Estimation of p(y&lt;x) in the gamma case

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

doi:10.1080/03610918608812513

Cited by 94 publications

(29 citation statements)

References 4 publications

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“…On the ratio X/(X + Y ) for Weibull and Lévy distributions, Journal of the Korean Statistical Society, 34,[11][12][13][14][15][16][17][18][19][20]2005.…”

Section: Discussionmentioning

confidence: 99%

“…Many papers have investigated estimation of P (Y < X) when X and Y arise from a specific distribution. For details, see Awad and Gharraf [4], Surles and Padgett [34] for the case X, Y are Burr distributed; Constantine et al [11], Ismail et al [20] for the case X, Y are gamma distributed; Obradovic et al [27] for the case X, Y are geometricPoisson distributed; Babayi et al [5] for the case X, Y are generalized logistic distributed; Kundu and Raqab [22] for the case X, Y are generalized Rayleigh distributed; Saracoglu et al [32] for the case X, Y are Gompertz distributed; Nadar et al [25] for the case X, Y are Kumaraswamy distributed; Downtown [14], Reiser and Guttman [30] for the case X, Y are normal distributed; Genc [17] for the case X, Y are Topp-Leone distributed; McCool [24] for the case X, Y are Weibull distributed. There are also semiparametric and nonparametric methods for estimating P (Y < X).…”

Section: Introductionmentioning

confidence: 99%

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Estimation of P(Y X) for the L´evy distribution

Najarzadegan¹,

Babaii²,

Rezaei³

et al. 2015

HJMS

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“…On the ratio X/(X + Y ) for Weibull and Lévy distributions, Journal of the Korean Statistical Society, 34,[11][12][13][14][15][16][17][18][19][20]2005.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Estimation of P(Y X) for the L´evy distribution

Najarzadegan¹,

Babaii²,

Rezaei³

et al. 2015

HJMS

View full text Add to dashboard Cite

show abstract

“…Estimation of the Pr(Y < X) when X and Y are normally distributed was considered by Govidarajulu (1967) and Church and Harris (1970). Constantine and Karson (1986) considered the estimation of P(Y < X), when X and Y are independent gamma random variables. In our case, the stress-strength parameter R is given by…”

Section: Estimation Of the Stress-strength Parametermentioning

confidence: 99%

A generalization of the exponential-Poisson distribution

Barreto‐Souza

Cribari‐Neto

2009

Statistics & Probability Letters

123

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The two-parameter distribution known as exponential-Poisson (EP) distribution, which has decreasing failure rate, was introduced by Kus (2007). In this paper we generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the ith order statistic. We derive the rth raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher's information matrix. Furthermore, expressions for the Rényi and Shannon entropies are given and estimation of the stress-strength parameter is discussed. Applications using two real data sets are presented.

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“…Many research results have been obtained for the estimation of AUC under popular survival models: Bi-exponential model [17], Bi-normal model [5,18], Bi-gamma model [19], Bi-Burr type X [20][21][22][23], Bi-generalized exponential [24], BiWeibull model [25]. However, all the popular survival models such as generalized exponential (GE), Weibull (WE), and Gamma do not allow nonmonotonic failure rates which often occur in real practice.…”

Section: Isrn Probability and Statisticsmentioning

confidence: 99%

Maximum Likelihood Estimator of AUC for a Bi-Exponentiated Weibull Model

Chang

Qian

2013

ISRN Probability and Statistics

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For a Bi-Exponentiated Weibull model, the authors obtain a general AUC formula and derive the maximum likelihood estimator of AUC and its asymptotic property. A simulation study is carried out to illustrate the finite sample size performance. ROC Curve and AUC for a Bi-Exponentiated Weibull ModelIn medical science, a diagnostic test result called a biomarker [1,2] is an indicator for disease status of patients. The accuracy of a medical diagnostic test is typically evaluated by sensitivity and specificity. Receiver Operating Characteristic (ROC) curve is a graphical representation of the relationship between sensitivity and specificity. The area under the ROC curve (AUC) is an overall performance measure for the biomarker. Hence the main issue in assessing the accuracy of a diagnostic test is to estimate the ROC curve and its AUC. Suppose that there are two groups of study subjects: diseased and nondiseased. Let be a continuous biomarker. Assume that the larger is, the more likely a subject is diseased. That is, a subject is classified as positive or diseased status if > and as negative or nondiseased status otherwise, where is a cutoff point. Let be the disease status:= 1 represents diseased population while = 0 represents nondiseased population. Sensitivity of is defined as the probability of being correctly classified as disease status and specificity as the probability of being correctly classified as nondisease status. That is, sensitivity ( ) = ( > | = 1) , specificity ( ) = ( < | = 0) .(1) Let = { | = 1} and = { | = 0} be the biomarkers for diseased and nondiseased subjects with survival functions 1 and 0 , respectively. Then sensitivity ( ) = 1 ( ) ,The ROC function (curve) is defined asNotice that ROC curve is a monotonic increasing curve in a unit square starting at (0, 0) and ending at (1, 1). A good biomarker has a very concave down ROC curve. We say a bivariate random vector ( , ) is a Bi-model if and are independent and are from the same distribution family but with different parameters. Pepe [3] summarizes ROC analysis and notices that among Bi-model, Bi-normal model has been the most popular one. The ROC curve method traces back to Green and Swets [4] in the signal detection theory. Bi-normal model assumes the existence of a monotonic increasing transformation such that the biomarker can be transformed into normally distributed for both diseased and nondiseased patients. Pepe [3, Results 4.1 and 4.4] shows that the ROC curve of a biomarker is invariant under a monotone increasing transformation. Hence, a good estimator of ROC curve should satisfy this invariance property.

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Estimation of p(y<x) in the gamma case

Cited by 94 publications

References 4 publications

Estimation of P(Y X) for the L´evy distribution

Estimation of P(Y X) for the L´evy distribution

A generalization of the exponential-Poisson distribution

Maximum Likelihood Estimator of AUC for a Bi-Exponentiated Weibull Model

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