Order statistics from overlapping samples: bivariate densities and regression properties

arXiv:1903.08047v1 [math.PR] 19 Mar 2019 ORDER STATISTICS FROM OVERLAPPING SAMPLES: BIVARIATE DENSITIES AND REGRESSION PROPERTIES F. L´ OPEZ-BL´ AZQUEZ1, NAN-CHENG SU2AND JACEK WESO LOWSKI3 1Departamento Estadistica e Investigati´on Operativa, Universidad de Sevilla, Spain 2Department of Statistics, National Taipei University, Taiwan 3Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland Abstract. In this paper we are interested in the joint distribution of two order statistics from overlapping samples. We give an explicit formula for the distribution of such a pair of random variables under the assumption that the parent distribution is absolutely continuous (with respect to the Lebesgue measure on the real line). The distribution is identified through the form of the density with respect to a measure which is a sum of the bivariate Lebesgue measure on R2and the univariate Lebesgue measure on the diagonal {(x, x) : x∈R}. We are also interested in the question to what extent conditional expectation of one of such order statistic given another determines the parent distribution. In particular, we provide a new characterization by linearity of regression of an order statistic from the extended sample given the one from the original sample, special case of which solves a problem explicitly stated in the literature. It appears that to describe the correct parent distribution it is convenient to use quantile density functions. In several other cases of regressions of order statistics we provide new results regarding uniqueness of the distribution in the sample. Nevertheless the general question of identifiability of the parent distribution by regression of order statistics from overlapping samples remains open. Keywords: characterization; order statistics; overlapping samples; linearity of regression; uniqueness theorem. MSC 2010 Subject Classifications: Primary: 60E05; secondary: 62E10 1. Introduction Properties of order statistics (os’s) X1:n≤X2:n≤ ··· ≤ Xn:nbased on the sample {X1, X2,···, Xn}of independent and identically distributed (iid) random variables with absolutely continuous distribution are widely known, see e.g. the monographs David and Nagaraja (2003) or Arnold, Balakrishnan and Nagaraja (2008) for excellent reviews. Much less is known for os’s which arise from different samples which have common elements. There are two special cases which until now have been studied in the literature: (1) moving os’s, when the subsequent samples are of the same size and have the same size of the overlap - see e.g. Inagaki (1980), David and Rogers (1983), Ishida and Kvedaras (2015) or Balakrishnan and Tan (2016); (let us mention that moving samples have a long history in quality control and time series analysis - in particular, the moving median is a simple robust estimator of location and the moving range is a current measure of dispersion complementing the moving average); (2) special cases of os’s from the original and extended sample which except the original sample contains a number of additional observations - see e.g. Siddiqui (1970), Tryfos and Blackmore (1985), Ahsanullah and Nevzorov (2000) or L´opez-Bl´azquez and Salamanca-Mi˜no (2014). E-mail address:2Corresponding author; [email protected]; Tel:886-2-86741111 ext 66778. Date: March 20, 2019. 1 2 In the latter paper the authors introduced a reference measure νwith respect to which the joint distribution of (Xn−k+1:n, Xn−k+2:n+1) (which they were interested in) has a density. This measure, ν, defined by (1) ν(B) = µ2(B) + µ1(π(B)), B ∈ B(R2), where π(B) = {x∈R: (x, x)∈B}and µiis the Lebesgue measure in Ri,i= 1,2, will be of special interest for us here since νwill serve as the reference measure for bivariate densities of os’s from overlapping samples. In this paper we consider iid random variables X1, X2,... with the cumulative distribution function (cdf) denoted by F, its tail denoted by ¯ F:= 1 −Fand the density with respect to µ1denoted by f. Let ∅ 6=A⊂ {1,2,...}be such that nA:= |A|<∞, where |A|denotes the number of elements in A. By Xi:Adenote the ith os from the sample {Xk, k ∈A},i= 1,...,nA. In case A={1,...,n}we have Xi:A=Xi:n,i= 1,...,n. Consider additionally ∅ 6=B⊂ {1,2,...