The Birnbaum-Saunders (BS) distribution was proposed by Birnbaum and Saunders (1969) motivated by problems of vibration in commercial aircrafts that caused fatigue in materials. Although, in principle, its origin is for modeling equipment lifetimes subjected to dynamic loads, the BS distribution has been widely studied and applied in many applied fields including, for example, engineering, business, economics, medicine, atmospheric contaminants, finance and quality control; see Jin and Kawczak (2003), Balakrishnan et al. (2007), Bhatti (2010), Villegas et al. (2011), Paula et al. (2012), Saulo et al. (2013), Marchant et al. (2013), Leiva et al. (2014), Leão et al. (2017a, b), and references therein. The interested reader on the BS distribution is refereed to Johnson et al. (1995); Leiva (2016)
. These works present a full review about this model. The BS distribution has been used quite effectively to model positively skewed data, especially lifetime data and crack growth data. This distribution is related to the normal distribution and has interesting properties.
Some works on reparameterized versions of the BS distribution were proposed by Ahmed et al. (2008), Lio et al. (2010) and Santos-Neto et al. (2012, 2016). In particular, the work of Santos-Neto et al. (2012) proposed several parameterizations of the BS distribution, which allow diverse features of data modeling to be considered. One of such parameterizations is indexed by the parameters and , where is a scale parameter and the mean of the distribution, whereas is a shape and precision parameter. The notation
is used when an random variable (RV) follows a reparameterized BS (RBS) distribution; more details about this model is found in Section2.
The bivariate version of the BS distribution was proposed by Kundu et al. (2010), where the authors discussed maximum likelihood (ML) estimation and modified moment (MM) estimation of the model parameters. Recently, Khosravi et al. (2015) observed that the bivariate BS model proposed by Kundu et al. (2010)
can be written as the weighted mixture of bivariate inverse Gaussian distribution(Kocherlakota, 1986) and its reciprocals. They also introduced a mixture of two bivariate BS distributions and discussed its various properties. Kundu et al. (2013) extended to the multivariate case the generalized BS distribution introduced by Díaz-García and Leiva (2005). Other bivariate and multivariate distributions related to the BS model can be found in Vilca et al. (2014a, b), Kundu (2015b, a) and Jamalizadeh and Kundu (2015).
In this context, the primary objective of this paper is to introduce a bivariate RBS (BRBS) distribution based on the RBS distribution proposed by Santos-Neto et al. (2012). The secondary objectives are: (i) to present the reliability measure for the BRBS model and obtain the monotonicity of the hazard rate (HR); (ii) to discuss some unimodality properties of the BRBS model; (iii) to derive the ML estimators and MM estimators of the unknown parameters as well as their asymptotic properties; (iv) to evaluate the performance of the ML and MM estimators via Monte Carlo (MC) simulations; and (v) to illustrate the potential applications by using real data.
The rest of the paper proceeds as follows. In Section 2, we describe briefly the BS and RBS distributions and their related properties. In Section 3, we introduce the BRBS distribution and discuss some of its properties. In Section 4, we present the ML and MM estimators of the unknown parameters and their corresponding asymptotic results. In Section 5, an evaluation of the ML and MM estimators using MC simulation is shown, as well as an illustrative example by using a real data set. Finally, in Section 5.2, we provide some concluding remarks and also point out some problems worthy of further study.
2 BS distributions
2.1 The BS distribution
The BS distribution is related to the normal distribution by means of the stochastic representation
where and are shape and scale parameters, respectively, is a RV following a standard normal distribution , such that is BS distributed with notation
. The probability density function (PDF) ofis given by
The mean and variance ofare given by and , respectively. The scale parameter is also the median of the distribution. The BS distribution holds the reciprocal property, that is, has the same distribution of with the parameter replaced by , , which implies
2.2 The RBS distribution
The RBS distribution is indexed by the parameters and , where and are the original BS parameters of (1), and are the scale and shape (precision) parameters, respectively. If , then its PDF, for , is given by
and its cumulative distribution function (CDF) is denoted by. From (2), the survival function (SF) and hazard rate (HR) function are given by
respectively, where is the CDF of the standard normal distribution. Figure 1 displays some shapes for the PDF and HR of .
Considering the function
and denoting , where and , expressions for the first, second and third derivatives of are given by
Note that where denotes the PDF of the standard normal distribution. The function is a bijection of to and has inverse, denoted by , given by
Some properties of the RBS distribution are the following. Let and . The RBS distribution satisfies the following properties:
The PDF of the RBS distribution has at most one mode, see Proposition 2.7 in Vila et al. (2017).
