 # Testing for exponentiality for stationary associated random variables

In this paper, we consider the problem of testing for exponentiality against univariate positive ageing when the underlying sample consists of stationary associated random variables. In particular, we discuss the asymptotic behavior of the tests by Deshpande (1983), Hollander and Proschan (1972) and Ahmad (1992) for testing exponentiality against IFRA, NBU and DMRL, respectively under association. A simulation study illustrates the effect of dependence on the asymptotic normality of the test statistics and on the size and power of the tests.

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## 1 Introduction

The need to test for exponentiality against various univariate ageing classes occurs in many fields of research, such as reliability and survival analysis, queueing theory and economics among others. Traditionally, for testing for exponentiality it was assumed that the the random variables of interest are independent and identically distributed . However, in many real applications the assumption of independence is seldom satisfied. The aim of this paper is to discuss the testing problem when the underlying random variables are associated.

In the following, we discuss the popular ageing classes studied in the paper, the concept of association and various examples of associated random variables occurring in the literature, and then finally the tests for exponentiality against the ageing classes under association.

In reliability analysis, interest often lies in studying the ageing concepts of the lifetime of a component or a system as these help to analyze how it improves or deteriorates with time. Let be the lifetime of the component/ system under consideration with the distribution function (, ), the survival function

, and the probability density function

. The failure rate function and the mean residual lifetime function associated with are defined as and , whenever , , respectively. The ageing concepts are often described via the characteristics of the functions , , and .

Depending on the behavior of the chosen ageing criteria, the lifetime distribution can be categorized into various ageing classes. “No ageing” is synonymous to the lifetime distribution being exponential. Positive (negative) ageing occurs when the system or component under consideration deteriorates (improves) over time. Some of the widely used classes of positive ageing include the class of “Increasing Failure Rate Average (IFRA)”, the class of “New Better than Used (NBU)”, and the class of “Decreasing Mean Residual Lifetime (DMRL)”. The negative dual of these classes are DFRA, NWU, and IMRL respectively. These classes are defined as follows.

###### Definition 1.1.

is said to be IFRA (DFRA) if is increasing (decreasing) in . This is equivalent to , , .

###### Definition 1.2.

is said to be NBU (NWU) if for all and strict inequality for some .

###### Definition 1.3.

is said to be DMRL (IMRL) if the Mean Residual Life (MRL) function is decreasing (increasing) in , i.e., for .

Optimal maintenance, replacement, and resource allocation policies can be separately designed for each family of distributions. The knowledge of the lifetime belonging to a particular class of distributions can be used to choose appropriate parametric or a constrained nonparametric model for the underlying ageing process.

Testing for exponentiality against different ageing alternatives is also useful in queueing theory. For example, the service times and inter-arrival times in the classical queueing model, , are assumed to come from mutually independent sequences of exponential random variables. It leads to analytically tractable expressions of the performance metrics, like the mean number of customers in the system, and the mean service and arrival rates. Extensions of the classical model include , , and , and among others. In all the models, the service times have a General distribution. Several queueing models also assume that the inter-arrival times have a general distribution. For example, the queueing model

. In most queueing models, the probability distribution of the service times and the inter-arrival times impact the output characteristics. Hence, the knowledge of the service time and the inter-arrival times belonging to a particular class of distributions is useful in developing a queueing model for the underlying system to determine its long term behavior. For example, in

Abramov (2006) stochastic inequalities for the number of losses for some single-server queueing models when the inter-arrival times or the services times are NBU or NWU have been derived.

The classification of distributions into various ageing classes is also of interest to researchers in economics. An application is in testing for the duration dependence (see Ohn et al. (2004)). Another possible application is in choosing the appropriate marginal distribution for modeling various time series data. For example, processes like and with heavy tailed marginal distributions have been used to model many financial time series.

Many tests exist in literature that test for exponentiality (or the assumption of constant failure rate) against different positive or negative ageing alternatives. A detailed discussion on the various classes of ageing along with their testing procedures and applications for random variables can be found in Deshpande and Purohit (2005) and Lai and Xie (2006). However, in many real applications, the random variables under consideration are dependent.

For example, in reliability analysis, the lifetimes of independent components in a reliability structure when the components share the same load or are subject to a shared environmental stress are dependent (see Barlow and Proschan (1975) and Li et al. (2011)

). Various autoregressive models with minification structures have positively correlated components. For example, let

be a non-degenerate and non-negative random variable, and be a sequence of independent and identically distributed , non-negative and non-degenerate random variables independent of . Then, the non-negative random variables

 Xn=kmin(Xn−1,ϵn) for all n∈N and for% some k>1,

are dependent. Minification processes have been used to model dependent lifetime data (for example, see Cordeiro et al. (2014)) and dependent service times (for example, see Livny et al. (1993)).

