global maximum and minimum are Ξ2:n(0) = 12 and Ξ2:n(v1(2)) < −12, respectively, where v1(2) is the unique zero of (2.2.7) in (0, 1). The bounds for the case r = n − 1 are derived in much the same way.
For 3 ≤ r ≤ n − 2, Lemma 1 and (2.2.8) assert that (2.2.3) is either consecutively increasing, decreasing and eventually increasing in (0, 1), or simply increasing in the whole interval. Relations Ξr:n(0) = a0(c(r)) = 12 and Ξr:n(1) = an−2(c(r)) = −12 contradict the latter possibility. Therefore (2.2.8) has two zeros in (0, 1). The first one u1(r) provides the global maximum of (2.2.3), and the other v1(r) gives the minimum. This ends the proof.
Table 2.1 presents numerical values of upper bounds for the single order statistics Xr:n, 3 ≤ r < n = 12. Every cell contains two numbers: the bound value for respective r and n (printed in bold), and the probability of the smaller point of the two-point distribution which attains the bound (printed in the standard type). We adhere to the above conventions presenting numerical evaluations later on. We have not included cases r = 1, r = 2 and r = n, because then the bounds have simple forms and do not have to be determined numerically.
As one can expect, the bound values increase in the consecutive columns, and decrease in the rows. The column increase accelerates, and the row decrease slows down. Analogous tendencies reveal the probability values. The conclusions would not change if we included case r = 2 into the analysis. The bound values amount to 0.5 then, and are attained in the limit as the probability of the smaller point tends to 0. Using the Table, we can deduce the lower bounds as well. The bound for the rth smallest order statistic Xr:n is identical with the negative of the upper one for the rth greatest order statistic Xn+1−r:n. The attainability parameter is obtained by subtracting the respective table value for Xn+1−r:n for one.
Table2.1:Upperboundsonexpectationsofsingleorderstatistics EXr:n−µ∆ for3≤r<n≤12.
n r34567891011
40.541670.1666750.514450.651460.059110.4072460.506830.571390.780080.027890.219860.5534770.503800.542590.646100.916200.015430.135820.356610.6453980.502330.528530.594110.728501.056130.009450.091780.248330.459120.7073090.501540.520530.567110.652960.814991.198280.006220.066030.183180.343460.536020.75145100.501070.515500.550870.612540.715790.903921.341840.004320.049740.141100.267410.420770.594980.78436110.500770.512140.540190.587590.661800.781060.994441.486350.003120.038790.112350.214890.340110.483520.641260.80976120.500580.509760.532720.570800.627980.713430.847931.086061.631540.002330.031100.091810.177090.281630.401830.534930.678410.82992
Furthermore
αi(c(r, s)) = αi(c(s)) − αi(c(r))
=
0, i = 0, . . . , r − 3,
(n − 2)ai+1(c(r, s)), i = r − 2,
(n − 2)[ai+1(c(r, s)) − ai(c(r, s))], i = r − 1, . . . , s − 3,
−(n − 2)ai(c(r, s)), i = s − 2,
0, i = s − 1, . . . , n − 3.
(2.3.2)
The first and last zeros disappear when r ≤ 2 and s ≥ n − 1, respectively. The second and fourth rows vanish for r = 1 and s = n, respectively. The third one does not appear for the spacings, i.e. when s = r + 1. We also check that under condition r − 1 ≤ i ≤ s − 3 yields ai+1(c(r, s)) < (=, >)ai(c(r, s)) iff i < (=, >)n−32 , respectively. This implies that all the elements of the third row are non-positive and non-negative iff s ≤ n+32 and r ≥ n−12 , respectively. They are first negative and ultimately positive when r < n−12 and s > n+32 . Presenting below the bounds on the order statistics differences, we consider two cases separately. In Proposition 2, we analyze the possibilities that the difference contains at least one of the sample extremes. The remaining ones 2 ≤ r < s ≤ n − 1 are treated in Proposition 3.
Proposition 2. (i) For the sample range we have 2 − 22−n ≤ EXn:n− X1:n
∆ ≤ n
2.
The lower bound is attained by the symmetric two-point distributions. The upper one is attained in the limit by the two-point distributions such that the probability of one point decreases to 0.
(ii) If either r = 1 < s < n or 1 < r < s = n, then 0 ≤ EXs:n− Xr:n
∆ ≤ n
2.
In the former (latter) case the lower and upper bounds are attained in the limit by the two-point distributions such that the probabilities of the smaller two-points tend to 1 and 0, respectively (0 and 1, respectively).
