Let
X 1,...,
X n,
n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis
H 0:
X 1,...,
X n is a randomsample on
X with Var
X <∞. We revisit the problem of nonparametric testing for
H 0 versus the “at most one change (AMOC) in the mean” alternative hypothesis
H A: there is an integer
k*, 1 ≤
k* <
n, such that
EX 1 = · · · = EX
k* ≠ EX
k*+1 = ··· =
EX n. A natural way of testing for
H 0 versus
H A is via comparing the sample mean of the first
k observables to the sample mean of the last
n -
k observables, for all possible times
k of AMOC in the mean, 1 ≤
k <
n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ
k = |
S k/
k - (
S n -
S k)/(
n -
k)|, where
S k:=
X 1 + ··· +
X k. Asymptotic distributions of these test statistics under
H 0 as
n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |
Z n(
t)|, where {
Z n(
t), 0 ≤
t ≤ 1}
n≥1 are the tied-down partial sums processes such that
$${Z_n}left( t right): = left( {{S_{leftlceil {left( {n + 1} right)t} rightrceil }} - left[ {left( {n + 1} right)t} right]{S_n}/n} right)/sqrt n $$
if 0 ≤
t < 1, and
Z n(
t):= 0 if
t = 1. In the present paper, we propose an alternative route to nonparametric testing for
H 0 versus
H A via sup-functionals of appropriately weighted |
Z n(
t)|. Simply considering max
1?k<n Δ
k as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |
Z n(
t)|/(
t(1 -
t)) under
H 0 and assuming also that
E|
X|
r < ∞ for some
r > 2. We believe the weight function
t(1 -
t) for sup-functionals of |
Z n(
t)| has not been considered before.