In all these cases, the random variables under consideration are associated - a concept defined by Esary et al. (1967) as follows.

###### Definition 1.4.

A finite collection of random variables is said to be associated, if for any choice of component-wise non-decreasing functions , , we have,

 Cov(h(X1,…,Xn),g(X1,…,Xn))≥0

whenever it exists. An infinite collection of random variables is associated if every finite sub-collection is associated.

Any set of independent random variables is associated (Esary et al. (1967)). Non-decreasing functions of associated random variables are associated, for example, order statistics corresponding to a finite set of independent random variables are associated (Esary et al. (1967)). Few other examples of associated random variables are: positively correlated normal random variables (Pitt (1982)); the components of Marshall and Olkin (1967)

multivariate exponential distribution, multivariate extreme-value distribution (

Marshall and Olkin (1983)) and Downton multivariate exponential distribution (Downton (1970)); the components of the moving average process , where , are independent random variables and , have the same sign. A detailed compilation of results and applications for associated random variables can be found in Bulinski and Shashkin (2007), Prakasa Rao (2012) and Oliveira (2012).

While the control of dependence in stochastic processes is generally given in terms of mixing conditions, an obvious drawback is that the mixing coefficients are defined using -fields. It makes these coefficients difficult to compute in practice. For associated random variables, the control of dependence is through the covariance structure of the random variables. The simplicity of the conditions under which the limit theorems can be proved gives an advantage over the popularly used mixing processes.

In this paper, we discuss the limiting behavior of some of the tests of exponentiality against univariate positive ageing based on U-statistics when the underlying random variables are stationary and associated. In particular, we look at tests by Deshpande (1983), Hollander and Proschan (1972) and Ahmad (1992) for testing exponentiality against IFRA, NBU and DMRL respectively. The kernels of the test statistics of the given tests belong to the class of kernels which are bounded (but are not of bounded variation). For tests based on U-statistics for

random variables, the test statistics can be shown to be asymptotically normally distributed using the results of

Hoeffding (1948). However, it is not possible to directly extend the theory of asymptotic normality for U-statistics based on dependent random variables. Hence, the asymptotic behavior of U-statistics for associated random variables needs to be looked into separately.

We first develop a central limit theorem for U-statistics based on the class of kernels discussed above for stationary associated random variables. We next use this result to obtain critical points, size and power for the given tests. This helps in analyzing the behavior of the considered tests under the dependent setup.

For the rest of the paper, assume is a stationary sequence of associated random variables with the distribution function of denoted by . We also assume that are uniformly bounded, there exists a , such that . For applications in reliability and survival analysis this assumption is reasonable.

The paper is organized as follows. In the next section, Section 2, we give a general theorem for the asymptotic distribution of U-statistics based on bounded kernels for stationary associated random variables. In Section 3, we apply this result to discuss the limiting behavior of the tests by Deshpande (1983), Hollander and Proschan (1972) and Ahmad (1992) under association. In Section 4, the asymptotic normality of the test statistics and the power of the tests under the discussed dependent setup is illustrated via simulations. Section 5 is a brief discussion on the applications of our results and our intended future work. Section 6 contains some preliminary results and the proofs of our technical results.

## 2 Central limit theorem for U-statistics based on bounded kernels

The main result of this section, Theorem 2.3, gives the central limit theorem for U-statistics based on a bounded kernel of degree , when the underlying sample is a sequence of stationary associated random variables. The extension of this theorem to U-statistics with kernels of a general finite degree k are also discussed. These results are applied in section 3 to show the asymptotic normality of the test statistics of the considered tests of exponentiality against positive ageing under the dependent setup. Proof of the results are postponed to section 6.

The central limit theorem for U-statistics discussed extends the results of Dewan and Prakasa Rao (2001, 2002, 2015), Garg and Dewan (2015, 2018b) to a wider class of kernels.

The U-statistic of degree based on with a symmetric kernel is defined as,

 Un(ρ)=(n2)−1∑1≤j1

Let . Define

 ρ1(x1)=∫Rρ(x1,x2) dF(x2),h(1)(x1)=ρ1(x1)−θ, and h(2)(x1,x2)=ρ(x1,x2)−ρ1(x1)−ρ1(x2)+θ.

