Inference for multicomponent stress–strength reliability based on unit generalized Rayleigh distribution

Abstract

This paper considers the problem of making inference for a multicomponent stress–strength (MSS) model under Type-II censoring. It is assumed that stress–strength components of the multicomponent system follow unit generalized Rayleigh (UGR) distributions. Inferences are derived when the strength and stress have a common unknown UGR parameters. We derive maximum likelihood estimator of MSS reliability based on observed censored data. In sequel, interval estimator is evaluated using delta method and asymptotic normality property. Pivotal quantities based inference upon MSS reliability are derived as well. In addition, we further consider the case when all the parameters of stress–strength model are unknown and obtain various inferences for the reliability. Equivalence of model parameters is considered based on likelihood ratio test. Assessment of different methods is evaluated from Monte Carlo simulations and remarks are presented for further discussion. Three numerical examples including a petroleum reservoirs data are analyzed for illustration purposes.

Appendices

A. Proof of Theorem 2

We recall that ${{{R}}}_{{{s}},{{k}}}$ is expressed as ${{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)$ as defined in (6). Also using Taylor’s theorem, ${{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)$ written as (e.g., Xu and Long (2005))

$${{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)={{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)+{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}\left(\widehat{\upgamma }-\upgamma \right)+\frac{1}{2}{\left(\widehat{\upgamma }-\upgamma \right)}^{{{T}}}\left[{\nabla }^{2}{{{R}}}_{{{s}},{{k}}}\left({\upgamma }^{*}\right)\right]\left(\widehat{\upgamma }-\upgamma \right)$$

$$\approx {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)+{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}\left(\widehat{\upgamma }-\upgamma \right)$$

(25)

where $\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)$ and ${\nabla }^{2}{{{R}}}_{{{s}},{{k}}}\left({\upgamma }^{*}\right)$ denote respective matrices. Note also ${\upgamma }^{*}$ lies between $\upgamma$ and $\widehat{\upgamma }$. Then expression (25) turns out to be ${{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)\to {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)$ as ${{n}}\to \infty$ as Theorem 1 implies that $\widehat{\upgamma }\to\upgamma$.

The variance of ${{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)$ is given by

$${{Var}}\left({{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)\right)\approx {{Var}}\left[{{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)+{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}\widehat{\upgamma }-{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}\upgamma \right]$$

$$={{Var}}\left[{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}\widehat{\upgamma }\right]={\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}{{Var}}\left(\widehat{\upgamma }\right)\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]$$

Therefore, we further get that

$${{{R}}}_{{{s}},{{k}}}\left(\widehat{\upgamma }\right)-{{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\stackrel{{{d}}}{\to }{{N}}\left(0,{\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]}^{{{T}}}{{Var}}\left(\widehat{\upgamma }\right)\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upgamma \right)\right]\right)$$

This completes the proof.

Proof of Theorem 3

Given ${{i}}={1,2},\ldots,{{m}}$, let ${Z}_{i1},{Z}_{i2},\ldots,{Z}_{is}$ be order statistics from UGR distribution. So ${R}_{s,k}\left(\widehat{\eta }\right)-{{{R}}}_{{{s}},{{k}}}\left(\upeta \right)\approx {\left[\nabla {{{R}}}_{{{s}},{{k}}}\left(\upeta \right)\right]}^{{{T}}}{X}_{ij}=-{\vartheta }_{1}{{log}}\left(1-{B}_{ij}\right),j={1,2},\cdots s$ denote censored samples from standard exponential distribution. We have.

$$ \begin{gathered} L_{i1} = kX_{i1} ,\;L_{i2} = \left( {k - 1} \right)\left( {X_{i2} - X_{i1} } \right) \hfill \\ \ldots \hfill \\ L_{{i\left( {s - 1} \right)}} = \left( {k - s + 2} \right)\left( {X_{{i\left( {s - 1} \right)}} - X_{{i\left( {s - 2} \right)}} } \right), \hfill \\ L_{is} = \left( {k - s + 1} \right)\left( {X_{is} - X_{{i\left( {s - 1} \right)}} } \right), \hfill \\ \end{gathered} $$

Are independent and identically distributed as standard exponential distribution (see, e.g., Lawless (2003)).

