Prediction intervals for future lifetime of three parameters Weibull observations based on generalized order statistics

In this paper, two pivotal statistics are introduced to construct prediction intervals for future lifetime of three parameters Weibull observations based on generalized order statistics, which can be widely applied in reliability theory and lifetime problems. The probability density functions as well as the explicit form of the distribution functions of our pivotal statistics are derived. Monte Carlo simulations are performed to demonstrate the efficiency of the proposed methods and a real data analysis is conducted for illustrative purposes.


Introduction
Prediction of unobserved or censored observations is an interesting topic, especially in the viewpoint of actuarial, biological science, physics, medical and engineering sciences. An authoritative review of developments on prediction problems has been prepared by Kaminsky and Nelson [8]. It is well known that quite often the survival data come with a special feature called censoring. Censoring occurs in life testing experiments, when exact survival times are known only for a portion of the individuals or items under study. The experimenter may not always be in a position to observe the life times of all the products (or items) were put on test either intentionally or unintentionally; this may be because of time limitation and/or other restrictions (such as money, mechanical or experimental difficulties, material resources, etc.); see, for example, Nelson [16] and Balakrishnan and Cohen [1].
The Weibull distribution is one of the most widely used distributions in reliability and survival analysis. Because of its various shapes of the probability density function and its convenient representation of the distribution/survival function, the Weibull distribution has been used very effectively for analyzing lifetime data, particularly when the data are censored, which is very common in most life testing experiments. Moreover, Weibull distribution without any doubt is one of the most important models in modern statistics because of its ability to fit data from various fields, ranging from life data to weather data or observations made in economics and business administration, in hydrology, in biology or in the engineering sciences. A commonly used model in reliability theory and lifetime studies is the three-parameter Weibull distribution, which was introduced by the Swedish statistician Waloddi Weibull for the first time in 1939 in connection with his studies on the strength of materials (for more details and applications of Weibull distribution see Rinne [17]).
In a wide subclass of gos which contains most of the important practical models when γ 1,n , . . . , γ n,n are assumed to be pairwise different, Kamps and Cramer [10] derived the marginal pdf of the r th gos and the joint pdf of the r th and the sth gos, which are given by where A random variable X is said to have three-parameter Weibull distribution, denoted by W (η, ξ, δ), if its probability density function (pdf) is given by where η ∈ R is a location parameter, ξ > 0 is a scale parameter and δ > 0 is a shape parameter. The corresponding distribution function (df) is given by In this paper, we modified two pivotal quantities to construct two exact prediction intervals for future observations from three-parameter Weibull distribution based on generalized order statistics. The rest of the paper is organized as follows: In Sect. 2 we present the main results. Section 3 include Monte Carlo simulation for some important models and an application of real lifetime data is given in Sect. 4.

The main results
The following lemma is needed in the proof of Theorem 2.3, which expresses an interesting fact that can be applied for solving other problems.

Lemma 2.1
Suppose that X (1, n,m, k), . . . , X (n, n,m, k) are the first n gos based on Weibull distribution with the pdf (4). Then the rv's are independent and identically distributed (iid) according to the standard exponential distribution.
Proof By noting that the Jacobian J , can be written in the form The joint pdf of X (1, n,m, k), . . . , X (n, n,m, k) based on Weibull distribution with pdf (4) can be written in the form where y i = (x i − η)/ξ. Therefore, we have the following equation: which by the Factorization Theorem implies the assertion of the lemma.
The main goal of this paper is to use the first observed r gos, X (1, n,m, k), . . . , X (r, n,m, k), to construct prediction intervals for the sth gos (1 ≤ r < s ≤ n), through the following two modified statistics: where Proof By Equations (3), (4) and (5), the joint pdf for the r th and sth gos takes the form By standard transformation methods, the joint pdf of the subrange It is not difficult to show that the joint pdf of U r,s = W r,s Y and Y can be written as Thus, we have Integrating (12) form 0 to u and simplifying the result, we obtain (9) which proves the theorem.
In the following theorem we derive the distribution of the pivotal statistic V r,s Proof In view of Lemma 2.1, the statistic T Integrating (14) and simplifying the result we obtain (13). This completes the proof of the theorem.

Remark 2.4
The (1−α)100% predictive conference intervals for the future unobserved value of X (s, n,m, k), based on the pivotal statistics U r,s and V r,s , respectively, are given by and where x r is an observed value of X (r, n,m, k), u α can be obtained from Eq. (9)

Simulation study
In this section, Monte Carlo simulations are conducted to investigate the efficiency of the obtained results in the preceding section. For this purpose an algorithm is constructed. In the simulation study, we generate 100,000 ordered random samples, for any value of s, each sample of size n from three-parameter Weibull distribution W (η, ξ, δ) for some values of η, ξ and δ. The coverage probability and the average interval width based on the two statistics U r,s and V r,s are computed for three special cases from gos.

An illustrative example
The order random variables play an important role for the lifetime prediction methods because if m items are put simultaneously in a life test, the weakest component will fail first, followed by the second weakest and so on until all have failed. For example, in manufacture we are interested in the time to failure after n units are put in a life test. In such cases, the observations arrive in ascending order of magnitude and do not have to be ordered after collection of the data. The practical importance of such experiments is evident. Moreover, the possibility is now open of terminating the experiment before its conclusion by stopping after a given time (Type I censoring) or after a given number of failures (Type II censoring). It may be of interest to predict  the time at which all the components will have failed or to predict the mean failure time of the unobserved lifetimes. In these cases, the interval or point predict are of interest. In this section, an example for real data is presented to demonstrate the importance of results obtained in Sect. 2. The data were given by Lawless [12, p. 189]. It consists of voltage levels at which failures occurred in a certain type of electrical cable insulation (Type 1 insulation) when specimens were subjected to an increasing voltage stress in a laboratory experiment. The test involved 20 specimens and the failure voltages in kilovolts per millimeter are given in Table 4.
For the purposed data, Gini statistic, (see [6]), as well as the max p value method, are applied to get the best fitting to Weibull distribution. Moreover, prediction intervals for the unobserved failures, x s , s = r + 1, r + 2, . . . , n are obtained.
We use Gini statistic and the max p value method to show that c = 9.1973 is very close to optimum value and the maximum p value = 0.999996, and in this case the maximum Likelihood estimate of b is b = 47.7383 which gives a good fitting to two-parameter Weibull distribution (a = 0.0) see Table 5.

Discussion and concluding remarks
Two pivotal quantities are modified to predict future observations from three-parameter Weibull distribution. Numerical results of these pivotal quantities for three different models are presented through simulation studies. Finally, an example has been given to illustrate the results discussed in this paper.
From the simulation studies (Tables 1, 2, 3) it is clear that 1. In all cases the coverage probability is close to 1 − α, α = 0.1, 0.5. 2. When s is fixed, the AIW of the PCI of X (s, n,m, k) decreases, with increasing r as expected, since more available data improved prediction results.