Recurrence relations for moments of record values

ELSEVIER

journal of statistical planning and inference

Journal of Statistical Planning and Inference 45 (1995) 225-234

Recurrence relations for moments of record values Udo Kamps

Institut fi~r Statistik und Wirtschaftsmathematik, Aachen University of Technology, Wiillnerstr. 3, D-52056 Aachen, Germany Received 10 December 1992; revised 28 September 1993

Abstract Recurrence relations for moments of order statistics have been investigated extensively in the literature, and there are also a few results concerning moments of record values. We consider a model of record values based on non-identically distributed random variables, which contains order statistics and ordinary record values as special cases. For this model, we propose a unified approach to some types of recurrence relations for moments.

A M S Subject Classification: Primary 62E10; secondary 62G30, 62E15 Key words: Order statistics; Record values; kth record values; Generalized order statistics; Recurrence relations; Moments; Characterizations

1. Introduction Let X1, X2, ... be a sequence of i.i.d, r a n d o m variables with a b s o l u t e l y c o n t i n u o u s d i s t r i b u t i o n function F a n d density function f The o r d e r statistics c o r r e s p o n d i n g to X1 . . . . . X . are d e n o t e d by XI,.~< ... ~
L(n+l)=min{j>L(n);Xj>XL(.)},

n>~l,

a n d (upper) record values XL~.), n >11. L o o k i n g at the successive kth largest values in the sequence, D z i u b d z i e l a a n d K o p o c i f i s k i (1976) i n t r o d u c e the m o d e l of kth record values, which can be viewed as o r d i n a r y record values (see above: k--- 1) based on the d i s t r i b u t i o n function G ( x ) = 1 - ( 1 - F ( x ) ) k. 0378-3758/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved. SSDI 0 3 7 8 - 3 7 5 8 ( 9 4 ) 0 0 0 7 3 - 5

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Order statistics and kth record values are included in Pfeifer’s (1982a, b) model for record values from non-identically distributed random variables, where the distribution of the underlying random variables may change after each record event. In this way we can model interventions such as modifying the situation after the occurrence of a record. This paper presents a unified approach to certain types of recurrence relations for moments of ordered random variables. For our purposes, we will consider Pfeifer’s model in a restricted version.

1.1. Let a constant

m and positive

a,=k+(n-r)(m+

integers

k and n be given satisfying

1
1)3 1,

Let IXr.j)l
be a double sequence of independent X,.. j, j 3 1, are identically distributed according to F,(x)=

1 -(l-F(x))“*,

Then the interrecord dr=l, and the record

random

variables

such that

1
times are defined

by

d,+,=min{j31;X,+1,j>Xl,d,},

l
values by Xr.d, = X(r, n, m, k), say.

Throughout this paper, the random variables X( 1, n, m, k), . . . , X(n, n, m, k) will be called generalized order statistics (cf. Kamps, 1992a). In reliability theory this structure may be used in shock models. Successive underlying distributions (i.e. distributions before and after any record event) are always stochastically ordered. Depending on the choice of m we get transitions to stochastitally larger or smaller distributions. The record values X(r, n, m, k), 1 t(X(r-l,n,m,k)=s)=

t>s,

F(s)
2Grdn.

(1)

Thus, order statistics (m =O, k = 1) and kth record values (m= - 1) are covered Pfeifer’s model. We obtain that the marginal density of X(r, n, m, k), 1 d r
x
by

(2)

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227

where (Pr,. is the following density on the unit interval (cf. K a m p s , 1992a): qo,,,(x)-- c~-1 ( l _ x ) a , - 1Ym-'-X,x,tJ, (r-- 1)!

(3)

x~(0, 1),

where

cr-l=c,-x(n,k)= f l al,

l <~r~n,

i=1

and

g,,(y)=f~(1-t)mdt,

0~
E x a m p l e 1.2. In the case of order statistics and kth record values we obtain (i) for m = 0 , k = l :

fx~ .... o, 1 ) ( x ) = r ( ~ ) F ' - l ( x ) ( 1 - F ( x ) ) " - ' f ( x ) ; (ii) for m = - 1, k e N :

fx( .... _l,k~(x)=(r--1) ! log

(l--F(x))k-l f(x)

(e.g. David, 1981; Chandler, 1952; Dziubdziela and Kopociflski, 1976). The distribution function ~r,. corresponding to the density function (p,.. is given by r--1

cI)~..(x)=l--C~-l(l--x) ~ ~, (j!cr_j_l)-lgj~(x),

XE(0,1).