Consider ${W}_{ij}=\sum_{r=1}^{j}{L}_{ir}={-\vartheta }_{1}\left\{\sum_{r=1}^{j}{{log}}\left(1-{B}_{ir}\right)+\left(k-j\right){{log}}\left(1-{B}_{ij}\right)\right\},$

$j={1,2},\ldots,$ s, and using Stephens (1986) and Viveros and Balakrishnan (1994), one further has that the quantities

$$ \begin{gathered} U_{i1} = \frac{{W_{i1} }}{{W_{is} }} = \frac{{\mathop \sum \nolimits_{r = 1}^{1} \log \left( {1 - B_{ir} } \right) + \left[ {k - 1} \right]\log \left( {1 - B_{i1} } \right)}}{{\mathop \sum \nolimits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left[ {k - s} \right]\log \left( {1 - B_{is} } \right)}} \hfill \\ U_{i2} = \frac{{W_{i2} }}{{W_{is} }} = \frac{{\mathop \sum \nolimits_{r = 1}^{2} \log \left( {1 - B_{ir} } \right) + \left[ {k - 2} \right]\log \left( {1 - B_{i2} } \right)}}{{\mathop \sum \nolimits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left[ {k - s} \right]\log \left( {1 - B_{is} } \right)}} \hfill \\ U_{{i\left( {s - 1} \right)}} = \frac{{W_{{i\left( {s - 1} \right)}} }}{{W_{is} }} = \frac{{\mathop \sum \nolimits_{r = 1}^{s - 1} \log \left( {1 - B_{ir} } \right) + \left[ {k - s + 1} \right]\log \left( {1 - B_{{i\left( {s - 1} \right)}} } \right)}}{{\mathop \sum \nolimits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left[ {k - s} \right]\log \left( {1 - B_{is} } \right)}} \hfill \\ \end{gathered} $$

denote order statistics from ${{U}}\left({0,1}\right)$ distribution. Also ${U}_{i1}<{U}_{i2}<\cdots {U}_{i\left(s-1\right)}$ being independent with ${W}_{is}{=-\vartheta }_{1}\left[\sum_{r=1}^{s}{{log}}\left(1-{B}_{ir}\right)+\left[k-s\right]{{log}}\left(1-{B}_{is}\right)\right]$.

We have

$$ \begin{aligned} D_{i1} \left( \varepsilon \right) & = - 2\mathop \sum \limits_{j = 1}^{s - 1} \log \left( {U_{ij} } \right) \\ & = 2\mathop \sum \limits_{j = 1}^{s - 1} {{log}}\left[ {\frac{{\mathop \sum \nolimits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left( {k - s} \right)\log \left( {1 - B_{is} } \right)}}{{\mathop \sum \nolimits_{r = 1}^{j} \log \left( {1 - B_{ir} } \right) + \left( {k - j} \right)\log \left( {1 - B_{ij} } \right)}}} \right], \\ \end{aligned} $$

is chi-square distributed with $2\left({{s}}-1\right)$ degrees of freedom and is independent of

$${E}_{i1}\left({\vartheta }_{1},\varepsilon \right)=2{W}_{is}=-2\sum_{i=1}^{m}\left\{\sum_{r=1}^{s}{{log}}\left(1-{B}_{ir}\right)+\left(k-s\right){{log}}\left(1-{B}_{is}\right)\right\}$$

is ${\chi }^{2}\left(2s\right)$ random variable.

Thus from independence of ${D}_{i1}\left(\varepsilon \right),{D}_{i2}\left(\varepsilon \right),\ldots,{D}_{im}\left(\varepsilon \right)$, it is seen that

$$ \begin{gathered} D_{1}^{Z} \left( \varepsilon \right) = \mathop \sum \limits_{i = 1}^{m} D_{i1} \left( \varepsilon \right) \hfill \\ = 2\mathop \sum \limits_{i = 1}^{m} \mathop \sum \limits_{j = 1}^{s - 1} {{log}}\left[ {\frac{{\mathop \sum \nolimits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left( {k - s} \right)\log \left( {1 - B_{is} } \right)}}{{\mathop \sum \nolimits_{r = 1}^{j} \log \left( {1 - B_{ir} } \right) + \left( {k - j} \right)\log \left( {1 - B_{ij} } \right)}}} \right] \hfill \\ \end{gathered} $$

is a $\chi^{2} \left( {2m\left( {s - 1} \right)} \right)$ variable

$$ \begin{gathered} E_{1}^{Z} \left( {\vartheta_{1} ,\varepsilon } \right) = \mathop \sum \limits_{i = 1}^{m} E_{i1} \left( {\vartheta_{1} ,\varepsilon } \right) \hfill \\ = - 2\vartheta_{1} \mathop \sum \limits_{i = 1}^{m} \left\{ {\mathop \sum \limits_{r = 1}^{s} \log \left( {1 - B_{ir} } \right) + \left( {k - s} \right)\log \left( {1 - B_{is} } \right)} \right\} \hfill \\ \end{gathered} $$

follows ${\chi }^{2}\left(2ms\right)$ distribution, and ${D}_{1}^{Z}\left(\varepsilon \right)$ and ${E}_{1}^{Z}\left({\vartheta }_{1},\varepsilon \right)$ are statistical independent. Thus proof is completed.

