Characterizations of probability distributions via bivariate regression of record values

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1 Metrika (2008) 68:51 64 DOI /s Characterizations of probability distribtions via bivariate regression of record vales George P. Yanev M. Ahsanllah M. I. Beg Received: 4 October 2006 / Pblished online: 17 Jly 2007 Springer-Verlag 2007 Abstract Bairamov et al. (Ast N Z J Stat 47: , 2005) characterize the exponential distribtion in terms of the regression of a fnction of a record vale with its adjacent record vales as covariates. We extend these reslts to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means. Keywords Characterization Non-adjacent record vales Exponential distribtion 1 Introdction and main reslts Let X 1, X 2,... be independent copies of a random variable X whose distribtion fnction is denoted by F. There is a nmber of stdies on characterizations of F by means of regression relations of a fnction of one record vale on one or two other record vales. For a recent paper on the sbject we refer to Pakes (2004) (see also G. P. Yanev (B) Department of Mathematics and Statistics, University of Soth Florida, Tampa, FL 33620, USA gyanev@cas.sf.ed M. Ahsanllah Department of Management Sciences, Rider University, Lawrenceville, NJ 08648, USA ahsan@rider.ed M. I. Beg Department of Mathematics and Statistics, University of Hyderabad, Hyderabad , India m_i_beg@yahoo.com

2 52 G. P. Yanev et al. Ahsanllah and Raqab 2006, Chapt. 6). Denote pper record times by L(1) 1 and, for n > 1, L(n) min{ j : j > L(n 1) and X j > X L(n 1) }, and the corresponding pper record vale by X (n) X L(n) ;seenevzorov (2001). Gpta and Ahsanllah (2004) stdy the characterization of F by means of the eqation E[ψ(X (n)) X (n k) z ϕ(z), for k 1 and k 2, where the fnctions ψ and ϕ satisfy certain reglarity conditions. Bairamov et al. (2005) consider a characterization of the exponential distribtion in terms of the regression on two adjacent record vales E[ψ(X (n)) X (n 1), X (n + 1) v (l F < <v<r F ), where l F inf{x : F(x) >0} and r F sp{x : F(x) <1} are the left and right extremities of F, respectively. The aim of this paper is to extend the reslts given in Bairamov et al. (2005) by stdying characterizations of F in terms of the regression on two non-adjacent record vales E[ψ(X (n)) X (n k), X (n + r) v (l F < <v<r F ), where 1 k n 1 and r 1. Frther on, for a given fnction h, we adopt the notation h(v) h() M(,v), i M j (,v) i+ j ( ) h(v) h() v i v j ( v), (1) v as well as i M(,v)and M j (,v)for the ith and jth partial derivative of M(,v)with respect to and v, respectively. Bairamov et al. (2005) characterize the exponential distribtion as follows. Theorem (Bairamov et al. 2005). Sppose F is absoltely continos with density f, that h is continos in [l F, r F and continosly differentiable in (l F, r F ), and that almost everywhere in this open interval h (x) M(l F, x). (2) Then E[h (X (n)) X (n 1), X (n + 1) v M(,v) (l F < <v<r F ) (3) holds if and only if l F >,r F and F(x) 1 e c(x l F ) (x l F ), where c > 0 is an arbitrary constant.

