IDENTIFIABILITY OF THE MULTIVARIATE NORMAL BY THE MAXIMUM AND THE MINIMUM

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1 Surveys in Mathematics and its Applications ISSN (electronic), (print) Volume 5 (200), IDENTIFIABILITY OF THE MULTIVARIATE NORMAL BY THE MAXIMUM AND THE MINIMUM Arunava Mukherjea Abstract. In this paper, we have discussed theoretical problems in statistics on identification of parameters of a non-singular multi-variate normal when only either the distribution of the maximum or the distribution of the minimum is known. Introduction Let X, X 2,..., X m, where X i = (X i, X i2,..., X in ) be m independent random n- vectors each with a n-variate non-singular density in some class F. Let Y, Y 2,..., Y p be another such independent family of random n-vectors with each Y i having its n- variate non-singular density in F. Suppose that X 0 = (X 0,..., X 0n ), where X 0j = max{x ij : i m}, and Y 0 = (Y 0,..., Y 0n ), where Y 0j = max{y ij : i p}, have the same n-dimensional distribution function. The problem is to determine if m must equal p, and the distributions of {X, X 2,... X m } are simply a rearrangement of those of {Y, Y 2,..., Y p }. This problem comes up naturally in the context of a supply-demand problem in econometrics, and as far as we know, was considered first in [2], where it was solved (in the affirmative ) for the class of univariate normal distributions, and also, for the class of bivariate normal distributions with positive correlations. In [8], it was solved in the affirmative for the class of bivariate normal distributions with positive or negative correlations. However, if such distributions with zero correlations are allowed, then unique factorization of product of such distributions no longer holds, and this can be verified by considering simply four univariate normal distributions 200 Mathematics Subject Classification: 62H05; 62H0; 60E05. Keywords: Multivariate normal distributions; Identification of parameters.

2 32 A. Mukherjea F (x), F 2 (x), F 3 (y), F 4 (y) and observing that for all x, y, we have the equality: [F (x)f 3 (y)][f 2 (x)f 4 (y)] = [F (x)f 4 (y)][f 2 (x)f 3 (y)]. In [9], it was shown in the general n-variate normal case, that when the n-variate normal distributions of each X i, i m, and each Y j, j p, have positive partial correlations (that is, the off-diagonal entries of the covariance matrix are all negative), then when X o and Y o, as defined earlier, have the same distribution function, we must have m = p, and the distributions of the X i, i m, must be a permutation of those of the Y j, j m. In [7], this problem was solved in the affirmative for the general n-variate case when the covariance matrices of all the n-variate normal distributions are of the form: Σ ij = ρσ i σ j for i j. As far as we know, the general problem discussed above is still open. The maximum problem above occurs in m-component systems where the components are connected in parallel. The system lifetime is then given by max{x, X 2,..., X m }, and this is observable. There are instances where this maximum is observable, but the individual X i s are not. It may also be noted that if the distribution function of Y is F (x), and if X and X 2 are two independent random variables each with distribution F (x), then max{x, X 2 } and Y have the same distribution function. In [8], a corresponding minimum problem was discussed in the context of a probability model describing the death of an individual from one of several competing causes. Let X, X 2,..., X n be independent random variables with continuous distribution functions, Z = min{x, X 2,..., X i } and I = k iff Z = X k. If the X i have a common distribution function, then it is uniquely determined by the distribution function of Z. In [8], it was shown that when the distribution of the X i are not all the same, then the joint distribution function of the identified minimum (that is, that of (I, Z)) uniquely determines each F i (x), the distribution function of X i, i =, 2,..., n. Since max{x, X 2,..., X n } = min{ X, X 2,..., X n }, the distribution of the identified maximum also uniquely determines the distribution of the X i. Notice that if we consider n = 2 and the case where the independent random variables X and X 2 are both exponential such that their density functions are given by and f (x) =λe λx, x > 0; =0, otherwise, f 2 (x) =µe µx, x > 0; =0, otherwise. Surveys in Mathematics and its Applications 5 (200), 3 320

