Evaluation for Confidence Interval of Reliability of Rolling Bearing Lifetime with Type I Censoring

: In the lifetime test of rolling bearings under type I censoring with a small sample, the confidence interval of reliability needs to be evaluated to ensure safe and reliable operation of a system like an aerospace system. Thus the probability density function of Weibull distribution parameters must be attained. Owing to very few test data and for lack of prior knowledge, it is difficult to take it out for prevailing methods like the moment method, the maximum likelihood method and the Harris method. For this end, the bootstrap likelihood maximum-entropy method is proposed by fusing the bootstrap method, the maximum likelihood method and the Harris method. The lifetime test data with the small sample are made into the simulated parameter data with the large sample to obtain the probability density function on the parameters. The confidence intervals of the Weibull distribution parameters are estimated and the confidence interval of reliability is calculated. The tests of the complete large-sample data, the complete small-sample data and the incomplete small-sample data are produced to prove effectiveness of the proposed method. Results show that the proposed method can assess the confidence interval of the reliability without any prior information on the Weibull distribution parameters.


INTRODUCTION
In the lifetime test of rolling bearings, the confidence interval of the reliability needs to be evaluated to ensure safe and reliable operation of a system like an aerospace system, a nuclear reactiondiffusion system and a weapon system (Xia et al., 2009;Chowdhury and Adhikari, 2011).This is a new indicator of reliability analysis for the rolling bearing lifetime.Such a requirement, in theory, is justified in a performance test because, according to metrology, the estimated value of a parameter has uncertainty, along with a confidence level and interval.The value of the lifetime reliability is estimated by the parameters of a lifetime probability distribution and it has necessarily indirect uncertainty (Jean-Francois and Joseph, 2010;Radoslav et al., 2011;Luxhoj and Shyur, 1995;Joarder et al., 2011;Jiang et al., 2010;Chen et al., 2009;Fu et al., 2010;Liu et al., 2009a).
The two-parameter Weibull distribution is one of the most commonly used functions in reliability analysis for the rolling bearing lifetime.To assess the confidence interval of the reliability, the Weibull distribution parameters, the shape parameter β and the scale parameter η, should be estimated and the probability density functions of the two parameters should also be obtained.
According to Bayesian statistics, β and η can be regarded as two independent stochastic variables.Thus, material data; based on two independent normal random variables, Barbiero (2011) proposed the procedure to get confidence intervals for the reliability in stress-strength models; and with the help of a gamma function model, Kimura (2008) focused on the generalization of several software reliability models and the derivation of confidence intervals of reliability assessment measures.Nevertheless, available findings rely on the priori information of the parameters of a probability distribution.For type I censoring with a small sample, how to assess the confidence interval of the reliability of the rolling bearing lifetime is still a puzzle under the condition of the lack of the priori information about the Weibull distribution parameters.
As is well known, the bootstrap method (Kimura, 2008;Othman and Musirin, 2011;Xia et al., 2010) and the maximum entropy method (Yonamoto et al., 2011;Li and Zhang, 2011) are two of the prevailing methods for data analysis and information processing.Based on a small sample and via the maximum likelihood method, the bootstrap method can be adopted to imitate a large number of β j and η j , but it requires the priori knowledge of γ (β) and ε (η) in advance.If lack of the priori knowledge of γ (β) and ε (η), the estimated intervals of β and η and the confidence interval of the reliability can be calculated hardly.So far the priori knowledge of γ (β) and ε(η), in fact, is reported rarely from the experimental investigation of the rolling bearing lifetime.For this reason, the maximum entropy method can be applied to structure the probability density functions γ (β) and ε (η), but it demands a large number of β j and η j .
Synthesizing the strong points of the bootstrap method, the maximum likelihood method and the maximum entropy method, a novel method called the bootstrap likelihood maximum-entropy method is proposed to evaluate the confidence interval of the reliability of rolling bearings under type I censoring with a small sample.The procedure is as follows: • Based on the test data with a small sample of size n, a large number of the data t ji (i = 1,2,…,n; j = 1,2,…,B) is generated by means of the bootstrap method, where t ji stands for the ith data in the jth bootstrap sample, n for the number of the test data obtained in a lifetime test and B for the number of the bootstrap samples • Based on t ji , β j and η j are calculated with the help of the maximum likelihood method • β j and η j are processed with the aid of the maximum entropy method and γ(β) and ε (η) can accordingly be acquired • The expected value β mean of β and the expected value η mean of η are computed via γ (β) and ε (η), respectively.
• Given a confidence level p, the estimated interval [β L , β U ] of β and the estimated interval [η L , η U ] of η are obtained by γ (β) and ε (η), respectively • Given a failure probability q, the lifetime L and its reliability function R (t) are calculated by β and η, where t is the stochastic variable of the lifetime It is easy to see from the procedure that the bootstrap likelihood maximum-entropy method proposed in this study is characterized by theoretically fusing the strong points of the bootstrap method, the maximum likelihood method and the maximum entropy method.It follows that the probability density functions of the Weibull distribution parameters can be simulated, the confidence intervals of the shape parameter and the scale parameter can be obtained and then the confidence interval of the reliability of rolling bearings can be assessed.Clearly, only by the test data with small sample sizes the bootstrap likelihood maximum-entropy method is able to analyse the reliability without any priori information of the shape parameter and the scale parameter.
In this study, the tests of complete large-sample data, the complete small-sample data and the incomplete small-sample data are produced to prove the effectiveness and practicability of the bootstrap likelihood maximum-entropy method.

