Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation

Yuan, Fuqing

doi:10.1002/qre.2276

Cited by 11 publications

(9 citation statements)

References 28 publications

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“…According to large sample theory for independent and identical data, if

E ((), (||, X^{k} Y^{l})) < \infty

, then

\sqrt{n} ((), \overset{μ}{true} - bold-italicμ) \to N ((), bold0, normalΣ)

. Yuan sets

Σ_{i j} = c o v ((), μ_{i}, μ_{j})

and claims that

\sqrt{n} (\hat{bold-italicθ} - θ) \to N (0, G Σ {boldG}^{'}) .

This is incorrect. Actually,

Σ = (\begin{matrix} v a r false(X false) & c o v ((), X, X^{2}) & c o v false(X, Y false) & c o v ((), X, Y^{2}) & c o v false(X, X Y false) \\ c o v ((), X^{2}, X) & v a r ((), X^{2}) & c o v ((), X^{2}, Y) & c o v ((), X^{2}, Y^{2}) & c o v ((), X^{2}, X Y) \\ c o v false(Y, X false) & c o v ((), Y, X^{2}) & v a r false(Y false) & c o v ((), Y, Y^{2}) & c o v false(Y, X Y false) \\ c o v ((), Y^{2}, X) \end{matrix})

…”

Section: Corrected Confidence Intervalsmentioning

confidence: 99%

“…After some calculations, the confidence interval for θ i reported in Yuan is as follows:

([], θ_{L}, θ_{U}) = ([], {\overset{θ}{true}}_{i} - \frac{z_{1 - \frac{α}{2}} \sqrt{V_{i, i}}}{n}, {\overset{θ}{true}}_{i} + \frac{z_{1 - \frac{α}{2}} \sqrt{V_{i, i}}}{n}) .

…”

Section: Corrected Confidence Intervalsmentioning

confidence: 99%

“…The estimates of the parameters in can be derived by equating E ( X ),

E ((), X^{2})

, E ( Y ),

E ((), Y^{2})

, and E ( X Y ) with the sample raw moments

\overset{X}{true} = \sum X_{i} false/ n

\overset{X^{2}}{true} = \sum X_{i}^{2} false/ n

\overset{Y}{true} = \sum Y_{i} false/ n

\overset{Y^{2}}{true} = \sum Y_{i}^{2} false/ n

and

\overset{X Y}{true} = \sum X_{i} Y_{i} false/ n

. The expectations as reported in Yuan are

\begin{matrix} E false(X false) & = \frac{normalΓ ((), 1 + \frac{1}{γ β_{1}})}{α_{1}}, \\ E ((), X^{2}) & = \frac{normalΓ ((), 1 + \frac{2}{γ β_{1}})}{α_{1}^{2}}, \\ E false(Y false) & = \frac{normalΓ ((), 1 + \frac{1}{γ β_{2}})}{α_{2}}, \\ E ((), Y^{2}) & = \frac{normalΓ ((), 1 + \frac{2}{γ β_{2}})}{α_{2}^{2}}, \\ E false(X Y false) & = normalΓ ((), 1 + \frac{1}{γ β_{1}} + \frac{1}{γ β_{2}}) normalΓ ((), \frac{1}{β_{1}} + 1) normalΓ (()) \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

“…The aim of this note is to correct errors in section 3.4 of Yuan about the definition of Σ in equation 18, the definition of V in equation 21, the confidence interval in equation 22, and the confidence interval in equation 23.…”

Section: Introductionmentioning

confidence: 99%

“…The contents of this note are organized as follows. Section states the results of confidence intervals in Yuan, and corrections are made for them. Section applies the corrected confidence intervals to the field data in Yuan …”

Section: Introductionmentioning

confidence: 99%

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A note on “Parameter estimation for bivariate Weibull distribution using generalized moment method for reliability evaluation”

Zhang

Nadarajah

2018

Quality & Reliability Eng

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Yuan [Quality and Reliability Engineering International, 2018, doi: 10.1002/qre.2276] proposed a generalized moment method for parameter estimation of a particular bivariate Weibull distribution and derived corresponding confidence intervals. The expressions given for the confidence intervals do not appear correct. In this note, we correct the confidence intervals and recompute them for two real data sets.

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“…According to large sample theory for independent and identical data, if