Page 14 - Contributed Paper Session (CPS) - Volume 3

CPS1923 Deemat C M. et al.
Next, we develop a family of test for testing exponentiality against
ℎ . We are interested to test the null hypothesis

∶ is exponential
∗
0
against the alternatives

∗
∶ is ℎ (and not exponential).
1

on the basis of a random sample , , … , : from an absolutely continuous
∗
∗
∗
2

1
distribution function .
∗
For the above testing problem Sepehrifar et al. (2015) proposed a non-
parametric test based on the departure measure ∆ ( ) defined by
∗
∗

∗
1 1 ∆( )
∗
∗
∗
∗
∗
∆ ( ) = (( , ) − ) =
∗ ∗ 1 2 2 1 ∗

where ∆( ) = (( , ) − ) and = ( ) . Based on
1
∗
∗
∗
∗
∗
∗
∗
1
1
2
1
2
U-statistics theory Sepehrifar et al. (2015) obtained the following test statistic
̂ ∗
∆ = Δ ̂ , (1)
̂ ∗

1
̂
̅
∗
∗
∗
∗
∗
∗
where = ∑ and ∆= 2 ∑ ∑ ℎ( , ) with ℎ( , ) =
=1 (−1) =1 <;=1 1 2
1
min( , ) − . Hence the test procedure is to reject the null hypothesis
∗
∗
∗
1
1
2
2
in favour of for large values of ∆ .
̂ ∗
0
1
Next we develop an exact test based on the test statistics ∆ and then
̂ ∗
calculate the critical values for different sample size. We use a result due to
Box (1954) to find the exact null distribution of the test statistic.

Theorem 2.1. Let be continuous non-negative random variable with
∗

−
() = . Let , , … be independent and identical samples from .
∗
∗
∗
∗
∗
2
1
2

Then for fixed

, −
̂ ∗
(∆ > ) = ∑ ∏ ( ) (, ),
− ,
=1 =1,≠ , ,
1 ≤
provided , ≠ , for ≠ , where (, ) = { 0 > and
,
(−2+1)
, = .
2(−1)

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