(4)

j=0 TO verify (4), the identities 1 - (1 - x)" + 1 = (m + 1)gin(x), m # - 1, and c,_ 1 = a, c,_ 2 are used to s i m p l i f y ~ ; , . ( x ) . Thus, for r ) 2 w e get

¢,,,(x)-¢,_,,.(x)=

C'~(1--x)aro~-I(x),

(r-l)!

Xe(0,1).

(5)

In view of

Fx~...... ,)(x)=~,,.(F(x)),

x
(6)

Eq. (4) yields a representation of the distribution function of X(r, n, m, k). Applying the pseudo-inverse function F - 1 of F, we find an expression for the expected value of any measurable function ~u of X(r, n, m, k)(subject to existence):

EtP(X(r,n,m,k))= I x T(F '(x))tp,,.(x)dx. 3o

(7)

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Finally, we give a representation for the difference of successive generalized order statistics, which can be proved via integration by parts by applying (7) and (5).

1.3. Let 0 # a s ~, and let the pseudo-inverse F - 1 be absolutely continuous such that the following regularity condition is fulfilled: F-l(t)>~O

for a l l t ~ ( 0 , 1 ) , i f c ~ > l a n d ~ ¢ N ,

F-l(t)>0

for a l l t ~ ( 0 , 1 ) , i f ~ < l and - ~ ¢ N ,

F

for all t~(0, 1) or F

a(t)>0

x(t)<0 for all tE(0, 1), if - c ~ N .

If the moments EX~(r, n, m, k) and E X ' ( r - 1, n, m, k) of generalized order statistics from F exist, then

EX~(r, n, m, k ) - E X ~ ( r - 1, n, m, k) = ~Cr-2

f~ (F l(t))~- I(F 1)'(t)(1--t)arg~m-l(t)dt,

r>~2.

Proofs of the above assertions, sufficient conditions for the existence of moments, and further results on distribution theory for generalized order statistics can be found in Kamps (1992a).

2. Recurrence relations for specific distributions There are numerous results in the literature on recurrence relations for moments of order statistics. In comparison with direct computations via explicit expressions, the application of such identities is much easier. Besides reducing the numerical effort, it may raise accuracy, in particular when calculating higher moments or when considering higher sample sizes. Moreover, recurrence relations provide some more insight into structural properties of the underlying distributions. Some results of this type for special distributions are given in David (1981), Arnold and Balakrishnan (1989) and Balakrishnan and Cohen (1991); in particular, we refer to the detailed survey article by Balakrishnan et al. (1988) and to Khan et al. (1983), where truncated distributions are considered. Usually, explicit expressions for moments are used to derive recurrence relations for special distributions. Often, the results are not as general as possible with respect to their parametrization, and similarly structured identities and relationships between several distributions remain hidden. In papers by Khan et al. (1983) and Lin (1988b) we find a more systematic treatment. Once having derived a certain representation for the difference of moments of successive order statistics, they plug in different distribution functions to obtain related recurrence relations. Moreover, characterization

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229

results are stated in Lin (1988b) for some specific distributions. This fact motivates a unified approach (cf. Kamps, 1991b). The starting point is some parameterized recurrence relation. Then a characterization result may lead to a corresponding parametrized family of distributions. Going backwards, the strong assumptions are dropped, and, under mild conditions, the relation is verified within this class of distributions. Performing these steps, we obtain families of distributions, which are related by the same type of recurrence relations for moments of order statistics. Isolated results known from the literature can be subsumed, and integrated within a general framework. Moreover, the identities can be generalized with respect to the parametrization of the underlying distribution as well as to non-integer moments, and new relations are found. Identities for kth record values appear only incidentally in the literature; namely as a by-product when characterizing equality in inequalities for moments of records (cf. Lin, 1988a; Gajek and Gather, 1991; Kamps, 1991a). By analogy with results for order statistics, some other relations are shown in Too and Lin (1989) and in Kamps (1992b). More general, the above method leads to a parametrized recurrence relation for moments of generalized order statistics based on corresponding parametrized distributions within a general family of distributions. Hence, the results for order statistics and kth record values are contained as special cases. Theorem 2.1. Let ~ , fl>~O with ~ + f l ~ 0 , and let X(r,n,m,k), X ( r - l , n , m , k ) and X ( r + p , n + q , m , k + s ) be generalized order statistics with 2<~r<~n, p,q, seT/, 1 <~r+p<<.n+q, k+s>~l. Let h be a function on (0, 1) given by ,

