Solution of the problem of the identified minimum for the tri-variate normal

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol., No. 4, November 0, pp c Indian Academy of Sciences Solution of the problem of the identified minimum for the tri-variate normal A MUKHERJEA, and M ELNAGGAR University of Texas-Pan American, Edinburg, TX , USA Corresponding Author. arunava.mukherjea@gmail.com MS received 4 March 0; revised 0 August 0 Abstract. Let X = (X, X, X 3 ) be a non-singular tri-variate normal vector with zero means. Let T = minx, X, X 3 },andi = i iff T = X i, i =,, 3. The problem of the identified minimum (I, T ) is then to find if its joint distribution determines uniquely X. This problem is solved here in the affirmative. To the best of our knowledge, it was first solved in the bivariate normal case (and partially in the tri-variate normal case) in 978 in [. Keywords. Tri-variate normal; identified minimum; identification of parameters.. Introduction In [, the following 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,...,X n be independent random variables with continuous distribution function T = minx, X,...,X n }, and I is defined as I = k if and only if T = X k.ifthe X i s have a common distribution function, then it is unique determined by the distribution function of T.In[, it was shown that when the distribution of X i are not all the same, then the joint distribution function of the identified minimum (that is, the distribution function of (I, T )) uniquely determined each individual distribution function F i (x) of X i, i =,,...,n. The problem whether the joint distribution function of the identified minimum (I, T ) uniquely determines the parameters of the distribution of (X, X,...,X n ), when the X i s are not necessarily independent, was solved in [, when n = and (X, X ) has a bivariate normal distribution and solved only partially in [, when n = 3 and X, X, X 3 has a tri-variate normal distribution. Here, in this paper, we solve the problem completely for the tri-variate normal distribution.. The pdf of the identified minimum For simplicity, we will assume in this paper that the tri-variate normal vector whose identified minimum is considered below has zero means, and distinct variances. The solution of our problem in the general case (when the means are not necessarily zeros and the variances are not necessarily distinct) is no more difficult. 645

2 646 A Mukherjea and M Elnaggar Let (X, X, X 3 ) be a tri-variate normal random vector with zero means, and variances σ,σ,σ 3 such that σ >σ >σ 3, and a non-singular covariance matrix, where ij = ρ ij σ i σ j for i = j, i = σi and i = j. LetT = minx, X, X 3 }, and I = i if and only if T = X i, i =,, 3. Let Then we have F(t, i) = P(T t, I = i). P(T t, I = ) = P(X t, X X, X X 3 ) t = P(X x, X 3 x X = x) f X (x)dx. Differentiating with respect to t we obtain f (t) = d dt [F(t, ) = f X (t)p(x t, X 3 t X = t). The conditional density of (X, X 3 ),givenx = t is a bivariate normal with means σ ρ σ σ t,ρ 3 3 σ t and variances σ ( ρ ), σ 3 ( ρ 3 ). Thus, f (t) = ( ) t ϕ P W ρ σ σ t, W 3 σ σ σ ρ 3 σ 3 σ t, ρ σ 3 ρ3 where (W, W 3 ) is a bivariate normal with zero means, variances each one, and correlation ρ 3. = ρ 3 ρ ρ 3 and ϕ is the standard normal density. Let us write ρ ρ 3 σ ρ σ = a, σ ρ 3 σ 3 = a 3 ; σ σ ρ σ σ 3 ρ 3 σ ρ σ = a, σ ρ 3 σ 3 = a 3 ; σ σ ρ σ σ 3 ρ 3 σ 3 ρ 3 σ = a 3, σ 3 ρ 3 σ = a 3. σ σ 3 ρ 3 σ σ 3 ρ 3 Then, we have f (t) = ( t ϕ σ ) P(W a t, W 3 a 3 t). (.) σ Similarly, using similar notations, we have f (t) = ( ) t ϕ P(W a t, W 3 a 3 t), (.) σ σ f 3 (t) = ( ) t ϕ P(W 3 a 3 t, W 3 a 3 t). (.3) σ 3 σ 3

