Confidence Interval Estimation of P(Y&lt;X) in the Gamma Case

Constantine, Kenneth; Karson, Marvin J.; Tse, Siu‐Keung

doi:10.1080/03610919008812854

Cited by 10 publications

(3 citation statements)

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“…Assuming normality of X 1 and X 2 , Hall (1984), Reiser and Guttman (1986), and Guo and Krishnamoorthy (2004) proposed approximate methods for computing confidence limits for R. Several authors considered the problem of estimating R when X 1 and X 2 are independent gamma random variables. Basu (1981) and Constantine, Karson, and Tse (1986, 1990) considered point and interval estimation of R. As pointed out by Constantine et al (1990), many investigators assume that the shape parameters are known and are integer-valued. If the shape parameters are known, then it is not difficult to obtain exact confidence limits for R (see Sec.…”

Section: Introductionmentioning

confidence: 98%

Normal-Based Methods for a Gamma Distribution

2008

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In this article we propose inferential procedures for a gamma distribution using the Wilson-Hilferty (WH) normal approximation. Specifically, using the result that the cube root of a gamma random variable is approximately normally distributed, we propose normal-based approaches for a gamma distribution for (a) constructing prediction limits, one-sided tolerance limits, and tolerance intervals; (b) for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations; and (c) assessing the reliability of a stress-strength model involving two independent gamma random variables. For each problem, a normal-based approximate procedure is outlined, and its applicability and validity for a gamma distribution are studied using Monte Carlo simulation. Our investigation shows that the approximate procedures are very satisfactory for all of these problems. For each problem considered, the results are illustrated using practical examples. Our overall conclusion is that the WH normal approximation provides a simple, easy-to-use unified approach for addressing various problems for the gamma distribution.

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Section: Introductionmentioning

confidence: 98%

Normal-Based Methods for a Gamma Distribution

2008

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“…Exponential case with common location parameter was examined by Baklizi and Quader El-Masri [2]. The gamma case was studied by Constantine and Karson [5], Ismail et al [12] and Constantine et al [6]. Kundu and Gupta considered generalized exponetial case [16].…”

Section: Introductionmentioning

confidence: 99%

Estimation of P(X Y) for Geometric-Poisson Model

Jovanovi

Milošević

Jevremović

et al. 2014

HJMS

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In this paper we estimate R = P {X ≤ Y } when X and Y are independent random variables from geometric and Poisson distribution respectively. We find maximum likelihood estimator of R and its asymptotic distribution. This asymptotic distribution is used to construct asymptotic confidence intervals. A procedure for deriving bootstrap confidence intervals is presented. UMVUE of R and UMVUE of its variance are derived and also the Bayes estimator of R for conjugate prior distributions is obtained. Finally, we perform a simulation study in order to compare these estimators.

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“…Tong [16] analyzed estimation of R when X and Y are exponential variables. Constantine and Karson [2], Ismail et al [7], and Constantine et al [3] estimated R when X and Y are from gamma distributions with known shape parameters. Reliability for logistic distribution is analyzed in Nadarajah [10], and for Laplace distribution in Nadarajah [11].…”

Section: Introductionmentioning

confidence: 99%