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

Save this PDF as:
 WORD  PNG  TXT  JPG

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, 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 It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of

2 Joural of Moder Applied Statistical Methods 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. 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Stat 200 -Testing Summary Page 1

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

More information

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

Chapter 13, Part A Analysis of Variance and Experimental Design

Chapter 13, Part A Analysis of Variance and Experimental Design Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of

More information

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

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

More information

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

Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution

Tolerance Limits on Order Statistics in Future Samples Coming from the Two-Parameter Exponential Distribution America Joural of Theoretical ad Applied tatistics 6; 5(-): -6 Published olie November 8 5 (http://wwwsciecepublishiggroupcom/j/ajtas) doi: 648/jajtass65 IN: 36-8999 (Prit); IN: 36-96 (Olie) Tolerace imits

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

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

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

A General Family of Estimators for Estimating Population Variance Using Known Value of Some Population Parameter(s) Rajesh Sigh, Pakaj Chauha, Nirmala Sawa School of Statistics, DAVV, Idore (M.P.), Idia Floreti Smaradache Uiversity of New Meico, USA A Geeral Family of Estimators for Estimatig Populatio Variace Usig

More information

BOOTSTRAP BIAS CORRECTION IN SEMIPARAMETRIC ESTIMATION METHODS FOR ARFIMA MODELS

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

More information

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

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

More information

Introducing Sample Proportions

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

More information

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

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

More information

NCSS Statistical Software. Tolerance Intervals

NCSS Statistical Software. Tolerance Intervals Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided

More information

Final Examination Solutions 17/6/2010

Final Examination Solutions 17/6/2010 The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:

More information

STATISTICAL method is one branch of mathematical

STATISTICAL method is one branch of mathematical 40 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, AUGUST 07 Optimizig Forest Samplig by usig Lagrage Multipliers Suhud Wahyudi, Farida Agustii Widjajati ad Dea Oktaviati

More information

Testing Statistical Hypotheses with Fuzzy Data

Testing Statistical Hypotheses with Fuzzy Data Iteratioal Joural of Statistics ad Systems ISS 973-675 Volume 6, umber 4 (), pp. 44-449 Research Idia Publicatios http://www.ripublicatio.com/ijss.htm Testig Statistical Hypotheses with Fuzzy Data E. Baloui

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

Chapter 1 (Definitions)

Chapter 1 (Definitions) FINAL EXAM REVIEW Chapter 1 (Defiitios) Qualitative: Nomial: Ordial: Quatitative: Ordial: Iterval: Ratio: Observatioal Study: Desiged Experimet: Samplig: Cluster: Stratified: Systematic: Coveiece: Simple

More information

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

Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation ; [Formerly kow as the Bulleti of Statistics & Ecoomics (ISSN 097-70)]; ISSN 0975-556X; Year: 0, Volume:, Issue Number: ; It. j. stat. eco.; opyright 0 by ESER Publicatios Some Expoetial Ratio-Product

More information

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

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

More information

Section 14. Simple linear regression.

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

More information

Simple Linear Regression

Simple Linear Regression Simple Liear Regressio 1. Model ad Parameter Estimatio (a) Suppose our data cosist of a collectio of pairs (x i, y i ), where x i is a observed value of variable X ad y i is the correspodig observatio

More information

Testing Statistical Hypotheses for Compare. Means with Vague Data

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

More information

Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogonal Moment Features Using Chebyshev Polynomials Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

More information

Statistical Test for Multi-dimensional Uniformity

Statistical Test for Multi-dimensional Uniformity Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Theses ad Dissertatios Uiversity Graduate School -0-20 Statistical Test for Multi-dimesioal Uiformity Tieyog Hu Florida Iteratioal Uiversity,

More information

Estimation of the Population Mean in Presence of Non-Response

Estimation of the Population Mean in Presence of Non-Response Commuicatios of the Korea Statistical Society 0, Vol. 8, No. 4, 537 548 DOI: 0.535/CKSS.0.8.4.537 Estimatio of the Populatio Mea i Presece of No-Respose Suil Kumar,a, Sadeep Bhougal b a Departmet of Statistics,

More information

UCLA STAT 110B Applied Statistics for Engineering and the Sciences

UCLA STAT 110B Applied Statistics for Engineering and the Sciences UCLA STAT 110B Applied Statistics for Egieerig ad the Scieces Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistats: Bria Ng, UCLA Statistics Uiversity of Califoria, Los Ageles,

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

Binomial Distribution

Binomial Distribution 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible

More information

TRACEABILITY SYSTEM OF ROCKWELL HARDNESS C SCALE IN JAPAN

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

More information

Regression with quadratic loss

Regression with quadratic loss Regressio with quadratic loss Maxim Ragisky October 13, 2015 Regressio with quadratic loss is aother basic problem studied i statistical learig theory. We have a radom couple Z = X,Y, where, as before,