}such that nB:= |B|<∞. Our aim is to study the joint distribution of (Xi:A, Xj:B), i= 1,...,nA,j= 1,...,nB. Of course, when A∩B=∅ the samples {Xk, k ∈A}and {Xk, k ∈B}are independent and the joint distribution of (Xi:A, Xj:B) is just a product of marginal distributions of Xi:Aand Xj:B. We will only consider the case when A∩B6=∅. Due to the permutation invariance of the distribution of (X1, X2,...) it suffices to take A={1,...,m} and B={r+ 1, r + 2,...,r+n}with r < m ≤n. Then we denote X(r) j:n:= Xj:B. In Section 2 we will study the joint density (with respect to the reference measure ν) of the pair (Xi:m, X(r) j:n). The case r= 0 is technically much simpler but the main idea of the approach is the same as in the general case. Therefore we first derive the joint distribution of (Xi:m, Xj:n) in Subsection 2.1 while the general case of an arbitrary r≥0 is considered in Subsection 2.2 (with some technicalities moved to Appendix). In Section 3 we are interested in regressions E(Xi:m|X(r) j:n), E(X(r) j:n|Xi:m) and related characterizations or identifiability questions. The main tools are representations of these regressions in terms of combinations of E(Xk:n+r|Xℓ:n+r), k, l ∈ {1,...,n+r}. Since for r > 0 such representations are rather complex, our considerations in this case will be restricted to the simplest cases of regressions of X1:2,X2:2 given X(1) 1:2 or given X(1) 2:2 . They are studied in Subsection 3.1. The case of r= 0 is much more tractable, though since m < n the analysis of each of two dual regressions E(Xi:m|Xj:n) and E(Xj:n|Xi:m) is quite different. In particular, Do legowski and Weso lowski (2015) (DW in the sequel) proved that (2) P(Xi:m=Xk:n) = (k−1 i−1)(n−k m−i) (n m)I{i,...,n−m+i}(k), and, consequently, obtained the following representation (3) E(Xi:m|Xj:n) = n−m+i X k=i (k−1 i−1)(n−k m−i) (n m)E(Xk:n|Xj:n). Here and everywhere below equations involving conditional expectations are understood in the P-almost sure sense. It was proved in DW with the help of (3) that the condition (4) E(Xi:m|Xj:n) = aXj:n+b, characterizes the parent distributions (exponential, Pareto and power) when j≤iand j≥n−m+i. The case of i=jhad been considered earlier in Ahsanullah and Nevzorov (2000) even for an arbitrary shape of the regression function. The characterization through condition (4) given in DW is a direct generalization of characterizations by linearity of E(Xi:n|Xj:n). Analysis of such problems has a long history - see e.g. references in DW, in particular, Ferguson (2002). In this case the complete answer was given in Dembi´nska and Weso lowski (1998) through an approach based on the integrated Cauchy functional equation (see also L´opez-Bl´azquez and Moreno-Rebollo (1997) who used instead differential 3 equations). Actually, the question of determination of the parent distribution by the (non-linear) form of regression E(Xi:n|Xj:n) for non-adjacent iand jhas not been completely resolved until now - see e.g. Bieniek and Maciag (2018). In Subsection 3.2 we investigate characterizations by linearity of regression (4) in the remaining unsloved cases, i.e. when i < j < n −m+i. In particular, we solve the easiest non-trivial open problem explicitly formulated in DW. The dual case of regressions of an os from the extended sample given an os from the original sample, i.e. E(Xj:n|Xi:m) is considered in Subsection 3.3. The main results in this subsection identify several new situations in which the shape of the regression function determines uniquely the parent distribution. Finally, some conclusions are discussed in Section 4. 2. Bivariate distribution of os’s from overlapping samples In this section we will derive joint distribution of the pair (Xi:m, X(r) j:n). This will be given through the density fXi:m, X(r) j:n with respect to the measure νintroduced in Section 1. This density will be expressed as a linear combination of densities of pairs of os’s (Xk:n+r, Xℓ:n+r), 1 ≤k, ℓ ≤n+r. The general formula is quite complicated technically as can be seen in Subsection 2.2, however the basic ideas are the same as in the simple case of r= 0 which, as a warm up, is considered first in Subsection 2.1. 2.1. Original sample and its extension - the case of r= 0.Let Rn 6=={x∈Rn:xi6=xjfor i6=j} and Rn ↑={x∈Rn:x1<···< xn}. A vector with increasingly sorted components of x= (x1,...,xn)∈ Rn 6=will be denoted by sortn(x) := (x1:n,...,xn:n)∈Rn ↑and σn(x) = τ∈ Sn(set of permutations of {1,...,n}) defined by τ(i) = jif xi:n=xj. The correspondence x∈Rn 6=↔(sortn(x), σn(x)) ∈Rn ↑×Sn is bijective. For x∈Rn 6=denote x(m):= (x1,...,xm)∈Rm 6=,m= 1,...,n. Then sortm(x(m)) = (x1:m,...,xm:m) and σm(x(m)), m= 1,...,n, are sequences of increasing lengths that keep track of the sorting up to the sequential observation of the m-th component of x. For instance, if x= (2.3,1.7,3.4,2.5,1.2) then