3 Identified minimum for the tri-variate normal 647 Statement of the problem. We assume that the functions f, f, f 3 in (.) (.3) are given, where the parameters (the variances and the correlations) σ,σ,σ 3,ρ,ρ 3,ρ 3 are all unknown. The problem is to find if there is a unique set of parameters σ,σ,σ 3,ρ,ρ 3,ρ 3 corresponding to the three given functions f, f and f 3. In the next section we prove some lemmas which will be needed for the solution of the problem above. In 4, we show that there is a positive solution to this problem (that is, there is a unique set of parameters corresponding to the given f, f and f 3 ). Thus, it makes sense to estimate these parameters by usual estimation methods such as the method of moments or the method of maximum likelihood. Before we go to the next section, let us state the well-known Mills ratio result for a standard normal random variable Z, namely that P(Z t) t π e t, as t, (that is, as t, the ratio of the sides ). We will use this result often. 3. Some useful lemmas If f (t) and f (t) are two real functions of the real variable t, then we say that f and f have the same order as t a, where a is either or, and write f f,ast a, if and only if t a f (t) f (t) =. In Lemma 3. below and all other lemmas, (X, Y ) is a non-singular bivariate normal vector with zero means. In only Lemma 3., the variances of (X, Y ) are σ and.inall other lemmas, the variances are each one, and the correlation is ρ. Lemma 3.. (see [3). Let be the covariance matrix of (X, Y ) and = (α,α ), where α > 0,α > 0, and is the vector (, ). Thus, as t, the following result holds: P(X t, Y t) e t ( T ) πα α t det. (3.) Lemma 3.. Let α>0,β >0,α β and ρ< β α. Then as t, we have P(X αt, Y βt) which is o(t e β t ) as t. [ ( ρ ) 3 α β α +β ραβ πt (α ρβ)(β ρα) e ρ t Proof. Note that when ρ: the correlation of (X, Y ), is less than β α, then is positive, where is the covariance matrix of ( X α, Y β ). Thus, this lemma follows from Lemma 3..

4 648 A Mukherjea and M Elnaggar Lemma 3.3. Let X, Y,α,β,ρ be as in Lemma 3.. Let ρ> β α. Then, as t, P(X >αt, Y >βt) βt π e β t. Proof. Notice that P(X αt, Y βt) = = = βt βt βt P(X αt Y = x) f Y (x) dx f Y (x) dx f Y (x) dx [ αt ( y ρx ) π ρ e ρ dy [ π u(x) e y dy, where u(x) = αt ρx. Since ρ> α ρ β,itfollowsthatforx >βt, αt ρx < t(α ρβ). This means that as t, u(x). Thus, as t, P(X αt, Y βt) P(Z βt) βt π e β t, by the well-known Mills ratios result, where Z is the standard normal random variable. Lemma 3.4. Let ρ = α β in the previous lemma, instead of the assumption ρ> α β. Then, for t > 0 we have P(Z βt) P(X αt, Y βt) P(Z βt). Proof. The proof of this lemma follows easily from that of the previous lemma. Lemma 3.5. Let α<0, β>0, α <β. Then, as t, (i) P(X αt, Y βt) βt π e β t, when ρ> α β ; e π α βρ βt (ii) when ρ< α β, P(X αt, Y βt) ρ (iii) when ρ = α β, t [ P(X αt, Y βt) P(Z βt) βt π e Proof. We prove (ii) first. As before, we can write P(X αt, Y βt) = βt f Y (x) dx [ π u(x) α +β ραβ ρ ; β t. e y dy,

