2.28 Exceedance statistics. Let X1, X2, . . . , Xm and Y1, Y2, . . . , Yn be two independent random samples from arbitrary continuous cdf’s FX and FY, respectively, and let Sm(x) and Sn(y) be the corresponding empirical cdf’s. Consider, for example, the quantity m[1Sm(Y1)], which is simply the count of the total number of X’s that exceed (or do not precede)

Y1 and may be called an exceedance statistic. Several nonparametric tests proposed in the literature are based on exceedance (or precedence)

statistics and these are called exceedance (or precedence) tests. We will study some of these tests later.

Let Y(1)

Answer parts

(a) through (h) assuming FX¼FY.

(a) Show that Sm(Yi), i¼1, 2, . . . , n, is uniformly distributed over the set of points (0, 1=m, 2=m, . . . , 1).

(b) Show that the distribution of Sm(Y(j))Sm(Y(k)), k

(Fligner and Wolfe, 1976)

(c) Show that the distribution of P(i)¼mSm(Y(i)) is given by P[P(i) ¼ j] ¼

m þ n  i  j m  j

 

i þ j  1 j

 

m þ n n

  j ¼ 0, 1, . . . ,m The quantity P(i) is the count of the number of X’s that precede the ith order statistic in the Y sample and is called the ‘‘placement’’

of Y(i) among the observations in the X sample. Observe that P(i)¼r1 þ    þ ri, where ri is the ith block frequency and thus ri¼P(i)P(i1).

(d) Show that E(P(i)) ¼
mi n þ 1 and var(P(i)) ¼
i(n  i þ 1)m(m þ n þ 1)
(n þ 1)2(n þ 2)
(Orban and Wolfe, 1982)

(e) Let T1 be the number of X observations exceeding the largest Y observation, that is, T1¼m[1Sm(Y(n))]¼mP(n). Show that P(T1 ¼ t) ¼
m þ n  t  1 m  t  
m þ n m  

(f) Let T2 be the number of X’s preceding (not exceeding) the smallest Y observation; that is, T2¼mSm(Y(1))¼P(1). Show that the distribution of T3¼T1þT2 is given by P(T3 ¼ t) ¼ (t þ 1)
m þ n  t  2 m  t  
m þ n m  
(Rosenbaum, 1954)
(g) Let T4 be the number of X’s in the interval I¼(Y(r), Y(nþ1r)], where Y(r) is the pth sample quantile of the Y’s. The interval I is called the interquartile range of the Y’s. Note that T4¼m[Sm(Y(nþ1r)) Sm(Y(r))]. Show that the distribution of T4 is given by P(T4 ¼ t) ¼
m þ 2r  i  1 m  t  
n þ t  2r t  
m þ n m   t ¼ 0, 1, . . . ,m (h) Show that E(T4) ¼
2m n þ 1 and var(T4) ¼
2m(n  1)(m þ n þ 1)
(n þ 1)2(n þ 2)
(Hackl and Katzenbeisser, 1984)
The statistics T3 and T4 have been proposed as tests for H0: FX¼FY against the alternative that the dispersion of FX exceeds the dispersion of FY.

(c) Show that the number of tiles n to be put on a future test such that all of the n measurements exceed X(1) with probability p is given by n ¼
m(1  p)
p