sort1(x(1)) = (2.3), σ1(x(1)) = (1) sort2(x(2)) = (1.7,2.3), σ2(x(2)) = (21) sort3(x(3)) = (1.7,2.3,3.4), σ3(x(3)) = (213) sort4(x(4)) = (1.7,2.3,2.5,3.4), σ4(x(4)) = (2143) sort5(x(5)) = (1.2,1.7,2.3,2.5,3.4), σ5(x(5)) = (52143). Observe that for m < n,σm(x(m)) is obtained from σn(x) by deletion of m+ 1, ...,n. Given a permutation τ∈ Sn, let us denote by τ(m)∈ Smthe permutation obtained from τby deletion of the elements m+ 1,...,n. For (fixed) values i, k, m, n such that 1 ≤i≤m, 1 ≤k≤nand m≤n, let us define Ai:m;k:n={τ∈ Sn:τ(k) = τ(m)(i)} ∈ Sn. Note that for any x∈Rn 6= (5) σn(x)∈Ai:m;k:n⇔xi:m=xk:n. For instance, in the previous example, σ5(x) = (52143) ∈A2:4; 3:5 because x2:4 =x3:5 = 2.3. Since (X1,...,Xn) has absolutely continuous distribution X∈Rn 6=P-a.s. Therefore, sortn(X) and σn(X) are well defined P-a.s. In particular, sortn(X) = (X1:n,...,Xn:n) are the os’s from the sample of size n. Lemma 2.1. Random elements sortn(X)and σn(X)are independent. The result follows immediately from the fact that the distribution of (X1,...,Xn) is invariant under permutation and that ties appear with probability zero. 4 For n≥1, and k6=jwith 1 ≤j, k ≤n, it is well known that (Xk:n, Xj:n) has a density with respect to µ2. This density, denoted here by fk,j:n, see, e.g. David and Nagaraja (2003), p.12 for the explicit expression in terms of F,¯ Fand f, satisfies (6) P(Xk:n≤x, Xj:n≤y) = ZZ (−∞,x]×(−∞,y] fk,j:n(s, t)dµ2(s, t) = ZZ (−∞,x]×(−∞,y] fk,j:n(s, t)dν(s, t), where for the last equality to hold we chose a version of the density fk,j:nsatisfying fk,j:n(s, s) = 0, s∈R. We also denote the density of Xj:nby fj:nfor more simplification. If k=j, the random vector (Xk:n, Xj:n) assumes values on the diagonal of R2so that it does not have a density with respect to µ2, but it has a density with respect to νof the form fj,j:n(s, t) = fj:n(s)δs,t (with δs,t the Kronecker’s delta). Indeed, we have P(Xj:n≤x, Xj:n≤y) = P(Xj:n≤min(x, y)) = Zmin(x,y) −∞ fj:n(s)dµ1(s) =ZZ (−∞,x]×(−∞,y] fj:n(s)δs,t dν(s, t) = ZZ (−∞,x]×(−∞,y] fj,j:n(s, t)dν(s, t).(7) Theorem 2.2. For integers 1≤i≤m,1≤j≤n,m≤n, the random vector (Xi:m, Xj:n)has an absolutely continuous distribution with respect to νand the density function is of the form (8) fXi:m, Xj:n(x, y) = i+m−n X k=ik−1 i−1m−k n−i m nfk,j:n(x, y). Proof. A consequence of (5) and (2) is (9) P(σn(X)∈Ai:m;k:n) = P(Xi:m=Xk:n) = (k−1 i−1)(n−k m−i) (n m)I{i,...,n−m+i}(k). Using Lem. 2.1, (9) and expressions (6) and (7), we get P(Xi:m≤x, Xj:n≤y) = i+n−m X k=i P(Xi:m≤x, Xj:n≤y, Xi:m=Xk:n) = i+n−m X k=i P(Xk:n≤x, Xj:n≤y, σn(X)∈Ai:m;k:n) = i+n−m X k=i P(Xk:n≤x, Xj:n≤y)P(σn(X)∈Ai:m;k:n) = i+n−m X k=ik−1 i−1m−k n−i m nP(Xk:n≤x, Xj:n≤y) =ZZ (−∞,x]×(−∞,y] i+m−n X k=ik−1 i−1m−k n−i m nfk,j:n(s, t)dν(s, t), which proves the assertion.  Note that for j /∈ {i, . . . , i+n−m}the distribution (Xi:m, Xj:n) is absolutely continuous with respect to the bivariate Lebesgue measure, µ2. On the contrary, for j∈ {i, . . . , i +n−m}, it has a singular part, so that there is no density function with respect to µ2. The advantage of the measure νintroduced in 5 Section 1 is that the joint distribution of (Xi:m, Xj:n) is absolutely continuous with respect to νin any case. Formula (8) implies that conditional distribution PXi:m|Xj:n=yhas the density, with respect to the measure νydefined by νy(B) = µ1(B) + δB(y), B∈ B(R), which reads (10) fXi:m|Xj:n=y(x) = i+n−m X k=i (k−1 i−1)(n−k m−i) (n m)fXk:n|Xj:n=y(x) where fXj:n|Xj:n=y(x) = I{y}(x). Consequently, the formula for the conditional expectation of Xi:mgiven Xj:nas given in (3) follows. 2.2. Overlapping samples - the general case of r≥0.In order to derive the formula for density of (Xi:m, X(r) j:n) in the general case, r≥0, we need first to do a little bit of combinatorics of permutations, which will allow us to find the probabilities P(Xi,m =Xk:n+r, X(r) j:n=Xℓ:n+r), k, ℓ ∈ {1,...,n+r}. Consider three disjoint sets A={1,...,r},B={r+ 1, r + 2,...,r+s},C={r+s+ 1, r +s+ 2,...,r+s+t}. Denote r+s+t=nand consider the set Snof permutations of {1,...,n}. We will be interested in the subset Dof permutations from Snfor which there are exactly ielements from the set Cat the first k positions and there are exactly jelements from the set Aat the first ℓ+kpositions. That is, D={σ∈ Sn:|σ({1,...,k})∩C| =iand |σ({1,...,k+ℓ})∩A| =j}. We assume i≤min{t, k}and j≤min{r, k +ℓ}, since otherwise D=∅. Lemma 2.3. Let Dr,s,t,k,ℓ,i,j =|D|, the number of elements in D. Then (11) Dr,s,t,k,ℓ,i,j =n! (n k,ℓ)t ir jmin{j,k−i} X m=max{0,j−ℓ}j m s k−i−ms+t+m−k ℓ+m−j. Proof. We denote (a)b=a(a−1) ...