APPLICATIONES MATHEMATICAE 9,3 (), pp. 85 95 Erhard Cramer (Oldenurg) Udo Kamps (Oldenurg) Tomasz Rychlik (Toruń) EVALUATIONS OF EXPECTED GENERALIZED ORDER STATISTICS IN VARIOUS SCALE UNITS Astract. We present sharp upper ounds for the deviations of expected generalized order statistics from the population mean in various scale units generated y central asolute moments. No restrictions are imposed on the parameters of the generalized order statistics model. The results are derived y comining the unimodality property of the uniform generalized order statistics with the Moriguti and Hölder inequalities. They generalize evaluations for specific models of ordered oservations.. Introduction. The derivation of ounds for moments of order statistics from an iid sample has received a great attention in the literature. Classical results are the Gumel (954) and Hartley and David (954) ounds for the expected sample maximum and the Moriguti (953) ounds for aritrary expected order statistics. Some refinements and improvements were proposed, e.g., y Balakrishnan (99, 993) and Balakrishnan and Bendre (993). The Moriguti (953) method, ased on the greatest convex minorant, was utilized y Raqa (997) for kth record values and y Balakrishnan et al. () for progressively type II censored order statistics. Since these results are also ased on the Cauchy Schwarz inequality, the ounds are given in terms of the mean and standard deviation. However, instead of the Cauchy Schwarz inequality it is near at hand to use its direct generalization, the Hölder inequality with p. This leads to ounds where the measurement units are powers of asolute central moments of the underlying Mathematics Suject Classification: Primary 6G3; Secondary 6E5. Key words and phrases: generalized order statistics, unimodal distriution, Moriguti inequality, Hölder inequality, sharp ound. [85]
86 E. Cramer et al. distriution: σ p (E X µ p ) /p. In the context of order statistics, Arnold (985) applied this method to the sample maximum. A generalization to L-statistics was proposed y Rychlik (998). Arnold (985) and Rychlik (993) presented analogous evaluations for the maximum and aritrary L-statistics of dependent identically distriuted samples, respectively. Recently Raqa and Rychlik () considered p-norm ounds for record statistics. We consider generalized order statistics as a unified approach to models of ordered random variales (cf. Kamps (995)). In this paper we show that the preceding results can e seen as particular results for generalized order statistics. No restrictions are imposed on the parameters of the generalized order statistics model. The ounds follow from the fundamental oservation that densities of uniform generalized order statistics are unimodal (cf. Cramer et al. ()), which is thoroughly discussed in Section. Section 3 contains the main results of the paper.. Preliminaries. Generalized order statistics ased on some distriution function G and parameters γ,..., γ n > were introduced y Kamps (995) via the quantile transformation X r G (U r ), r n, where U,..., U n are generalized order statistics ased on the (standard) uniform distriution and the parameters γ,..., γ n. The joint density function of U,..., U n is given y f U,...,U n (u,..., u n ) k ( n )( n ) γ j ( u j ) γ j γ j+ ( u n ) γ n j on the cone {(u,... u n ) : u... u n < } R n with parameters n N, γ,..., γ n >. It was shown in Cramer and Kamps () that the marginal density of the rth uniform generalized order statistic can e written in terms of a particular Meijer G-function, i.e., ( r ) f r (t) f U r (t) γ i G r, r,r i j [ t γ,..., γ r γ,..., γ r ], t [, ). The marginal cumulative distriution function of the rth uniform generalized order statistic is denoted y F r. Hence, the cumulative distriution function of X r G (U r ) ased on some distriution function G is given y G r F r G. Since only the marginal distriution is considered in the following calculations, and this does not depend on the ordering of the parameters