3 Characterizations via bivariate regression of record vales 53 Next reslt is a generalization of the above theorem to the case of regression on a pair of non-adjacent record vales. Namely, (3) is extended for 1 k n 1 and r 1to E[h (k+r 1) (X (n)) X (n k), X (n + r) v (k + r 1)! (k 1)!(r 1)! r 1 M k 1 (,v) (l F < <v<r F ). (4) We consider two cases with respect to the spacings from the right record vale in the condition as follows: X (n) is one spacing away (there is a gap of size one); or X (n) is two or more spacings away (there is a gap of size two or more). The techniqes of the proofs in these two cases differ. Theorem 1 Sppose F is absoltely continos and h is continos in [l F, r F and h (k+r 1) (x) is continos in (l F, r F ) for 1 k n 1 and r 1. A. Let r 1, 1 k n 1 and Then (4) holds if and only if M k (l F,v) 0 (l F <v<r F ). (5) F(x) 1 e c(x l F ) (x l F ) and l F >, r F, (6) where c > 0 is an arbitrary constant. B. Let r 2, 2 k n 1 and r M k 1 (l F,v) 0(l F <v<r F ). Then both (4) and E[h (k+r 1) (X (n)) X (n k + 1) 2, X (n + r) v (k + r 2)! (k 2)!(r 1)! r 1 M k 2 ( 2,v) (l F < < 2 <v<r F ), (7) where M (,v)[h (v) h ()/(v ), hold if and only if (6) is tre. Remarks (i) If k r 1, then Theorem 1A coincides with Theorem 1 in Bairamov et al. (2005) becase, sing (9) below, the assmption (2) can be written as M 1 (l F,v) 0. (ii) The statement in Theorem 1A holds tre when k 1 and r 1 as well [with r M(l F,v) 0 instead of (5) and can be proved along the same lines, differentiating with respect to instead of v. The following reslt is an extension of Theorem 2 in Bairamov et al. (2005) to regression on a pair of non-adjacent covariates. As we will see in the next section, it follows from Theorem 1 choosing h(x) x k+r /(k + r)!. Theorem 2 A. Let r 1. For1 k n 1 E[X (n) X (n k), X (n + r) v r + kv k + 1 (l F < <v<r F ) (8) if and only if the continos r.v. X has the exponential distribtion (6).

4 54 G. P. Yanev et al. B. If r 2 and 2 k n 1, then both (8) and for l F < < 2 <v<r F E[X (n) X (n k + 1) 2, X (n + r) v r 2 + (k 1)v r + k 1 hold if and only if the continos r.v. X has the exponential distribtion (6). The proofs of Theorems 1 and 2 are given in Sect. 2. In Sect. 3 we consider monotone transformations which extend the reslts in the previos sections to a more general setting. Illstrations are given in terms of characterizations involving arithmetic, geometric, and harmonic means. 2 Proofs of Theorems 1 and 2 Frther on we will need some recrrent relations for the derivatives of M(,v)given in the following lemma. Lemma 1 Let h(x) be a given fnction and for integer i, j 1 define M(,v), i M(,v),M j (,v), and i M j (,v) as in (1). Ifh(x) has a continos derivative of order max{i, j} over the interval (a, b), then for a < <v<b M j (,v) h( j) (v) jm j 1 (,v), j M(,v) j j 1M(,v) h ( j) () v v and i M j (,v) i i 1M j (,v) j i M j 1 (,v), (10) v where M 1 (,v) and 1 M(,v) are given in (9) and 1 M 1 (,v) (M 1 (,v) 1M(, v))/(v ). Proof It is not difficlt to prove (9) by indction. We will proceed with the proof of (10). One can check (10) fori 1, 2 and j 1, 2. Assme that (10) holds for some (i, j). Fixing i we shall prove it for (i, j + 1), i.e., i 1M j+1 (,v) ( j + 1) i M j (,v) (v ) i M j+1 (,v) Indeed, sing the indction assmption, we have i i 1 M j+1 (,v) ( j + 1) i M j (,v) [ ii 1 M j (,v) j i M j 1 (,v) v i M j (,v) [ (v ) i M j (,v) i M j (,v) v (v ) i M j+1 (,v). Similarly, assming (10) for an i and fixed bt arbitrary j, one can prove that it holds for (i + 1, j). The lemma is proved. (9)