(a−b+ 1), where bis positive integer, and (a)0= 1. Moreover, we follow the rule: a b= 0 if b < 0 or a < b. To obtain σ∈Dwe perform the following four steps: (1) Choose ipositions out of {1,...,k}in k iways and fill these positions with elements from Cin (t)iways. (2) For any m= 0,...,j choose mout of remaining k−ipositions in {1,...,k}in k−i mways and fill them with elements of Ain (r)mways. Remaining k−i−mpositions out of {1,...,k}fill with elements of Bin (s)k−i−mways. (3) Choose j−mpositions for elements of Afrom {k+ 1,...,k+ℓ}in ℓ j−mways and fill them with elements of Ain (r−m)j−mways. Remaining ℓ−j+mpositions out of {k+ 1,...,ℓ}fill with elements of B ∪C in (s−k+i+m+t−i)ℓ−j+m= (s+t+m−k)ℓ−j+mways. (4) The remaining n−k−ℓpositions fill with the rest of the elements of A∪B ∪C in (n−k−ℓ)! ways. Combining these four steps we get |D|=k i(t)i j X m=0 k−i m(r)m(s)k−i+mℓ j−m(r−m)j−m(s+t+m−k)ℓ−j+m!(n−k−ℓ)!. The formula (11) follows by simple transformations involving e.g. (r)m(r−m)j−m= (r)j. 6 Remark 2.1. Since the subset of permutations Das defined above can alternatively be written as D={σ∈ Sn:|σ({k+ℓ+ 1,...,n})∩A| =r−jand |σ({k+ 1,...,n})∩C| =t−i}, we have an equivalent formula for the number of elements in D: (12) |D|=Dt,s,r,n−k−ℓ,ℓ,r−j,t−i. In the next result we give explicit forms for P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r) for all possible configurations of parameters i, m, k, n, r, j, ℓ. Proposition 2.4. Let A={1,...,r},B={r+ 1,...,m}and C={m+ 1,...,n+r}. Probabilities pr,(i,m,k),(j,n,ℓ):= P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r). are non-zero only if i≤k≤i+n+r−mand j≤ℓ≤j+r. Then (i) for k < ℓ pr,(i,m,k),(j,n,ℓ)=(n−j+1)(|A|D|A|−1,|B|,|C|,k−1,ℓ−k−1,k−i,ℓ−j−1+|B|D|A|,|B|−1,|C|,k−1,ℓ−k−1,k−i,ℓ−j) (r+n−ℓ+1)(r+n)! ; (ii) for k=ℓ pr,(i,m,k),(j,n,k)=|B|D|A|,|B|−1,|C|,k−1,0,k−i,k−j (r+n)! ; (iii) for k > ℓ pr,(i,m,k),(j,n,ℓ)=(m−i+1)(|C|D|C|−1,|B|,|A|,ℓ−1,k−ℓ−1,ℓ−j,k−i−1+|B|D|C|,|B|−1,|A|,ℓ−1,k−ℓ−1,ℓ−j,k−i) (r+n−k+1)(r+n)! . Proof of Prop. 2.4, due to its computational complexity, is given in Appendix. Now we are ready to derive the formula for the density of fXi:m,X(r) j:n . The independence property given in Lem. 2.1 allows to write the density of (Xi:m, X(r) j:n) as a linear combination of densities of bivariate os’s from the sample (X1,...,Xn+r). Theorem 2.5. (13) fXi:m,X(r) j:n (x, y) = n+r X k,ℓ=1 pr,(i,m,k),(j,n,ℓ)fk,ℓ:n+r(x, y) with coefficients pr,(i,m,k),(j,n,ℓ)as given in Prop. 2.4. Proof. Note that P(Xi:m≤x, , X(r) j:n≤y) = n+r X k,ℓ=1 P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r, Xk:n+r≤x, Xℓ:n+r≤y). From the proof of Prop. 2.4 (see Appendix) it follows that the event {Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r}is a union of events of the form {Xσ(1) ≤...≤Xσ(n+r)}, where the union is with respect to permutations from special subsets of Sn+r(these subsets are different in each of three cases: k < ℓ,k=ℓand k > ℓ). By Lem. 2.1 it follows that P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r, Xk:n+r≤x, Xℓ:n+r≤y) =P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r)P(Xk:n+r≤x, Xℓ:n+r≤y). Therefore the density fXi:m,X(r) j:n of (Xi:m, X(r) j:n) with respect to the measure ν(introduced in Section 1) assumes the form fXi:m, X(r) j:n = n+r X k,ℓ=1 P(Xi:m=Xk:n+r, X(r) j:n=Xℓ:n+r)fk,ℓ:n+r. 7 Now the result follows by inserting in the above expression correct forms of probabilities P(Xi:m= Xk:n+r, X(r) j:n=Xℓ:n+r) which are given in Prop. 2.4  Below we derive joint densities (with respect to ν) of (Xi:m, X(r) j:n) in several cases of special interest. (i) Order statistics from the original and extended samples. Without any loss of generality we can assume that m≤n. Since (14) P(Xi:m=Xk:n) = P(Xi:m=Xk:n, X(0) j:n=Xj:n), Prop. 2.4 with r= 0, j=ℓ,|A|= 0, |B|=m,|C|=n−mapplies and since the left hand side of (14) does not depend on ℓwe can choose the case k=ℓ. Therefore, P(Xi:m=Xk:n) = mD0,m−1,n−m,k−1,0,k−i,0 n!=m(n−k)!(k−1)! n!n−m k−im−1 i−1=(k−1 i−1)(n−k m−i) (n m), and thus the formula for the density of (Xi:m, Xj:n) agrees with (8). (ii) Moving maxima. We consider (Xn:n, X(r) n:n). From Prop. 2.4 we get P(Xn:n=Xk:n+r, X(r) n:n=Xn+r:n+r) = (k−1 n−1) (n+r r), k =n, n + 1,...,n+r−1, P(Xn:n=Xn+r:n+r, X(r) n:n=Xn+r:n+r) = n−r n+r, P(Xn:n=Xn+r:n+r, X(r) n:n=Xℓ:n+r) = (ℓ−1 n−1) (n+r r), ℓ =n, n + 1,...,n+r−1. Consequently, Th. 2.5 gives fXn:n,X(r) n:n(x, y) =          Pn+r−1 k=n (k−1 n−1) (n+r r)fk,n+r:n+r(x, y), x < y, n−r n+rfn+r:n+r(x), x =y, Pn+r−1 k=n (k−1 n−1) (n+r r)fk,n+r:n+r(y, x), x > y. (iii) Moving minima. We consider (X1:n, X(r) 1:n). Then from Prop. 2.4 we get P(X1:n=X1:n+r, X(r) 1:n=Xℓ:n+r) = (n+r−ℓ n−1) (n+r r), ℓ = 2,3,...,r+ 1, P(X1:n=X1:n+r, X(r) 1:n=X1:n+r) = n−r n+r, P(X1:n=Xk:n+r, X(r) 1:n=X1:n+r) = (n+r−k n−1) (n+r r), k = 2,3,...,r+ 