Generalized order statistics 87 γ,..., γ r, we assume without loss of generality that the parameters are decreasingly ordered, i.e., γ... γ r >. Let X e a random variale with distriution function G. Susequently, we tacitly assume that the expectation µ EX and the pth asolute central moment σ p p E X µ p G (u) µ p du, p <, exist and are finite whenever they are used. Moreover, if the support of X is ounded, we denote the essential supremum of its deviation from the mean y σ ess sup X µ sup{ G (u) µ : u (, )}. In the following, the ehavior of f r at the endpoints of the interval (, ) is important. For r, the density function f (t) γ ( t) γ, t (, ), has the limits If r, we have lim f (t) γ >, t + lim f r(t), t + lim f r(t) t lim f (t) t { + if γ <, if γ, if γ >. { + if γr < or γ r γ r, f r () (, ) if γ r < γ r, if γ r >. The value of f r () in the the case γ r < γ r can e calculated explicitly (see Cramer and Kamps ()). Susequently, we denote the limits of f r at the endpoints of (, ) y f r () and f r (), respectively. The following theorem is fundamental in deriving ounds on expected generalized order statistics (cf. Cramer et al. ()). Proposition. The density function of each uniform generalized order statistic is unimodal. For r, the density function is strictly increasing, constant and strictly decreasing for γ <,, and >, respectively. For r the following result holds: If γ r, then the density function is strictly increasing. Otherwise it is strictly unimodal with a mode in (, ). Four cases are distinguished in the following considerations: (A) f r, i.e., r, γ ; (B) f r is strictly decreasing, i.e., r, γ > ; (C) f r is strictly increasing with
88 E. Cramer et al. (i) f r () <, i.e., r, γ r > γ r ; (ii) f r (), i.e., r, γ r < or r, γ r γ r ; (D) f r is strictly unimodal with a mode z r (, ) and f r () f r (), i.e., r, γ r >. 3. Bounds via a comination of the Moriguti and Hölder inequalities. We aim now at deriving the projections P f r of the density functions f r onto the family of nondecreasing functions in the space L ([, ), du) of square integrale functions on the standard unit interval. Considering the preceding set-ups (A) (D) and Proposition., we come up with the following conclusions. Proposition 3. (i) P f r in cases (A) and (B); (ii) P f r f r in case (C); (iii) P f r ( ) f r (min(, )) in case (D), where < < z r is the unique solution of the equation () ( x)f r (x) F r (x), x (, ). Proof. The projection P f r of f r onto the convex cone of nondecreasing functions in L ([, ), du) is derived y means of Moriguti s greatest convex minorant method. This is the derivative of the greatest convex minorant of the antiderivative F r of the density function f r. A formal justification can e found in Rychlik (, pp. 4 6). In cases (A) and (C), f r is actually nondecreasing, and coincides with the projection P f r. In case (B), F r (t) ( t) γ is concave increasing from F r () to F r (). Therefore its greatest convex minorant is the linear function F r (t) t, t [, ], with the derivative F r P f r. The only nontrivial case is (D), where F r is strictly convex increasing on (, z r ) and strictly concave increasing on (z r, ). The greatest convex minorant F r of F r is F r (t) for t [, ], () Fr (t) F r () (t ) + for t [, ], for some z r. The case is excluded, ecause f r (), and F r lies elow the line joining (, ) and (, ) in a neighorhood of, i.e., F r (t) < t for t (, ε). Looking for, we consider the linear functions l x (t) f r (x)(t x) + F r (x), t R, tangent to F r at x (, z r ] and evaluate them for t. Since l x () is a
Generalized order statistics 89 continuous function in x, the values l x () strictly increase from l () to l zr () f r (z r )( z r ) + F r (z r ) z r f r (u) du + z r f r (z r ) du > Consequently, there exists a unique (, z r ) such that l () F r () + ( )f r () f r (u) du F r (). (cf. ()). Comining this property with () we otain the derivative { (3) P f r (t) F r(t) fr (t) for t [, ], f r () for t [, ], which has the desired representation. In consequence, we otain the Moriguti inequality (cf. Rychlik (, p. 34)) (4) EX r G (u)f r (u) du G (u)p f r (u) du. Equality holds in (4) if G is constant on each interval where the distriution function F r is greater than its greatest convex minorant. In cases (A) and (C), there is no restriction since f r P f r so that (4) holds with equality. In case (B), G is constant on (, ), which means that G is concentrated at the single point µ. In the remaining case (D), G is constant on the interval (, ), implying that G has a jump of proaility at least at the right endpoint of its support. Applying the equalities P f r(u) du and [G (u) µ] du we find for aritrary c R the inequality (5) EX r µ [G (u) µ]p f r (u) du [G (u) µ][p f r (u) c] du. Choosing c in cases (A) and (B), we can evaluate the right hand side of (4) directly: EX r µ [G (u) µ][p f r (u) ] du. We have equality in case (A), and the simple relation EX r µ holds for every distriution function G. In case (B), the ound EX r µ is attained for