1

h (t)=~(1-t)(q-P)t"+l)+S-lg~(t)

a.e. on (0, 1),

d > 0 , such that the quantity (flh(t)) lip is well defined for fl>0. If the underlying distribution function of the generalized order statistics is determined by

F _ l l D = ~ e x p { h ( t ) } , fl=O, "" [(flh(t)) l/a, fl>O, with ElX'+#(r,n,m,k)l, E [ X ' + # ( r - l , n , m , k ) l , E l X ' ( r + p , n + q , m , k +s)l< ~ , then we have the followin 9 recurrence relation: E X ~ + # ( r , n , m , k ) - E X ~ + # ( r - l,n,m,k )=(c~ + fl)C EX~(r + p,n + q,m,k + s), with C-

1 cr-2(n,k) (r+p-1)! d c r + p - l ( n + q , k + s ) ( r - 1)!

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Proof. If/3=0, we have (F 1)'(t)=exp{h(t)}h'(t)=F-l(t)h'(t), and if/~>0, we find ( F - a( t ) ) p - l ( F - 1),(t)= h'(t). Thus, using 1.3 and the representation of h'(t), we obtain the desired result.

[]

Remark 2.2. The constant C in Theorem 2.1 is equal to the expectation of some difference of successive generalized order statistics (subject to existence). More precisely: If the expected values E Y(r,n,m,k) and E Y ( r - 1 , n , m , k ) of generalized order statistics with an underlying distribution function G exist, where

G-l(t)=h(t),

te(0, 1)

(with h as in Theorem 2.1), we get

C=EY(r,n,m,k)--EY(r-l,n,m,k). In the case of ordinary order statistics, i.e. choosing m = 0, k = 1 and s = 0, numerous results known from the literature are contained in Theorem 2.1 as special cases. We obtain e.g. recurrence relations for moments of order statistics from Pareto, power function, Weibull, logistic, Burr XII distributions, and reflected versions of them (see K a m p s (1991b, 1992b) for explicit examples), Putting m = - 1, we get recurrence relations for moments of kth record values (cf. Kamps, 1992b) based on Pareto, power function, Weibull and Burr XII distributions, and some others. For order statistics and fixing some parameters it is shown in K a m p s (1991b) that the recurrence relation is a characteristic property of the corresponding distribution. Working with appropriate complete sequences of functions (see e.g. Hwang and Lin, 1984; Huang, 1989; Lin, 1989), analogous characterization results can also be obtained for generalized order statistics.

3. Recurrence relations for arbitrary distributions The most important and often used recurrence relation for moments of order statistics from a continuous distribution is given by Cole (1951):

(n-r)EX~,,,+rEX~+a,,=nEX~,,_I,

l<~r<~n-1.

Using the representation in (7) this relation is seen to be valid for arbitrary distributions. Other identities for arbitrary distributions can be found in David (1981), Arnold and Balakrishnan (1989), Balakrishnan and Cohen (1991) and in the detailed review of Malik et al. (1988). The above important identity can be extended to generalized order statistics. In the special case of kth records ( m = - 1) the identity becomes trivial, however.