1. Consequently, Th. 2.5 gives fX1:n,X(r) 1:n(x, y) =          Pr+1 k=2 (n+r−k n−1) (n+r r)f1,k:n+r(x, y), x < y, n−r n+rf1:n+r(x), x =y, Pr+1 k=2 (n+r−k n−1) (n+r r)f1,k:n+r(y, x), x > y. (iv) Moving ith os’s. We consider (Xi:m, X(1) i:m). Then from Prop. 2.4 we get P(Xi:m=Xi:m+1, X(1) i:m=Xi:m+1) = (m−i+1)(m−i) (m+1)m, P(Xi:m=Xi:m+1, X(1) i:m=Xi+1:m+1) = i(m−i+1) (m+1)m, P(Xi:m=Xi+1:m+1, X(1) i:m=Xi:m+1) = i(m−i+1) (m+1)m, P(Xi:m=Xi+1:m+1, X(1) i:m=Xi+1:m+1) = i(i−1) (m+1)m. 8 Consequently, Th. 2.5 gives fXi:m,X(1) i:m (x, y) =      i(m−i+1) (m+1)mfi,i+1:m+1(x, y), x < y, (m−i+1)(m−i) (m+1)mfi:m+1(x) + i(i−1) (m+1)mfi+1:m+1(x), x =y, i(m−i+1) (m+1)mfi,i+1:m+1(y, x), x > y. 3. Regression of overlapping os’s From Prop. 2.4 we know that pr,(i,m,k),(j,n,l)are non-zero only if i≤k≤i+n+r−mand j≤l≤j+r. This together with (13) implies fXi:m,X(r) j:n (x, y) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l)fk,ℓ:n+r(x, y). Consequently, the conditional distribution PXi:m|X(r) j:n=yhas a density with respect to νy(dx) = µ1(dx) + δy(dx) of the form fXi:m|X(r) j:n=y(x) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l)fXk:n+r|Xℓ:n+r=y(x)fℓ:n+r(y) fj:n(y) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l) ℓ(n+r ℓ) j(n j)Fℓ−j(y)¯ Fj+r−ℓ(y)fXk:n+r|Xℓ:n+r=y(x). and the conditional distribution PX(r) j:n|Xi:m=xhas a density with respect to νx(dy) = µ1(dy) + δx(dy) of the form fX(r) j:n|Xi:m=x(y) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l)fXℓ:n+r|Xk:n+r=x(y)fk:n+r(x) fi:m(x) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l) k(n+r k) i(m i)Fk−i(x)¯ Fn+r−m−k+i(x)fXℓ:n+r|Xk:n+r=x(y). Therefore, (15) E(Xi:m|X(r) j:n=y) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l) ℓ(n+r ℓ) j(n j)Fℓ−j(y)¯ Fj+r−ℓ(y)E(Xk:n+r|Xℓ:n+r=y) and (16) E(X(r) j:n|Xi:m=x) = n+r−m+i X k=i j+r X ℓ=j pr,(i,m,k),(j,n,l) k(n+r k) i(m i)Fk−i(x)¯ Fn+r−m−k+i(x)E(Xℓ:n+r|Xk:n+r=x). That is, both regressions we are interested in are represented through rather complicated expressions (15) and (16). Thus characterizations or identifiability questions for parent distributions through the form of E(Xi:m|X(r) j:n) or E(X(r) j:n|Xi:m) seems to be a difficult task in such a general framework. Therefore we will concentrate rather on the special cases of r= 0 (when m < n) distinguishing two quite different subcases: in Subsection 3.2 we will consider characterizations by linearity of E(Xi:m|Xj:n) while in Subsection 3.3 we will study identification through E(Xj:n|Xi:m). For r > 0 we will consider only the simplest case of r= 1 and m=n= 2 in Subsection 3.1 below. 9 3.1. Identifiability through regression functions when r= 1 and m=n= 2.Here we only consider the simplest case of os’s from overlapping samples (X1, X2) and (X2, X3), that is the case of r= 1, m=n= 2. Then (15) gives (i) E(X2:2|X(1) 2:2 =y) = y 2F(y) + R∞ yxf(x)dx +1 F(y)Ry −∞ xF(x)f(x)dx, (ii) E(X1:2|X(1) 1:2 =y) = y 2¯ F(y) + Ry −∞ xf(x)dx +1 ¯ F(y)R∞ yx¯ F(x)f(x)dx, (iii) E(X1:2|X(1) 2:2 =y) = y 2¯ F(y) + 1 2F(y)Ry −∞ xf(x)dx +Ry −∞ xf(x)dx −1 F(y)Ry −∞ xF(x)f(x)dx, (iv) E(X2:2|X(1) 1:2 =y) = y 2F(y) + 1 2¯ F(y)R∞ yxf(x)dx +R∞ yxf(x)dx −1 ¯ F(y)R∞ yx¯ F(x)f(x)dx. We will show that each of these four regressions determines uniquely the parent distribution. Note that for Yi=−Xiand u=−ywe have E(X1:2|X(1) 1:2 =y) = −E(Y2:2|Y(1) 2:2 =u) and E(X1:2|X(1) 2:2 =y) = −E(Y2:2|Y(1) 1:2 =u). Consequently, (i) and (ii) as well as (iii) and (iv) above are equivalent. Theorem 3.1. Let the parent distribution be absolutely continuous distribution with the interval support (a, b). Then regression function E(X2:2|X(1) 2:2 =y)(alternatively, E(X1:2|X(1) 1:2 =y)), y∈(a, b), determines uniquely the distribution of Xif (a, b)( R. If (a, b) = Rit determines the distribution up to a shift. Proof. It suffices to consider only the case of E(X2:2|X(1) 2:2 ). Let us denote K(y) = E(X2:2|X(1) 2:2 =y), y∈(a, b). Performing integration by parts in (i) we get K(y) = y+Zb y ¯ F(x)dx −1 2F(y)Zy a F2(x)dx. Consequently, if K(y) is the same for two distribution functions Fand G, which are strictly increasing on (a, b) and thus have differentiable quantile functions QFand QG, then 2tZ1 t (1 −w)dQF(w)−Zt 0 w2dQF(w) = 2tZ1 t (1 −w)dQG(w)−Zt 0 w2dQG(w). For H:= (QF−QG)′we obtain (17) L:= 2tZ1 t (1 −w)H(w)dw =Zt 0 w2H(w)dw =: R, t ∈(0,1). Differentiating (17) with respect to ttwice we obtain 4(1 −t)H(t) + t(2 −t)H′(t) = 0, t ∈(0,1). Consequently, there exists a constant csuch that H(t) = c (2−t)2t2, t ∈(0,1). Now, assuming that c6= 0 we plug Hback into (17). After canceling cat the left hand side we get L= 2tZ1 t 1−w w2(2−w)2dw < 2tZ1 t w−2dw = 2(1 −t), while at the right hand side we have R=Zt 0 (2 −w)−2dw > t 4. Therefore, for any t∈(0,1) we have t < 8(1 −t), which is impossible for tsufficiently close to 1. Consequently, c= 0, and thus QF−QG=Bfor some constant B. It implies that either B= 0 or a=−∞ and b=∞. In the latter case for any y∈Rthere exists unique x∈Rsuch that y=QG(F(x)). Therefore G(y) = F(x) = F(QF(F(x))) = F(QG(F(x)) + B) = F(y+B). 