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
Ahmadi J, Arghami NR (2003) Tolerance intervals from record values data. Stat Pap 44:455–468
Ahmadi J, Balakrishnan N (2005) Distribution-free confidence intervals for quantile intervals based on current records. Stat Probab Lett 75:190–202
Ahsanullah M (2013) Inferences of type II extreme value distribution based on record values. Appl Math Sci 7:3569–3578
Arnold BC, Balakrishnan N, Nagaraja HN (1998) Records. Wiley, New York
Amini M, Balakrishnan N (2013) Nonparametric meta-analysis of independent samples of records. Comput Stat Data Anal 66:70–81
Balakrishnan N, Ahsanullah M, Chan PS (1995) On logistic record values and associated inference. J Appl Stat Sci 2:233–248
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
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
Balakrishnan N, Li T (2008) Ordered ranked set samples and applications to inference. J Stat Plan Inf 138:3512–3524
Balakrishnan N, Rao CR (2003) Some efficiency properties of best linear unbiased estimators. J Stat Plan Inf 113:551–555
Barlow RE, Hunter L (1960) Optimum preventive maintenance policies. Oper Res 8:90–100
Beutner E, Cramer E (2010) Nonparametric meta-analysis for minimal-repair systems. Aust N Z J Stat 52:383–401
Beutner E, Cramer E (2011) Confidence intervals for quantiles in a minimal repair set-up. Int J Appl Math Stat 24:86–97
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
Cramer E, Kamps U (2003) Marginal distributions of sequential and generalized order statistics. Metrika 58:293–310
Doostparast M (2009) A note on estimation based on record data. Metrika 69:69–80
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
Dorado C, Hollander M, Sethuraman J (1997) Nonparametric estimation for a general repair model. Ann Stat 25:1140–1160
Goldberger AS (1962) Best linear unbiased predictors in the generalized regression model. J Am Stat Assoc 57:369–375
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
Gulati S, Padgett WJ (1994b) Nonparametric quantile estimation from record-breaking data. Aust J Stat 36:211–223
Hollander M, Presnell B, Sethuraman J (1992) Nonparametric methods for imperfect repair models. Ann Stat 20:879–896
Kamps U (1995a) A concept of generalized order statistics. Teubner, Stuttgart
Kamps U (1995b) A concept of generalized order statistics. J Stat Plan Inf 48:1–23
Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26:89–102
Last G, Szekli R (1998) Asymptotic and monotonicity properties of some repairable systems at imbedded failure epochs. Adv Appl Probab 30:1089–1110
Mann NR (1969) Optimum estimators for linear functions of location and scale parameters. Ann Math Stat 40:2149–2155
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
Samaniego FJ, Whitaker LR (1988) On estimating population characteristics from record-breaking observations. II: nonparametric results. Nav Res Logist Quart 35:221–236
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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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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 \(1\le r <s\le n\),
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
and
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:
and
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 \(r<s\le t\),
for \(r+1\le t<s\),
for \(t=r < s\),
for \(0<t<r<s\),
and finally,
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
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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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DOI: https://doi.org/10.1007/s00180-014-0552-8