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Pooled parametric inference for minimal repair systems

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Abstract

Consider two independent and identically structured systems, each with a certain number of observed repair times. The repair process is assumed to be performed according to a minimal-repair strategy. In this strategy, the state of the system after the repair is the same as it was immediately before the failure of the system. The resulting pooled sample is then used to obtain best linear unbiased estimators (BLUEs) as well as best linear invariant estimators of the location and scale parameters of the presumed parametric families of life distributions. It is observed that the BLUEs based on the pooled sample are overall more efficient than those based on one sample of the same size and also than those based on independent samples. Furthermore, the best linear unbiased predictor and the best linear invariant predictor of a future repair time from an independent system are also obtained. A real data set of Boeing air conditioners, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes, is used to illustrate the inferential results developed here.

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References

  • Agustin ZA, Peña EA (1999) Order statistic properties, random generation, and goodness-of-fit testing for a minimal repair model. J Am Stat Assoc 94:266–272

    Article  MATH  Google Scholar 

  • Ahmadi J, Arghami NR (2003) Tolerance intervals from record values data. Stat Pap 44:455–468

    Article  MATH  MathSciNet  Google Scholar 

  • Ahmadi J, Balakrishnan N (2005) Distribution-free confidence intervals for quantile intervals based on current records. Stat Probab Lett 75:190–202

    Article  MATH  MathSciNet  Google Scholar 

  • Ahsanullah M (2013) Inferences of type II extreme value distribution based on record values. Appl Math Sci 7:3569–3578

    MathSciNet  Google Scholar 

  • Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New York

    Book  MATH  Google Scholar 

  • Amini M, Balakrishnan N (2013) Nonparametric meta-analysis of independent samples of records. Comput Stat Data Anal 66:70–81

    Article  MathSciNet  Google Scholar 

  • Balakrishnan N, Ahsanullah M, Chan PS (1995) On logistic record values and associated inference. J Appl Stat Sci 2:233–248

    MATH  MathSciNet  Google Scholar 

  • Balakrishnan N, Chan PS (1995) On the normal record values and associated inference. Technical report, McMaster University, Hamilton, Ontario, Canada

  • Balakrishnan N, Cohen AC (1991) Order statistics and inference: estimation methods. Academic Press, Boston

    MATH  Google Scholar 

  • Balakrishnan N, Li T (2006) Confidence intervals for quantiles and tolerance intervals based on ordered ranked set samples. Ann Inst Stat Math 58:757–777

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan N, Li T (2008) Ordered ranked set samples and applications to inference. J Stat Plan Inf 138:3512–3524

    Article  MATH  MathSciNet  Google Scholar 

  • Balakrishnan N, Rao CR (2003) Some efficiency properties of best linear unbiased estimators. J Stat Plan Inf 113:551–555

    Article  MATH  MathSciNet  Google Scholar 

  • Barlow RE, Hunter L (1960) Optimum preventive maintenance policies. Oper Res 8:90–100

    Article  MATH  MathSciNet  Google Scholar 

  • Beutner E, Cramer E (2010) Nonparametric meta-analysis for minimal-repair systems. Aust N Z J Stat 52:383–401

    Article  MathSciNet  Google Scholar 

  • Beutner E, Cramer E (2011) Confidence intervals for quantiles in a minimal repair set-up. Int J Appl Math Stat 24:86–97

    MathSciNet  Google Scholar 

  • Cramer E, Kamps U (2001) Sequential k-out-of-n systems. In: Balakrishnan N, Rao CR (eds) Handbook of statistics-20. Elsevier, Amsterdam, pp 301–372

    Google Scholar 

  • Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58:293–310

    Article  MATH  MathSciNet  Google Scholar 

  • Doostparast M (2009) A note on estimation based on record data. Metrika 69:69–80

    Article  MathSciNet  Google Scholar 

  • Doostparast M, Akbari MG, Balakrishnan N (2011) Bayesian analysis for the two-parameter Pareto distribution based on record values and times. J Stat Comput Simul 81:1393–1403

    Article  MATH  MathSciNet  Google Scholar 

  • Dorado C, Hollander M, Sethuraman J (1997) Nonparametric estimation for a general repair model. Ann Stat 25:1140–1160

    Article  MATH  MathSciNet  Google Scholar 

  • Goldberger AS (1962) Best linear unbiased predictors in the generalized regression model. J Am Stat Assoc 57:369–375

