Page 229 - Contributed Paper Session (CPS) - Volume 2

CPS1844 Reza M.

1
̂
() = ∑ ∑ ( () ≤ ()), (1)

=1 =+1

We obtain a simultaneous plot of

((), (())) , ((), (())) , ((), (())) , = 1, … . . , . (2)
̂
̂
̂

With applications in statistical decision theory, clinical trials, experimental
design, and portfolio analysis, stochastic orderings of random vectors is of
interest when comparing several multivariate distributions. While there is a
rich literature on testing for stochastic ordered among univariate distributions
the literature dealing with inference on nonparametric multivariate
distributions is sparse. Giovagnoli and Wynn (1995) define the multivariate
d
dispersive order (weak D-ordering) for two random vectors X and Y in R as
follows. Define X ≤D Y if and only if Dx ≤St Dy, where ≤St is the stochastic ordering
between two random variables. One can show that Dx ≤St Dy holds if and only
if () ≥ () for all t ∈ℝ. Giovagnoli and Wynn (1995) show that Dx ≤St Dy if

and only if the average probability content of balls of a given radius with center
having the same distribution (independently), is more for X than for Y.
Under K0 : X ≤D Y, it follows that Dx ≤St Dy with probability 1. One can estimate
P(Dx ≤ Dy) and reject K0 for small values of this probability. It is not difficult to
show that the estimated probability is a U-statistic with a normal limiting
distribution for large sample sizes. The hypothesis K0 : X ≤D Y is equivalent to
′ : () ≥ () for all t ∈R. It follows from X ≤D Y, that E(r(Dx)) ≤E(r(Dy)) for

0
all non-decreasing functions r on [0,∞) and ((X)) ≤ ((Y)).
Furthermore, the dispersion order ≤D is location and rotation free. If Dx ≤St Dy,
then X + a ≤D ΛY + b for all orthogonal matrices and Λ and for all vectors
a and b.

Lemma 1 When the and groups differ in location µ only; i.e. () =
1
( − µ), we have = ≥ with equality holding if and only if µ = 0.

2
If µ = 0, then the result follows since G1 = G2 is equivalent to = = .

One can verify that the within sample IPDs are invariant with respect to
location shift so that = . Moreover, the between sample IPDs will

become larger than within sample IPDs when µ ≠ 0 since X−Y is no longer
centered at zero. Hence, = ≥ . As the following example shows,

when ECDFs () and () are closer together than the ECDF (); i.e.
̂
̂
̂

= ≥ , one can say that the two distributions G1 and G2 have a location

shift difference.
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