Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example

Dey, Sanku; Raheem, Enayetur; Mukherjee, Saikat; Ng, Hon Keung Tony

doi:10.1080/09720510.2016.1228261

Cited by 18 publications

(14 citation statements)

References 26 publications

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“…In this subsection, the LSEs and WLSEs are obtained. The LSEs and WLSEs were used by Swain et al 18 [19][20][21][22] Let x (11) , · · ·, x (n k k) be the order statistics of a random sample of size N from the ER distribution under constant ALT. The LSEs denoted byâ LSE ,b LSE , and̂L SE can be obtained by minimizing the following function:…”

Section: Least Square and Weighted Least Square Methodsmentioning

confidence: 99%

“…This method was first introduced by Kao 23 for estimating Weibull parameters. Many authors have used this method of estimation, including Kundu and Raqab 4 and Dey et al [19][20][21][22] From (2), we can write the cdf of the ER distribution under constant ALT as follows:…”

Section: Methods Of Percentile Estimationmentioning

confidence: 99%

“…Recently, some authors have used the method of estimation in their work. See, for example, Kundu and Raqab and Dey et al() Let

x_{false(11 false)}, ⋯, x_{false(n_{k} k false)}

be the order statistics of a random sample of size N from the ER distribution under constant ALT. The LSEs denoted by

{\hat{a}}_{L S E}, {\overset{b}{true}}_{L S E}

, and

{\overset{β}{true}}_{L S E}

can be obtained by minimizing the following function:

S false(a, b, β false) = true \sum_{j = 1}^{k} true \sum_{i = 1}^{n_{j}} {([], {false[1 - e^{- β x_{false(i j false)}^{2}} false]}^{e^{a + b S_{j}}} - \frac{i}{n_{j} + 1})}^{2} .

…”

Section: Estimation Of the Unknown Parametersmentioning

confidence: 99%

“…This method was first introduced by Kao for estimating Weibull parameters. Many authors have used this method of estimation, including Kundu and Raqab and Dey et al() From , we can write the cdf of the ER distribution under constant ALT as follows:

G (x_{i j}) = {[1 - e^{- β x_{i j}^{2}}]}^{e^{a + b S_{j}}};

therefore,

x_{i j} = {([], - \frac{\ln false(1 - p_{i j}^{e^{- false(a + b S_{j} false)}} false)}{β})}^{0.5},

where

p_{i j} = \frac{i}{n_{j} + 1}

. The PCEs of a , b , and β denoted by

{\hat{a}}_{P C E}, {\overset{b}{true}}_{P C E}

, and

{\overset{β}{true}}_{P C E}

are obtained by minimizing the function:

P false(a, b, β false) = true \sum_{j = 1}^{k} true \sum_{i = 1}^{n_{j}} {({}, x_{false(i j false)} - {([], - \frac{\ln false(1 - p_{i j}^{e^{- false(a + b S_{j} false)}} false)}{β})}^{0.5})}^{2} .

The PCEs can be obtained by solving the following nonlinear equations:

\begin{matrix}  \end{matrix}

…”

Section: Estimation Of the Unknown Parametersmentioning

confidence: 99%

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Different estimation methods for exponentiated Rayleigh distribution under constant‐stress accelerated life test

Nassar

Dey²

2018

Quality & Reliability Eng

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The accelerated life testing is an efficient approach and has been used in several fields to obtain failure time data of test units in a much shorter time than testing at normal operating conditions. In this article, we consider 6 frequentist estimation methods, namely, method of maximum likelihood estimation, method of least square estimation, method of weighted least square estimation, method of percentile estimation, method of maximum product of spacing estimation, and method of Cramér‐von‐Mises estimation to estimate the parameters and the reliability function of the exponentiated Rayleigh distribution under normal use conditions using constant accelerated life testing. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. The performances of the estimators have been compared in terms of their mean squared error using simulated samples. Simulation results showed that percentile method gives the best results among other estimation methods in terms of mean squared error. Finally, a real data set is analyzed for illustrative purposes.

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Section: Least Square and Weighted Least Square Methodsmentioning

confidence: 99%

Section: Methods Of Percentile Estimationmentioning

confidence: 99%

“…Recently, some authors have used the method of estimation in their work. See, for example, Kundu and Raqab and Dey et al() Let

x_{false(11 false)}, ⋯, x_{false(n_{k} k false)}

be the order statistics of a random sample of size N from the ER distribution under constant ALT. The LSEs denoted by

{\hat{a}}_{L S E}, {\overset{b}{true}}_{L S E}

, and

{\overset{β}{true}}_{L S E}

can be obtained by minimizing the following function:

S false(a, b, β false) = true \sum_{j = 1}^{k} true \sum_{i = 1}^{n_{j}} {([], {false[1 - e^{- β x_{false(i j false)}^{2}} false]}^{e^{a + b S_{j}}} - \frac{i}{n_{j} + 1})}^{2} .