5 Characterizations via bivariate regression of record vales 55 Denote R(x) ln(1 F(x)). Using the Markov dependence of record vales, one can show (e.g., Ahsanllah 2004) that the conditional density of X (n) given X (n k) (1 k n 1) and X (n + r) v(r 1) is (k + r 1)! (k 1)!(r 1)! [ R(t) R() R(v) R() k 1 [ R(v) R(t) r 1 R (t) R(v) R() R(v) R(), (11) where < t <v. We will need the following lemma, which is of independent interest as well. Lemma 2 Let h(x) be a continos in [l F, r F fnction sch that h (k+r 1) (x) is continos in (l F, r F ) for 1 k n 1 and r 1. If F(x) 1 e c(x l F ) (x l F ) and l F >, r F, where c > 0 is an arbitrary constant, then E[h (k+r 1) (X (n)) X (n k), X (n + r) v (k + r 1)! (k 1)!(r 1)! r 1 M k 1 (,v) (l F < <v<r F ). (12) Proof It is not difficlt to verify (12)fork r 1. Let s prove it for r 1 and any 1 k n 1, i.e., E[h (k) (X (n)) X (n k), X (n + 1) v km k 1 (,v) (13) for 1 k n 1. Assming that (13) istrefork i (1 i n 1), we will prove it for k i + 1. Indeed, making se of (11) with R(x) c(x l F ) and the indction assmption, we obtain E[h (i+1) (X (n)) X (n i 1), X (n + 1) v i + 1 (v ) i+1 h (i+1) (t)(t ) i dt i + 1 h (i) (v ) i+1 (v)(v ) i i h (i) (t)(t ) i 1 dt i + 1 (v ) [ i+1 h (i) (v)(v ) i (v ) i E[h (i) (X (n)) X (n i), X (n + 1) v i + 1 [ (v ) i+1 h (i) (v)(v ) i i(v ) i M i 1 (,v) i + 1 [ h (i) (v) im i 1 (,v) v (i + 1)M i (,v),

6 56 G. P. Yanev et al. where the last eqality follows from (9). This proves (13)fork i + 1 and ths (12) is tre for r 1 and any 1 k n 1. Similarly one can prove (12) fork 1 and any r 1, i.e., E[h (r) (X (n)) X (n 1), X (n + r) v r r 1 M(,v) (14) To complete the proof of the lemma we need to show (12) forr 2 and 2 k n 1. Let s assme (12) forr j and any 2 k n 1. We will prove it for r j + 1 and any 2 k n 1. Since the left-hand side of (12) forr j + 1is E[h (k+ j) (X (n)) X (n k), X (n + j + 1) v (k + j)!(v ) (k+ j) (k 1)! j! h (k+ j) (t)(t ) k 1 (v t) r dt, to prove (12) forr j + 1 we need to show that for 2 k n 1 I (k, j + 1) h (k+ j) (t)(t ) k 1 (v t) j dt (v ) k+ j j M k 1 (,v) (15) nder the indction assmption that for any 2 k n 1 I (k, j) h (k+ j 1) (t)(t ) k 1 (v t) j 1 dt (v ) k+ j 1 j 1M k 1 (,v) (16) Integrating by parts, we have for 2 k n 1 I (k, j) 1 j h (k+ j 1) (t)(t ) k 1 (v t) j 1 dt + k 1 j h (k+ j) (t)(t ) k 1 (v t) j dt h (k+ j 1) (t)(t ) k 1 (v t) j dt

7 Characterizations via bivariate regression of record vales 57 Let (x) n be the falling factorial, i.e., (x) n x(x 1) (x n + 1) (n 1) and (x) 0 1. After iterating, we have for 2 k n 1 I (k, j + 1) ji(k, j) (k 1)I (k 1, j + 1) k 2 j ( 1) i (k 1) (i) I (k i, j) i0 Observe that (14) and (9) lead to +( 1) k 1 (k 1) (k 1) h ( j+1) (t)(v t) j dt h ( j) ()(v ) j + j h ( j+1) (t)(v t) j dt (17) h ( j) (t)(v t) j 1 dt (v ) j { } h ( j) ()+E[h ( j) (X (n)) X (n 1), X (n + j) v (v ) j [ h ( j) () + j j 1 M(,v) (v ) j [ h ( j) () + (v ) j M(,v)+ h ( j) () (v ) j+1 j M(,v) (18) Using the indction assmption (16) and (18) we write (17) as k 2 I (k, j + 1) (v ) j+1 j ( 1) i (k 1) (i) (v ) k 2 i j 1M k 1 i (,v) i0 Now, applying (10) and iterating, we obtain ( 1) k 2 (k 1) (k 1) j M(,v) (19) k 3 I (k, j + 1) (v ) j+1 j ( 1) i (k 1) (i) (v ) k 2 i j 1M k 1 i (,v) i0 +( 1) k 2 [ (k 1) (k 2) j j 1 M 1 (,v) j M(,v) k 3 j ( 1) i (k 1) (i) (v ) k 2 i j 1M k 1 i (,v) i0 ( 1) k 3 (k 1) (k 2) (v ) j M 1 (,v) k 4 j ( 1) i (k 1) (i) (v ) k 2 i j 1M k 1 i (,v) i0 +( 1) k 3 (k 1) (k 3) (v ) [ j j 1 M 2 (,v) 2 j M 1 (,v)

