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Sharp Bounds for Generalized Order Statistics via Logarithmic Moments a
M. Kaluszka & A. Okolewski
a
a
Institute of Mathematics , Technical University of Lodz , Lodz, Poland Published online: 02 Sep 2006.
To cite this article: M. Kaluszka & A. Okolewski (2005) Sharp Bounds for Generalized Order Statistics via Logarithmic Moments, Communications in Statistics - Theory and Methods, 34:9-10, 1911-1923, DOI: 10.1080/03610920500200790 To link to this article: http://dx.doi.org/10.1080/03610920500200790
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Communications in Statistics—Theory and Methods, 34: 1911–1923, 2005 Copyright © Taylor & Francis, Inc. ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610920500200790
Order Statistics
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Sharp Bounds for Generalized Order Statistics via Logarithmic Moments M. KALUSZKA AND A. OKOLEWSKI Institute of Mathematics, Technical University of Lodz, Lodz, Poland We present sharp bounds for expectations of generalized order statistics with random indices. The bounds are expressed in terms of logarithmic moments EX a log max1 Xb of the underlying observation X. They are attainable and provide characterizations of some non trivial distributions. No restrictions are imposed on the parameters of the generalized order statistics model. Keywords Generalized order statistics; Lomax distribution; Moriguti’s inequality; Order statistics; Pareto distribution; Progressive Type II censored order statistics; Records. Mathematics Subject Classification 62G30; 62H10.
1. Introduction Distribution-free bounds for moments of order statistics from an i.i.d. sample were first derived by Hartley and David (1954), Gumbel (1954), Moriguti (1953), and Ludwig (1960). The bounds are expressed in terms of the mean and standard deviation of the underlying distribution and they provide characterizations of some distributions. Analogous evaluations for records and kth records were established by Nagaraja (1978), Grudzien´ and Szynal (1985), and Raqab (1997). Extensions of these results to progressive Type II censored order statistics and generalized order statistics are given in Balakrishnan et al. (2001) and Kamps (1995). In Gajek and Gather (1991), p-norm bounds for order and record statistics were determined. The bounds for generalized order statistics based on inequalities of Diaz and Metcalf and Pólya and Szegö were derived by Kamps (1995). In Gajek and Okolewski (2000a), some evaluations for generalized order statistics were obtained by the approach which is equivalent to the combination of the Moriguti inequality and the Steffensen inequality. The results proved by the Steffensen inequality alone are given in Gajek and Okolewski (2000b). In the case of restricted families of distributions there are known improvements of Moriguti-type bounds for order Received June 2, 2004; Accepted April 15, 2005 Address correspondence to M. Kaluszka, Institute of Mathematics, Technical University of Lodz, ul. Wolczanska 215, 93-005 Lodz, Poland; E-mail:
[email protected]
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and record statistics determined by applying projections of elements of functional Hilbert spaces onto convex cones (see Rychlik, 2001). A summary of known bounds for generalized order statistics is presented in Kamps (1995). The results for order and record statistics are presented, e.g., in David and Nagaraja (2003), Arnold and Balakrishnan (1989), Arnold et al. (1998), and Rychlik (1998, 2001). The majority of the mentioned bounds is not universal in the sense that they require the existence of some moments although it is not necessary for the existence of the evaluated quantities. For example, the Grudzien´ and Szynal (1985) bounds for expected record values EYrk hold on condition that the observations have finite variance. Nagaraja (1978) showed, however, that the existence of EYr1 is equivalent to the existence of the logarithmic moment EXlog Xr−1 of 1 the underlying observation X. A universal bound for the expectation EY2 was established in Kaluszka and Okolewski (2003). Our attempts to determine such bounds for EYr1 r ≥ 3 expressed in EXlog Xr−1 , were not successful. It appears, however, that some universal bounds can be formulated in terms of the logarithmic moments of the form EXlogb+ X or more generally, EX a logb+ X where a b ≥ 0 and logb+ x = log max1 xb One can easily check that for non negative random variables the existence of EX a logb X is equivalent to the existence of EX a logb+ X Such moments are used in different contexts. For example, the assumption that EXlogb+ X < is used, e.g., in the Zygmund multivariate ergodic theorem and in the Hardy and Littlewood, and Wiener moment inequalities (see Kallenberg, 2002, Theorem 