Then, the Hoeffding-decomposition for is , where is the U-statistic of degree based on the kernel , . When the observations are , .

Similarly, the Hoeffding’s decomposition for U-statistics of a finite degree can be obtained.

### 2.1 Central limit theorem

Before proceeding, we need to define the following.

Let denote the , and let

denote the characteristic function of random vector

, respectively.

###### Definition 2.1.

The of the random vector Z is said to satisfy a Lipshitz condition of order 1, if for every and some finite constant ,

 |fZ(x + u)−fZ(x )|≤Ck∑j=1|uj|. (2.2)
###### Definition 2.2.

(Newman (1984)) If and are two real-valued functions on , then iff and are both coordinate-wise non-decreasing.

If , then will be coordinate-wise non-decreasing.

We next define the conditions and that will be needed to prove Theorem 2.3.

In the following, let be a sequence of random variables independent of the sequence , with the marginal distribution function of being .

• For all distinct , such that and ,

• , , and are bounded and satisfy the Lipshitz condition of order 1 and

• , , and are absolutely integrable.

• For any 3 distinct indices from , such that and ,

• and , are bounded and satisfy the Lipshitz condition of order 1 and,

• and are absolutely integrable.

###### Theorem 2.3.

Let be the U-statistic based on a symmetric kernel which is bounded ( , for some for all ). Define ,
and for all .

Assume the following.

• ;

• and hold; and

• and .

Then

 Var(Un(ρ))=4σ2Un+o(1n), % where σ2U=σ21+2∞∑j=1σ21j. (2.3)

Further, if and there exists a function , such that and

 ∞∑j=1Cov(~ρ1(X1),~ρ1(Xj))<∞, (2.4)

then

 √n(Un(ρ)−θ)2σUL→N(0,1)asn→∞. (2.5)
###### Remark 2.4.

Theorem 2.3 can be easily extended to a U-statistic based on a kernel of any finite degree . Let be the U-statistic based on the symmetric kernel which is bounded.

Let be a sequence of random variables independent of the sequence , with the marginal distribution function of being .

Assume for all the following are true.

• For all distinct indices , such that and ,

• , , and are bounded and satisfy the Lipshitz condition of order 1, and

• , , and are absolutely integrable.

• For distinct indices from , such that and ,

• and are bounded and satisfy the Lipshitz condition of order 1, and

• and are absolutely integrable.

Further, if , and , then

 Var(Un(ρ))=k2σ2Un+o(1n). (2.6)

If and there exists a function , such that and

 ∞∑j=1Cov(~ρ1(X1),~ρ1(Xj))<∞, (2.7)

then

 √n(Un(ρ)−θ)kσUL→N(0,1)asn→∞. (2.8)
###### Remark 2.5.

The results can be extended to non-uniformly bounded random variables under stricter covariance restrictions, by using the standard truncation technique and putting appropriate assumptions on the moments of the underlying random variables.

## 3 Tests for ageing

Deshpande (1983), Hollander and Proschan (1972) and Ahmad (1992) had proposed tests for testing exponentiality against IFRA, NBU and DMRL respectively, for a sample of observations. In this section, we prove the asymptotic normalty of the test statistics of these tests when the underlying sample consists of stationary associated random variables.

Let in the following.

### 3.1 Testing Exponentiality against IFRA alternatives

Our aim is to test

 H0:F(x)=1−exp(−x/μ),x≥0, μ>0, against
 H1:F is IFRA but not exponential.

The test statistic, , of the test proposed by Deshpande (1983) is

 J(n,b)=1(n2)∑1≤i

where

 ρ(x,y)=h1(x,y)+h1(y,x)2,

and

 h1(x,y)={1ifx>by0otherwise.

When are , the asymptotic distribution of under , as discussed in Deshpande (1983) is

 √n(J(n,b)−1b+1)2√ξ1L→N(0,1), as n→∞, (3.2)

where

 ξ1=Var(ρ1(X1))=14{1+b2+b+12b+1+2(1−b)(1+b)−2b(1+b+b2)−4(b+1)2},

and

 ρ1(x)=¯F(bx)+F(xb)2,x≥0. (3.3)

We now obtain a limiting distribution for when the observations are associated.

###### Theorem 3.1.

Let be a sequence of stationary associated random variables, such that , for some , for all . Assume that conditions of Theorem 2.3 are satisfied. Then, the limiting distribution of under is

 D(n,b)=√n(J(n,b)−1b+1)2σDL→N(0,1)asn→∞,

where .

###### Proof.

By Hoeffding’s decomposition, .

