On the upper truncated Weibull distribution and its reliability implications

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Abstract

The characteristics and application of the truncated Weibull distribution are studied in this paper. This distribution is applicable to the situation where the test data are bounded in an interval because of test conditions, cost and other restrictions. An important property of the truncated Weibull distribution is that it can have bathtub-shaped failure rate function. In this paper, the parametric analysis and parameter estimation methods of the distribution are investigated. Both the graphical approach and the maximum likelihood estimation are considered. The applicability of this distribution to modeling lifetime data is illustrated by an example and the results of comparisons to other competitive models in modeling the given data are also presented. Moreover, the possible application of the distribution to modeling component or system failure is discussed.

Introduction

Truncated distributions are applicable to the situation when the range of a random variable is bounded from below and/or above for some reasons. This is a common situation in reliability applications. For example, failures during the warranty period may not be counted. Items may also be replaced after certain time following the replacement policy, so that failures of the item are ignored. Another example is when failure or reliability life test data are gathered by an automatic recording device; the lifetimes below L and above T (L<T) may not be measured at all because of the resolving power of the device or other environmental effects. Another case to use truncated distribution is in description on the development of the pit depths on a water pipe in which the pit depth is ranged in the interval [0, h0] with h0 being the thickness of the pipe [1].

Since Weibull distribution is widely utilized in industry, the truncated Weibull distribution is also of interest in application.

The truncated Weibull distribution has been found being applied in several engineering fields. Zutter et al. [2] utilized the truncated two-parameter Weibull to analyze diameter data of trees truncated at a specific threshold level. Maltamo et al. [3] used the left-truncated Weibull to predict the height distribution of small trees based on incomplete laser scanning data. Palahi et al. [4] applied the truncated Weibull functions to modeling the diameter distribution of forest. Deperrois et al. [5] presented high-cycle fatigue strength prediction methods. In their research, material (fatigue) parameters are declared to be most accurately described by the truncated Weibull distribution. Pereira et al. [6] developed a parametric model to appropriately characterize the observed Portuguese fire size distribution during the period 1980–1997. Their results have shown that the best models are those based on the Weibull and the truncated Weibull distributions. The truncated Weibull distribution has also been applied to seismological data analysis for earthquakes, see Martinez [7] and Nishenko [8].

Several researchers have also studied the parameter estimation of the truncated Weibull distribution and its application. Wingo [9] proposed point estimation of parameters for a doubly truncated Weibull distribution. Mittal and Dahiya [10] investigated the problem of existence of the MLE for the truncated Weibull distribution, where the truncation point is assumed to be known. Martinez [7] studied the MLE of parameters of the upper truncated Weibull distribution. Shalaby and El-Yoursef [11] presented Bayesian estimates of parameters for a doubly truncated Weibull distribution and Shalaby [12] discussed the Bayesian risk of the estimation. Seki and Yokoyama [13] developed a simple and robust estimation method for the Weibull and truncated Weibull parameters. This method has good properties for parameter estimation in several cases such as small sample size, failure to obtain the early observations and presence of outliers. Yan et al. [14] analyzed patent and latent failures known to exist in many products and systems with bathtub shaped failure rate model. The resulting failure distribution is a mixed Weibull distribution that is truncated. They presented model equations but did not give the parameter estimates of the models. Glaser [15] investigated the case of Weibull accelerated testing with unreported early failures, the example of left truncation. Navarro and Ruiz [16] presented Kaplan & Meier’s nonparametric estimator for censored data. Based on their work, Betensky and Martin [17] applied nonparametric MLE to a doubly truncated sample of times to brain tumor progression. Efron and Petrosian [18] presented nonparametric analysis in detail for doubly truncated data. Ahmad and Fawzy [19] derived recurrence relations for moments of generalized order statistics within a class of doubly truncated distributions. Jawitz [20] carried out study on moment expressions of truncated or incomplete distributions. Teodorescu and Paniatescu [21] studied the parameter estimation for the truncated composite Weibull–Pareto model.

It can be seen that there are numerous applications of the truncated Weibull distribution in industry and its parameter estimation methods have been presented by different authors. Its characteristics as a whole, however, need to be elaborated further and the parameter estimation by graphical approach has not been reported. Nevertheless, the potential application of the truncated Weibull distribution with comparison to other competitive Weibull models to modeling the failure time data has not been studied. These are very interesting research points that trigger the motivation of study as given in this paper.

The purpose of this paper is to study the characteristics of the upper truncated Weibull distribution and parameter estimation by graphical approach. It gives the parametric analysis of the model’s failure rate function and then the parameter estimation methods which include the graphical approach and MLE. The applicability of this distribution to modeling failure data is illustrated by an example and the results of comparisons to other models in modeling the given data are also presented. Moreover, the possible application of the distribution to system reliability modeling is discussed.

The remainder of this paper is organized as follows. Section 2 presents the distributional properties of the truncated Weibull distribution. Section 3 is involved in parameter estimation in which both the graphical method and the maximum likelihood estimation are considered. Section 4 is devoted to application of the distribution to lifetime data analysis with an illustrative example, where some comparative studies are also included. In Section 5, its reliability implications are discussed. Finally, the paper is concluded as given in Section 6.

Section snippets

The truncated Weibull distribution

A doubly truncated Weibull distribution is given asG(t)=F(t)F(t0)F(T)F(t0),t0tT.where F(t) is the usual two-parameter Weibull, F(t)=1−exp[−(t/η)β]. When t0=0 and T→∞, it becomes the two-parameter Weibull. When t0=0, it is the upper truncated Weibull and when T→∞, it is the lower truncated Weibull. From Eq. (1), the reliability function is obtained asR(t)=1G(t)=F(T)F(t)F(T)F(t0),t0tT.and the probability density function isg(t)=β/η(t/η)β1exp[(t/η)β]F(T)F(t0),t0tT.Based on the

Estimation of model parameters

Parameter estimation is important and usually it is a difficult problem as methods like the maximum likelihood estimation cannot yield a close form solution in general. The numerical calculation and iteration method are needed. There are different methods which can be applied to model parameter estimation. Among them, the graphical approach such as WPP method and MLE are of the commonly used ones. They are presented in this section.

An application example

A set of real test data representing time-to-failure of turbocharger of one type of engine is shown in Table 2 [31]. Here, as an illustration, this set of data is treated as complete data without censoring. As described in graphical approach, the WPP of this set of data is generated first, which is shown in Fig. 3. Then, analyze the property of the plot and try one or more models for this set of data. Based on the property of the plot, we have fitted the data with the upper truncated Weibull,

Reliability modeling

In industry, it is of interest to know if a system has a bathtub-shaped failure rate behavior. Manufacturers expect their products to be reliable in the designated service life while some of them might not be still interested in the higher reliability of the products after the designed service life. It is of great interest to design a product that has very low failure rate from initial to its end of service but it will fail very soon after the designated service life. That could be a trend in

Conclusions

In this paper, the property of the truncated Weibull distribution is studied. This distribution gives bathtub-shaped failure rate function when the shape parameter β is less than 1. A failure rate function with very nice bathtub shape can be found with selected combinations of η and T when β is less than 1. When β≥1, this distribution has increased failure rate. Moreover, the density function is decreasing in time t when β≤1 and unimodal when β>1 and the truncation point is large enough, which

Acknowledgement

This research was partially supported by NSFC under the contract number 70828001. The authors would like to thank the two anonymous referees for their constructive comments on an early version of this paper.

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