5 Identified minimum for the tri-variate normal 649 where u(x) = αt ρx.ifρ< α (α βρ)t ρ β, then ρ<0and for x βt, u(x) ρ t. This shows that as t, ( ) (α βρ)t P(X αt, Y βt) P Z P (Z βt), ρ,as and (ii) follows, by Mills ratio result. The result in (iii) also follows from here since α βρ = 0, when ρ = β α. To prove (i), notice that P(X αt, Y βt) = P(Y βt) P( X ( α)t, Y βt). The result in (i) then follows from Lemma 3., when the correlation ρ of ( X, Y ) is less than β α (or when ρ>α β ), as t. Remark to Lemma 3.5. Suppose now that t and α > β > 0, α < 0in Lemma 3.5. Then we can write P(X αt, Y βt) = P(Y ( β)( t), X ( α)( t)). Thus, when t, t, and Lemma 3.5 again applies with (α, β) in Lemma 3.5 now replaced by ( β, α). We can thus state the following lemma. Lemma 3.5A. Let α<0, β>0, α >βand t. Then the following hold: (i) when ρ> β α, P(X αt, Y βt) αt π e α t ; [ (ii) when ρ< β α, P(X αt, Y βt) ρ e t α +β ραβ ρ ; π αρ β αt (iii) when ρ = β α, P(X αt, Y βt) P(Z αt) αt π e α t. Lemma 3.6. Let α<0, β>0, α >β, and t. Then P(X αt, Y βt) βt π e β t. Lemma 3.6A. Let α<0, β>0, α <β, and t. Then P(X αt, Y βt) αt π e α t. We prove only Lemma 3.6. The proof of Lemma 3.6A will follow just like that of Lemma 3.5A followed from Lemma 3.5.

6 650 A Mukherjea and M Elnaggar Proof of Lemma 3.6. Write P(X αt, Y βt) = P(Y βt) P( X ( α)t, Y βt). By Lemma 3., as t, when ρ < β α (that is, when ρ>β α ), P(X αt, Y βt) P(Y βt) βt π e β t. Also, by Lemmas 3.3 and 3.4, when ρ β α,ast, ( ) P( X ( α)t, Y βt) = o βt π e β t. The lemma follows. Lemma 3.7. Let α>0 or α<0. Then as t (or as t ), P(X αt, Y ( α)t) Proof. Let α<0 and t. Then αt π e α t. P(X αt, Y ( α)t) = P(Y ( α)t) P( X ( α)t, Y ( α)t). The proof follows from Lemma 3.. Lemma 3.8. Let α<0, β<0, α β, and t. Then (i) ρ< α β, P(X αt, Y βt) ( ρ ) 3 α β e [ t α +β ραβ ρ πt (α ρβ)(β ρα) (ii) ρ> β α, P(X αt, Y βt) βt π e β t ; (iii) ρ = β α, P(Z βt) P(X αt, Y βt) P(Z βt). Proof. The proof follows immediately from Lemmas 3., 3.3 and 3.4 by observing that P(X αt, Y βt) = P(X ( α)( t), Y ( β)( t)). Lemma 3.9. Let β>0and t. Then we have P(X 0, Y βt) βt π e β t, if ρ>0; ( ) P(Z βt)p Z ρβt, if ρ<0; ρ βt π e β t, if ρ = 0. Here Z is the standard normal random variable. ;

7 Identified minimum for the tri-variate normal 65 Proof. Observe that as in earlier results, P(X 0, Y βt) ( ) = f Y (x)dx y ρx π ρ e ρ dy = βt βt ρx ρ 0 [ f Y (x)dx u(x) e y dy, π where u(x) =.Nowifx βt, ρ>0, then ρx ρβt, so that u(x) as t. It is clear that P(X 0, Y βt) P(Y βt) βt π e β t, when ρ>0. The rest of the lemma is now clear. Lemma 3.0. Let α>0, β>0,α β. Then as t, P(X αt, Y βt) α t π e α t, if ρ< β α (when α β), and also if ρ β α (when α<β). Ifα = β, as t, P(X αt, Y βt) α t π e Proof. For a proof, see [4. α t. Lemma 3.. Let α>0, β>0, α β. Then, as t, P(Z αt) P(X αt, Y βt) βt π e β t,when ρ< α β. Proof. Write P(Z αt) P(X αt, Y βt) =P(X αt, Y βt) =P(Y βt) P(X αt, Y βt) =P( Y β( t)) P( X α( t), Y β( t)). The lemma now follows from Lemma 3.. Let us finally state without proof the following lemma which follows from our earlier lemmas. Lemma 3.. Let α<0, β>0. Let us define s = sup s } t est P(X αt, Y βt) = 0.

