A Simple Normal Approximation for Weibull Distribution with Application to Estimation of Upper Prediction Limit

Abstract

We propose a simple close-to-normal approximation to a Weibull random variable (r.v.) and consider the problem of estimation of upper prediction limit (UPL) that includes at least l out of m future observations from a Weibull distribution at each of r locations, based on the proposed approximation and the well-known Box-Cox normal approximation. A comparative study based on Monte Carlo simulations revealed that the normal approximation-based UPLs for Weibull distribution outperform those based on the existing generalized variable (GV) approach. The normal approximation-based UPLs have markedly larger coverage probabilities than GV approach, particularly for small unknown shape parameter where the distribution is highly skewed, and for small sample sizes which are commonly encountered in industrial applications. Results are illustrated with a real dataset for practitioners.

1. Introduction

Weibull distribution is widely used in reliability and survival analysis due to its flexible shape and ability to model a wide range of failure rates. It can be derived theoretically as a form of extreme value distribution, governing the time to occurrence of the “weakest link” of many competing failure processes. Its special case with shape parameter 𝑏=2 is the Rayleigh distribution which is commonly used for modeling the magnitude of radial error when x and y coordinate errors are independent normal variables with zero mean and the same standard deviation while the case 𝑏=1 corresponds to the widely used exponential distribution.

Let 𝑋 follow a Weibull distribution with scale parameter 𝑎 and shape parameter 𝑏. The pdf of 𝑋 is given by 𝑓_𝑋()=𝑏𝑥;𝑎,𝑏𝑎𝑥𝑎^𝑏−1−𝑥exp𝑎^𝑏,𝑥>0,𝑎>0,𝑏>0,0,otherwise. (1.1) If a random sample of size 𝑛 is given, an important statistical problem is to construct a (1−𝛼)100% UPL such that 𝑙 of 𝑚 future sample values are below the UPL at each of 𝑟 locations (or time periods). This problem for Weibull-distributed data is less attended in the literature. Such prediction limits are very much useful in monitoring and control problems during the operation of a production process in industries, particularly when the characteristic of interest is smaller the better type. UPLs are also useful in groundwater quality monitoring in the vicinity of hazardous waste management facilities (HWMF). For example, to monitor ground water quality, a series of 𝑚 samples, that is, measurements of a pollutant like vinyl chloride from each of 𝑟 monitoring wells in the vicinity of an HWMF are often compared with UPL based on a sample of 𝑛 measurements obtained from one or more upgradient sampling locations of the facility. If at least 𝑙 samples out of these 𝑚 samples from each of 𝑟 locations are less than the UPL, then the facility is considered to be within compliance. If this requirement is not met, then monitoring of contaminants like vinyl chloride is much more needful. Bhaumik and Gibbons [1] have discussed applications of UPLs in the fields like molecular genetics and industrial quality control. Davis and McNichols [2] obtained UPL assuming normality for the parent distribution. Bhaumik and Gibbons [1] and Krishnamoorthy et al. [3] proposed approximate methods for constructing UPL for Gamma distribution. Krishnamoorthy et al. [4] used the GV approach for constructing UPL for Weibull distribution.

In this paper, we propose a simple close to normal transformation for Weibull distribution when the shape parameter 𝑏 is known. The transformation is based on two key features of the normal distribution, namely, symmetry and the tail behaviour. This transformation and the well-known Box-Cox transformation are used to obtain approximate UPLs for Weibull distribution when the shape parameter 𝑏 is known. For unknown shape parameter 𝑏, replacing it by its maximum likelihood estimator (mle) gave equally good results. A simulation-based comparison of the proposed UPLs with existing ones revealed that the proposed UPLs outperform their competitors even for small sample sizes, and more prominently for small shape parameters which are frequently encountered in many real applications.

The article is organized as follows. Section 2 provides a brief review of the GV approach-based UPL developed by Krishnamoorthy et al. [4]. Section 3 describes the proposed normal approximation method and develops UPL for Weibull distribution based on the proposed and the Box-Cox transformations. Section 4 reports the comparison of the proposed UPLs with the GV approach-based UPL with respect to the simulated expected coverages and expected lengths. Section 5 illustrates the methods using a real dataset. Section 6 provides overall conclusions and recommendations.