If , the moments about the origin of are given by
and in general
where . Then,
Since , the th moment always exists. As mentioned, can be interpreted as a precision parameter, that is, for fixed values of , when , the variance of tends to zero. Also, for fixed , if , then .
If , it is possible to show that , with , and . On the other hand, if , since the function is strictly increasing, it follows that
Let and be two RVs statistically independent such that represents “strength” and represents “stress”, then the reliability of a component is given by Since , with , and for each , by the independence of the RVs, we have that
Then, whenever .
3 The bivariate RBS distribution
3.1 Density and shape analysis
The bivariate random vectoris said to follow a BRBS distribution with parameters , , , , , denoted by , if the joint CDF of and can be expressed as
, , , , , , , and is the standard bivariate normal CDF with correlation coefficient . It follows that the joint PDF associated with (5) is given by
where is a normal joint PDF given by
Following the notation in (3) and considering , with and , for , note that
where , for
From now on, we will use the following notation
Some important properties of the function are:
is an increasing function for all .
If (), the function is increasing (decreasing) for all .
whenever , and .
whenever , and .
Let . For and , we have
Let . For and , we have
The joint PDFs and HRs of are unimodal and the surface plots for some values of the parameters are presented in Figure 2.
Next, some results on the unimodality properties of BRBS distribution are obtained. We will consider the following hypothesis:
Hypothesis 1. Let , and , for .
, where .
Under Hypothesis 1 there is an unique constant such that the point is critical for
Theorem 3.1 (Unimodality).
Under Hypothesis 1 there is an unique constant in the interval such that the point is a mode for That is, under Hypothesis 1 the BRBS distribution is unimodal.
3.2 Properties of the BRBS distribution
Proposition 3.2 (Marginal functions).
Let . The marginal PDFs, denoted , and the marginal CDFs, denoted , are given by
respectively, where and are the PDF and the CDF of the RBS distribution defined in (2). That is, .
Proposition 3.3 (Reliability function).
If , the reliability can be expressed as
where is defined in (7).
If and are identically distributed RVs following a RBS distribution, then
where , and if in addition , then
is the bivariate SF. An analogous procedure to Proposition 3.4 shows that
Therefore, we get
The function is decreasing in both and .
For the BRBS distribution,
if , and ; then the function is increasing in .
if , and ; then the function is increasing in .
if and ; then the function is increasing in .
Let and be the reliability. Then,
Let , and . Then,
Let and be RVs, the covariance and correlation of and , as usual, are denoted by and respectively. The following result tells us that two RVs with BRBS distribution are associated and correlated positively.
Let , and , for . Then,
Let be a bivariate positive RV with bivariate SF (defined in (8)) and with finite. If we assume that the sampling probability of is proportional to , the recurrence times are , where and , where and
have independent uniform distributions inand are independent of . The vector is known as the size biased random vector and has PDF defined by
Proposition 3.9 (Equilibrium distribution).
For the BRBS distribution, the function is decreasing in both and .
Let and , for . Then,
3.3 Failure rate with presence of dependence
A real function , which is defined on and , and are linearly ordered sets, is said to be totally positive of order two () and reverse rule of order two () in and if
We also define a local dependence function (LDF)
The LDF, , can be defined for any positive and mixed differentiable function , which does not need to be a density function. See Holland and Wang (1987) for definition and properties of the local dependence function.
(Holland and Wang, 1987) A function is if and only if .
It can be verified that
whenever . Then, by Theorem 3.4 the joint PDF is an example of a function when . The LDF will be used for studying the monotonicity of certain HRs. If is a bivariate random vector distributed according to , then one can consider (the CDF of given that ). The conditional HR (CHR) of given and mean residual function (MRF), using obvious notation, are defined by
denotes the conditional SF (CSF).
We know a lot about the normal distribution. For example, if has a distribution we have (Feller (1968), Section 7.1)
Clearly the inequalities are useful only for larger , because as decreases to zero the lower bound goes to and the upper bound goes to . The inequality above is known as the Gaussian tail inequality.
Theorem 3.5 (Monotonicity).
Let us assume that has a distribution and consider and for . Then,
The CHR: is decreasing for all and for all , whenever .
The MRF: is increasing for all and for all , whenever .
The CSF : is increasing for all and for all