3 Identifiability of the multivariate normal 33 where λ and µ are both positive, then even though the joint distribution of their identified minimum (that is, that of (I, min{x, X 2 }) uniquely determines the parameters λ and µ, the min{x, X 2 }, by itself, does not identify uniquely the parameters λ and µ. Thus, one natural question comes up: when does the minimum of a n-variate random vector uniquely determine the parameters of the distribution of the random vector? In what follows, in the rest of this section, we discuss this problem. As far as we know, the problem on identification of parameters of a random vector (X, X 2,..., X n ) by the distribution of the minimum (namely, min{x i : i n}) is still unsolved even in the case of a n-variate normal vector. In the bivariate normal case, the problem was solved in the affirmative (in the natural sense) in [5] and [6] independently. The problem was considered also in the tri-variate normal case in [5] in the context of an identified minimum, and solved partially. The general minimum problem in the n-variate normal case, in the case of a common correlation, was solved in [9], and in the case of the tri-variate normal with negative correlations was solved in [2]. In the next section, we present asymptotic orders of certain tail probabilities for a multi-variate normal random vector that are useful in the context of the problems mentioned above. The purpose of this note here is to present some essential results which are useful in solving the identified minimum problem in the general tri-variate normal case. 2 Tail probabilities of a multivariate normal Let (X, X 2,..., X n ) be a n-variate normal random vector with a symmetric positive definite covariance matrix Σ such that the vector Σ = (α, α 2,..., α n ), where = (,,..., ), is positive (that is, each α i is positive). Then it was proven in [9] that as t, the tail probability P (X > t, X 2 > t,..., X n > t) is of the same (asymptotic) orders as that of where C exp( 2 t2 [Σ T ]), c = (2π) n 2 detσ (α α 2... α n )t n. Here we consider two functions f(t) and g(t) having the same order as t if =. Thus, we can state, when n = 2, the following lemma. lim t f(t) g(t) Surveys in Mathematics and its Applications 5 (200), 3 320

4 34 A. Mukherjea Lemma. Let (X, X 2 ) be a bivariate normal with zero means and variances σ 2, σ2 2 (where σ 2 σ2 2, and correlation ρ, ρ <, such that ρ < σ 2 σ. Then, we have: as t, ( P (X > t, X 2 > t) C exp 2 t2 [ σ2 + σ2 2 2ρσ ) σ 2 σ 2σ2 2 ( ], ρ2 ) where C = 2π( ρ2 ) 3 2 t 2 (σ 2 ρσ )(σ ρσ 2 ). Let us now consider the case ρ > σ 2 σ for the general bivariate normal (with zero means, for simplicity) considered in Lemma In this case, we no longer have Σ > 0, and thus, we need to use another idea. We can write where = = = P (X > t, X 2 > t) t t t P (X > t X 2 = x)f X2 (x)dx ( ) f X2 (x)dx exp y 2 ρσ x σ 2 t 2πσ ρ 2 2 σ ρ 2 dy f X2 (x)dx exp ( 2 ) z2 dz, g(x,t) 2π g(x, t) = t ρσ x σ 2 σ ρ 2. Since ρ > σ 2 σ, for t sufficiently large (for a pre-assigned positive δ), we can write: P (X > t, X 2 > t) > ( δ) It follows easily that as t, when ρ > σ 2 σ, we have: P (X > t, X 2 > t) σ [ 2 exp 2π t 2 t f X2 (x)dx. t 2 σ2 2 ] (2.) When ρ = σ 2 σ (< ) for the bivariate normal considered above, we have, similarly, for any ɛ > 0 and all sufficiently large t, [ ( ɛ)σ 2 exp 2 2π t 2 t 2 σ2 2 ] P (X > t, X 2 > t) ( + ɛ)σ 2 2π t [ exp 2 t 2 σ2 2 ] (2.2) Surveys in Mathematics and its Applications 5 (200), 3 320

5 Identifiability of the multivariate normal 35 Lemma 2. Let (X, X 2 ) be a bivariate normal with zero means, variances each, and correlation ρ, ρ <. Let α > 0, β > 0 and ρ < α β. Then we have: P (X αt, X 2 βt) [ C exp ( α 2 + β 2 )] 2ραβ 2 t2 ρ 2, as t, which is o(exp [ 2 β2 t 2] ) and also o(exp [ 2 α2 t 2] ), where C = 2πt 2 (α ρβ)(β ρα)( ρ 2 ) 3 2 α 2 β 2. Lemma 3. Let (X, X 2 ) be as in Lemma 2.2 letα > 0, β > 0, α β and ρ > α β. Then we have P (X > αt, X 2 > βt) exp [ 2 ] β2 t 2 as t. 2π βt Both Lemmas 2 and 3 follow from Lemma and equation (2.) above. In Lemma 3, if we take ρ = α β, then given ɛ > 0 and for all all sufficiently large t, [ 2 ] β2 t 2 ɛ 2 exp 2π βt P (X > αt, X 2 > βt) + ɛ exp [ 2 ] β2 t 2. 2π βt (2.3) Lemma 4. Let (X, X 2 ) be as in Lemma 2. Let α > 0, β > 0. Let ρ < min{ α β, β α }. Then as t, P (X > αt, X 2 > βt) exp [ 2 ] µ2 t 2, µ = min{α, β}. 2π µ t Proof. Notice that P (X > αt, X 2 > βt) =P (X αt, X 2 βt) + P (X αt, X 2 βt) + P (X αt, X 2 > βt) ( The first term on the right hand side is, by Lemma 2, o exp [ 2π µ t 2 µ2 t 2]) as t. The second term there can be written as P (X 2 βt) P ( X > α( t), X 2 > β( t)) P (X 2 βt) as t, and similarly, the third term is P (X αt) as t. The lemma is now clear. Surveys in Mathematics and its Applications 5 (200), 3 320

