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Testing exponentiality against HNBUE
Umeå University, Faculty of Science and Technology, Mathematical statistics.
1980 (English)Report (Other academic)
Abstract [en]

Let F be a life distribution with survival function F = 1 - F and00 —finite mean y = /q F(x)dx. Then F is said to be harmonic new better00 —than used in expectation (HNBUE) if / F(x)dx < y exp(-t/y) for t > 0. If the reversed inequality is true F is said to be HNWUE (W = worse). We develop some tests for testing exponentiality against the HNBUE (HNWUE) property. Among these is the test based on the cumulative total time on test statistic which is ordinarily used for testing against the IFR (DFR) alternative. The asymptotic distributions of the statistics are discussed. Consistency and asymptotic relative efficiency are studied. A small sample study is also presented.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 1980. , 26 p.
Keyword [en]
Life distribution, HNBUE, HNWUE, exponential distribution, TTT-transform, hypothesis testing, efficiency, consistency, power
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-78995OAI: oai:DiVA.org:umu-78995DiVA: diva2:638412
Projects
digitalisering@umu
Note

This is a revised version of Sections 5 and 6 in Statistical Research Report 1979-9, Department of Mathematical Statistics, University of Umeå.

There are some occurring misspellings in the formulas in the abstract on this webpage. Read the abstract in the full-text document for correct spelling in formulas, see the downloadable file.

Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved
In thesis
1. Properties and tests for some classes of life distributions
Open this publication in new window or tab >>Properties and tests for some classes of life distributions
1980 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

A life distribution and its survival function F = 1 - F with finitemean y = /q F(x)dx are said to be HNBUE (HNWUE) if F(x)dx < (>)U exp(-t/y) for t > 0. The major part of this thesis deals with the classof HNBUE (HNWUE) life distributions. We give different characterizationsof the HNBUE (HNWUE) property and present bounds on the moments and on thesurvival function F when this is HNBUE (HNWUE). We examine whether theHNBUE (HNWUE) property is preserved under some reliability operations andstudy some test statistics for testing exponentiality against the HNBUE(HNWUE) property.The HNBUE (HNWUE) property is studied in connection with shock models.Suppose that a device is subjected to shocks governed by a counting processN = {N(t): t > 0}. The probability that the device survives beyond t isthen00H(t) = S P(N(t)=k)P, ,k=0where P^ is the probability of surviving k shocks. We prove that His HNBUE (HNWUE) under different conditions on N and * ^orinstance we study the situation when the interarrivai times between shocksare independent and HNBUE (HNWUE).We also study the Pure Birth Shock Model, introduced by A-Hameed andProschan (1975), and prove that H is IFRA and DMRL under conditions whichdiffer from those used by A-Hameed and Proschan.Further we discuss relationships between the total time on test transformHp^(t) = /q ^F(s)ds , where F \t) = inf { x: F(x) > t}, and differentclasses of life distributions based on notions of aging. Guided by propertiesof we suggest test statistics for testing exponentiality agains t IFR,IFRA, NBUE, DMRL and heavy-tailedness. Different properties of these statisticsare studied.Finally, we discuss some bivariate extensions of the univariate properties NBU, NBUE, DMRL and HNBUE and study some of these in connection with bivariate shock models.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 1980. 16 p.
Keyword
Life distribution, survival function, exponential distribution, IFR, IFRA, NBUE, DMRL, HNBUE, shock model, total time on test transform, testing of exponentiality
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-78990 (URN)
Public defence
1980-10-17, Samhällsvetarhuset, hörsal D, Umeå universitet, Umeå, 09:15
Projects
digitalisering@umu
Note

There are some occurring misspellings in the formulas in the abstract on this webpage. Read the abstract in the full-text document for correct spelling in formulas, see the downloadable file.

Available from: 2013-07-30 Created: 2013-07-30 Last updated: 2013-07-30Bibliographically approved

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