16 Consequently, E(Xi:m+1|Xi:m=x) = x¯ F(x) + Zx −∞ tP(Xi−1:m< t|Xi:m=x)f(t)dt +E(Xi−1:mF(Xi−1:m)|Xi:m=x) =x¯ F(x) + Zx −∞ tF(t) F(x)i−1f(t)dt +Zx −∞ yF(y) (i−1)Fi−2(y) Fi−1(x)f(y)dy and this yields the assertion (iii). Finally, the assertion (iv) follows from E(Xj:n|X1:1 =x) = E(Xj:n|Xn=x) and Xj:n=Xj:n−1I{Xn>Xj:n−1}+XnI{Xj−1:n−1<Xn<Xj:n−1}+Xj−1:n−1I{Xn<Xj−1:n−1}. The proof is completed.  Our main objective in this subsection is to show that the shape of the regresion curves studied in Lem. 3.7 determine the parent distribution F. In the remaining part of this subsection we assume that the density f=F′is strictly positive on (a, b) := {x∈R: 0 < F(x)<1}with −∞ ≤ a < b ≤ ∞. First we will consider the cases (i) and (ii) of Lem. 3.7. It appears that under mild conditions E(X1:n|X1:m) and E(Xn:n|Xm:m) determine the parent cdf F. Theorem 3.8. Suppose that gis a differentiable function on (a, b). (A) If E|X1:n|<∞and (32) E(X1:n|X1:m) = g(X1:m), then (A.1) limx→a+g(x) = a; (A.2) g′is a decreasing function with limx→a+g′(x) = 1 and limx→b−g′(x) = 0; (A.3) F(x) = 1 −n−m pg′(x), for x∈(a, b). (B) If E|Xn:n|<∞and (33) E(Xn:n|Xm:m) = g(Xm:m), then (B.1) limx→b−g(x) = b; (B.2) g′is an increasing function with limx→a+g′(x) = 0 and limx→b−g′(x) = 1; (B.3) F(x) = n−m pg′(x), for x∈(a, b). Proof. Since there is an obvious duality between the two cases (it suffices to consider negative of the original observations to move between (A) and (B)) we provide only the proof of (A). From (i) of Lem. 3.7 and (32) we obtain the equation (34) x¯ Fn−m(x) + (n−m)Zx a t¯ Fn−m−1(t)f(t)dt =g(x), x ∈(a, b), then (A.1) follows easily by taking limits x→a+in both sides of (34). Differentiating (34), after elementary algebra, we get ¯ Fn−m(x) = g′(x), x ∈(a, b), then (A.2) follows from the well-known properties of a cdf and (A.3) is immediate.  17 Note that linearity of regression in (32) (or (33)) is impossible since it would lead to Fbeing constant. As an illustration of Th. 3.8 we provide two examples: •if either E(X1:n|X1:m=x) = 1−(1 −x)n−m+1 n−m+ 1 ,0≤x≤1, or E(Xn:n|Xm:m=x) = xn−m+1 +n−m n−m+ 1 ,0≤x≤1. then the parent distribution is uniform on (0,1); •if E(X1:n|X1:m=x) = 1−exp(−(n−m)x) n−m, x ≥0, then the parent distribution is standard (i.e. with mean 1) exponential. Now we will consider case (iii) of Lem. 3.7. Theorem 3.9. Suppose E|Xi:m+1|<∞for certain integers 2≤i≤mand let hbe a differentiable function on (a, b). If E(Xi:m+1|Xi:m) = Xi:m−h(Xi:m). Then (35) F(x) = h−1 i−1(x) h−1 i−1(b−) + 1 i−1Zb x h−i i−1(t)dt , x ∈(a, b). Proof. Using Lem. 3.7(iii) we have xF(x) + i Fi−1(x)Zx a tFi−1(t)f(t)dt =x−h(x), x ∈(a, b) and, after simple algebra, Fi−1(x)h(x) = xF i(x)−iZx a tFi−1(t)f(t)dt, x ∈(a, b). Integrating by parts (observe that if a=−∞then limx→a+xFi(x) = 0 due to the hypothesis E|Xi:m+1|< ∞and Lem. 3.3), (36) Fi−1(x)h(x) = Zx a Fi(t)dt, x ∈(a, b), from which we conclude that h(x)>0 for x∈(a, b). Let G(x) = Rx aFi(t)dt,x∈(a, b], so (37) F(x) = G′(x)1 i, x ∈(a, b), and, after some algebra, (36) yields G−i i−1(x)G′(x) = h−i i−1(x), x ∈(a, b). Therefore, Zb x G−i i−1(t)G′(t)dt =Zb x h−i i−1(t)dt, x ∈(a, b) or equivalently G−1 i−1(x)−G−1 i−1(b) = 1 i−1Zb x h−i i−1(t)dt, x ∈(a, b). 18 From (36) it follows that h(b−) = G(b), and thus (38) G(x) = h−1 i−1(b−) + 1 i−1Zb x h−i i−1(t)dt!−(i−1) , x ∈(a, b) (if h(b−) = ∞, which is possible, we take h(b−)−1 i−1= 0). The result follows now easily by (37).  As an illustration of Th. 3.9 we provide some examples: •if for α > 0 E(Xi:m+1 |Xi:m=x) = x−xα+1 iα+1 ,0≤x≤1, then the parent distribution is power with the cdf F(x) = xα,x∈[0,1]. •if for A > 0 and r > 0 such that ri > 1 E(Xi:m+1 |Xi:m=x) = x−(1+A(b−x))−r+1 A(ri−1) , x ≤b, then the parent distribution is (negative) Type IV Pareto distribution with the cdf F(x) = (1 + A(b−x))−r,x∈(−∞, b]; •if we specialize the above example by fixiing r= 1 we obtain characterization of cdf F(x) = 1 1+A(b−x),x≤b, by the linearity of regression E(Xi:m+1|Xi:m=x) = x−1 A(i−1), x ≤b; •if for λ > 0 E(Xi:m+1 |Xi:m=x) = x−exp(λx) iλ , x ≤0. then the parent distribution is negative exponential with the cdf F(x) = exp(λx), x∈(−∞,0]. Characterization by E(X1:1|Xj:n) (of course, X1:1 =X1) was studied in WG and Balakrishnan and