8 58 G. P. Yanev et al. k 4 j ( 1) i (k 1) (i) (v ) k 2 i j 1M k 1 i (,v) i0 ( 1) k 4 (k 1) (k 3) (v ) 2 j 1M 2 (,v)... j (v ) k 2 j 1M k 1 (k 1)(v ) k 2 j 1M k 2 (,v) (v ) k 1 j 1M k 1 (,v). This implies (15). Similarly assming (12) fork i (2 i n 2) and any r 2 one can prove it for k i + 1 and r 2. The lemma is proved. 2.1 Proof of Theorem 1A Assme (4). Setting r 1in(11), we obtain from (4) M k 1 (,v)[r(v) R() k h (k) (t)[r(t) R() k 1 R (t)dt. Letting l + F and noting that the integrand is continos and lim l + F R() lim l + F ( ln(1 F())) 0 we simplify to M k 1 (l F,v)[R(v) k h (k) (t)[r(t) k 1 R (t)dt. l F Differentiating both sides of the above eqation with respect to v, we obtain km k 1 (l F,v)[R(v) k 1 R (v) + M k (l F,v)[R(v) k h (k) (v)[r(v) k 1 R (v) Rearranging and taking into accont (9), R (v) R(v) M k (l F,v) h (k) (v) km k 1 (l F,v) M k (l F,v) (v l F )M k (l F,v) 1 v l F, (provided that M k (l F,v) 0) and hence (6) holds. It follows that l F > and c > 0, and the continity of F implies that r F. The converse statement follows from Lemma 2. The proof is complete.

9 Characterizations via bivariate regression of record vales Proof of Theorem 1B Assme both (4) and (7) are tre. Formla (11) together with (4)imply r 1M k 1 (,v)[r(v) R() k+r 1 h (k+r 1) (t) [R(v) R(t) r 1 [R(t) R() k 1 R (t)dt. Since the integrand is continos, differentiating both sides of the above eqation with respect to, we obtain r M k 1 (,v)[r(v) R() k+r 1 (k + r 1) [R(v) R() k+r 2 R () r 1 M k 1 (,v) (k 1)R () On the other hand, (7) and (11) lead to h (k+r 1) (t) [R(v) R(t) r 1 [R(t) R() k 2 R (t)dt. (20) r 1M k 2 ( 2,v)[R(v) R( 2 ) k+r 2 2 h (k+r 1) (t) [R(v) R(t) r 1 [R(t) R( 2 ) k 2 R (t)dt. (21) Therefore, letting 2 + in (21) and rearranging terms, we write (20) as R () R() R(v) r M k 1 (,v) (k 1) r 1 M k 2 (,v) (k + r 1) r 1M k 1 (,v) (22) provided that the denominator in the right-hand side is not 0 (This is eqivalent to r M k 1 (,v) 0, as we will see below). Since r 1M k 2 (,v) for the denominator in (22) wehave k+r 3 r 1 v k 2 [ h (v) h () v k+r 3 r 1 v k 2 [M 1(,v)+ 1 M(,v) r 1 M k 1 (,v)+ r M k 2 (,v), (k 1) r 1 M k 2 (,v) (k + r 1) r 1M k 1 (,v) (k 1)[ r 1 M k 1 (,v)+ r M k 2 (,v) (k + r 1) r 1 M k 1 (,v) (k 1) r M k 2 (,v) r r 1 M k 1 (,v). (23)