…”

Section: Estimation Of the Unknown Parametersmentioning

confidence: 99%

G (x_{i j}) = {[1 - e^{- β x_{i j}^{2}}]}^{e^{a + b S_{j}}};

therefore,

x_{i j} = {([], - \frac{\ln false(1 - p_{i j}^{e^{- false(a + b S_{j} false)}} false)}{β})}^{0.5},

where

p_{i j} = \frac{i}{n_{j} + 1}

. The PCEs of a , b , and β denoted by

{\hat{a}}_{P C E}, {\overset{b}{true}}_{P C E}

, and

{\overset{β}{true}}_{P C E}

are obtained by minimizing the function:

P false(a, b, β false) = true \sum_{j = 1}^{k} true \sum_{i = 1}^{n_{j}} {({}, x_{false(i j false)} - {([], - \frac{\ln false(1 - p_{i j}^{e^{- false(a + b S_{j} false)}} false)}{β})}^{0.5})}^{2} .

The PCEs can be obtained by solving the following nonlinear equations:

\begin{matrix}  \end{matrix}

…”

Section: Estimation Of the Unknown Parametersmentioning

confidence: 99%

See 2 more Smart Citations

Different estimation methods for exponentiated Rayleigh distribution under constant‐stress accelerated life test

Nassar

Dey²

2018

Quality & Reliability Eng

Self Cite

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“…The uniqueness of this study comes from the fact that thus far, no attempt has been made to compare all these estimators for the twoparameter Gompertz distribution along with statistical properties. Different estimation methods were compared for generalized Rayleigh distributions by Kundu and Raqab (2005); for generalized logistic distributions by Alkasasbeh and Raqab (2009); for the Weibull distribution by Teimouri et al (2013) and Dey et al (2014Dey et al ( , 2015Dey et al ( , 2016Dey et al ( , 2017aDey et al ( , 2017bDey et al ( , 2017cDey et al ( , 2017d for the two-parameter Rayleigh, weighted exponential, twoparameter Maxwell, exponentiated-Chen, Dagum, transmuted-Rayleigh, two parameter exponentiated-Gumbel, new extension of generalized exponential and NH distributions. Maximum likelihood estimates using different experimental schemes studied by Azzam (1994) and asymptotic normality of maximum likelihood estimators for non-parametric Markov chains was studied by AL-Eideh (1996).…”

Section: Introductionmentioning

confidence: 99%

Statistical properties and different methods of estimation of Gompertz distribution with application

Dey¹,

Moala

Kumar

2018

Journal of Statistics and Management Systems

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This article addresses the various properties and different methods of estimation of the unknown parameters of Gompertz distribution. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of the Gompertz distribution (such as quantiles, moments, moment generating function, hazard rate, mean residual lifetime, mean past lifetime, stochasic ordering, stressstrength parameter, various entropies, Bonferroni and Lorenz curves and order statistics) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, moments estimators, pseudo-moments estimators, modified moments estimators, L-moment estimators, percentile based estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, minimum spacing

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On a new extension of Weibull distribution: Properties, estimation, and applications to one and two causes of failures

Nassar

Afify²,

Shakhatreh

et al. 2020

Quality & Reliability Eng

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In this paper, we try to supplement the distribution theory literature by introducing a new distribution, called the logarithmic transformed Weibull (LTW) distribution by logarithm transformation method. The proposed distribution exhibits constant, decreasing, increasing, unimodal and unimodal then bathtub-shaped hazard rates. Although our main focus is on the estimation from the frequentist point of view, in addition, we derive some useful structural and statistical properties of the proposed LTW distribution. The LTW parameters are estimated by eight frequentist methods of estimation. We perform extensive simulation experiments to show the performances of the proposed estimators. Applications of the model are presented by reanalyzing two real data sets, and comparisons are made with the fit attained by some other well-known distributions for illustrative purposes. As an illustration, one data set is analyzed for competing risks to demonstrate the flexibility of the proposed model.

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Two parameter exponentiated Gumbel distribution: properties and estimation with flood data example

Cited by 18 publications

References 26 publications

Different estimation methods for exponentiated Rayleigh distribution under constant‐stress accelerated life test

Different estimation methods for exponentiated Rayleigh distribution under constant‐stress accelerated life test

Statistical properties and different methods of estimation of Gompertz distribution with application

On a new extension of Weibull distribution: Properties, estimation, and applications to one and two causes of failures

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