Proof of Lemma 1

From the definition of $R\left(h\right)$, it is noted that,${{li}}{m}_{h\to 0}R\left(h\right)$ is zero upon zero form. We use ${{L}}$ Hospital's rule then.

$$\underset{h\to 0}{{{lim}}}R\left(h\right)=\frac{exp{\left[h{{log}}c\right]}^{2}}{exp{\left[h{{log}}d\right]}^{2}}\Rightarrow \underset{h\to 0}{{{lim}}}R\left(h\right)=1$$

For $0<{{c}}<{{d}}<1$, since

$${{log}}\left(d\right)>{{log}}\left(c\right),\Rightarrow {\left(-{{log}}\left(c\right)\right)}^{2}>{\left(-{{log}}\left(d\right)\right)}^{2},\Rightarrow \underset{h\to \infty }{{{lim}}}R\left(h\right)=\infty $$

This completes the proof of first part. By computation, the limitation results can be obtained directly. Therefore, the assertion is completed.

Proof of Corollary 1

Considering ${D}_{1}^{Z}$ and ${D}_{1}^{T}$, for $i={1,2},\ldots,{{m}}$ and ${{j}}={1,2},\ldots,{{s}}-1$ or${{m}}-1$, we have

$$\frac{\sum_{r=1}^{s}\left({{log}}\left(1-{B}_{ir}\right)+\left(k-s\right){{log}}\left(1-{B}_{is}\right)\right)}{\sum_{r=1}^{j}\left({{log}}\left(1-{B}_{ir}\right)+\left(k-j\right){{log}}\left(1-{B}_{ij}\right)\right)}=$$

$$1+\frac{\sum_{r=j+1}^{s}\frac{\left({{log}}\left(1-{B}_{ir}\right)\right)}{\left({{log}}\left(1-{B}_{ij}\right)\right)}+\left(k-s\right)\frac{\left({{log}}\left(1-{B}_{is}\right)\right)}{\left({{log}}\left(1-{B}_{ij}\right)\right)}-\left(k-j\right)}{\sum_{r=1}^{j}\frac{\left({{log}}\left(1-{B}_{ir}\right)\right)}{\left({{log}}\left(1-{B}_{ij}\right)\right)}+\left(k-j\right)}$$

and

$$\frac{\sum_{r=1}^{m}\left({{log}}\left(1-{A}_{\left(r\right)}\right)\right)}{\sum_{r=1}^{j}\left({{log}}\left(1-{A}_{\left(r\right)}\right)\right)+\left(m-j\right)\left({{log}}\left(1-{A}_{\left(j\right)}\right)\right)}$$

$$=1+\frac{\sum_{r=j+1}^{m}\frac{\left({{log}}\left(1-{A}_{\left(r\right)}\right)\right)}{\left({{log}}\left(1-{A}_{\left(j\right)}\right)\right)}-\left(m-j\right)}{\sum_{r=1}^{j}\frac{\left({{log}}\left(1-{A}_{\left(r\right)}\right)\right)}{\left({{log}}\left(1-{A}_{\left(j\right)}\right)\right)}+\left(m-j\right)}$$

we see that numerators of above two terms increase in ε and denominators tend to decrease in ε. So ${D}_{1}^{Z}\left(\varepsilon \right)$ and ${D}_{1}^{T}\left(\varepsilon \right)$ are increasing function on $\upvarepsilon$. We further have.

$\underset{\varepsilon \to 0}{{{lim}}}{D}_{1}^{Z}\left(\varepsilon \right)=0,\underset{\varepsilon \to +\infty }{{{lim}}}{D}_{1}^{Z}\left(\varepsilon \right)=+\infty$,and $\underset{\varepsilon \to 0}{{{lim}}}{D}_{1}^{T}\left(\varepsilon \right)=0,\underset{\varepsilon \to +\infty }{{{lim}}}{D}_{1}^{T}\left(\varepsilon \right)=+\infty$. Hence the result is shown.