10 60 G. P. Yanev et al. Finally, from (22) to(23) and applying (10), we obtain R () R() R(v) r M k 1 (,v) (k 1) r M k 2 (,v) r r 1 M k 1 (,v) r M k 1 (,v) r M k 1 (,v)( v) 1 v. Integrating both sides with respect to from l F to v, we obtain ln[r(v) R(l F )ln(v l F ) + ln c (c > 0) and ths R(v) c(v l F ) and (6) follows. The converse statement in the theorem follows from Lemma Proof of Theorem 2 Let h(x) x k+r /(k + r)! and ths h (k+r 1) (x) x. We shall prove that, with this choice of h,(4) becomes E[X (n) X (n k), X (n + r) v r + kv k + r (l F < <v<r F ). (24) Indeed, 1 v k+r k+r M(,v) (k + r)! v vk+r 1 + +v k r 1 + v k 1 r + + k+r 1 (k + r)! and differentiating r 1 and k 1 times with respect to and v, we obtain (k + r 1)! (k 1)!(r 1)! r 1 M k 1 (,v) (k + r 1)! (r 1)!k!v + r!(k 1)! (r 1)!(k 1)! (k + r)! r + kv k + r, (25) which proves (24). Now, if r 1 [note that M k (l F,v) l F /(k + 1) 0 the claim in Theorem 2A follows from Theorem 1A. Similarly to (25) one can see that (7) becomes

11 Characterizations via bivariate regression of record vales 61 (k + r 2)! E[X (n) X (n k + 1) 2, X (n + r) v (k 2)!(r 1)! r 1 M k 2 ( 2,v) r 2 + (k 1)v. (26) k + r 1 Theorem 1B, (25) and (26) imply Theorem 2B. The proof is complete. 3 Monotone transformations and some particlar cases In this section, following Bairamov et al. (2005), we give a formal generalization of Theorem 1, involving a monotone transformation of X. Let Y be a random variable with distribtion fnction G. The corresponding pper record vales are denoted by Y (n). The following extension of Theorem 1A holds. The proof is similar to that of Theorem 3 in Bairamov et al. (2005) and it is omitted here. Denote for i, j 0 i M j (T (s), T (t)) i+ j ( ) h(y) h(x) x i y j xt (s), yt (t) (y x). y x Theorem 3 Sppose that (i) Y has a continos distribtion fnction G on [l G, r G ; (ii) the fnction T is continos and strictly increasing in (l G, r G ), τ T (l G+ )> and T (r G ) ; and (iii) h (k+r 1) (x) is continos in (τ, ) for 1 k n 1 and r 1. A. Let r 1 and M k (τ, T (t)) 0 (l G < t < r G ). Then for 1 k n 1 and l G < s < t < r G E[h (k) (T (Y (n))) Y (n k) s, Y (n + 1) t km k 1 (T (s), T (t)) (27) if and only if G(y) 1 e c[t (y) τ (l G < y < r G ) (28) where c > 0 is an arbitrary constant. B. Let r 2 and 2 k n 1, and r M k 1 (τ, T (t)) 0 where l G < t < r G. Then both (27) and for l G < s < s 2 < t < r G E[h (k+r 1) (T (Y (n))) Y (n k + 1) s 2, Y (n + r) t (k + r 2)! (k 2)!(r 1)! r 1 M k 2 (T (s 2), T (t)) hold if and only if (28) is tre. Remark An analog of Theorem 3 when T is a strictly decreasing fnction holds as well; for the case k r 1seeBairamov et al. (2005), Theorem 3. Different choices of fnctions h and T in the above theorem yield many characterization reslts. The corollary below gives a characterization that involves a weighted arithmetic mean.

12 62 G. P. Yanev et al. Corollary 1 Let (i) and (ii) of Theorem 3 hold. If 1 k n 1, then for a strictly increasing fnction g E[g(Y (n)) Y (n k) s, Y (n + 1) t kg(t) + g(s) k + 1 (l G < s < t < r G ) holds if and only if G(y) 1 e c[g(y) g(τ) (l G < y < r G ), where c > 0 is an arbitrary constant. Proof The corollary follows from Theorems 2A and 3 setting T (y) g(y) and h(x) x k+1 /(k + 1)!. Next corollary presents characterizations involving a geometric mean as a special case. Corollary 2 Let (i) and (ii) of Theorem 3 hold. If 1 k n 1, then for a strictly decreasing fnction g and l G < s < t < r G E[g(Y (n)) Y (n k) s, Y (n + 1) t [g(t) k/(k+1) [g(s) r/(k+1) (29) holds if and only if G(y) 1 e c {[g(y) 1/(k+1) [g(τ) 1/(k+1)} (l G < y < r G ), where c > 0 is an arbitrary constant. Proof We will show that if h(x) x 1, then for j 1, 2,... M j (x, y) ( 1) j j!. (30) xyj+1 Indeed, one can check that M 1 (x, y) 1/(xy 2 ). Assming that (30) istrefor j, we will prove it for j + 1. Using (9)wehave M j+1 (x, y) 1 [ y x 1 y x ( 1) [ ( 1) j+1 ( j + 1)! ( 1) xy j+2 j+2 ( j + 1)! y j+2 ( j + 1)( 1) j j! j+1 ( j + 1)!(y x) xy j+2 and ths (30) follows by indction. Now, let s set for 1 k n 1 xy j+1 h(x) ( 1)k k!x and ths h (k) (x) 1 x k+1.