    Article  MATH  MathSciNet  Google Scholar 

  • Gulati S, Padgett WJ (1994a) Smooth nonparametric estimation of the distribution and density functions from record-breaking data. Commun Stat Theor Methods 23:1259–1274

    Article  MATH  MathSciNet  Google Scholar 

  • Gulati S, Padgett WJ (1994b) Nonparametric quantile estimation from record-breaking data. Aust J Stat 36:211–223

    Article  MATH  MathSciNet  Google Scholar 

  • Hollander M, Presnell B, Sethuraman J (1992) Nonparametric methods for imperfect repair models. Ann Stat 20:879–896

    Article  MATH  MathSciNet  Google Scholar 

  • Kamps U (1995a) A concept of generalized order statistics. Teubner, Stuttgart

    Book  MATH  Google Scholar 

  • Kamps U (1995b) A concept of generalized order statistics. J Stat Plan Inf 48:1–23

    Article  MATH  MathSciNet  Google Scholar 

  • Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26:89–102

    Article  MATH  MathSciNet  Google Scholar 

  • Last G, Szekli R (1998) Asymptotic and monotonicity properties of some repairable systems at imbedded failure epochs. Adv Appl Probab 30:1089–1110

    Article  MATH  MathSciNet  Google Scholar 

  • Mann NR (1969) Optimum estimators for linear functions of location and scale parameters. Ann Math Stat 40:2149–2155

    Article  MATH  Google Scholar 

  • Ren H (2011) Empirical Bayes estimation in exponential model based on record values under asymmetric loss. Knowl Eng Manag Adv Intell Soft Comput 123:659–667

    Google Scholar 

  • Samaniego FJ, Whitaker LR (1988) On estimating population characteristics from record-breaking observations. II: nonparametric results. Nav Res Logist Quart 35:221–236

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors thank two anonymous referees and the associate editor for their useful comments and suggestions on an earlier version of this manuscript which resulted in this improved version.

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Correspondence to Morteza Amini.

Appendices

Appendix 1: Formal proofs

Proof of Lemma 1

The proof uses the same method as in Cramer and Kamps (2001). To obtain the joint cdf of \(X(r,n,{\gamma _r}^{(t)})\) and \(X(s,n,{\gamma _s}^{(t)})\) for \(1\le r<s \le n\), first we obtain the joint cdf of \(Y(r,n,{\gamma _r}^{(t)})\) and \(Y(s,n,{\gamma _s}^{(t)})\), where \(Y(1,n,{\gamma _1}^{(t)}),\ldots ,Y(n,n,{\gamma _n}^{(t)}), t\ge 0\), is a GOS sample from the standard exponential distribution \((Exp(1))\), with \({\gamma _1}^{(t)}=\cdots ={\gamma _t}^{(t)}=2\), \({\gamma _{t+1}}^{(t)}=\cdots ={\gamma _n}^{(t)}=1\) for \(t\ge 1\), and \({\gamma _1}^{(t)}=\cdots ={\gamma _n}^{(t)}=1\) for \(t=0\). Indeed, we have \(X(r,n,{\gamma _r}^{(t)})=F^{-1}(1-e^{-Y(r,n,{\gamma _r}^{(t)})})\), for \(1\le r\le n\). Let \(D_1={\gamma _1}^{(t)}Y(1,n,{\gamma _1}^{(t)})\) and \(D_r={\gamma _r}^{(t)}(Y(r,n,{\gamma _r}^{(t)})-Y(r-1,n,\gamma _{r-1})),\; 2\le r\le n\). It has been shown by Kamps (1995a, p. 8) that \(D_1,\ldots ,D_n\) are iid as \(Exp(1)\). Then, we have for \(x<y\) and \(1\le r <s\le n\),

$$\begin{aligned}&P(Y(r,n,{\gamma _r}^{(t)})\le x,Y(s,n,{\gamma _s}^{(t)})\le y)\nonumber \\&\quad = \int _0^x P\left( \sum _{j=r+1}^s\frac{D_j}{\gamma _j}\le y-z\right) f_{\sum _{j=1}^r\frac{D_j}{\gamma _j}}(z)dz . \end{aligned}$$
(13)