8 65 A Mukherjea and M Elnaggar Then the following holds: (i) If α >β,then s α. (ii) If α β, then s = α. 4. Solution of the identified minimum problem In this section, we show that there can only be a unique set of parameters σ,,σ 3,ρ, ρ 3,ρ 3 (the three variances and the three correlations) corresponding to the given pdf funtions f, f and f 3 in (.), (.) and (.3). We will use the following notations: A j = min a ij, a kj }, B j = max a ij, a kj }, j =, or3, i, j, k} =,, 3}, ρ i ρ jk.i, the correlation of(w ji, W ki ). We have assumed, for simplicity, that σ >σ >σ 3. (4.) Note that we can always make the assumption σ σ σ3 by simply renaming X, X, X 3 appropriately. Also the proofs for the cases σ = >σ 3 and other similar cases, not considered here, should be clear from our proofs given here under the assumption (4.). We show below that the six parameters σ,σ,σ 3,ρ, ρ 3,ρ 3 can be determined uniquely, when the functions f, f, f 3 are given or known, through iting processes. Step. In this step, we identify σ, A, B. Notice that because of assumption (4.), a > 0 and a 3 > 0. As a result, it follows that = sup s } t est f (t) = 0. (4.) σ This identifies σ. To identify A and B, we consider s o = sup s } t est f (t) = 0. (4.3) It follows from Lemma 3. that and t t e s ot f (t) is non-zero and finite if ρ < A (4.4) B t tes ot f (t) is non-zero and finite if ρ A B,

9 and in this case, Identified minimum for the tri-variate normal 653 s o = σ + B. (4.5) In case (4.4) occurs, it follows from Lemma 3.0 that as t, ( ) t ϕ f (t) σ σ A t π e A ( ) t t ϕ. (4.6) σ σ Thus, (4.6) identifies A since σ + A = sup s [ ( ) } t t est ϕ f (t) = 0. σ σ In case (4.5) occurs, (4.3) along with (4.5) identifies B. Then, by Lemma 3.0, as t, ( ) ( ) t t c ϕ f (t) ϕ σ σ σ A t π e A t, where c is a constant. This identifies A. Thus, in case of (4.5), σ, A and B, all three, have been identified. However, in case of (4.4), we have identified only σ and A. To identify B in this case, we consider, as t, the function g(t) defined by g(t) = ( ) t ϕ P(Z A t) f (t). σ σ In this case (see (4.4)), Lemma 3. applies, and we have, as t, sup s } [ t est g(t) = 0 = + B. This identifies B, completing Step. σ Step. We are given the function f, and we need to identify σ, A and B using our knowledge of f, and that of σ, A and B identified in Step. This step is more complicated since (4.) does not imply that a > 0, though a 3 > 0. Let us define s = sup s } t est f (t) = 0. (4.7) Then we have (i) If a 0, then s =. Thus, for a σ 0, < σ 3 + a 3 = σ + a 3.

10 654 A Mukherjea and M Elnaggar When a > 0, < + a = σ + a, and this means, when a > 0, [ s < + A. (4.8) σ However when a = 0, [ = s = σ [ + a = σ + A. (4.9) (ii) If a < 0, then by Lemmas 3.5, 3.6A and 3.7, we have [ [ s + a + A. (4.0) Thus when a < 0, by (4.0), we have two possibilities: [ (i) s > + A, (ii) s = [ σ + a. σ Incaseof(i),by(4.7), [ σ +A t t f (t) = 0. (4.) In case of (ii), by Lemmas 3.5, 3.6A and 3.7, we also have [ [ σ +A t f (t) = σ +a t t t f (t) = 0, (4.) because of a factor t in the denominators of the right-hand sides of Lemmas 3.5, 3.6A and 3.7. However, when a = 0, it follows from (4.9) that [ σ +A t t f (t) = 0. (4.3) Thus (4.8), (4.), (4.) and (4.3) help us determine when a > 0, a < 0, and a = 0. Since a 3 > 0, when a > 0, we can use the method of Step to identify σ, A and B.Ifa = 0, σ is already identified by (4.9). By Lemma 3.9, as t,wehave f (t) c ( ) t ϕ σ σ a 3 t π e a3 t, (4.4)