Proof. (i) By the comments following (2.3.2), function Ξ1,n:n is first decreasing and then increasing. Due to Remark 2, this is symmetric about 12. Therefore
0≤u≤1max Ξ1,n:n(u) = Ξ1,n:n(0) = Ξ1,n:n(1) = n 2,
0≤u≤1min Ξ1,n:n(u) = Ξ1,n:n
1 2
= 2 − 22−n,
because
Ξ1,n:n(u) = Pn−1
i=1 Bi,n(u)
B1,2(u) = 1 − (1 − u)n− un 2u(1 − u) .
(ii) Suppose that r = 1 and s < n. If s ≤ n+32 , then function Ξ1,s:n is decreasing by the VDP of its derivative. We have Ξ1,s:n(0) = n2 which is the upper bound, and Ξ1,s:n(1) = 0 which is the lower one. If s > n+32 , then Ξ1,s:n is decreasing in some neighborhoods of 0 and 1, but it is also possible that it is increasing in some inner subinterval (v0, u0), say, of (0, 1). Then the right end-point of the subinterval is a possible candidate for the global maximum. Suppose that Ξ1,s:n(u0) > Ξ1,s:n(0) = n2. Therefore for some two-point distribution we have
Ξ1,s:n(u0) = EXs:n− X1:n
∆ ≤ EXn:n− X1:n
∆ ≤ n
2,
which contradicts part (i) of the proposition. Moreover, point v0 cannot provide the global minimum less than Ξ1,s:n(1) = 0 either, because
E
Xs:n− X1:n
∆ = Ξ1,s:n(v0) < Ξ1,s:n(1) = 0
is impossible for the expectation of non-negative random function Xs:n− X1:n. The proof for 1 < r < s = n is similar.
Proposition 3. Assume that 2 ≤ r < s ≤ n − 1.
(i) If either r ≥ n−12 or s ≤ n+32 , polynomial
Ξ0r,s:n(u) =
n−3
X
i=0
αi(c(r, s))Bi,n−3(u),
0 ≤ u ≤ 1, see (2.3.2) has a unique zero u1 = u1(r, s), say, in (0, 1). Also,
0 ≤ EXs:n− Xr:n
∆ ≤ Ξr,s:n(u1).
(ii) If r < n−12 and s > n+32 , the polynomial has either one zero u1 ∈ (0, 1) or three zeros u1 < v1 < u2, where u1 and u1, u2 are local maxima of Ξr,s:n in the former and latter cases, respectively. In these cases we have
0 ≤ EXs:n− Xr:n
∆ ≤ Ξr,s:n(u1), 0 ≤ EXs:n− Xr:n
∆ ≤ max{Ξr,s:n(u1), Ξr,s:n(u2)},
respectively.
The lower bounds in the inequalities of statements (i) and (ii) are attained in the limit by the two-point distributions with the contributions of one point decreasing to 0. If the maximum of Ξr,s:n(u) , 0 ≤ u ≤ 1, is attained at a single point ui, then the upper bound is attained by the two-point distribution with the probability mass of the smaller point equal to ui. If the global maximum is attained simultaneously at u1 and u2, then the bound is attained by the parent distribution functions of the form
F (x) =
0, x < x0, u1, x0 ≤ x < x1, u2, x1 ≤ x < x2, 1, x ≥ x2, for arbitrary x0 ≤ x1 ≤ x2 > x0.
Existence of two global maxima is possible, e.g. for the quasi-ranges with s = n + 1 − r. Then Ξr,s:n(u) is symmetric about 12. Moreover, for r located relatively close to 1 and far from n2, the polynomial is bimodal. Clearly, the lower and upper bounds for Xr:n− Xs:n, r < s, are the negatives of the upper and lower bounds for Xs:n− Xr:n.
Proof. Under the assumptions of point (i), the derivative Ξ0r,s:n(u) changes its sign once, from + to −. Therefore Ξr,s:n(u) is unimodal with the global maximum at the zero point of its derivative. The lower bound coincides with
Ξr,s:n(0) = Ξr,s:n(1) = a0(c(r, s)) = an−2(c(r, s)) = 0.
In case (ii), Lemma 1 asserts that possible sequence of signs of Ξ0r,s:n are either + − +− or +−. In the latter one, we can repeat the arguments of part (a). Otherwise, there are two local maxima in (0, 1), and both are appropriate candidates for the global one. Notice that the local minimum point v1 located between them cannot beat the interval endpoints in the contest for the global minimum, because it is impossible that for any non-degenerate parent distribution
EXs:n− Xr:n
∆ = Ξr,s:n(v1) < 0 = Ξr,s:n(0) = Ξr,s:n(1).