9 E. Cramer et al. the one-point distriution with mass at µ. These trivial cases are excluded from the further study. In the remaining cases (C) and (D), the Hölder inequality is applied to evaluate the upper ound (5). Note that P f r has the common form (3) with under assumption (C). Moreover, P f r f r, and it suffices to evaluate the middle term in (4). The cases p, < p <, and p are considered separately in the susequent sections. (6) 3.. Case p. The relations (5) yield EX r µ [G (u) µ][p f r (u) c] du sup u (,) P f r (u) c G (u) µ du B (c)σ. The constant c is chosen so that it minimizes the factor B (c). Theorem 3.. With defined in (), we have EX r µ in case (C)(ii), sup B f r ()/ < in case (C)(i), G σ f r ()/ < in case (D). The ounds are not attained, ut there are sequences of distriutions, e.g., three-point distriutions, which attain the ounds in the limit. Proof. Since P f r is nondecreasing, P f r (), and P f r () f r () <, P f r (), and P f r () f r () < in cases (C)(i), (C)(ii), and (D), respectively, the upper ounds are immediately determined y calculating B inf B (c) inf c R c R sup u (,) P f r (u) c. We show that the ounds are attained. In case (C), for < ε < /, we define a random variale X with the distriution P ( X µ σ ε ) ( P X µ + σ ) ε, P (X µ) ε. ε It is easily checked that EX µ and E X µ σ. Moreover, EX r µ σ G (u) µ f r (u) du [ σ ε ε f r( ε) f r (ε) f r() f r (u) du B as ε. ε ] f r (u) du
P Generalized order statistics 9 In case (D), for < ε <, we choose X so that ( X µ σ ) ( ε, P (X µ) ε, P X µ + σ ) ε ( ). This distriution satisfies the moment conditions, and has a jump of height at the right end of its support. Therefore EX r µ σ G (u) µ σ P f r (u) du f r() ε ε f r (u) du f r() f r (ε) f r() B as ε. 3.. Case < p <. Applying the Hölder inequality and setting q p/(p ) we conclude that (7) EX r µ [G (u) µ][p f r (u) c] du G (u) µ p P f r c q P f r c q σ p B p (c)σ p, where the ounds B p (c) P f r c q depend on the constants c R. Theorem 3.3. Let in case (C) and let solve equation () in case (D). Then (8) sup G where B p p a EX r µ σ p B p, [f r (a) f r (u)] q du + + ( )[f r () f r (a)] q, and a (, ) is the unique solution of the equation (9) a [f r (a) f r (u)] q du a a [f r (u) f r (a)] q du [f r (u) f r (a)] q du + ( )[f r () f r (a)] q. A distriution function G that attains the ound (8) is given y ( if t α, () G(t) fr f r (a) + sgn(t µ) B ) p t µ p/q if α t < β, σ p if t β,
9 E. Cramer et al. for α µ σ p [ fr (a) B p ] q/p [ ] q/p fr () f r (a) and β µ + σ p. Proof. First we determine the constant c that minimizes the ound in (7). The function P f r is continuous nondecreasing with image [, f r ()]. It is ovious that the optimal c fulfils c [, f r ()] so that it can e represented as c f r (a) for a unique a [, ]. Define the function D p (x) B p p(f r (x)) P f r f r (x) q q Its derivative is given y D p(x) qf r(x) x x [f r (x) f r (u)] q du + x B p [f r (u) f r (x)] q du + ( )[f r () f r (x)] q, x (, ). { x [f r (x) f r (u)] q du [f r (u) f r (x)] q du ( )[f r () f r (x)] q }. Both q and f r(x) for x (, ) are positive. Moreover, the first term in races is equal to at, and it strictly increases in x. The sum of the second and third terms is negative, strictly increasing and vanishing at. Therefore there exists a unique a (, ) at which D p changes sign from minus to plus, and thus D p attains its minimum. Equivalently, the point is uniquely determined y equation (9). Now we show that () is the unique distriution function that satisfies oth the moment conditions and provides equalities in (7) with the optimal constant. Equalities in the Hölder inequality and in the moment conditions hold if G (x) µ σ p P f r(x) f r (a) q/p sgn{p f r (x) f r (a)}, x (, ). P f r f r (a) q/p q This is a nondecreasing function which defines the distriution function (). Note that in case (D), it has an atom with mass at the right endpoint of its support, and so the Moriguti inequality is attained as well. In case (C), P f r f r, and the application of the Moriguti inequality is redundant. In the particular case p q, the representation of the ounds simplifies consideraly. First of all, from P f r c P f r + (c )