The underlying kernel is not continuous and not of local bounded variation. However, it is bounded. Also, is a Lipschitz function (i.e , for all and for some ). From Theorem 2.3,

 D(n,b)L→N(0,1)asn→∞.

Rejection criteria: Since is unknown, we use the following test statistic

 ^D(n,b)=√n(J(n,b)−1b+1)2^σD,

where

is a consistent estimator for

. Reject at a significance level if , where is percentile of .

### 3.2 Testing exponentiality against NBU alternatives

Our aim is to test

 H0:¯F(s+t)=¯F(s)¯F(t),s,t≥0, (i.e F is % exponential) against
 H2:¯F(s+t)≤¯F(s)¯F(t),s,t≥0; with strict % inequality for some s,t.

The test by Hollander and Proschan (1972) rejects for small values of the statistic, , defined by , where

 Nn=6n(n−1)(n−2)∑1≤i

and

 ϕ(x1,x2,x3)={1if x1>x2+x3,0otherwise.

When are , the asymptotic distribution of can be obtained by the central limit theorem for U-statistics as discussed in Hoeffding (1948). In particular, under , we get

 √n(Nn−14)√5/432L→N(0,1)asn→∞. (3.5)

The kernel is of degree .

 ρ1(x)=13(∫x0F(x−z)dF(z)+∫∞0¯F(x+z)dF(z)+∫∞xF(z−x)dF(z)) (3.6)

for .

###### Theorem 3.2.

Let be a sequence of stationary associated random variables, such that , for some , for all . Under the conditions discussed in Remark 2.4 (k =3) and

 HPn=√n(Nn−14)3σHPL→N(0,1)asn→∞,

where .

Rejection criteria: Since is unknown, we use the following test statistic

 ^HPn=√n(Nn−14)3^σHP,

where is a consistent estimator for . Reject at a significance level if , .

### 3.3 Testing exponentiality against DMRL alternatives

Assume the MRL function is differential and as . Our aim is to test

 H0:μ(x) is constant (i.e F is exponential), against
 H3:dμ(x)dx≤0 or f(x)∫∞x¯F(u)du≤¯F2(x),x≥0.

A test by Ahmad (1992) rejects in favor of for large values of , where

 δFn=1(n2)∑1≤i1

and . Here,

 ϕ(x1,x2)={(3x1−x2)if x2>x1,0otherwise.

When are , the asymptotic distribution of can be obtained by the central limit theorem for U-statistics as discussed in Hoeffding (1948). In particular, under , we get,

 √nδFnμ√1/3L→N(0,1)asn→∞. (3.8)

The kernel is of degree .

 ρ1(x) =12(∫∞x(3x−y)dF(y)+∫x0(3y−x)dF(y))=2x¯F(x)−x2+3μ2−2∫∞xydF(y). (3.9)

The statistic can be made scale invariant by considering (). The limiting distribution of under follows using and the Slutsky’s theorem, i.e

 √nδFn¯Xn√1/3L→N(0,1)asn→∞. (3.10)
###### Theorem 3.3.

Let be a sequence of stationary associated random variables, such that , for some , for all . Assume that conditions of Theorem 2.3 are satisfied. Then, under ,

 An=√nδFn2σAL→N(0,1)asn→∞,

where .

Rejection criteria: Since is not known, we use the following test statistic

 ^An=√nδFn2^σA,

where is a consistent estimator for . Reject at a significance level if .

###### Remark 3.4.

The kernels of the test statistics discussed are discontinuous and not of local bounded variation. The existing results on U-statistics by Garg and Dewan (2015, 2018b) cannot be used to obtain the limiting distribution of the statistics discussed under the dependent setup.

###### Remark 3.5.

The test statistics , , and (under appropriate rejection criteria) can also be used for testing exponentiality against DFRA, NWU and IMRL respectively.

## 4 Simulations

We assessed the performance of IFRA, NBU and DMRL tests based on , , and when the underlying observations are stationary and associated via simulations. We generated associated random variables using the property that non-decreasing functions of independent random variables are associated. We used the statistical software R (R Core Team (2016)) for our simulations.

• We investigated the asymptotic normality of the statistics under . The marginal distribution of was taken as , , i.e we take (the 3 tests discussed do not depend on the choice of ). The samples were generated as follows.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• We also calculated the empirical power of the above tests for the following alternatives.

• The marginal distribution of was taken as , , . We took . The samples were generated as follows.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• The marginal distribution of was taken as , , . We took . The samples were generated as follows.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.

• , where were pseudo-random numbers from generated using function in R.