The functions Φsp(F∗1,F∗2,F∗3,t), and Φps(F∗1,F∗2,F∗3,t), based on sub-distributions F∗1, F∗2, and F∗3, areThe SDF F∗1, F∗2, and F∗3 determine (uniquely) the DF F2 of an SPS for t≤t∗ by F2(t)=Φsp(F∗1,F∗2,F∗3,t), and the DF F2 of a PSS for t≤t∗ by F2(t)=Φps(F∗1,F∗2,F∗3,t). This section presents two examples to demonstrate the estimation steps and to show the quality of the Bayesian nonparametric estimator. For the jth component, the sample from the posterior can be expressed as (β(1)j,β(2)j,…,β(np)j), (η(1)j,η(2)j,…,η(np)j) and (μ(1)j,μ(2)j,…,μ(np)j). 2, Appendix B. 6. DefineandThe main result of this study is given in the following paragraph.
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Let Ω be a sample space, α1,…,αm be finite positive measures defined over Ω, and ρ=(ρ1,…,ρm) be a random vector having a Dirichlet distribution with parameters (α1(Ω),…,αm(Ω)). 3 and 1. In reliability, a component under test may be used at an earlier moment and the beginning of the test is not necessarily the beginning of the component’s life. The estimation steps for the PSS are very similar to those for the SPS, and for the sake of brevity, we have omitted them. In the context of SPS and PSS, consider Ω=(0,∞), ρj=P(δ=j) and Pj(t)=P(T≤t|δ=j), for j=1,2,3.
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5 presents the estimates of the five distribution functions: components 1 through 4 and the system. For simplicity, consider the case of m=2. Besides, δi=j if Ti=Xji, for i=1,…,n and j=1,2,3. Simulate λ3j from π(λ3j∣t,δj,υj,d(b)j,θ(b)j,λ(b−1)1j,λ(b−1)2j) in whichthat is, λ3j∣(t,δj,υj,dj,θj,λ1j,λ2j)∼Beta(∑i: υji=1d3ji+1,nl+1), in which nl is the number of systems in which component j is observed to be left-censored. 3.
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Using (6), we obtain the estimate for component 1 DF; and from (6), we obtain the estimate for component 2 DF. 1) can be estimated using the PSS result of Theorem right here which has a simple solution. Also, the integral part of the estimator can be solved by using a numerical procedure, such as the Simpson’s rule. For component 2, even for values of t that the posterior mean is more distant from the true values of the reliability, the upper limit of HPD interval is very close to the true curve. 3). Table 4.
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Since the components in si are right censored or responsible for system failure, λ3j=0, for j=1,2,3, and no hard-drive is subject of left censored failure time. A series of examples illustrates the flexibility of the approach. First consider a parallel system with m components. For instance, system ID=4 failed at time 2. We obtained 100 observations of SPS presented in Figure 2.
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9) does not work for the case with jump points.
Considering an augmented data procedure (latent variable), define d1ji=1 if the masked observation is not censored or d1ji=0 otherwise, d2ji=1 if the masked observation is right censored or d2ji=0 otherwise, and d3ji=1 if the masked observation is left censored or d3ji=0 otherwise. There were three possible causes of failure: eletronic hard (component j=1), head flyability (component j=2) and head/disc magnetics (component j=3). Besides, the three-parameter Weibull family is a very rich family since most real situations will have random aspects that can be represented by an element of the family. Figure 2.
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communitiesAlready have an account? Login hereLogin to DeepAIDon’t have an account? Signup hereSHARE© 2022 Deep AI, Inc. 17), we havewhich are the estimators of SDF. However, we do not have information about the cases 4 to 6, check my blog when the data is masked, we do not know if that component was censored or not. 60% of systems and the remaining 19. Note that in our proposal approach the configuration of set s is not a big deal, once the important information for estimation of jth component reliability is if j belongs to s or not.
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We chose the exponential distribution with mean 1 as the prior guess for each of the three components DF. Journal of Multivariate AnalysisSign up for DeepAIJoin one of the world’s largest A. In this case, for a fixed t, we have that a natural prior choice is the Dirichlet distribution, and for any t, we have the Dirichlet multivariate process. 5) leads toFinally, (2. The dataset is available in Flehinger et al. .