13 Characterizations via bivariate regression of record vales 63 It is not difficlt to see that, with this choice of h,(27) and (30) yield [ 1 1 E Y (n k) s, Y (n + 1) t [T (Y (n)) k+1 T (s)[t (t) k. Setting in the last eqation T (x) [g(x) 1/(k+1), leads to the statement of the corollary. It is worth noting that if k r 1, then the right-hand side in (29) is the geometric mean of g(s) and g(t). Next corollary is a characterization in terms of a harmonic mean. Corollary 3 Let (i) and (ii) of Theorem 3 hold. For a strictly decreasing fnction g, E[g(Y (n)) Y (n 1) s, Y (n + 1) t 2g(s)g(t) g(s) + g(t) (l G < s < t < r G ). holds if and only if G(y) 1 e c {[g(y) 2 [g(τ) 2} (l G < y < r G ) where c > 0 is an arbitrary constant. Proof The reslt follows from Theorem 3 setting h(x) 2x 1/2 and T (y) [g(y) 2. Finally, let s note that Theorem 3 and its corollaries yield many special cases. In particlar, one can easily adjst to or more general setting the examples given in Bairamov et al. (2005). We present here only two examples making se of Corollary 2. Example 1 (Weibll distribtion). Let l G 0,r G, and g(y) y α(k+1). Then, according to Corollary 2, Y has the Weibll distribtion with G(y) 1 exp{ cy α } if and only if for 1 k n 1 E [ [Y (n) α(k+1) Y (n k) s, Y (n + 1) t t αk s α, where 0 < s < t <. In particlar, a random variable Ỹ has the Inverse Weibll distribtion with G(y) exp{ cy 1/2 } if and only if E [Ỹ (n) Ỹ (n 1) s, Ỹ (n + 1) t st. Example 2 (Pareto distribtion). Let l G a > 0, r G, and g(y) [log y (k+1). Then, Y has the Pareto distribtion with G(y) 1 (a/y) c (y a) if and only if for 1 k n 1, E [ [log Y (n) (k+1) Y (n k) s, Y (n + 1) t [log t k [log s 1, where a s < t <. Note that the regression relation is independent of a.

14 64 G. P. Yanev et al. Acknowledgments The athors are gratefl to the referees for carefl reading and constrctive sggestions, which were helpfl in improving sbstantially the presentation. References Ahsanllah M (2004) Record vales theory and applications. Lanham, University Press of America, MD Ahsanllah M, Raqab M (2006) Bonds and characterizations of record statistics. Nova Science Pblishers, New York Bairamov I, Ahsanllah M, Pakes A (2005) A characterization of continos distribtions via regression on pairs of record vales. Ast N Z J Stat 47: Gpta RC, Ahsanllah M (2004) Some characterization reslts based on the conditional expectation of a fnction of non-adjacent order statistic (record vale). Ann Inst Statist Math 56: Nevzorov VB (2001) Records: mathematical theory. Providence, American Mathematical Society, RI Pakes AG (2004) Prodct integration and characterization of probability laws. J Appl Statist Sci 13:11 31

CHARACTERIZATIONS OF EXPONENTIAL DISTRIBUTION VIA CONDITIONAL EXPECTATIONS OF RECORD VALUES. George P. Yanev

Pliska Std. Math. Blgar. 2 (211), 233 242 STUDIA MATHEMATICA BULGARICA CHARACTERIZATIONS OF EXPONENTIAL DISTRIBUTION VIA CONDITIONAL EXPECTATIONS OF RECORD VALUES George P. Yanev We prove that the exponential