10.12 and Proposition 10.10). The Doob’s fundamental “maximal inequality in L1 ” for submartingales makes use of the moment EXlog+ X for X ≥ 0 (see Shiryaev, 1996, Theorem 2, p. 493). Similar moments are employed to ensure the convergence of some series and to develop strong laws of large numbers. i Grinceviˇcius (1974) showed that the series i=1 Xi k=1 Yk where Xi and Yk are sequences of i.i.d. random variables, is convergent with probability one if and only if Elog+ X1 < . Martikainen (1995) proved the strong law of large numbers for sums of parwise independent identically distributed random variables with the finite expectation EX1 a logb+ X1 Logarithmic moments are also related to autoregression processes and convergence of conditional expected values. The aim of this article is to derive sharp bounds for expectations of generalized order statistics Xr n m ˜ k (see Definition 2.1 below) with random parameters r and n In the particular case of order statistics, random r and n appear naturally, e.g., in the stochastic scenario of relaxation (see Jurlewicz and Weron, 2002). The bounds are expressed in terms of the logarithmic moments of the underlying distribution and are universal for record values Yr1 They provide characterizations of some distributions, which enable, e.g., constructing goodnessof-fit tests for given distribution types (see Morris and Szynal, 2001, 2002). They can also be utilized to show the existence of the generalized order statistic expectations, which is crucial for developing strong laws of large numbers. Another possible application is evaluating the asymptotic behavior of the moments of generalized order statistics and their functions, e.g., the rate of grow in change of some parameter (see Kaluszka and Okolewski, 2003, Example 7). Furthermore, the bounds for EXr n m ˜ k combined with, e.g., Markov’s inequality, lead to estimates for probabilities of some events. Eventually, the bounds afford possibilities for constructing confidence intervals for moments of generalized order statistics. For example, suppose that 0 TX1 Xn is a 1 − confidence interval for EtX and that the following inequality holds: EXr n m ˜ k ≤ EtX + drk m.
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Then the interval 0 TX1 Xn + drk m is at least a 1 − confidence interval for EXr n m ˜ k One can similarly construct tests regarding moments of generalized order statistics.
2. The Result Let X X1 X2 be i.i.d. random variables with a common distribution function F . Let Xr n denote the rth order statistic from the sample X1 Xn and let Yrk r = 1 2 be the kth record statistics, i.e.,
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Yrk = XLk r Lk r+k−1 r = 1 2 k = 1 2 where Lk 1 = 1, Lk r + 1 = minj XLk r Lk r+k−1 < Xj j+k−1 for r = 1 2 (cf. Dziubdziela and Kopocinski, 1976). Define the quantile function F −1 t = ´ infs ∈ R Fs ≥ t t ∈ 0 1 The generalized order statistics are defined by Kamps (1995) as follows. Definition 2.1. Let n ∈ N k > 0 m ˜ = m1 mn−1 ∈ Rn−1 be parameters such n−1 that r = k + n − r + j=r mj > 0 for all r ∈ 1 n If the random variables Ur n m ˜ k r = 1 n possess a joint density function of the form f
U1nmkUnn ˜ mk ˜
n−1 n−1 mi u1 un = k j 1 − ui 1 − un k−1 j=1
i=1
on the cone 0 ≤ u1 ≤ · · · ≤ un < 1 of Rn then they are called uniform generalized order statistics. The random variables ˜ k r = 1 n Xr n m ˜ k = F −1 Ur n m are called generalized order statistics based on the distribution function F . If m1 = · · · = mn−1 = m say, the random variables Xr n m ˜ k are denoted by Xr n m k In the case of m = 0 and k = 1 the Xr n m k reduces to the Xr n from the sample X1 Xn for continuous F m = −1, and k ∈ N we obtain Yrk based on the sequence X1 X2 while for absolutely continuous F mi = Ri where Ri ∈ 0 1 2 i = 1 2 n are such that R1 + · · · + Rn + n = M and k = M − n−1 ˜ k recovers the progressive censored i=1 mi − n + 1 the Xr n m RM type II order statistic Xr n with the censored scheme R = R1 Rn (see e.g., Balakrishnan and Aggarwala, 2000; Balakrishnan et al., 2001). Denote = r n ∈ N × N 1 ≤ r ≤ n and m ˜ = m1n mn−1n ∈ Rn−1 n ≥ 2 Let R N be a random vector with values in independent of Xr n m ˜ k r n ∈ Random parameters R and N are the natural ones for both order and record statistics. Define
t =
rn∈
rn tprn
(1)
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where prn = PR = r N = n r n ∈ and
rn t =
t r 0
j=1
1 − s j Gr0 rr
1 r ds t ∈ 0 1 1 − 1 r − 1
(2)
Gr0 rr denotes a particular Meijers G-function (e.g., see Mathai, 1993). Note that rn ( respectively) is the distribution function of Ur n m ˜ k (UR N m ˜ k). Let and be the greatest convex minorant and the smallest concave majorant of and let and ¯ be the right-hand side derivatives of and respectively. Recall that X stands for a random variable with the distribution function F .