Generalized order statistics 93 we deduce that c is the optimal constant. This yields B P f r P f r f r (u) du + ( )f r (). The ound is attained y the distriution function if t µ σ, B ( G(t) fr + B ) (t µ) if µ σ t < µ + σ [f r () ], σ B B () if t µ + σ B [f r () ]. 3.3. Case p. In this set-up we otain the upper ound EX r µ [G (u) µ][p f r (u) c] du sup u [,] G (u) µ with B (c) P f r c depending on the real c. P f r (u) c du B (c)σ Theorem 3.4. Let or e the solution to (), in cases (C) and (D), respectively. Then { EX r µ fr () if /, () sup B G σ F r (/) if > /. The ound is attained y, e.g., the two-point distriution defined y (3) P (X µ σ ) min{, /} ( ) min{, /} P X µ + σ. min{, /} Proof. First, we derive c y minimizing B (c) in (). An argument similar to that in the proof of Theorem 3.3 shows that the optimal c can e written as c f r (a) for some a [, ]. In order to determine the optimal a, we consider (4) D (x) x P f r (u) f r (x) du [f r (x) f r (u)] du + + ( )[f r () f r (x)] x [f r (u) f r (x)] du
94 E. Cramer et al. (x )f r (x) F r (x) + F r () + ( )f r () (x )f r (x) F r (x) +. The last relation follows from (). Furthermore D (x) (x )f r(x), x (, ). Since f r(x) is positive for every x (, ), the only zero of the derivative is a / suject to the condition that / <. Otherwise, the minimum is attained at the right endpoint of the domain, i.e., for a. Hence, a min{, /} minimizes (4) and provides the est ound in (). If / < (in case (C), in particular), then In case (D) with < /, we have B D (/) F r (/). B D () f r (), due to () again. We now claim that the ound in () is attained y the distriution (3). First we oserve that EX µ and ess sup X µ σ. Moreover, the distriution function of X has a jump of height at least at the right endpoint, and so it provides equality in the Moriguti inequality (see (5)). If /, then Otherwise EX r µ EX r µ This ends the proof. [G (u) µ][p f r (u) f r (/)] du σ P f r (u) f r (/) du D (/)σ B σ. [G (u) µ][p f r (u) f r ()] du σ P f r (u) f r () du D ()σ B σ. Acknowledgements. The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG) for supporting their joint work. The third author was also partially supported y the KBN (Polish State Committee for Scientific Research) Grant No. 5 P3A.
Generalized order statistics 95 References B. C. Arnold (985), p-norm ounds on the expectation of the maximum of possily dependent sample, J. Multivariate Anal. 7, 36 33. N. Balakrishnan (99), Improving the Hartley David Gumel ound for the mean of extreme order statistics, Statist. Proa. Lett. 9, 9 94. N. Balakrishnan (993), A simple application of inomial-negative inomial relationship in the derivation of sharp ounds for moments of order statistics ased on greatest convex minorants, iid. 8, 3 35. N. Balakrishnan and S. M. Bendre (993), Improved ounds for expectations of linear functions of order statistics, Statistics 4, 6 65. N. Balakrishnan, E. Cramer and U. Kamps (), Bounds for means and variances of progressive type II censored order statistics, Statist. Proa. Lett. 54, 3 35. E. Cramer and U. Kamps (), Marginal distriutions of sequential and generalized order statistics, sumitted. E. Cramer, U. Kamps and T. Rychlik (), Unimodality of uniform generalized order statistics, with applications to mean ounds, sumitted. E. J. Gumel (954), The maxima of the mean largest value and of the range, Ann. Math. Statist. 5, 76 84. H. O. Hartley and H. A. David (954), Universal ounds for mean range and extreme oservation, Ann. Math. Statist. 5, 85 99. U. Kamps (995), A Concept of Generalized Order Statistics, Teuner, Stuttgart. S. Moriguti (953), A modification of Schwarz s inequality, with applications to distriutions, Ann. Math. Statist. 4, 7 3. M. Z. Raqa (997), Bounds ased on greatest convex minorants for moments of record values, Statist. Proa. Lett. 36, 35 4. M. Z. Raqa and T. Rychlik (), Sharp ounds for the mean of the kth record value, Comm. Statist. Theory Methods, to appear. T. Rychlik (993), Sharp ounds for expectations of L-estimates from dependent samples, iid., 53 68; Correction: iid. 3, 35 36. T. Rychlik (998), Bounds for expectations of L-estimates, in: Order Statistics: Theory & Methods, N. Balakrishnan and C. R. Rao (eds.), Handook of Statistics Vol. 6, North-Holland, Amsterdam, 5 45. T. Rychlik (), Projecting Statistical Functionals, Lecture Notes in Statist. 6, Springer, New York. Department of Mathematics University of Oldenurg 6 Oldenurg, Germany E-mail: cramer@mathematik.uni-oldenurg.de kamps@mathematik.uni-oldenurg.de Institute of Mathematics Polish Academy of Sciences Chopina 87- Toruń, Poland E-mail: trychlik@impan.gov.pl Received on 5.4.; revised version on 8.7. (65)