Optimality

246 H. El Barmi and H. Mukerjee
The confidence bands could always be improved by the following consideration.
The 100(1−α)% simultaneous confidence bands, [Li, Ui], for Fi, 1≤i≤k, in the
unrestricted case obey the following probability inequality
P(Fi∈ [Li, Ui] : 1≤i≤k)≥1−α.
Under our model, F1≤ F2≤···≤Fk, this probability is not reduced if we replace
[Li, Ui] by [L∗ i , U ∗ i ], where
L ∗ i = max{Lj : 1≤j≤ i} and U ∗ i = min{Uj : i≤j≤ k},1≤i≤k.
6. Hypotheses testing
Let H0 : F1 = F2 =··· = Fk and Ha : F1≤ F2≤···≤Fk. In this section we
propose an asymptotic test of H0 against Ha− H0. This problem has already been
considered by El Barmi et al. [3] when k = 2, and the test statistic they proposed
is Tn = √ n sup x≥0[ ˆ F2(x)− ˆ F1(x)]. They showed that under H0,
(6.1)
lim
n→∞ P(Tn > t) = 2(1−Φ(t)), t≥0,
where Φ is the standard normal distribution function.
For k > 2, we use an extension of the sequential testing procedure in Hogg [7]
for testing equality of distribution functions based on independent random samples.
For testing H0j : F1 = F2 =··· = Fj against Haj− H0j, where Haj : F1 = F2 =
··· = Fj−1≤ Fj,j = 2, 3, . . . , k, we use the test statistic sup x≥0 Tjn(x) where
Tjn = √ n √ cj[ ˆ Fj− Av[ˆF; 1, j− 1]],
with cj = k(j− 1)/j. We reject H0j for large values Tjn, that may be also written
as
Tjn = √ cj[Zjn− Av(Zn; 1, j− 1)],
where Zn = (Z1n, Z2n, . . . , Zkn) T . By the weak convergence result in (3.1) and the
continuous mapping theorem, (T2n, T3n, . . . , Tkn) T converges weakly to (T2, T3, . . . ,
Tk) T , where
Tj = √ cj[Zj− Av[Z; 1, j− 1]].
A calculation of the covariances shows that the Tj’s are independent. Also note
that
d
= Bj(F), 2≤j≤ k,
Tj
where the Bj’s are independent standard Brownian motions and F = �k i=1 Fi =
kF1 under H0. We define our test statistic for the overall test of H0 against Ha−H0
by
Tn = max Tjn(x).
2≤j≤k sup
x≥0
By the continuous mapping theorem, Tn converges in distribution to T, where
T = max
2≤j≤k sup
x≥0
Tj(x).