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Proposition 2.1. Suppose that PX ≥ 0 = 1 a ≥ 1 b > 1 and c d > 0. If t = 0 a.e., then EXR N m ˜ k ≤ c−1 da−1 EX a logb+ dX + uc d
(3)
where uc d =
b + a − 1 log gct 1 1 tgct dt d 0 b + a log gct
(4)
and gy > 1 is the unique solution of the equation y = g a−1 log gb−1 b + a log g y > 0
(5)
Equality in (3) is attained if and only if ct = dF −1 ta−1 logdF −1 tb−1 b + a logdF −1 t 0 < t < 1
(6)
Proof. It was shown in Kamps (1995) and Cramer et al. (2002) that the expected value of Xr n m ˜ k can be represented as follows: EXr n m ˜ k =
0
1
F −1 td rn t
where rn is given by (2). By Fubini’s theorem we have EXR N m ˜ k =
rn∈
1 0
F −1 td rn tprn =
1
F −1 td t
0
with defined by (1). We shall now apply Moriguti’s lemma. Let us recall it for completeness of the presentation. Lemma 2.1 (Moriguti, 1953). Let , and a b → R be continuous, non decreasing functions such that a = a = a, b = b = b and t ≤
t ≤ t for every t ∈ a b. Then the following inequalities hold
b
b (i) a xtd t ≤ a xtd t,
b
b (ii) a xtd t ≥ a xtd t
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for any non decreasing function x a b → R for which the corresponding integrals exist. The equality in (i) holds if and only if x is constant on each connected interval from the set t ∈ a b t < t. The equality in (ii) holds if and only if x is constant on each connected interval from the set t ∈ a b t > t. By the lemma we get
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EXR N m ˜ k ≤
1
F −1 ttdt
(7)
0
One can easily check that the function 0 ≤ x → xa logb+ x is convex, and strictly convex for x ≥ 1 Applying the inequality fx ≥ fg + f gx − g to the function fx = xa logb+ x we get xa logb+ x ≥ g a log gb + g a−1 log gb−1 b + a log gx − g
(8)
for any g > 1 Equality occurs in (8) iff x = g The function gy given by (5) is well defined because 1 < g → g a log gb b + a log g is strictly increasing. Of course, g0+ = 1 g = , and gy > 1 for every y > 0 By the definition of g and (8), for arbitrary x ≥ 0 and y > 0 xy ≤ xa logb+ x + ygy
b + a − 1 log gy b + a log gy
(9)
The equality is attainable iff gy = x Putting x = dF −1 t, y = ct with c d > 0 integrating both sides of (9) and dividing by c and d we obtain
1 0
F −1 ttdt ≤ cd−1
1 0
+ cd−1
dF −1 ta logb+ dF −1 tdt
1
ctgct 0
b + a − 1 log gct b + a log gct
dt
(10)
Combining (10) with (7) completes the proof. Remark 2.1. Explicit formulae for g defined by (5) can be determined in a few special cases only. For example, if a = 1 and b = 2 then gy = exp y + 1 − 1, while for a = 1 and b = 3 gy is given by the Cardano formula (e.g., see Varadarajan, 1998). Remark 2.2. The coefficients uc d defined by (4) can be derived numerically provided that they are finite. The finiteness of uc d is crucial for establishing the existence of the evaluated moments. We now give an upper bound for uc d. Of course, log gct > 0 so uc d ≤
1 1 tgctdt d 0
(11)
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Observe that for a > 1, g a−1 log gb−1 b + a log g ≥ ag a−1 log gb = a ≥a
b a−1
b
g
a−1 b
b a−1
b
g
a−1 b
log g
a−1 b
b
b −1
Therefore
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gy ≤