Akhundov (2003) in the linear case. In case (iv) of Lem. 3.7 we will consider the dual conditional expectation E(Xj:n|X1:1). We are not able to express the parent distribution in terms of this regression function in this case. Instead we solve a more modest question of identifiability of the distribution of X1. Theorem 3.10. Suppose E|Xj:n|<∞for certain integers 1≤j≤n. Then the conditional expectation E(Xj:n|X1:1)uniquely determines the parent cdf F. Proof. Denote h(x) = E(Xj:n|X1:1 =x), x∈(a, b). From (iv) of Lem. 3.7, h(x) = Zx −∞ tfj:n−1(t)dt +xn−1 j−1Fj−1(x)¯ Fn−j(x) + Z∞ x tfj−1:n−1(t)dt, x ∈(a, b). Then differentiating the above equation with respect to xwe get (39) h′(x) = n−1 j−1Fj−1(x)¯ Fn−j(x), x ∈(a, b). Let us assume that the His not unique, that is there exist two different distribution functions Fand Gwith the same support such that His the same for Fand G. Hence F j−1 n−j(x)¯ F(x) = G j−1 n−j(x)¯ G(x), x ∈(a, b), which can be rewritten as (40) ZF(x) G(x)(j−1)t j−1 n−j−1−(n−1)t j−1 n−jdt = 0, x ∈(a, b). 19 Note that, for 0 < t < j−1 n−1,the integrand in (40) is strictly positive. Therefore F(x) = G(x) in a right neighbourhood of the left end of the support. Consequently, we have x0= sup{x≥a:F(x) = G(x)}> a and by continuity, F(x0) = G(x0). Let us prove that F(x0) = G(x0)≥j−1 n−1. Assume the opposite, F(x0) = G(x0)<j−1 n−1. Then, by continuity of Fand G, there exists ǫ > 0 such that F(x0+ǫ)<j−1 n−1and G(x0+ǫ)<j−1 n−1. Hence again the integrand in (40) is strictly positive and we get F(x0+ǫ) = G(x0+ǫ) which contradicts the definition of x0. Therefore, F(x0) = G(x0)≥j−1 n−1. Consider now an arbitrary x > x0. Since Fand Gare strictly increasing on (a, b) we see that F(x)>j−1 n−1and G(x)>j−1 n−1. But for t > j−1 n−1the integrand in (40) is strictly negative. Consequently F(x) = G(x).  Note that due to (39) the derivative of the regression function can be useful for determining the parent cdf. In particular, it follows from (39) that (a) if E(X1:n|X1:1 =x) = g(x) is differentiable then F(x) = 1 −n−1 pg′(x); (b) if E(Xn:n|X1:1 =x) = g(x) is differentiable then F(x) = n−1 pg′(x). Actually, these results are also covered by Th. 3.8 for m= 1. Finally, we use (39) to derive two new characterizations of the logistic distribution. Corollary 3.11. Assume that either E(X2:3|X1:1 =x) = 2ex 1+ex, x ∈R, or with unknown parent cdf F E(X2:3|X1:1 =x)∝F(x), x ∈R, or E(X2:3|X1:1 =x)∝¯ F(x), x ∈R. Then X1,X2and X3have the logistic distribution. Proof. In the first case (39) implies F(x)¯ F(x) = ex (1+ex)2. There are two solutions of the above quadratic equation in the unknown F(x). Only one of them, F(x) = ex 1+ex,x∈R, gives the valid (logistic) distribution function. In the remaining two cases (39) yields f∝F¯ F, i.e. we obtain a distribution from CB(1,1) family, which is the family of logistic laws - see e.g. Galambos (1991).  The last two cases in the above corollary can be easily generalized: •if E(Xj:n|X1:1 =x)∝Fs(x), s∈R, then the parent distribution belongs to CB(j−s, n −j); •if E(Xj:n|X1:1 =x)∝¯ Fs(x), s∈R, then the parent distribution belongs to CB(j−1, n−j−s+1). 4. Conclusion The aim of this paper is two-fold: (1) derivation of bivariate distribution of os’s from overlapping samples in the general overlapping scheme; (2) investigations of regression properties of os’s from overlapping samples, in particular, extension of characterizations by linearity of regression of os’s or identifiability results to the overlapping situation. Throughout the paper we assumed that the original observations are iid and their common distribution is absolutely continuous with respect to the Lebesque measure. The first task was fully resolved. Though the general formula is quite complicated, in several important special cases it gives quite transparent formulas and can be useful e.g. in studying moving order statistics or analyzing conditional structure of os’s from overlapping samples. Regarding the second task we identified new settings in which linearity of regression or the general form of the regression function characterizes 20 the parent distributions, in several other cases uniqueness results were obtained instead. However, the issue of characterizing of the parent cdf Fby using a general relation EXi:m|X(r) j:n=h(X(r) j:n), where h:R→Ris a Borel function, remains open and seems to be rather difficult to settle. Finally let us mention that in the special case of E(Xi:m|Xj:n), due to (3), the problem we studied embeds naturally in the question of characterization of the parent distribution by regression of L-statistics of the form Pn i=1 aiXi:non a single os