TWO-PARAMETER WEIBULL DISTRIBUTION
Suppose the lifetime of rolling bearings is of the two-parameter Weibull distribution with the probability density function: where, f (t) = The probability density function of the twoparameter Weibull distribution t = The stochastic variable of the lifetime β = The shape parameter η = The scale parameter.
The probability distribution function is given by: where, F(t) is the probability distribution function of the two-parameter Weibull distribution.

TEST DATA
Under type I censoring, the test data include the failure data and the truncated data.In terms of the reliability theory, the test data of the type may be classified as two categories, the complete data and the incomplete data.The former consists simply of the failure data and the latter consists of both the failure data and the truncated data.
Suppose n product units are randomly selected for a time censored test and the number of failure product units is r (0<r ≤ n).The sample of the failure data is geven by: where, T F = The sample of the failure data r = The number of the failure data The sample of the truncated data is given by: ) ,..., , ( ) ,..., , ( totaling where, T C = The sample of the truncated data n = The number of the test data s = n-r = The number of the truncated data Bootstrapt method: To facilitate the description, β and η are noted by θ and γ (β) and ε (η) are noted by ξ (θ).The bootstrap method is to simulate the large sample Tj about the failure data with the bootstrap resampling from the sample TF, the maximum likelihood method is to calculate the simulated value θj of the parameter θ via the large sample Tj and the maximum entropy method is to establish the probability density function ξ (θ) by θ j .
Assume the jth bootstrap sampling is conducted.A sampling datum t j1 can be obtained by an equiprobable sampling with replacement from T F .Resampling so r times, the r sampling data which are regarded as a sample (t j1 , t j2 ,…, t jr ) can be obtained.Considering the s truncated data in equation ( 4), the jth bootstrap sample of the incomplete data in the time truncated test is structured, as follows: where, B is the number of the bootstrapt samples.In Eq. ( 5), the jth smaple of the truncated data is:

MAXIMUM LIKELIHOOD METHOD
According to the maximum likelihood method for simulating β and η, there are: The jth simulated value β j of β and the jth simulated value η j of η can be calculated by Eq. ( 7) and ( 8), respectively and the B simulated results can hence be obtained, which are noted by: ) ,..., ,..., , ( where, θ j stands for the jth simulated value of θ and Θ for a sample of the simulated values of θ. The sample Θ can be applied to establish ξ (θ) by means of the maximum entropy method deduced in detail below.

MAXIMUM ENTROPY METHOD
According to the information entropy theory, the information entropy H of ξ (θ) is defined as: where, H stands for the information entropy of ξ (θ) and for the integral domain.A basic idea of the maximum entropy method is that in all the feasible solutions to a problem, the solution of maximizing the information entropy is the most unbiased solution.Accordingly, let: The constraint condition is: where, k stands for the sequence number of the order of the origin moment, m k for the kth origin moment and m for the highest order of the origin moment.
With the help of the histogram principle in statistics, θ j is rearranged from small to large order and is divided into Z groups.As a result, the kth origin moment m k is given by: where, θ z = The median in the zth group Ξ z = The frequency at θ z According to the Lagrange method of multipliers, the probability density function of satisfying Eq. ( 10) to ( 13) is: where λ k is the kth Lagrange multiplier and should meet Eq. ( 15): The Lagrange multiplier λ 0 is given by: Given the confidence level p, the estimated interval of θ can be obtained, as follows: where, θ L and θ U are, respectively, the lower bound and the upper bound of θ and there are:

EVALUATION FOR RELIABILITY
Let the failure probability be q, then lifetime L q is defined as Harris (1991): The reliability function R (t) is defined as: In this study, the expected lifetime L meanq based on the bootstrap likelihood maximum-entropy method is defined as: where, β mean is the expected value of β and η mean is the expected value of η.
Let the lifetime interval be [L Lq , L Uq ] based on the bootstrap likelihood maximum-entropy method, where L Lq is the lower bound of L meanq and is given by: and L Uq is the upper bound of L meanq and is given by: where, β L and β U are, respectively, the lower bound and the upper bound of β and η L and η U are, respectively, the lower bound and the upper bound of η.
The mid-value lifetime L mq based on the bootstrap likelihood maximum-entropy method is defined as: where β 0.5 is the estimated mid-value of β and η 0.5 is the estimated mid-value of η.
The expected reliability function ) ( mean t R based on the bootstrap likelihood maximum-entropy method is defined as: At the given confidence level p, let the estimated interval of R mean (t) be [R L (t), R U (t)] based on the bootstrap likelihood maximum-entropy method, where R L (t) is the lower bound which is given by: and R U (t) is the upper bound which is given by: The mid-value reliability function R m (t) based on the bootstrap likelihood maximum-entropy method is defined as:

EXPERIMENTS AND DISCUSSION
In this section, different types of experiments (three cases) are done to test the soundness and effectiveness of the bootstrap likelihood maximum-entropy method proposed in this study, along with a comparison with the existing methods.

Case study of complete large-sample data:
Case 1: This is a case of evaluation for the complete large-sample data, along with a comparison with the moment method, the maximum likelihood method and Harris method.Let β = 2.5 and η = 200, which can be regard as the true values of Weibull distribution parameters, then the failure data are imitated, as follows (n = r = 50) (Jiang and Zhou, 1999) The results estimated for β and η are listed in Table 1.For β, the relative errors of the four methods are less than 5%, having a good effect.For η, the relative error of the proposed method is less than 10%, but the relative errors of the existing methods are more than 10%.

Case study of complete small-sample data:
Case 2: This is a case of evaluation for the complete small-sample data, along with a comparison with the Harris method employed frequently in reliability analysis of the rolling bearing lifetime test with a small sample.The failure data obtained by Harris (1991) are cited, as follows (n = r = 10): 14.01 15.38 20.94 29.44 31.15 36.72 40.32 48.61 56.42 56.97With the help of the bootstrap likelihood maximumentropy method, γ (β) and ε (η) are estimated, as shown in Fig. 3 and 4. The results of the Weibull distribution parameters and the rolling bearing lifetime are shown in Table 3 and 4. It is easy to see from Table 3 and 4 that the bootstrap likelihood maximum-entropy method is able to obtain the estimated interval of the reliability, but the Harris method is unable to do that.

Case study of incomplete small-sample data:
Case 3: This is a case of evaluation for the incomplete small-sample data.

Item
Confidence level, p% -----------------------------------------------------------------------90 95 99 Failure probability, q/% 10 5 1 Expected value of shape parameter, βmean The results estimated by the bootstrap likelihood maximum-entropy method are shown in Table 5 and  Fig  decrease, their heights increase and their peaks move to the left and to the right, respectively.This indicates that the number of the truncated data has an effect upon the probability density functions of the Weibull distribution parameters.In addition, the feature of the left peak of γ (β) and of the approximatively symmetrical peak of ε (η) is irrelative to the number of the truncated data according to Fig. 8 and 9 and is also irrelative to the number of the failure data and the category of the test data according to Fig. 1 to 6.

CONCLUSION
Under type I censoring, without any priori information beyond the test data to be dealt with, the bootstrap likelihood maximum-entropy method is able to calculate the estimated interval of the reliability of the rolling bearing lifetime based on the two-parameter Weibull distribution.
The tests of the complete large-sample data, the complete small-sample data and the incomplete smallsample data prove the effectiveness of the bootstrap likelihood maximum-entropy method.
The probability density function of the shape parameter is a curve with one left peak and the probability density function of the scale parameter is a curve with one approximatively symmetrical peak.The feature of the two curves is irrelative to the number of the truncated data, the number of the failure data and the category of the test data.
mid-value θ 0.5 of θ is defined as:

Fig. 6 :Fig. 7 :Fig. 8 :
Fig. 6: Probability density function ε(η) of scale parameter η in Case 3 (bootstrap likelihood maximum-entropy method) Feature of probability density function of weibull distribution parameter: It can be seen from Fig. 1 to 3 that γ (β) is a curve with one left peak and ε (η) is a curve with one approximatively symmetrical peak.In order to study γ (β) and ε (η), in Case 3, their change with s, the number of the truncated data, is presented In Fig.8 and 9.As s increases, their widths

Table 1 :
Comparison between estimated results of Weibull distribution parameters in case 1

Table 2 :
Comparison between results of lifetime in case 1 (q = 10%)

Table 5 :
Result estimated by bootstrap likelihood maximum-entropy method in case 3 (B = 10000)