a−1 b
b b b1 a−1 y 1 +1 ≤ Ay a−1 + B a
where A and B are some constants; the second inequality is satisfied since x + 1p ≤ 2maxp1 xp + 1 for all x p > 0 Combining this with (11) we get uc d ≤
1 1 1 tC0 t a−1 + D0 dt d 0
(12)
where C0 and D0 are some constants. Another evaluation for uc d can be derived if a = 1 Then log gb−1 b + log g ≥ log gb so 1
gy ≤ expy b and consequently, uc d ≤
1 1 1 t expct b dt d 0
(13)
Bounds (12) and (13) may be used to show that uc d < . Proposition 2.2. Suppose that PX ≥ 0 = 1 a ≥ 1 and c d > 0 Then EXR N m ˜ k ≤ c−1 da−1 EX a log+ dX − c−1 EX + uc d
(14)
1 + a − 1 log gct 1 1 1 uc d = t + gct dt d 0 c 1 + a log gct
(15)
where
and gy ≥ 1 is the solution of the equation y = g a−1 1 + a log g − 1 y ≥ 0
(16)
Equality in (14) is attained if and only if ct + 1 = dF −1 ta−1 1 + a logdF −1 t 0 < t < 1
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Proof. We only prove an auxiliary inequality which is a counterpart of (8). By the convexity of 0 ≤ x → xa log+ x for a ≥ 1 we have xa log+ x ≥ g a log g + g a−1 1 + a log gx − g g ≥ 1
(17)
The equality holds iff x = g Let g be the unique solution of Eq. (16). From (17) we get xy ≤ xa log+ x − x + y + 1gy
1 + a − 1 log gy x y ≥ 0 1 + a log gy
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with the equality iff x = gy. The rest of the proof runs as before.
(18)
Remark 2.3. For a > 1 the function g defined by (16) can be written as 1 a−1 a−1 1 W y + 1 exp − gy = exp a−1 a a a where W denotes the Lambert function, i.e., y = z expz iff z = Wy (e.g., see Corless et al., 1996). Proposition 2.3. Suppose that PX > 0 = 1 a ∈ 0 1/2 and c d > 0 If 1− < 1/c then EXR N m ˜ k ≤ c−1 d−1 dEX − da EX a log+ dX + E1 − dX+ + uc d (19) where uc d =
0
1
gctct − 1
1 + a − 1 log gct 1 + a log gct
dt
(20)
and gy ≥ 1 is the solution of the equation y = 1 − g a−1 a log g + 1 y ∈ 0 1
(21)
Equality in (19) is attained if and only if 1 − ct = dF −1 ta−1 a logdF −1 t + 1, 0 < t < 1 Proof. We restrict our attention to the proof of a basic inequality. Define la x = xa log+ x − 1 − x+ , where z+ = max0 z and 0 < a ≤ 1/2 It is easy to check that the function 0 ≤ x → la x is concave. Therefore for any g ≥ 1 la x ≤ g a log g + g a−1 a log g + 1x − g and the equality holds iff x = g. Let g be the unique solution of the equation y = 1 − g a−1 a log g + 1 y ∈ 0 1. Then xy ≤ x − la x + g a y log gy − gy + ygy = x − la x + gyy − 1
1 + a − 1 log gy 1 + a log gy
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The expectation EXR N m ˜ k can also be evaluated via logarithmic moments EX a logb 1 + X, where logb x = log xb . Observe that EX a logb 1 + X < iff EX a logb+ X < since the functions xa logb 1 + x and xa logb+ x are continuous on 0 2 and xa logb+ x ≤ xa logb 1 + x ≤ 2b xa logb+ x x ≥ 2 Proposition 2.4. Suppose that PX ≥ 0 = 1 a b ≥ 1 and c > 0. Then EXR N m ˜ k ≤
1 EX a logb 1 + X + uc c
(22)
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where uc =
1 0
1 tgct − gcta logb 1 + gct dt c
(23)
and gy ≥ 0 is the solution of the equation bg y = g a−1 logb−1 1 + g a log1 + g + y ≥ 0 1+g
(24)
Equality in (22) is attained if and only if −1
