Provläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE

Provläsningsexemplar / Preview TECHNICAL REPORT INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE TECHNICAL REPORT CISPR 16-4-3 2004 AMENDMENT 1 2006-10 INTERNATIONAL SPECIAL COMMITTEE ON RADIO INTERFERENCE Amedmet 1 Specificatio for radio disturbace ad immuity measurig apparatus ad methods Part 4-3:

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

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie

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

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

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen)

Goodness-of-Fit Tests and Categorical Data Analysis (Devore Chapter Fourteen) Goodess-of-Fit Tests ad Categorical Data Aalysis (Devore Chapter Fourtee) MATH-252-01: Probability ad Statistics II Sprig 2019 Cotets 1 Chi-Squared Tests with Kow Probabilities 1 1.1 Chi-Squared Testig................

More information

SYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA

SYSTEMATIC SAMPLING FOR NON-LINEAR TREND IN MILK YIELD DATA Joural of Reliability ad Statistical Studies; ISS (Prit): 0974-804, (Olie):9-5666 Vol. 7, Issue (04): 57-68 SYSTEMATIC SAMPLIG FOR O-LIEAR TRED I MILK YIELD DATA Tauj Kumar Padey ad Viod Kumar Departmet

More information

This chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples.

This chapter focuses on two experimental designs that are crucial to comparative studies: (1) independent samples and (2) matched pair samples. Chapter 9 & : Comparig Two Treatmets: This chapter focuses o two eperimetal desigs that are crucial to comparative studies: () idepedet samples ad () matched pair samples Idepedet Radom amples from Two

More information