To obtain the distribution of \(\sum _{j=r+1}^s\frac{D_j}{\gamma _j}\) and \(\sum _{j=1}^r\frac{D_j}{\gamma _j}\) in (13), let \(G_n\) and \(G'_m\) denote two independent random variables with \(Gamma(n,1)\) and \(Gamma(m,1)\) distributions, possessing the cdfs \(F_{G_n}\) and \(F_{G'_{m}}\) and pdfs \(f_{G_n}\) and \(f_{G'_{m}}\), respectively. We can easily conclude that

$$\begin{aligned} \sum _{j=1}^r\frac{D_j}{\gamma _j}\mathop {=}\limits ^\mathrm{d} \left\{ \begin{array}{ll} G_r/2,&{}\quad for \quad r\le t\\ G_t/2+G'_{r-t},&{}\quad for\quad 0<t<r\\ G_r,&{}\quad for\quad t=0\\ \end{array}\right. \end{aligned}$$
(14)

and

$$\begin{aligned} \sum _{j=r+1}^s\frac{D_j}{\gamma _j}\mathop {=}\limits ^\mathrm{d} \left\{ \begin{array}{ll} G_{s-r}/2,&{}\quad for \quad s\le t\\ G_{t-r}/2+G'_{s-t},&{}\quad for\quad r+1\le t<s\\ G_{s-r},&{}\quad for \quad t<r+1\le s, \end{array}\right. \end{aligned}$$
(15)

where \(\mathop {=}\limits ^\mathrm{d}\) stands for “identically distributed”. The cdf and pdf of \(Z=\frac{G_n}{2}+G'_m\) can be easily obtained, respectively, as follows:

$$\begin{aligned} F_Z(z)&=1-\sum _{j=0}^{m-1}\frac{e^{-z}z^j}{j!}-\sum _{j=0}^{n-1}\sum _{k=0}^{j}{j\atopwithdelims ()k} \frac{2^j(-1)^{k}(k+m-1)!}{j!(m-1)!}e^{-2 z}z^{j-k}\nonumber \\&\times \,\left\{ (-1)^{k+m}-e^z\sum _{l=0}^{k+m-1}\frac{(-1)^{k+m-l}z^l}{l!}\right\} \\&= 1-\sum _{j=0}^{m-1}\frac{e^{-z}z^j}{j!}-\sum _{j=0}^{n-1}\sum _{k=0}^{j}{k+m-1\atopwithdelims ()k} 2^{k-1}(-1)^k f_{G_{j-k+1}/2}(z)\Lambda _{k+m-1}(z),\quad {\text{ say, }}\nonumber \end{aligned}$$
(16)

and

(17)

where \(\Lambda _k(z)=\int _{0}^{z}x^k\;e^x\;dx/k!=(-1)^{k+1}-e^z\sum _{l=0}^{k}\frac{(-1)^{k-l+1}z^l}{l!}\), and \(G_r\) is a \(Gamma(r,1)\) random variable. Substituting the expressions in (16) and (17), using (14) and (15), into Eq. (13), we obtain for \(r<s\le t\),

$$\begin{aligned}&F_{Y(r,n,{\gamma _r}^{(t)}),Y(s,n,{\gamma _s}^{(t)})}(x,y)\\&\quad = F_{G_r/2}(x)-\sum _{j=0}^{s-r-1}\sum _{k=0}^{j}{k+r-1 \atopwithdelims ()k}(-1)^{k}2^{r+k-1}\frac{x^{k+r}}{(k+r)!}f_{G_{j-k+1}/2}(y); \end{aligned}$$

for \(r+1\le t<s\),

$$\begin{aligned}&F_{Y(r,n,{\gamma _r}^{(t)}),Y(s,n,{\gamma _s}^{(t)})}(x,y) \\&\quad = F_{G_r/2}(x)-\sum _{j=0}^{s-t-1}\sum _{k=0}^{j}{k+r-1\atopwithdelims ()k}2^{r}(-1)^{k}\\&\quad \quad \times \, f_{G_{j-k+1}}(y)F_{G_{k+r}}(x)\\&\quad \quad -\,\sum _{j=0}^{t-r-1}\sum _{k=0}^{j}\sum _{m=0}^{j-k}{k+s-t-1 \atopwithdelims ()k}{m+r-1 \atopwithdelims ()m} 2^{r+m+k}\\&\quad \quad \times \,(-1)^{s-t+m}f_{G_{j-m-k+1}/2}(y)\frac{x^{m+r}}{(m+r)!}\\&\quad \quad +\, \sum _{j=0}^{t-r-1}\sum _{k=0}^{j}\sum _{m=0}^{l+j-k}\sum _{l=0} ^{k+s-t-1}{k+s-t-1 \atopwithdelims ()k}{m+r-1 \atopwithdelims ()m}{l+j-k \atopwithdelims ()l} \\&\quad \quad \times \,2^{r+j}(-1)^{m+s-t-l}f_{G_{l+j-m-k+1}}(y)F_{G_{m+r}}(x); \end{aligned}$$