6 36 A. Mukherjea Lemma 5. Let (X, X 2 ) be as in Lemma 2. Let α < 0, β > 0, α > β. Then P (X > αt, X 2 > βt) exp [ 2 ] β2 t 2 2π βt as t Proof. Let us write: P (X > αt, X 2 > βt) =P (X 2 > βt) P ( X > ( α)t), X 2 > βt) Let t. Since the correlation of ( X, X 2 ) is ρ, it follows from Lemma 2 that when ρ < β α, ( P ( X > ( α)t, X 2 > βt) = o exp [ 2 ]) β2 t 2. Also, it follows from Lemma 3 and inequalities in (2.3) that when ρ β α, as t, P ( X > ( α)t, X 2 > βt) C(t) exp [ 2 ] α2 t 2 2π αt ( =o exp [ 2 ]) β2 t 2. where 2 C(t). The lemma now follows easily. Lemma 6. Let (X, X 2 ) be as in Lemma 2. Let α < 0. Then as t, P (X > αt, X 2 > ( α)t) exp [ 2 ] α2 t 2 2π α t Proof. It is enough to observe that P (X > αt, X 2 > ( α)t) =P (X 2 > ( α)t) P ( X > ( α)t, X 2 > ( α)t) and then Lemma 2 applies. Surveys in Mathematics and its Applications 5 (200), 3 320

7 Identifiability of the multivariate normal 37 3 The pdf of the identified minimum Let (X, X 2, X 3 ) be a tri-variate normal random vector with zero means and variances σ 2, σ2 2, σ2 3, and a non-singular covariance matrix Σ, where Σ ij = ρ ij σ i σ j, i j, Σ ii = σ 2 i. We assume, with no loss of generality, that σ 2, σ2 2, σ2 3. Let Y = min{x, X 2, X 3 }. We define the random variable I by I = i iff Y = X i, i =, 2, 3. Let F (y, i) be the joint distribution of (Y, I) such that F (y, ) = P (Y y, I = ), F (y, 2) = P (Y y, I = 2), F (y, 3) = P (Y y, I = 3). Then we have: P (Y y, I = ) =P (X y, X X 2, X X 3 ) = y P (X 2 x, X 3 x X = x )f X (x )dx Now by differentiating with respect to y, we obtain: f (y) = d F (y, ) dy =f X (y)p (X 2 y, X 3 y X = y) Notice that the conditional density of (X 2, X 3 ), given X = y, is a bivariate normal with means ρ 2 ( σ 2 σ )y, ρ 3 ( σ 3 σ )y, and variances σ2 2( ρ 2 2 ), σ3 2( ρ 3 2 ). Thus, we can write: ( ) ( ) f (y) = ( ) y ρ σ2 2 σ ρ σ3 3 ϕ P W 2 σ σ σ y, W σ 3 2 ρ2 2 σ y (3.) 3 ρ3 2 where (W 2, W 3 ) is a bivariate normal with zero means, variances each one, and ρ correlation ρ 23. = 23 ρ 2 ρ 3, and ϕ is the standard normal density. Similarly, ρ2 2 ρ 2 3 we also get the expressions for f 2 (y) and f 3 (y), where f 2 (y) = f 3 (y) = d dy [F (y, 3)] given in (3.2) and (3.3) below. We have f 2 (y) = ( ) y ϕ P σ 2 σ 2 d dy [F (y, 2)] and ( ) ( ) ρ σ 2 σ 2 ρ σ3 23 W 2 σ y, W σ 2 32 ρ2 2 σ y (3.2) 3 ρ23 2 Surveys in Mathematics and its Applications 5 (200), 3 320

