4.7
Conclusions
In this chapter we presented a novel statistical method of imprecise semi-parametric inference for ALT data. In this chapter, we do not assume a failure time distribution at each stress level. The proposed method applies the use of the log-rank test to compare the survival distribution of pairwise stress levels, in combination with the Arrhenius model to find the interval of γ values. We developed imprecision through the use of nonparametric tests for the parameter of the link function between different stress levels, which enabled us to transform the observations at increased stress levels to interval-valued observations at the normal stress level and achieve robustness. The main findings drawn from this chapter are: we obtain an interval for the parameter of the link function, which is assumed at each stress level, by applying classical hypothesis testing between the pairwise stress levels to determine the level of imprecision. We showed why, in our method, we use the imprecision from combined pairwise log-rank tests, and not from a single log-rank test on all stress levels together. The latter would lead to less imprecision if the model fits poorly, while our proposed method leads to more imprecision. We have found that the end resulting [γ
i, γi] intervals get wider when we have more censored observations.
Throughout this research, we have presented two main contributions. First, Chapter 3 presented a new imprecise statistical method for ALT data with im- precision based on the likelihood ratio test to define the interval of values of the parameter γ of the Arrhenius link function. Secondly, Chapter 4, presented a simi- lar method, but we defined the interval of values of the parameterγ of the Arrhenius link function based on the log-rank test. Comparing these two scenarios, the results in the examples and the simulations show that we have more imprecision when we apply the nonparametric test than when we apply the likelihood ratio test with the assumption of a Weibull distribution at each stress level.
As with any novel statistical method developed for real-world applications, the real value of our method should be shown in practical applications. To implement the methods, no more is needed on the modelling side than for the classic inference methods with the same model assumptions, rather the main question is how one can use the resulting lower and upper survival functions to support real-world decisions.
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Further, we investigate this important aspect in the context of warranty contracts, which will be discussed in the next chapter.
Chapter 5
Study of warranties with ALT data
5.1
Introduction
In this chapter, we will illustrate a possible application of our new method using both approaches presented in Chapters 3 and 4. In particular, we focus on warranties and illustrate how our predictive inference can be used for inference on expected costs of warranty contracts. This section briefly introduces basic warranties considered in this chapter.
Products which include a warranty incur added costs to the manufacturer (or the consumer on occasion) for honouring the terms of the warranty: the warranty cost. This cost is related to a number of factors; the reliability of the product being the key factor. Products which fail within the warranty period entail the manufacturer taking responsibility for honouring the warranty, usually either by refunding or replacing faulty goods [57]. Generally, a warranty guarantees that a given product will provide reliable service for a defined period of time [69]. A warranty represents a contractual relationship between the manufacturer and the consumer that a specific product will provide reliable service and is absent of material or manufacturing defects, and, that if such defects cause the product to fail, it will be refunded, repaired or replaced at the manufacturer’s expense [13, 14]. However, a warranty is non-binding if the product has been used outside of certain specified conditions and manufacturers have no obligation to service the product in this case. Further, a warranty also outlines the limits of the manufacturers’ liability when a product is
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not used as intended.
Warranties are of equal importance to both consumers and manufacturers [54]. For example, consumers want to be confident the product they have purchased will function well. Warranties reassure consumers a product is of suitable quality and un- likely to develop a fault due to standardization issues, design faults or workmanship. On the other hand, manufacturers or distributors use warranties to safeguard their reputation and increase sales [54]. For example, providing a warranty lowers cus- tomers’ sense of risk in buying a product and encourages trust in the manufacturer’s products. Warranties can also increase sales by offering guaranteed reliability [54]. Offering a replacement or a refund of the customer’s original purchase price is an effective way of promoting a brand and increasing consumer demand [54].
Warranties also provide manufacturers with a level of protection against unfair demands for a refund or replacement by stating their responsibilities [54]. For ex- ample, while the manufacturer guarantees the consumer will receive a particular standard of performance from a product, this reduces unreasonable consumer de- mands that cause a financial loss. Finally, warranties also help manufacturers to gather consumer information for use in marketing and identify potential quality or workmanship issues [54].
Accelerated life testing (ALT) plays a key role in the manufacturing industry in terms of product design and development processes [45]. Indeed, the growth in competition within design innovation and the drive to slash product development timescales also underline how important ALT-based approaches are in product de- sign and development [45]. At present, products are checked under hard conditions to cause the types of failures that occur in real-life applications [45]. This produces an amount of data including failure mechanisms, causes, and aspects of probability distributions of failure times which indicate a product’s reliability in the field under normal use. These data can also be useful for highlighting further design modi- fications to enhance reliability [45]. However, determining a product’s reliability under normal conditions from the ALT data requires extrapolation in the form of a life-stress relationship [45, 60, 76], as described in the introduction of Chapter 1 and Section 2.2.
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ALT is widely used for reliability testing, predicting warranty cost, and assess- ment, and the comparison of a range of different approaches to solving product design issues [45]. Here, a well-rounded knowledge of statistical data analysis and validation techniques are key; indeed, the complexity of statistical models has led to the recruitment of researchers from a wide range of related fields, and this has be- come multidisciplinary with computational mathematics, statistics, and engineering all taking part [45].
A literature on warranties is available with focus on different perspectives. For example, Blischke and Murthy [14] provide an overview of warranty cost analysis. Various methods for analyzing and pricing warranty contracts can be found in review articles [39, 55, 56, 69]. Recently, researchers have been considering pricing warranty contracts based on ALT data. Yang [78] presented a design for accelerated life testing plans to predict warranty costs, assuming that the manufacturer offers a free replacement warranty policy [78]. He developed a test plan to minimize the analysis of the asymptotic variance of the maximum likelihood estimate of the warranty cost [78]. Meeker et al. [51] propose a simple use-rate model to predict the failure time distribution for a future component using accelerated life tests results. Zhao and Xie [82] use ALT data to predict warranty cost and risk warranty under imperfect repair. Their goal is to predict the expected warranty cost and provide confidence intervals for it [82].
Generally, in imprecise probability theory [10], lower and upper expectations of
a real-valued random quantity X, denoted by EX and EX respectively, can be
interpreted in terms of prices as follows. The lower expectation can be regarded as the maximum buying price for X, meaning that one would be willing to pay any amount up toEXin order to receive the random amountX. The upper expectation can be regarded as the minimum selling price for X, meaning that one would be willing to sell the random amount X for any price greater than EX. Whilst these interpretations may sometimes be somewhat difficult to link to reality, in our setting of warranties, one can use them and consider them, for example, as insurance prices. If a producer takes on a warranty with a random cost X, then they would prefer to pay a fixed cost up to the lower expectation of X instead of having to pay the