Page 399 - Contributed Paper Session (CPS) - Volume 6

CPS1995 Daniel B. et al.
represent the counts for the right-censored intervals. Thus, the total count of
the observations falling in a tail interval ( , ∞) is given by

= ∑ ( + −1 ) = ∑ ( + −1 ).

0

=+1
=+1
For the problem of testing that the underlying distribution is equal to F0,
we consider alternative distributions F which are asymptotically close. For
these class of distributions, an asymptotically uniformly most powerful (UMP)
invariant test for : =0 vs. : ≠0 is given by the test: Reject H if
2
2
1
1
= ∑ 0 ( 0 − ) + ∑ −1 0 ( − ) { > 0} (2)
=1

=1
0
0
0

0
] and
is sufficiently large, where is the number of observations in ( −1,
0
0 = ( ) − ( −1 ), = 1, … , ; 0 = ∑ 0 represents the

0
0
=+1
cumulative proportion of observations in ( , ∞) and =

1 ∑ ( + ) the corresponding estimator, = 1, … , −
0 +∑ =+1 −1 =+1 0 −1
1; and { > 0} is an indicator function signifying that the corresponding

term will be included only if the tail interval count is non-zero. The critical value
∞
C is determined by ∫ () where (. ) is the chi-square probability
2
2

density function with k-1 degrees of freedom. The asymptotic power of this
∞ 2
test is given by = ∫ () , where the non-centrality parameter v is
−1, 2
equal to ∑ =1 (( )−( −1 )) .
0
0 ( )− 0 ( −1 )
The asymptotic distribution of under H follows from the fact that
2
2
∑ 1 ( 0 − ) → −1 . Also, the second term in (2) can be shown to
0

0
=1
0 0
2
1
satisfy ∑ −1 0 ( − ) { > 0} → 0 . Thus the distributional result is

=1
0
obtained via the direct application of Slutsky’s theorem.
The asymptotic optimality of the test follows directly from the property of
rejection regions constructed on the basis of quadratic forms with underlying
multivariate normal distribution. That is, if ( , … , )′~MVN(η, ∑) then there

1
exists a uniformly most powerful invariant (under a suitable group G of linear
−1
̂
̂
transformations) with rejection region of the form ∑ (̂ − ̂ ) (̂ − ̂ ) >

,

̂
where ̂ minimizes the quadratic form under a linear space K, ̂ minimizes

−1
the quadratic form for H ⊂ K and Σ −1 = ( )[see Lehmann, E.L. (1986)].
Implementation of the test for parametric families (; )proceeds as
̂
follows. If is the MLE obtained by maximizing an appropriate log-likelihood
̂
̂
as given in Section 2, then substituting () = ( ; ) − ( −1 ;) in place

0
0
0
()
of 0 and 0 | = ∑ () in place of 0 in equation (2) yields an
0
=+1
asymptotically UMP invariant test for : = 0 vs. : ≠ 0 . Asymptotic
invariance of the test follows directly from the √ −consistency of MLE’s.

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