Remark 4. The upper bounds for the spacings (s = r + 1) and second spacings (s = r + 2) can be calculated analytically, because then equations Ξ0r,s:n(u) = 0 can be simplified to linear and quadratic equations, respectively. Note that the solutions to the cases r = 1 and s = n
were precisely described in Proposition 2(b). Otherwise we write
EXr+1:n− Xr:n
∆ ≤ Ξr,r+1:n r − 1
n − 2
= 1
2
n r
(r − 1)r−1(n − r − 1)n−r−1 (n − 2)n−2 , EXr+2:n− Xr:n
∆ ≤ Ξr,r+2:n(u1(r, r + 2)), where
u1(r, r + 2) = ( 1
2, if r = n−12 ,
n−3+2r(r−1)−√
a(r,n)
2(n−2)(2r−n−1) , otherwise.
and a(r, n) = [n − 3 + 2r(r − 1)]2− 4(r2− 1)(n − 2)(2r − n − 1).
Remark 5. The conclusions of Proposition 3(b) can be specified more precisely for the quasi-ranges, i.e. the differences of rth largest and smallest order statistics, 2 ≤ r ≤ n2. By Remark 2, function Ξr,n+1−r:n is symmetric about 12, and Ξ0r,n+1−r:n is antisymmetric. Hence the latter vanishes at 12. If
Ξ00r,n+1−r:n 1 2
= n!
(n − 4)!2n−4
n−r
X
i=r
n i
[(2i − n)2− n + 2] < 0, (2.3.3)
then Ξr,n+1−r:n has a local maximum at 12. Since it is positive on (0, 1), and vanishes at 0 and 1, and has either one or three local extremes in (0, 1), the latter possibility is excluded, and Ξr,n+1−r:n 12 is the global maximum. Summing up, the inequality in (2.3.3) implies
EXn+1−r:n− Xr:n
∆ ≤ Ξr,n+1−r:n 1
2
= 21−n
n−r
X
i=r
n i
, (2.3.4)
which can be also written as (2 − 22−n)Pr−1 i=0
n
i. The equality is attained by the symmetric two-point distributions.
If the inequality in (2.3.3) is reversed, then Ξr,n+1−r:n has a local minimum at 12 and two global maxima attained at two arguments 0 < u1 < 12 < u2 = 1 − u1 < 1 located symmetrically about 12. Accordingly,
E
Xn+1−r:n− Xr:n
∆ ≤ Ξr,n+1−r:n(u1) = Ξr,n+1−r:n(u2) ,
where u1 is the unique solution to Ξ0r,n+1−r:n(u)
n(n − 1)(n − 2) = Br−2,n−3(u) − Bn−r−1,n−3(u) r(n − r)
+
n−r−2
X
i=r−2
(2i + 3 − n)Bi,n−3(u)
(i + 1)(i + 2)(n − i − 1)(n − i − 2) = 0
in 0,12. This bound is attained by the families of two-point distributions where the probability of one point is u1, and symmetric three-point distributions where the probabilities of the extreme points are equal to u1, and the middle point has probability u2− u1 = 1 − 2u1.
Note that for n−
√n−2
2 ≤ r ≤ n2 we have (2i − n)2 < n − 2 for all r ≤ i ≤ n − r, and (2.3.4) follows. However, the assumption is very restrictive, and (2.3.4) holds true for much greater range of 2 ≤ r ≤ n2.
Table 2.2 contains numerical approximations of upper bounds for the differences of orders statistics Xs:n− Xr:n, 1 < r < s < n from the samples of size n = 12. Clearly, the bounds increase with respect to s and decrease with respect to r. The attainability parameters increase both in the columns and rows. The differences are hardly visible if r is small and s is large. However, if we added the extra row for r = 1 and column for s = n = 12, we would note a rapid rise of the bound value to n2 = 6. These bounds are attained in the limit by the two-valued distributions as the probability mass concentrates at one of them. It is seemingly surprising that in the same way we obtain the zero lower bounds for all the cases except for the sample range for which the strictly positive bound (here 2 − 2−10≈ 1.99902) is attained by the symmetric two-point distributions. On the other hand, the distributions provide the upper bounds for the rth quasi-ranges X13−r:n− Xr:n r = 3, 4, 5, 6. The bound for the second quasi-range X11:12− X2:12 is the unique among Xs:12− Xr:12, 1 ≤ r < s ≤ 12, which was determined by maximizing a bimodal function (the other ones were unimodal). Since the function is symmetric as well its maximum was attained at two arguments. This implies that this bound is attained by three families of distributions: one symmetric three-point, and two two-valued. We also observe that the bound values are symmetric about the opposite diagonal of the Table, and the respective probability mass parameters sum up to 1. This is an implication of the analytic identity Ξr,s:n(u) = Ξn+1−s,n+1−r:n(1 − u).