2. GV Method for Obtaining UPL for Weibull (𝑎,𝑏) Distribution

Krishnamoorthy et al. [4] proposed (1−𝛼)100% UPL that includes at least 𝑙 out of 𝑚 future observations from a Weibull(𝑎,𝑏) distribution at each of 𝑟 locations as exp𝜂+𝑢_{𝑛,𝑟,𝑙,𝑚}𝛽, (2.1) where 𝜂=log(𝑎) and 𝛽=1/𝑏̂. They computed value for 𝑢_{𝑛,𝑟,𝑙,𝑚} using the following simulation study.

For the given values of 𝑛,𝑟,𝑙,𝑚, and 1−𝛼, the following procedure is repeated 𝑁 (say 100000) times.(I)𝑛 independent and identically distributed (i.i.d.) random variables 𝑦^∗₁,𝑦^∗₂,…,𝑦^∗_𝑛 are generated from the extreme-value (0,1) distribution and mles 𝜂^∗, 𝛽^∗ are computed.(II)For the given 𝜂^∗, 𝛽^∗ following procedure is repeated 𝑟 times.(i)𝑚 i.i.d. random variables 𝑦^∗_𝑖1,𝑦^∗_𝑖2,…,𝑦^∗_𝑖𝑚;𝑖=1,2,…,𝑟 are generated from the extreme-value (0,1) distribution.(ii)𝑙th order statistic based on these samples 𝑦^∗_𝑖(𝑙);𝑖=1,2,…,𝑟 are computed.(III)The quantities 𝑦^∗_{𝑟,𝑚,𝑙}=max{𝑦^∗_𝑖(𝑙);𝑖=1,2,…,𝑟} and 𝑢=(𝑦^∗_{𝑟,𝑚,𝑙}−𝜂^∗)/𝛽^∗ are computed.

Then 100(1−𝛼)th percentile of the generated 𝑁 values of 𝑢 is the estimate of 𝑢_{𝑛,𝑟,𝑚,𝑙}. Note that the distribution of the pivotal quantity based on which UPL is developed does not depend on any unknown parameters, thus it is an exact method.

3. The Proposed Close-To-Normal Power Transformation-Based UPL

3.1. The Proposed Close-To-Normal Power Transformation

The proposed transformation is based on the two key features governing normality, namely, the symmetry and tail behaviour of the normal distribution.

Let 𝑋 follow a two parameter Weibull (𝑎,𝑏) distribution and the shape parameter 𝑏 is known. We consider a transformation 𝑌=𝑋^𝑝 for the r.v. 𝑋 where the power 𝑝 is chosen so that the distribution of the transformed variable 𝑌 has very small deviation from symmetry and simultaneously has tail behaviour very close to that of the normal distribution with the same mean and variance. Straightforward calculations show that the skewness of the distribution of 𝑌 is given by 𝛾(𝜃)=Γ((((𝜃Γ(((Γ((𝜃𝑝,𝑏=𝛾1+3𝜃))−3Γ1+))1+2𝜃))+21+)))³Γ(1+(2𝜃))−(Γ(1+(𝜃)))^23/2, (3.1) which is a function of the ratio 𝜃=𝑝/𝑏, and does not depend on the scale parameter 𝑎. Treating 𝛾(𝑝,𝑏) as a function of 𝜃, a solution for 𝛾(𝜃)=0 is 𝜃₁=0.2776, for which the distribution of the variable 𝑌=𝑋^𝑝 where 𝑝=𝑏𝜃, is exactly symmetric. To achieve control over the tail behaviour, it is noted that the 𝑘th central moment of the transformed r.v. 𝑌 is 𝐸(𝑌^𝑘)=𝐸(𝑋^𝑘𝑝)=𝑎^𝑘𝑝Γ(1+𝑘𝜃) leading to the mean and standard deviation of 𝑌 given by 𝜇_𝑦=𝑎^𝑝Γ(1+𝜃),𝜎_𝑦=𝑎^𝑝Γ(1+2𝜃)−(Γ(1+𝜃))^20.5. (3.2) Furthermore, the 𝛼th quantile of a normal distribution with mean 𝜇_𝑦 and standard deviation 𝜎_𝑦 is given by 𝜉_𝛼=𝜇_𝑦+𝜎_𝑦𝑍_𝛼=𝑎^𝑝Γ(1+𝜃)+Γ(1+2𝜃)−(Γ(1+𝜃))^20.5𝑍_𝛼. (3.3) Similarly, if 𝑥_𝛼 is the 𝛼th quantile of the Weibull (𝑎,𝑏) distribution that is, 𝑥_𝛼=𝑎(−log(1−𝛼))^1/𝑏, it easily follows that the 𝛼th quantile 𝑦_𝛼 of the distribution of 𝑌 is given by 𝑦_𝛼=𝑥^𝑝_𝛼=𝑎^𝑝−log(1−𝛼)^𝜃. (3.4) To make the tail behavior of the distribution of 𝑌 very close to that of the normal distribution with same mean and standard deviation as that of 𝑌, we solve the equation 𝜉_𝛼−𝑦_𝛼=0 for the commonly used choice of 𝛼=0.025 and 𝛼=0.975 for a two-sided interval leading to the solutions respectively 𝜃₂=0.2698 and 𝜃₃=0.2994. To control the symmetry and tail behaviour of the distribution of transformed r.v. 𝑌 simultaneously close to the normal distribution, we suggest taking 𝑝=𝜃𝑏 where 𝜃=(𝜃₁+𝜃₂+𝜃₃)/3=0.2823 as the power of a Weibull r.v. 𝑋. From (3.3) and (3.4) it follows that for this choice of 𝜃, the difference between the two quantiles 𝜉_𝛼 and 𝑦_𝛼 is given by (𝑎^𝑏)^0.2823𝑐(𝛼) where 𝑐(𝛼)=(Γ(1+𝜃)+(Γ(1+2𝜃)−(Γ(1+𝜃))²)^0.5𝑍_𝛼)−(−log(1−𝛼))^𝜃 is a constant depending on 𝛼. The values of 𝑐(𝛼) for various commonly used choices of 𝛼 are given in Table 1.

