Page 13 - Contributed Paper Session (CPS) - Volume 3

CPS1923 Deemat C M. et al.
The rest of the paper is organized as follows. In Section 2, we propose an
exact test for testing exponentiality against RIMRL ℎ class and then
calculate the critical values for different sample sizes. The asymptotic normality
and the consistency of the proposed test statistic are proved in Section 3. The
Pitman’s asymptotic efficacy value is also given in this section. In Section 4, we
report the result of the simulation study carried out to assess the performance
of the proposed test. Finally, in Section 5 we give the conclusions of our study.

2. Exact Test
Let be the lifetime of a device which has absolutely continuous
distribution function (. ) . Suppose () = ( > ) denotes the survival
̅
∞
function of at . Also let = () = ∫ () < ∞. Assume that the
̅
0
device under consideration is experiencing a random shock. Suppose ()
denotes the total number of shocks up to time with probability mass

function ( () = )= () − +1 (), = 0, 1, 2 …. Suppose that the random
variable , = 0, 1, 2, quantify the amount of hidden lifetime absorbed by the

th shock with = 0 and having common distribution function
0
() = ( ≤ ). The total cumulative life damage up to time is defined as

() = ∑ () with the cumulative distribution function () = (() ≤

=0

) = ∑ ∞ () [ () − (+1) ()]. It is assumed that the unit fails when the
=0
total life-damage exceeds a pre-specified level > 0. We refer to Glynn and
Whitt (1993), Roginsky (1994) and Sepehrifar et al. (2015) for discussion related
this framework. Let = − ()be the residual lifetime of an operating
∗
device with cumulative damage () . Note that the realizations of is
∗
available to us for further analysis. Consider a device subjected to ()
number of shocks up to time . Given that such a device is in an operating
situation at time instant after installation, the MRL function of denoted by
∗
() is defined by () = ( − | ≥ ). Note that the total life-damage
∗
∗
∗
∗
will not exceed the threshold level . From the definitions it is evident that the
random variables and () are independent.
∗

Definition 2.1. The mean residual life of a device under shock model
( ℎ ) at time t is defined as
1 ∞
∗
() = ∫ ̅() ,
̅()

where ̅() = ∫ ( + )().
̅
0

Definition 2.2. The random variable belongs to the ℎ class if the
function () is a non-decreasing function for all > 0.
∗

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