8 38 A. Mukherjea ( ) ( ) f 3 (y) = ( ) y ρ σ 3 σ 3 ρ σ2 23 ϕ P W 3 σ 3 σ 3 σ y, W σ 3 23 ρ3 2 σ y (3.3) 2 ρ23 2 where (W 2, W 32 ) and (W 3, W 23 ) are both bivariate normals with zero means and ρ variances all ones, and correlations given, respectively, by ρ 3.2 = 3 ρ 2 ρ 23 ρ2 2 ρ 2 23 ρ and ρ 2.3 = 2 ρ 3 ρ 23. Now the problem of identified minimum is the following: we will assume that the functions f (y), f 2 (y) and f 3 (y) are given, and we need ρ3 2 ρ 2 23 to prove that there can be only a unique set of parameters σ 2, σ2 2, σ2 3, ρ 2, ρ 23, ρ 3 that can lead to the same three given functions f, f 2 and f 3. The proof, besides other arguments, uses mainly the lemmas given in section 2. The proof is rather involved and will appear elsewhere in full. References [] T. Amemiya, A note on a Fair and Jaffee model, Econometrica 42 (974), [2] T. W. Anderson and S. G. Ghurye, Identification of parameters by the distribution of a maximum random variable, J. of Royal Stat. Society 39B (977), MR ( ). Zbl [3] T. W. Adnerson and S. G. Ghurye, Unique factorization of products of bivariate normal cumulative distribution functions, Annals of the Institute of Statistical Math. 30 (978), MR Zbl [4] A. P. Basu, Identifiability problems in the theory of competing and complementary risks: a survey in Statistical Distributions in Scientific Work, vol. 5, (C.Taillie, et. al, eds.), D. Reidel Publishing Co., Dordrecht, Holland, 98, MR Zbl [5] A. P. Basu and J. K. Ghosh, Identifiability of the multinormal and other distributions under competing risks model, J. Multivariate Analysis 8 (978), MR0526. Zbl [6] A. P. Basu and J. K. Ghosh, Identifiablility of distributions under competing risks and complementary risks model, Commun. Statistics A9 (4) (980), MR Zbl [7] T. Bedford and I. Meilijson, A characterization of marginal distributions of (possibly dependent) lifetime variables which right censor each other, The Annals of Statistics 25(4) (997), MR Zbl Surveys in Mathematics and its Applications 5 (200), 3 320

9 Identifiability of the multivariate normal 39 [8] S. M. Berman, Note on extreme values, competing risks and semi-markov processes, Ann. Math. Statist. 34 (963), MR Zbl [9] M. Dai and A.Mukherjea, Identification of the parameters of a multivariate normal vector by the distribution of the maximum, J. Theoret. Probability 4 (200), MR Zbl [0] H. A. David, Estimation of means of normal population from observed minima, Biometrika 44 (957), [] H. A. David and M. L. Moeschberger, The Theory of Competing Risks, Griffin, London, 978. MR Zbl [2] J. Davis and A. Mukherjea, Identification of parameters by the distribution of the minimum, J. Multivariate Analysis 9 (2007), MR Zbl [3] M. Elnaggar and A. Mukherjea, Identification of parameters of a tri-variate normal vector by the distribution of the minimum, J. Statistical Planning and Inference 78 (999), MR Zbl [4] R. C. Fair and H. H. Kelejian, Methods of estimation for markets in disequilibrium: a further study, Econometrica 42 (974), MR Zbl [5] F.M. Fisher, The Identification Problem in Econometrics, McGraw Hill, New York, 996. [6] D. C. Gilliand and J. Hannan, Identification of the ordered bi-variate normal distribution by minimum variate, J. Amer. Statist. Assoc. 75(37) (980), MR Zbl [7] J. Gong and A. Mukherjea, Solution of the problem on the identification of parameters by the distribution of the maximum random variable: A multivariate normal case, J. Theoretical Probability 4 (4) (99), MR3238. Zbl [8] A. Mukherjea, A. Nakassis and J. Miyashita, Identification of parameters by the distribution of the maximum random variable: The Anderson-Ghuiye theorem, J. Multivariate Analysis 8 (986), MR Zbl [9] A. Mukherjea and R. Stephens, Identification of parameters by the distribution of the maximum random variable: the general multivariate normal case, Prob. Theory and Rel. Fields 84 (990), MR Zbl Surveys in Mathematics and its Applications 5 (200), 3 320

10 320 A. Mukherjea [20] A. Mukherjea and R. Stephens, The problem of identification of parameters by the distribution of the maximum random variable: solution for the trivariate normal case, J. Multivariate Anal. 34 (990), MR Zbl [2] A. Nadas, On estimating the distribution of a random vector when only the smallest coordinate is observable, Technometrics 2 (4) (970), Zbl [22] A. A. Tsiatis, A non-identifiability aspect of the problem of computing risks, Proc. Natl. Acad.Sci. (USA) 72 (975), MR Zbl [23] A. A. Tsiatis, An example of non-identifiability in computing risks, Scand. Actuarial Journal 978 (978), Zbl Arunava Mukherjea Department of Mathematics, The University of Texas-Pan American, 20 West University, Edinburg, Tx, 7854, USA. arunava.mukherjea@gmail.com Surveys in Mathematics and its Applications 5 (200), 3 320