More information

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

Statisticians use the word population to refer the total number of (potential) observations under consideration 6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space

More information

V. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany

V. Nollau Institute of Mathematical Stochastics, Technical University of Dresden, Germany PROBABILITY AND STATISTICS Vol. III - Correlatio Aalysis - V. Nollau CORRELATION ANALYSIS V. Nollau Istitute of Mathematical Stochastics, Techical Uiversity of Dresde, Germay Keywords: Radom vector, multivariate

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

Bayesian Estimation of the Parameters of Two- Component Mixture of Rayleigh Distribution under Doubly Censoring

Bayesian Estimation of the Parameters of Two- Component Mixture of Rayleigh Distribution under Doubly Censoring Joural of Moder Applied Statistical Methods Volume 3 Issue Article 4-04 Bayesia Estimatio of the Parameters of Two- Compoet Mixture of Rayleigh Distributio uder Doubly Cesorig Tahassum N. Sidhu Quaid-i-Azam

More information

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

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

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

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

STA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to: STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio

More information

Nonparametric Tests for Two Factor Designs

Nonparametric Tests for Two Factor Designs Uiversity of Wollogog Research Olie Applied Statistics Educatio ad Research Collaboratio (ASEARC) - Coferece apers Faculty of Egieerig ad Iformatio Scieces 011 Noparametric Tests for Two Factor Desigs

More information

Lecture 4. Random variable and distribution of probability

Lecture 4. Random variable and distribution of probability Itroductio to theory of probability ad statistics Lecture. Radom variable ad distributio of probability dr hab.iż. Katarzya Zarzewsa, prof.agh Katedra Eletroii, AGH e-mail: za@agh.edu.pl http://home.agh.edu.pl/~za

More information

Element sampling: Part 2

Element sampling: Part 2 Chapter 4 Elemet samplig: Part 2 4.1 Itroductio We ow cosider uequal probability samplig desigs which is very popular i practice. I the uequal probability samplig, we ca improve the efficiecy of the resultig

More information

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

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

More information

SINGLE-CHANNEL QUEUING PROBLEMS APPROACH

SINGLE-CHANNEL QUEUING PROBLEMS APPROACH SINGLE-CHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT E-mail: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper

More information

DISCRIMINATING BETWEEN NORMAL AND GUMBEL DISTRIBUTIONS

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

More information

Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions

Non-Homogeneous Poisson Processes Applied to Count Data: A Bayesian Approach Considering Different Prior Distributions Joural of Evirometal Protectio, 202, 3, 336-345 http://dx.doi.org/0.4236/jep.202.3052 Published Olie October 202 (http://www.scirp.org/joural/jep) No-Homogeeous Poisso Processes Applied to Cout Data: A

More information

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

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

More information

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

(all terms are scalars).the minimization is clearer in sum notation: 7 Multiple liear regressio: with predictors) Depedet data set: y i i = 1, oe predictad, predictors x i,k i = 1,, k = 1, ' The forecast equatio is ŷ i = b + Use matrix otatio: k =1 b k x ik Y = y 1 y 1

More information

1036: Probability & Statistics

1036: Probability & Statistics 036: Probability & Statistics Lecture 0 Oe- ad Two-Sample Tests of Hypotheses 0- Statistical Hypotheses Decisio based o experimetal evidece whether Coffee drikig icreases the risk of cacer i humas. A perso

More information

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn

Review Questions, Chapters 8, 9. f(y) = 0, elsewhere. F (y) = f Y(1) = n ( e y/θ) n 1 1 θ e y/θ = n θ e yn Stat 366 Lab 2 Solutios (September 2, 2006) page TA: Yury Petracheko, CAB 484, yuryp@ualberta.ca, http://www.ualberta.ca/ yuryp/ Review Questios, Chapters 8, 9 8.5 Suppose that Y, Y 2,..., Y deote a radom

More information

ORF 245 Fundamentals of Engineering Statistics. Midterm Exam 2

ORF 245 Fundamentals of Engineering Statistics. Midterm Exam 2 Priceto Uiversit Departmet of Operatios Research ad Fiacial Egieerig ORF 45 Fudametals of Egieerig Statistics Midterm Eam April 17, 009 :00am-:50am PLEASE DO NOT TURN THIS PAGE AND START THE EXAM UNTIL

More information

A Risk Comparison of Ordinary Least Squares vs Ridge Regression

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

More information

Confidence Intervals for the Population Proportion p

Confidence Intervals for the Population Proportion p Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:

More information

Statistical inference: example 1. Inferential Statistics

Statistical inference: example 1. Inferential Statistics Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either