for \(t=r < s\),

$$\begin{aligned}&F_{Y(r,n,{\gamma _r}^{(t)}),Y(s,n,{\gamma _s}^{(t)})}(x,y)\\&\quad = F_{G_r/2}(x)-\sum _{j=0}^{s-r-1}\sum _{k=0}^{j}{k+r-1 \atopwithdelims ()k}2^r(-1)^kf_{G_{j-k+1}}(y)F_{G_{k+r}}(x); \end{aligned}$$

for \(0<t<r<s\),

$$\begin{aligned}&F_{Y(r,n,{\gamma _r}^{(t)}),Y(s,n,{\gamma _s}^{(t)})}(x,y) \\&\quad = F_{G_{r-t}}(x)-\sum _{j=0}^{t-1}\sum _{k=0}^{j}{k+r-t-1 \atopwithdelims ()k}\\&\quad \quad \times \, 2^{k-1}(-1)^{k}f_{G_{j-k+1}}(x)\Lambda _{k+r-t-1}(x)\\&\quad \quad -\,\sum _{j=0}^{s-r-1}\sum _{k=0}^{t-1}\sum _{l=0}^{j}{k+r-t-1 \atopwithdelims ()k}{l+t-k-1 \atopwithdelims ()l} 2^t\\&\quad \quad \times \,(-1)^{l+r-t}f_{G_{j-l+1}}(y)F_{G_{l+t-k}}(x)\\&\quad \quad +\, \sum _{j=0}^{s-r-1}\sum _{k=0}^{t-1}\sum _{l=0}^{j}\sum _{m=0}^{k+r-t-1}{k+r-t-1 \atopwithdelims ()k}{m+l+t-k-1 \atopwithdelims ()l,\;m,\;t-k-1} 2^t\\&\quad \quad \times \,(-1)^{l+r-t-m}f_{G_{j-l+1}}(y)\frac{x^{m+l+t-k}}{(m+l+t-k)!}; \end{aligned}$$

and finally,

$$\begin{aligned}&F_{Y(r,n,{\gamma _r}^{(t)}),Y(s,n,{\gamma _s}^{(t)})}(x,y)\\&\quad =F_{G_{r}}(x)-\sum _{i=0}^{s-r-1}\sum _{k=0}^{i}{r+k-1 \atopwithdelims ()k}(-1)^k f_{G_{i-k}}(y)\frac{x^{r+k}}{(r+k)!}. \end{aligned}$$

The required result is obtained by utilizing the transformation \(X(r,n,{\gamma _r}^{(t)})=F^{-1}(1-e^{-Y(r,n,{\gamma _r}^{(t)})})\). \(\square \)

Appendix 2

See Tables 2, 3, 4 and 5.

Table 2 Coefficients of the BLUE for the scale parameter based on the ordered pooled sample and its variance for different values of \(n\) and different sampling schemes for the scaled exponential distribution
Table 3 Relative efficiencies (RE) of the BLUEs for different values of \(n\) and different sampling schemes for exponential (Exp.) and two-parameter exponential (Two-Exp.) distributions (Dist.)
Table 4 Coefficients for the location (Loc.) and scale parameters (Par.) of the BLUEs based on the ordered pooled sample and their covariance matrix (Cov.) for different values of \(n\) and different sampling schemes for the two-parameter exponential distribution
Table 5 Coefficients of the BLUE based on ordered pooled values of inter-failure times for the data set following a two-parameter Rayleigh distribution

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Amini, M., Balakrishnan, N. Pooled parametric inference for minimal repair systems. Comput Stat 30, 605–623 (2015). https://doi.org/10.1007/s00180-014-0552-8

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