Table 1 

We note that the constant 𝑐(𝛼) is considerably small for commonly used level of significance 𝛼=0.05, and further numerical study revealed that the accuracy of the proposed transformation is very good for small values of 𝑏 (say 𝑏<4) and for small to moderate values of 𝑎 (say 𝑎<100), which covers a reasonable subset of the parameter space and commonly encountered real situations. We recall that the choice of 𝑝 is uniform for all 𝑎>0 since 𝛾 is free from 𝑎. When 𝑏 is unknown, we take 𝑝=𝜃𝑏̂, replacing 𝑏 by its mle 𝑏̂. In the sequel we refer to this transformation as the close to normal power transformation (CNPT).

3.2. UPL Based on the CNPT

Let 𝑌₁,𝑌₂,…,𝑌_𝑛 be a random sample of size 𝑛 from a normal distribution with mean 𝜇 and standard deviation 𝜎. Let 𝑌 and 𝑆_𝑦 be the sample mean and sample standard deviation. Then Davis and McNichols [2] suggested UPL that includes at least 𝑙 out of 𝑚 future observations from the same normal distribution at each of 𝑟 locations as 𝑌+𝑘_𝑢𝑆_𝑦, (3.5) where the value of 𝑘_𝑢 for selected values of (𝑛,𝑟,𝑙,𝑚) and level 𝛼 is the solution to the following equation ¹₀𝑁_𝑡√𝑛𝑘_𝑢;𝑛−1,√𝑛Φ⁻¹(𝑥)𝑟(𝐼(𝑥;𝑙,𝑚+1−𝑙))^𝑟−1𝑥^𝑙−1(1−𝑥)^𝑚−𝑙𝛽(𝑙,𝑚+1−𝑙)𝑑𝑥=𝛼, (3.6) where 𝑁_𝑡(𝑥;𝑣,𝛿) is the cumulative distribution function (Cdf) of noncentral 𝑡 r.v. with 𝑣 df and noncentrality parameter 𝛿, Φ⁻¹(𝑥) is inverse Cdf of the standard normal distribution at 𝑥, 𝛽(𝑚,𝑛) is the usual beta function and 𝐼(𝑥;𝑚,𝑛) is the Cdf of a beta distribution with parameters 𝑚 and 𝑛.