More information

Complex Algorithms for Lattice Adaptive IIR Notch Filter

Complex Algorithms for Lattice Adaptive IIR Notch Filter 4th Iteratioal Coferece o Sigal Processig Systems (ICSPS ) IPCSIT vol. 58 () () IACSIT Press, Sigapore DOI:.7763/IPCSIT..V58. Complex Algorithms for Lattice Adaptive IIR Notch Filter Hog Liag +, Nig Jia

More information

On Bayesian Shrinkage Estimator of Parameter of Exponential Distribution with Outliers

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

More information

On Marshall-Olkin Extended Weibull Distribution

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

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

CTL.SC0x Supply Chain Analytics

CTL.SC0x Supply Chain Analytics CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos

More information

5.1 A mutual information bound based on metric entropy

5.1 A mutual information bound based on metric entropy Chapter 5 Global Fao Method I this chapter, we exted the techiques of Chapter 2.4 o Fao s method the local Fao method) to a more global costructio. I particular, we show that, rather tha costructig a local

More information

Varanasi , India. Corresponding author

Varanasi , India. Corresponding author A Geeral Family of Estimators for Estimatig Populatio Mea i Systematic Samplig Usig Auxiliary Iformatio i the Presece of Missig Observatios Maoj K. Chaudhary, Sachi Malik, Jayat Sigh ad Rajesh Sigh Departmet

More information

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

Research Article Health Monitoring for a Structure Using Its Nonstationary Vibration

Research Article Health Monitoring for a Structure Using Its Nonstationary Vibration Hidawi Publishig Corporatio Advaces i Acoustics ad Vibratio Volume 2, Article ID 69652, 5 pages doi:.55/2/69652 Research Article Health Moitorig for a Structure Usig Its Nostatioary Vibratio Yoshimutsu

More information

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions

Assessment and Modeling of Forests. FR 4218 Spring Assignment 1 Solutions Assessmet ad Modelig of Forests FR 48 Sprig Assigmet Solutios. The first part of the questio asked that you calculate the average, stadard deviatio, coefficiet of variatio, ad 9% cofidece iterval of the

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

Formation of A Supergain Array and Its Application in Radar

Formation of A Supergain Array and Its Application in Radar Formatio of A Supergai Array ad ts Applicatio i Radar Tra Cao Quye, Do Trug Kie ad Bach Gia Duog. Research Ceter for Electroic ad Telecommuicatios, College of Techology (Coltech, Vietam atioal Uiversity,

More information

An Alternative Goodness-of-fit Test for Normality with Unknown Parameters

An Alternative Goodness-of-fit Test for Normality with Unknown Parameters Florida Iteratioal Uiversity FIU Digital Commos FIU Electroic Teses ad Dissertatios Uiversity Graduate Scool -4-04 A Alterative Goodess-of-fit Test for Normality wit Ukow Parameters Weilig Si amadasi335@gmail.com

More information

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

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

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

Estimation of Parameters of Johnson s System of Distributions

Estimation of Parameters of Johnson s System of Distributions Joural of Moder Alied tatistical Methods Volume 0 Issue Article 9 --0 Estimatio of Parameters of Johso s ystem of Distributios Florece George Florida Iteratioal iversity, fgeorge@fiu.edu K. M. Ramachadra

More information

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

o <Xln <X2n <... <X n < o (1.1) Metrika, Volume 28, 1981, page 257-262. 9 Viea. Estimatio Problems for Rectagular Distributios (Or the Taxi Problem Revisited) By J.S. Rao, Sata Barbara I ) Abstract: The problem of estimatig the ukow

More information

Decoupling Zeros of Positive Discrete-Time Linear Systems*

Decoupling Zeros of Positive Discrete-Time Linear Systems* Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical

More information

Probability, Expectation Value and Uncertainty

Probability, Expectation Value and Uncertainty Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such

More information

Gamma Distribution and Gamma Approximation

Gamma Distribution and Gamma Approximation Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract

More information

A Slight Extension of Coherent Integration Loss Due to White Gaussian Phase Noise Mark A. Richards

A Slight Extension of Coherent Integration Loss Due to White Gaussian Phase Noise Mark A. Richards A Slight Extesio of Coheret Itegratio Loss Due to White Gaussia Phase oise Mark A. Richards March 3, Goal I [], the itegratio loss L i computig the coheret sum of samples x with weights a is cosidered.

More information

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss

ECE 901 Lecture 12: Complexity Regularization and the Squared Loss ECE 90 Lecture : Complexity Regularizatio ad the Squared Loss R. Nowak 5/7/009 I the previous lectures we made use of the Cheroff/Hoeffdig bouds for our aalysis of classifier errors. Hoeffdig s iequality

More information

Lainiotis filter implementation. via Chandrasekhar type algorithm

Lainiotis filter implementation. via Chandrasekhar type algorithm Joural of Computatios & Modellig, vol.1, o.1, 2011, 115-130 ISSN: 1792-7625 prit, 1792-8850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais

More information