By N. Balakrishnan, C. R. Rao
Hardbound. the world of Reliability has turn into a vital and lively zone of study. this is often basically obvious from the big physique of literature that has been constructed within the kind of books, volumes and examine papers considering the fact that 1988 while the former guide of data in this zone was once ready by means of P.R. Krishnaiah and C.R. Rao. this is why we felt that this can be certainly the appropriate time to commit one other quantity within the guide of records sequence to spotlight a few contemporary advances within the region of Reliability. With this objective in brain, we solicited articles from major specialists operating within the quarter of Reliability from either academia and undefined. This, in our opinion, has ended in a quantity with a pleasant mix of articles (33 in overall) facing theoretical, methodological and utilized concerns in Reliability. For the ease of readers, now we have divided this quantity into thirteen components as follows: Reli
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Basu and S. E. Rigdon It can be shown that if n _> 1 then Because of the condition n _> 1, the estimator B N-lB_ N N- 1 ~N=I log(t/ti) is called the conditionally unbiased estimator for fl. Confidence intervals and tests of hypotheses for/? are based on the result that conditioned on N = n, the quantity 2nf/fi has a chi-square distribution with 2n degrees of freedom. Following the same procedure as the failure truncated case, we find that a 100(1 - «)% confidence interval for fi is given by X2_«/2(2n)/} Z2~/2(2n)/~ < fl < 2n (14) 2n Since the power law process becomes a homogeneous Poisson process when B = 1, it is often of interest to see whether the confidence interval contains the value fi = 1.
In the case of rare events direct simulation is not efficient since, in order to get an idea about the probability of the event, we have to hit it several times and thus use a large number of samples. For instance, if we want a 20% error and 95% confidence interval for the estimation of a probability of the order of 10 6, we will certainly need at least 108 realizations. 26 N. Balakrishnan, N. Limnios and C. Papadopoulos Given that direct simulation is too expensive in terms of the number of samples needed in order to obtain a given confidence level, other simulation methods have appeared to face this problem.
E. Schrage (1987). A Guide to Simulation. Springer, Berlin. Basic probabilistic models in reliability 41 ~inlar, E. (1969). Markov renewal theory. Adv. Appl. Probab. 1, 123 187. Crane, M. A. and D. L. Iglehart (1975). Simulating stable stochastic systems III, regenerative processes and discrete event simulation. Oper. Res. 23, 3345. Fishman, G. S. (1996). Monte Carlo. Concepts, Algorithms and Applications. Springer Series in Operations Research, Springer, New York. Fox, B. L. and P. W. Glynn (1986).