ct = F t
a−1
log
b−1
bF −1 t −1 1 + F t a log1 + F t + 1 + F −1 t −1
(25)
for t ∈ 0 1. Proof. Since 0 ≤ x → xa logb 1 + x a b ≥ 1 is strictly convex, we have for every g≥0 bg xa logb 1 + x ≥ g a logb 1 + g + g a−1 logb−1 1 + x a log1 + g + x − g 1+g with the equality iff x = g If g is the unique solution of (24), then xy ≤ xa logb 1 + x + ygy − gya logb 1 + gy
(26)
The equality holds iff x = gy Proposition 2.5. Suppose that PX > 0 = 1 c > 0 and a b ∈ 0 1 are such that a + b < 1. If t ¯ = 0 a.e., then EXR N m ˜ k ≥
1 EX a logb 1 + X + uc c
(27)
where uc is defined by (23) with replaced by ¯ and gy > 0 is the solution of Eq. (24) for y > 0 Equality in (27) is attained if and only if (25) with replaced by ¯ holds.
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Proof. For a b ∈ 0 1 such that a + b < 1 the function 0 < x → xa logb 1 + x is strictly concave. The rest of the proof is similar to that of Proposition 2.4. Proposition 2.6. Suppose that PX + c > 0 = 1 c ∈ R If t ¯ = 0 a.e., then 1 EXR N m ˜ k ≥ exp E logX + c + log tdt ¯ −c
(28)
0
with the equality if and only if F −1 t = d/t ¯ − c 0 < t < 1 for some constant d > 0
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Proof. Combining lower Moriguti’s inequality with Jensen’s inequality we get EXR N m ˜ k ≥
1 0
F −1 t + ctdt ¯ −c
1 0
tdt ¯
1
explogF −1 t + c + log tdt ¯ −c 1 ≥ exp logF −1 t + c + log tdt ¯ − c
=
0
0
Remark 2.4. The bounds given in Propositions 2.1–2.5 are not generalizations of the results of Kaluszka and Okolewski (2003).
3. Examples In this section we rewrite particular bounds for order statistics, records, and progressive censored Type II order statistics, and present the distributions characterized by the corresponding attainability conditions. 3.1. Order Statistics Choose m = 0 and k = 1. If PR = 1 N = n = 1 then t ¯ = n1 − tn−1 and bound (28) takes the form EX1 n ≥ n expE logX + c − n + 1 − c where n ≥ 2 c ∈ R. The bound is attained for the shifted Lomax distribution (e.g., see Childs et al., 2001) Ft = 1 −
d nc + t
1 n−1
t∈
d − c d > 0 n
Let PR = N = 1 and let N have the Poisson distribution with mean . Conventionally, we set X0 0 = 0 Then t = e−1−t . By (3), EXN N ≤ c−1 da−1 EX a logb+ dX + uc d
(29)
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in which a ≥ 1 b > 1 c d > 0 and uc d is given by (4). Equality occurs in (29) iff dta−1 1 b−1 log dtb + a logdt Ft = 1 + log c for t ∈ d−1 gc exp− d−1 gc, where gy is the unique solution of (5). From (19) we obtain
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EXN N ≤ c−1 d−1 dEX − da EX a log+ dX + E1 − dX+ + uc d
(30)
where a ∈ 0 1/2 c < 1/ d > 0 and uc d is defined by (20). Equality holds in (30) iff 1 1 − dta−1 1 + a logdt Ft = 1 + log c for t ∈ d−1 gc exp− d−1 gc, where gy is the solution of (21). Now we consider bounds for EX1 N . The first-order statistic from a sample of random length is used by Jurlewicz and Weron (2002) to model relaxation phenomenon. Namely, the distribution of the relaxation time of the entire system is determined by the first passage of the system from its initial state, so P ≥ t = Pmin1N0 NN0 ≥ t where N0 denotes the system size, N is