## 1 Introduction

In business and industry, there is great interest in predicting lifetimes of specimens that operate under stress because of safe life and warranty problems [1, 2]. The detection of potentially influential observations and their handling are relevant subjects in the calculation of percentiles for diverse types of analysis, for instance, in warranty. A bad estimation of the lower percentiles of a life distribution can produce significant monetary losses in commercial organizations because of an excessive amount of warranty claims. Then, the potential influence of a case on the estimation of lower percentiles should be considered in the design of structures and in the prediction of warranty claims.

To carry out life tests at levels of low stress can be very time consuming. To avoid this, failures of specimens at levels of high stress are observed, and then the failure times at low levels are predicted. This kind of testing is known as accelerated life (AL) test, and the models used in these tests are named AL models. Generally, these models are based on the physical or chemical nature that is behind the failure mechanism. To establish a parametric AL model, the following questions arise: (A1) What distribution can be assumed under regular stress conditions? (A2) Does the supposed distribution remain at stress levels different from the normal (regular) level? (A3) If the distribution remains, how are its parameters modified? (A4) What is the functional relationship between stressors and parameters? The relationship mentioned in (A4) can be described by an AL model, such that the interest parameter vector (** θ**) and the stressors of durability (

**) are linked by a function**

*x***(**

*θ***). Habitually, the stressors**

*x***identify load, pressure, temperature, vibration or voltage. Usually, the parameter**

*x***represents the mean or the scale of the distribution. This is because often, an increment in the stress level modifies the location or the scale but not the shape of the life distribution, making the distributional family remain unalterable; for more details about life distributions, see [3]. Thus, it is possible to predict the lifetime (**

*θ**T*) at different stress levels by using

**(**

*θ***). This functional relationship involves unknown additional parameters that need to be estimated. The usual AL models are: (B1) Arrhenius; (B2) exponential; (B3) Eyring; (B4) linear; and (B5) power law; for more details, see [4-6].**

*x*One distribution that has been used in reliability and quality studies, and notably in material fatigue, is the Birnbaum–Saunders (BS) distribution; see the seminal articles in [7, 8] and the more recent works on quality studies in [9, 10], and [11]. The BS model corresponds to the distribution of the random variable , when *Z* ∼ N(0,1). We use the notation *T* ∼ BS(*α*,*β*), where *α* is a shape parameter and *β* is both a scale parameter and the median of the distribution. The BS model shows that the failure of a specimen is due to the development and growth of a dominant crack produced by stress. The BS distribution is a natural model in many situations where an accumulation of damage produced by stress forces a quantity to exceed a critical threshold provoking the failure of a specimen. For more details about the BS distribution, see [12], pp. 651–658 and [2]. Several generalizations and extensions of the BS distribution have been made by [13-26]. Implementations in the R software of BS distributions can be found in [27-29]. These packages are available from the authors upon request. Although the BS distribution has its origin in engineering, it has been applied successfully to business, environmental and medical data [20-23, 26, 30-37].

If *T* ∼ BS(*α*,*β*), then:

- (C1)The PDF of
*T*iswhere*ϕ*( ⋅ ) denotes the N(0,1) PDF; - (C2) so that
*Z*^{2}∼*χ*^{2}(1); - (C3)
*b**T*∼ BS(*α*,*b**β*), with*b*> 0; - (C4)1 ∕
*T*∼ BS(*α*,1 ∕*β*); - (C5)The CDF of
*T*is , where Φ( ⋅ ) denotes the N(0,1) CDF; - (C6)The quantile function (QF) of
*T*is , where*z*(*q*) is the N(0,1) QF and is the inverse CDF of*T*; - (C7)The mean and variance of
*T*are E[*T*] =*β*[1 +*α*^{2}∕ 2] and V[*T*] =*β*^{2}*α*^{2}[1 + 5*α*^{2}∕ 4], respectively, and - (C8)The failure rate of
*T*is given by

It is possible to note that *h*_{T}(*t*) as defined in (C8) is unimodal for any *α*, that is, it increases until its change point (*t*_{c}) and decreases for *t* > *t*_{c}. In addition, *h*_{T}(*t*) approaches 1 ∕ [2*α*^{2}*β*] as *t* → ∞ , and it tends to be increasing as *α* → 0, such as shown in [7]. BS and lognormal models have similar properties and shape characteristics, but (i) the lognormal PDF has lighter tails than the BS PDF and (ii) the lognormal failure rate decreases until zero and it does not become stabilized at a positive constant greater than zero, as in the BS case. Then, there are reasons for preferring one of these models instead of the other one. For instance, [7] mentioned that fatigue life data usually have unimodal failure rates, which also occurs with environmental, financial duration and survival data [22, 34, 38-40]. In consequence, if one is interested in a warranty analysis on the basis of fatigue life data, as is the case of our application example, one should postulate the BS distribution as a candidate model to describe this type of data.

Accelerated life models have been considered for the BS distribution, but only some particular cases have been analyzed. For instance, [41] developed a BS power law AL (BSPAL) model. It is possible to present a more general methodology to develop an AL model on the basis of the BS distribution, which we call BS accelerated life (BSAL) models.

Influence diagnostic analysis is an important step in data modeling, which is carried out after the parameter estimation step for assessing the stability of this estimation. The analysis can be conducted by global and/or local influence diagnostic methods. These methods have been frequently employed in normal linear regression models and largely studied in the statistical literature. Global influence methods give emphasis to studying the effect of eliminating observations on the results from the fitted model; for more details on global influence [42, 43]. Cook [44] proposed an alternative method, known as local influence, for detecting the effect of small perturbations in the model and/or data on the maximum likelihood (ML) estimates. For more details about the use of the global and local influence methods in more general models than the normal one, see [30, 36, 45-49]. Diagnostic methods are useful in complex statistical models. However, these methods, specially the concept of generalized leverage (GL), are also useful in simpler models, such as in the case of the experimental design model employed in our application example [43, 50]. Specifically, in this example, we use a real fatigue life data set presented in [8] for illustrating the proposed methodology by means of which we conduct exploratory, confirmatory and diagnostic analyses. Once we choose a suitable model for these data, we estimate lower percentiles to predict warranty claims. The obtained results are implemented in the R software and available from the authors upon request.

The aim of this article is to introduce a methodology based on BSAL models that includes GL and influence diagnostic analyses. BSAL models and their estimation and inference based on likelihood methods are presented in Section 2. The global and local influence and GL methods are derived in Section 3. An application example with real data is discussed in Section 4. Some conclusions and an appendix with technical results are provided in the final sections.