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L e m m a 3.1. Let 1 <~r <~n - 1. The following identities are valid for moments o f generalized order statistics (subject to their existence): (i) (k + (n - r - 1)(m + 1)) E X ' ( r , n, m, k) + r(m + 1) E X ' ( r + 1, n, m, k)

= (k + (n-- 1)(m + 1)) EX~(r, n-- 1, m, k), (ii) (k + ( n - r - 1)(m + 1)) ( E X ' ( r + 1, n, m, k ) - E X ' ( r , n, m, k)) = (k + (n - 1)(m + 1)) ( E X ' ( r + 1, n, m, k ) - E X ' ( r , n - 1, m, k)), (iii) (k + (n - 1)(m + 1)) ( E X ~(r, n, m, k ) - EX~(r, n - 1, m, k)) = r(m + 1) ( E X ' ( r , n, m, k ) - E X ' ( r + 1, n, m, k)).

Proof. We use (7) and corresponding recurrence relations for the function ¢p,.. introduced in (3). If 1 -%
1)(m+ 1))(¢p~+ x , . ( x ) - ¢p~,.(x))

= ( k Av (n - - 1 ) ( m -[- 1))( ~or + 1, n ( x ) - - q)r, n - 1 ( x ) ) ,

(iii) ( k + ( n - 1)(m+ 1))(q~,.(x)-¢p,,._ l ( x ) ) = r ( m + 1)(~0r,.(x)-¢p,+ 1,.(x)). (i) If m = - l , Noticing that

then q~,,.(x)=cp,,.-l(x) and the equation is trivial. Let m # - l .

c, = cr(n) = (k + (n - r - 1)(m + 1))c,_ a (n)

and cr(n) = (k + ( n - 1)(m + 1 ) ) c r - l ( n - l)

we have (k+(n-r-

c,(n)

- -

-(r--l)!

1)(m+ 1))tpr,.(x)+r(m+ 1)~or+ 1,.(x) (1 - x ) k + t " - ~ - a)t.,+ 1)- 1 g ~ - l ( x ) ( ( 1 _ x ) , . + 1 + (m + 1 ) g , . ( x ) )

= (k + (n - 1)(m + 1))q~,,._ l(X)(ii) and (iii) are directly obtained from (i).

[]

The above identities may also be used to modify the relations in the preceding section.

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4. Characterizations of distributions by two moments

In order to characterize distributions by means of some recurrence relation, strong assumptions are needed in general (cf. Lin, 1988b; Kamps, 1991b); the validity of the relation is required for a whole sequence of indices. Too and Lin (1989) show identities for moments of order statistics and record values, which turn out to determine certain power function and Weibull distributions without any further assumptions. We give a generalization of this result to record values X(r, n, m, k), and by this characterizations of Burr XII and Weibull distributions. Theorem 4.1. Let pc~, k+(n+2p-1)(m+l)>O, and let X(r,n,m,k), X(r + p, n + p, m, k) be generalized order statistics with finite second and first moment, respectively. Then we have

(r-- 1)! E X 2 ( r , n , m , k ) _ 2 ( r + p - 1)! c, l(n, k) cr+p-l(n+p,k) x EX(r+p,n+p,m,k)+

(r + 2 p - 1)! =0, c,+ 2p_ l(n+ 2p, k)

iff the underlying distribution function is given by ( F(x) = g£, X(xl/P)= l

1--(1 --(m + 1)xa/P)1/{m+1},

( 1 --exp{ - x l/P},

{ xe(0, 1), m > - - l , xe(0, oe), m < - - l , x6(0, oe), m = - - l .

Proof. Using (7) and

(r+p-1)! ( r - 1)! E X 2 ( r , n , m , k ) _ 2 c,_l(n,k) cr+p-l(n+p,k) x E X ( r + p , n+p, m, k)+

(r+2p-1)! cr+ 2p- l(n + 2p, k)

f£ (F - l ( t ) - g~(t)) 2 (1 -- t) k +{.-,)tin+ 1)- 1g~- l(Q dt, the assertion follows noticing g,71(x)= j" 1 - - ( 1 - ( m + l)x) 1/t"+t), 1 - e -x, with the above restrictions concerning x.

mg=-1, m = --1, []

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233

Thus, the results of Too and Lin (1989) for ordinary order statistics and record values are included in a result valid for generalized order statistics. Fixing m, each generalized order statistic X(r, n,m, k) satisfies a corresponding recurrence relation, which is parametrized with respect to r, n and k. Analogously to Too and Lin (1989), we refer to the case r = n = 1, p = 1, which leads to the equation 2

EXZ(l'l'm'k)

2

k+m+~EX(2'2'm'k)+(k+m+l)(k+2m+2)

=0"

The distributions, which are characterized by this identity, are determined by

1--(1--(m--kl)x) /(m+U, m # - 1 , Fro(x)=

l_e_X,

m=--l,

with the above restrictions concerning x. Hence, the second moment of X(1, 1,m, k) is given by

EXZ(1,1,m,k) =k

(F2,1(t))2(1-t)k-ldt=(k+m+l)(k+2m+2),

if we assume k + 2 m + 2 > 0 (for m < - l ) . gives the following corollary.