Xj:n. Acknowledgments. The authors are greatly indebted to the referees for valuable suggestions which helped significantly in improving the final version of the paper. Support for this research was provided in part by the Ministry of Science and Technology, Taiwan, Grant No. 102-2118-M-305-003 and 104-2118- M-305-003, by Ministerio de Economia, Industria y Competitividad, Spain, Grant MTM2016-74983 and by the National Science Center, Poland, Grant 2016/21/B/ST1/00005. Appendix Proof of Prop. 2.4. (i) k < l.Note that (41) {Xi:m=Xk:n+r, X(r) j:n=Xl:n+r}=[ α∈A∪B β∈B∪C α6=β {Xi:m=Xk:n+r=Xα, X(r) j:n=Xl:n+r=Xβ} and the sets under Sat the right-hand side are pair-wise disjoint. Moreover, for any distinct α∈A∪B and β∈B∪C (42) {Xi:m=Xk:n+r=Xα, X(r) j:n=Xl:n+r=Xβ}=[ σ∈S(α,β){Xσ(1) ≤...≤Xσ(r+n)}, where S(α, β) = {σ∈ Sn+r:σ(k) = α, σ(l) = β, |σ({1,...,k−1})∩(Aα∪Bα,β)|=i−1, |σ({1, . . . , l −1}\{k})∩(Bα,β ∪Cβ)|=j−1−IB(α)}. Here and in the sequel we denote Ux1,...,xK:= U\{x1,...,xK}for any set U. Since the sets Aα,Bα,β and Cβare disjoint and Aα∪Bα,β ∪Cβ= (A∪B∪C)α,β it follows that S(α, β) = {σ∈ Sn+r:σ(k) = α, σ(l) = β, |σ({1,...,k−1})∩Cβ|=k−i, |σ({1,...,l−1}\{k})∩Aα|=l−j−1 + IB(α)}. Therefore, by Lem. 2.3 |S(α, β)|=D|A|−IA(α),|B|−IB(α)−IB(β),|C|−IC(β), k−1, l−k−1, k−i, l−j−IA(α). Note that the sets under the Ssign in (42) are pair-wise disjoint P-a.s. and each of them has probability 1/(n+r)! Therefore P(α, β) := P{Xi:m=Xk:n+r=Xα, X(r) j:n=Xl:n+r=Xβ}=|S(α,β)| (n+r)! . 21 There are four possible cases for the triplet (Aα, Bα,β, Cβ): (Aα, Bα,β, Cβ) =        (Aα, Bβ, C),if α∈A, β ∈B, (Aα, B, Cβ),if α∈A, β ∈C, (A, Bα,β, C),if α, β ∈B, (A, Bα, Cβ),if α∈B, β ∈C. That is, following (41) we obtain P{Xi:m=Xk:n+r, X(r) j:n=Xl:n+r =X α∈A, β∈B P(α, β) + X α∈A, β∈C P(α, β) + X α, β∈B P(α, β) + X α∈B, β∈C P(α, β) =|A||B||S(r,m)|+|A||C||S(r,n+r)|+|B|(|B|−1)|S(m−1,m)|+|B||C||S(m,n+r)| (r+n)! =|A||B|D|A|−1,|B|−1,|C|, k−1, l−k−1, k−i, l−j−1 (r+n)! +|A||C|D|A|−1,|B|,|C|−1, k−1, l−k−1, k−i, l−j−1 (r+n)! +|B|(|B|−1)D|A|,|B|−2,|C|, k−1, l−k−1, k−i, l−j (r+n)! +|B||C|D|A|,|B|−1,|C|−1, k−1, l−k−1, k−i, l−j (r+n)! . Denote numerators in subsequent four fractions above by I1,I2,I3and I4, respectively. Note that I1+I2=|A|[|B|D|A|−1,|B|−1,|C|, k−1, l−k−1, k−i, l−j−1+|C|D|A|−1,|B|,|C|−1, k−1, l−k−1, k−i, l−j−1] =|A|(|A|+|B|+|C|−2)! (|A|+|B|+|C|−2 k−1,l−k−1)|A|−1 l−j−1"|C| k−il−j−1 X m=1 l−j−1 m|B||B|−1 i−m−1|B|+|C|+m−k j+m−k +|C||C|−1 k−il−j−1 X m=1 l−j−1 m |B| i−m−1|B|+|C|+m−k j+m−k#. We will use several times the following elementary identity (43) ss−1 r= (s−r)s r. Applying (43) at the right hand side above we get I1+I2=|A|(|A|+|B|+|C|−2)! (|A|+|B|+|C|−2 k−1,l−k−1)|A|−1 l−j−1|C| k−i × l−j−1 X m=0 l−j−1 m |B| i−m−1|B|+|C|+m−k j+m−k(|B|+ 1 + m+|C|−k) =|A|(|B|+|C|−j+ 1) (|A|+|B|+|C|−2)! (|A|+|B|+|C|−2 k−1,l−k−1)|A|−1 l−j−1|C| k−i × l−j−1 X m=0 l−j−1 m |B| i−m−1|B|+|C|+m−k+ 1 j+m−k =|A|(|B|+|C|−j+1) |A|+|B|+|C|−l+1 D|A|−1,|B|,|C|,k−1,l−k−1,k−i,l−j−1. Similarly to I1+I2, we can also obtain a explicit form of I3+I4. Then combining the expressions for I1+I2and I3+I4we get the final formula in this case. (ii) k=l.Then (44) {Xi:m=Xk:n+r=X(r) j:n}=[ α∈B{Xi:m=Xk:n+r=X(r) j:n=Xα}=[ α∈B[ σ∈S(α){Xσ(1) ≤...≤Xσ(n+r)}, 22 where S(α) = {σ∈ S(n+r) : σ(k) = α, |σ({1,...,k−1})∩(A∪Bα)|=i−1, |σ({1,...,k−1})∩(Bα∪C)|=j−1} ={σ∈ S(n+r) : σ(k) = α, |σ({1,...,k−1})∩C|=k−i, |σ({1,...,k−1})∩A|=k−j}. By Lem. 2.3 it follows that |S(α)|=D|A|,|B|−1,|C|,k−1,0,k−i,k−j. Since the right-hand side of (44) is the union of pair-wise disjoint sets having the same probability 1/(n+r)! we get immediately the final formula in this case. (iii) k > l.Note that in this case (41) and (42) remain formally valid however this time the set S(α, β) is different: S(α, β) = {σ∈ Sn+r:σ(k) = α, σ(l) = β, |σ({1,...,l−1})∩(Bα,β ∪Cβ)|=j−1, |σ({1, . . . , k −1}\{l})∩(Aα∪Bα,β)|=i−1−IB(β)}. Consequently, S(α, β) = {σ∈ Sn+r:σ(k) = α, σ(l) = β, |σ({1,...,l−1})∩Aα|=l−j, |σ({1,...,k−1}\{l})∩Cβ)|=k−i−1 + IB(β)}. Therefore, according to Lem. 2.3 |S(α, β)|=D|Cβ|,|Bα,β|,|Aα|,l−1,k−l−1,l−j,k−i−IC(β). Thus, analogously as in Case (i) we obtain PXi:m=Xk:n+r, X(r) j:n=Xl:n+r =|B||C||S(m,n+r)| (r+n)! +|A||C||S(r,n+r)| (n+r)! +|B|(|B|−1)|S(m−1,m)| (n+r)! +|A||B||S(r,m)| (n+r)! =|B||C|D|C|−1,|B|−1,|A|, l−1, k−l−1, l−j, k−i−1 (r+n)! +|A||C|D|C|−1,|B|,|A|−1, j−1, k−l−1, l−j, k−i−1 (r+n)! +|B|(|B|−1)D|C|,|B|−2,|A|, l−1, k−l−1, l−j, k−i (r+n)! +|A||B|D|C|,|B|−1,|A|−1, l−1, k−l−1, l−j, k−i (r+n)! . This formula is the analogue of the respective one from Case (i) with the roles of |A|vs. |C|,kvs. land ivs. jbeing exchanged. 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