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be a random sample of size 𝑛 from a Weibull (𝑎,𝑏) distribution. Let 𝑈 be the normal based UPL obtained using (3.5), based on 𝑌_𝑖=𝑋^𝑝_𝑖; where 𝑝=0.2823𝑏;𝑖=1,2,…,𝑛. Then (𝑈)^1/𝑝 is the proposed UPL that is expected to include at least 𝑙 out of 𝑚 future observations from the Weibull (𝑎,𝑏) distribution at each of 𝑟 locations for known shape parameter 𝑏, with probability 1−𝛼. When the shape parameter 𝑏 is unknown, we suggest to replace it by its mle 𝑏̂ and the proposed UPL is (𝑈)^1/𝑝 where 𝑝=0.2823𝑏̂. It is noted that 𝑝 is a consistent estimator for 𝑝. Hence for large samples (𝑈)^1/𝑝 is expected to be close to (𝑈)^1/𝑝. Small sample behavior of (𝑈)^1/𝑝 is studied through simulation.

3.3. Box-Cox Transformation and Kullback-Leibler Information-(BCKL-) Based UPL

Hernández and Johnson [5] proposed the transformation 𝑋𝑌=^𝜆−1𝜆(𝑋)if𝜆≠0,logif𝜆=0, (3.7) to Weibull r.v. 𝑋 for approximating distribution of 𝑌 to normal and used a solution 𝜆=0.2654𝑏 for known shape parameter 𝑏 that minimizes the Kullback-Leibler information between the distribution of 𝑌 and the normal distribution with the same mean and variance as that of 𝑌. This transformation was used by Yang et al. [6] for obtaining prediction interval for a single future observation from Weibull (𝑎,𝑏) distribution. Using this transformation for the UPL problem under consideration, a (1−𝛼)100% UPL for Weibull distribution is (1+𝜆𝑈)^1/𝜆 where 𝑈 is the normal based UPL obtained using (3.5). As before an unknown value of 𝑏 will be replaced by its mle 𝑏̂. This also enjoys the large sample properties mentioned above and its small sample behavior is studied through simulation.

4. Comparison

In this section we compare the performance of above two proposed UPLs with the UPL based on GV method-based on a simulation study with respect to expected lengths and expected coverages of the UPLs.

For fixed values of the parameters 𝑎, 𝑏, and sample size 𝑛, we generate 𝑛 random numbers 𝑋_𝑖;𝑖=1,2,…,𝑛 from the Weibull (𝑎,𝑏) distribution, and set 𝑌_1𝑖=𝑋^𝑝_𝑖, 𝑌_2𝑖=𝑋^𝜆_𝑖−1/𝜆;𝑖=1,2,…,𝑛 where 𝑝=0.2823𝑏 and 𝜆=0.2654𝑏. Normal based UPLs, 𝑈₁ and 𝑈₂ are obtained using (3.5) based on the transformed samples 𝑌_1𝑖 and 𝑌_2𝑖,𝑖=1,2,…,𝑛 respectively. Then (𝑈₁)^1/𝑝 and (𝑈₂)^1/𝜆 are the UPLs based on the proposed CNPT and the BCKL transformation for Weibull (𝑎,𝑏) distribution with known shape parameter 𝑏. When the shape parameter 𝑏 is unknown, we suggest to replace it by its mle 𝑏̂. For the same sample 𝑋₁,𝑋₂,…,𝑋_𝑛, the GV method based UPL is obtained using (2.1) and the procedure described in Section 2. Next we generate 𝑟 sets of 𝑚 random numbers say 𝑋_𝑖𝑗;𝑖=1,2,…,𝑟;𝑗=1,2,…,𝑚 from Weibull (𝑎,𝑏) distribution, and set 𝑋^∗=max{𝑋_1(𝑙),𝑋_2(𝑙),…,𝑋_𝑟(𝑙)} where 𝑋_𝑖(𝑙) is the 𝑙th order statistic from 𝑋_𝑖𝑗 for 𝑖=1,2,…,𝑟. This procedure is repeated 100000 times. Then the proportions of events 𝑋^∗<𝑈₁^1/𝑝, 𝑋^∗<𝑈₂^1/𝜆 and 𝑋^∗<exp(𝜂+𝑢_{𝑛,𝑟,𝑙,𝑚}𝛽), in these 100000 repetitions are the simulated coverage probabilities of the UPLs based on normal approximation, BCKL transformation, and GV method, respectively. Average of 100000 UPLs based on each of the three approaches discussed above are the simulated expected lengths of the corresponding UPLs. The simulated expected lengths and expected coverages for 𝑛=6,10,20, 𝑎=0.1,1,5,50,100, and 𝑏=0.5,1,3 are reported in Tables 2 and 3 respectively. The combinations (𝑟=4,𝑙=2,𝑚=6), (𝑟=8,𝑙=1,𝑚=5), and (𝑟=16,𝑙=2,𝑚=5) are chosen. The values of 𝑘_𝑢 for these combinations are computed using (3.6) for 𝛼=0.05.