Using EX(2,2,m,k)=2/(k+2m+2),

this

Corollary 4.2. Let k + 2m + 2 > O. Then the moment equations

EX2(1, 1, m, k)=(k+m +

2 1)(k+2m+2)

and EX(2, 2, m, k ) =

k+2m+2

are valid iff the underlying distribution has the distribution function Fm.

Acknowledgement The author would like to thank the referees for their helpful comments and suggestions.

References Arnold, B.C. and N. Balakrishnan (1989). Relations, Bounds, and Approximations for Order Statistics. Springer, Berlin. Balakrishnan, N. and A.C. Cohen (1991). Order Statistics and Inference. Academic Press, Boston. Balakrishnan, N., H.J. Malik and S.E. Ahmed (1988). Recurrence relations and identities for moments of order statistics, II: Specific continuous distributions. Comm. Statist. Theory Methods 17, 2657 2694. Chandler, K.N. (1952). The distribution and frequency of record values, J. Roy. Statist. Soc. B 14, 220-228.

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Cole, R.H. (1951). Relations between moments of order statistics. Ann. Math. Statist. 22, 308-310. David, H.A. (1981). Order Statistics, 2nd ed. Wiley, New York. Dziubdziela, W. and B. Kopocifiski (1976). Limiting properties of the k-th record values. Appl. Math. 15, 187-190. Gajek, L. and U. Gather (1991). Moment inequalities for order statistics with applications to characterizations of distributions. Metrika 38, 357-367. Huang, J.S. (1989). Moment problem of order statistics: a review, lnternat. Statist. Rev. 57, 59 66. Hwang, J.S. and G.D. Lin (1984). Characterizations of distributions by linear combinations of moments of order statistics. Bull. Inst. Math. Acad. Sinica 12, 179-202. Kamps, U. (1991a). Inequalities for moments of order statistics and characterizations of distributions. J. Statist. Plann. Inference 27, 397-404. Kamps, U. (1991b). A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations. Metrika 38, 215 225. Kamps, U. (1992a). A concept of generalized order statistics. Habilitationsschrift, Aachen University of Technology. Kamps, U. (1992b). Identities for the difference of moments of successive order statistics and record values. Metron 50, 179 187. Khan, A.H., M. Yaqub and S. Parvez (1983). Recurrence relations between moments of order statistics. Naval Res. Logist. Quart. 30, 419-441. Corrigendum 32, 693 (1985). Lin, G.D. (1988a). Characterizations of uniform distributions and of exponential distributions. Sankhya Ser. A 50, 64 69. Lin, G.D. (1988b). Characterizations of distributions via relationships between two moments of order statistics. J. Statist. Plann. Inference 19, 73 80. Lin, G.D. (1989). Characterizations of distributions via moments of order statistics: a survey and comparison of methods. In: Y. Dodge, Ed., Statistical Data Analysis and lnJ~rence. North Holland, Amsterdam, 297 307. Malik, H.J., N. Balakrishnan and S.E. Ahmed (1988). Recurrence relations and identities for moments of order statistics, I: Arbitrary continuous distribution. Comm. Statist. Theory Methods 17, 2623 2655. Pfeifer, D. (1982a). Characterizations of exponential distributions by independent non-stationary record increments. J. Appl. Probab. 19, 127 135. Correction 19, 906. Pfeifer, D. (1982b). The structure of elementary pure birth processes. J. Appl. Probab. 19, 664 667. Too, Y.H. and G.D. Lin (1989). Characterizations of uniform and exponential distributions. Statist. Probab. Lett. 7, 357 359.