a random number of dipoles taking essentially part in the relaxation process, and variables iN0 i = 1 N represent the random waiting times of the particular responding dipoles for the initial state transition. There are several theories concerning limit distribution of (see Jurlewicz and Weron, 2002). Propositions 2.1–2.6 provide some evaluations on expected relaxation time expressed in terms of logarithmic moments. Suppose that PR = 1 = 1 Let N have the binomial distribution with p ∈ 0 1, M ≥ 2 Then t ¯ = pM 1 − ptM−1 , and bound (27) can be rewritten as EX1 N ≥
1 EX a logb 1 + X + uc c
in which c > 0 a b ∈ 0 1 are such that a + b < 1 and uc is defined by (23) with replaced by ¯ The equality holds iff Ft =
a−1 1 1 t logb−1 1 + ta log1 + t + bt/1 + t M−1 1− p cpM
for t ∈ d−1 gcpM d−1 gcpM1 − pM−1 , where gy is the solution of Eq. (24) for y > 0 If N has the Poisson distribution with mean then t ¯ = e−t , and bound (28) takes the form EX1 N ≥ exp E logX + c − − c (31) 2 where c ∈ R Equality is attained in (31) iff X has the shifted Pareto distribution c + t d d 1 t∈ − c e − c d > 0 Ft = log d
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If N has the geometrical distribution with mean 1/p, i.e., PN = n = p1 − pn−1 n = 1 2 p ∈ 0 1, then t ¯ = p/1 − 1 − p1 − t2 From (28) we get EX1 N ≥ p expE logX + c + 2p log p/1 − p + 2 − c with the equality iff F is the shifted Pareto distribution function 1 d pc + t 2 1 t ∈ dp − c − c 1− Ft = 1 − 1−p d p
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in which c ∈ R d > 0 3.2. Record Statistics Set m = −1 and k = 1 If R = r with probability one, then t = − log1 − tr−1 Applying (22) we have EYr1 ≤
1 r−1!
1 EX a logb 1 + X + uc c
where r ≥ 2 a b ≥ 1 c > 0 and uc is defined by (23). The bound is attained iff Ft = 1 − exp
−
1 r−1 r − 1! a−1 bt t logb−1 1 + t a log1 + t + c 1+t
for t ≥ 0 Let R have the geometrical distribution with mean 1/p Then t = p/1 − t1−p . Using (3) we obtain EYR ≤ c−1 da−1 EX a logb+ dX + uc d 1
(32)
in which a ≥ 1 b > 1 c d > 0 and uc d is given by (4). Equality holds in (32) iff Ft = 1 −
cp
1 1−p
dta−1 logb−1 dtb + a logdt
for t ∈ d−1 gcp , where gy is the unique solution of (5). 3.3. Progressive Censored Type II Order Statistics Set mi = Ri where Ri ∈ 0 1 2 i = 1 2 n are such that R1 + · · · + Rn + n = M and k = M − n−1 i=1 mi − n + 1 Let PR = N = n = 1 for fixed n ≤ M Suppose that n = 1 Then (see Balakrishnan et al., 2001) t = cn−1
n i=1
ain 1 − t i −1
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where cn−1 =
n j=1
j air =
n
1 − i j=1j=i j
RM The bound (14) for EXn n with a = 1 takes the form
RM ≤ c−1 EX log+ dX − c−1 EX + uc d EXn n
(33)
in which c d > 0 and uc d is defined by (15). Equality is attained in (33) iff n 1 i −1 t ∈ 0 1 ain 1 − t F t = exp ccn−1 d i=1
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−1
Note that describing the distribution via the quantile function simplifies generating pseudo-random samples from this distribution.
Acknowledgment The authors would like to thank the referees for their valuable comments which led to improvements in the article.
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