Table 2 

Simulated expected lengths of 95% UPL that contain at least l of m future observations at each of r locations from Weibull (𝑎,𝑏) distribution for sample sizes 𝑛=6, 10, 20 using GV, CNPT, and BCKL methods*.

Table 3 

Percentage simulated coverage probabilities of 95% UPL that contain at least l of 𝑚 future observations at each of r locations from Weibull (𝑎,𝑏) distribution for sample sizes 𝑛=6, 10, 20 using GV, CNPT, and BCKL methods*.

4.1. Results of the Simulation Study

Following prominent facts are clearly visible from Table 2.(1)CNPT-based UPLs have uniformly excellent coverage probabilities even for small sample sizes as small as 𝑛=6, for all 𝑎>0, 𝑏>0 and for all examined combinations of 𝑟, 𝑙, and 𝑚. The coverages are uniformly a little larger than those based on GV method, and the expected lengths are a little shorter than the same.(2)As mentioned in Section 2, GV method is exact and this fact is reflected in simulation study since its coverages are very close to the nominal coverage probability.(3)BCKL transformation-based UPLs have close to nominal coverage probabilities.

Based on these observations we recommend the proposed CNPT- and BCKL transformation-based UPLs that include at least 𝑙 out of 𝑚 future observations from a Weibull (𝑎,𝑏) distribution at each of 𝑟 locations.

5. Illustrative Example

Nowadays vinyl chloride is one of the fifty most produced chemicals in the world. Its production almost doubled in the last 20 years and currently estimated to be about 27 million tons/year worldwide. A high concentration of vinyl chloride in water can cause cancer and and liver damage. Therefore being toxic and carcinogenic to humans, more attention has to be given to vinyl chloride as a groundwater contaminant. In this section we illustrate the methods discussed in Sections 2 and 3 with a real dataset.

The data used here are vinyl chloride concentrations collected from clean upgradient monitoring wells. Krishnamoorthy et al. [4] showed an excellent fit of these data to a Weibull distribution. We computed various Weibull UPLs and compared them with those given in Krishnamoorthy et al. [4]. The a dataset representing the vinyl chloride concentration in micrograms per liter of water (𝜇g/𝐿), that is, number of parts per billion (ppb), from clean upgradient monitoring wells is 5.1, 2.4, 0.4, 0.5, 2.5, 0.1, 6.8, 1.2, 0.5, 0.6, 5.3, 2.3, 1.8, 1.2, 1.3, 1.1, 0.9, 3.2, 1.0, 0.9, 0.4, 0.6, 8.0, 0.4, 2.7, 0.2, 2.0, 0.2, 0.5, 0.8, 2.0, 2.9, 0.1, 4.0.

The Kolmogorov-Smirnov test to above dataset for fitting Weibull distribution resulted respective P value (2-tail) 0.94 indicating that Weibull is a good model for above dataset. Here 𝑎=1.89,𝑏̂=1.01, sample mean is 1.88 and sample standard deviation is 1.95. The mle 𝑏̂ indicates that the above dataset is moderately skewed. In order to compare the proposed 95% UPLs with those of Krishnamoorthy et al. [4], we chose the various combinations of 𝑟,𝑙, and 𝑚 and are given in Table 4.

Table 4 

95% Upper prediction limits for the vinyl chloride data.

From Table 4, it seems that the proposed UPLs are little less than those of Krishnamoorthy et al. [4]. We also notice that all the UPLs are well above the nominal range of vinyl chloride concentration (2.0–2.4) suggested by US Environmental Protection Agency (USEPA) indicating that future vinyl chloride concentrations are likely to be larger than the nominal level and hence monitoring of these wells is necessary.

6. Overall Conclusion

The proposed normal approximation exhibits markedly well performance even for small sample sizes for almost all parameter combinations for estimation of UPL that includes atleast 𝑙 out of 𝑚 future observations from Weibull distribution at each of 𝑟 locations. The superiority of normal approximation is much more strong for small shape parameters and small sample sizes which are commonly observed in real situations. It has an added advantage of being computationally simple, which is important from practitioners point of view.