11 Identified minimum for the tri-variate normal 655 where c = ifρ > 0 and c = if ρ = 0. It is also important to note that if ρ < 0, then as t,wehavebylemma3.9again, ( )[ t ϕ P(W 0, W 3 a 3 t) σ = ( ) t ϕ [P(W 0, W 3 a 3 ( t)) σ σ ( ) t ϕ σ σ a 3 t π e a 3 t. Thus, it is clear that in case a = 0, we can identify a 3 as well as σ. Next, we consider the case when a < 0. In this case, we use Lemmas 3.5, 3.5A, 3.6 and 3.6A. We observe that exactly one of the following three possibilities must then occur. Each one of these possibilities identifies σ and A = min a, a 3 }. () If a < a 3, with s as defined in (4.7), we must have by Lemmas 3.5 and 3.6A, [ s = + a, (4.5) π t t es t f (t) = = 0, (4.6) σ a t tes t f (t) = 0. (4.7) Also, defining s = sup s } t est f (t) = 0, (4.8) it follows from Lemma 3.5 that when the correlation ρ of (W, W 3 ) is greater than a a 3, then t tes t f (t) = 0 (4.9) and in this case (when (4.9) occurs and ρ > a a 3 ), by Lemma 3.5, we have s = σ + a 3. (4.0) Also, for ρ a a 3, by Lemma 3.5, we have s σ + a3. (4.) () If a > a 3, we must have by Lemmas 3.6 and 3.5A, [ s = + a3, (4.)

12 656 A Mukherjea and M Elnaggar t πtes t f (t) = = 0, (4.3) σ a 3 t tes t f (t) = 0. (4.4) It also follows from Lemma 3.5A that when ρ > a 3 a, then t tes t f (t) = 0 (4.5) and in this case, by Lemma 3.5A, when ρ > a 3 a, s = Also, for ρ a 3 a, by Lemma 3.5A, s + a. (4.6) + a. (4.7) (3) If a =a 3, then by Lemma 3.7, s = s and the first two results in each of () and () above then must hold. Thus, identification of σ, A and B are immediate. Note that when possibility () above occurs (that is, when s > s by (4.5), (4.0) and (4.)), σ, A are both identified by the equations (4.5) and (4.6), and when ρ > a a 3 and A = a, B = a 3 is identified by (4.0). Thus, we must still identify B in the case when ρ a a 3 and A = a is identified. To do this, we consider the function h defined by h(t) = P(Z a t) P(W a t, W 3 a 3 t) = P(W ( a )( t), W 3 a 3 ( t)) and if we define s = sup s } t est h(t) = 0, then, by Lemma 3., when ρ a a 3 (that is, when ρ a a 3 ), s = a 3, which identifies B. Similarly, when possibility () occurs, we can also identify σ, A and B. This completes Step. Step 3. In this step, we first show that in all situations except when we have σ + A = + A and σ + B = + B, (4.8)

13 Identified minimum for the tri-variate normal 657 we can identify uniquely a, a 3, a and a 3. In case of the above two equalities, we shall show that we must have a3 = a 3. In the proof of the above assertions, the main argument is the fact that σ + a = σ + a. (4.9) Let us also observe that in each of the following remaining cases (leaving aside (4.8) above) (i) σ + A = + A, σ + B = + B, (4.30) (ii) (iii) (iv) (v) σ σ σ σ + A = + A = + A = + A = + A, σ + B, σ + B, σ + B, σ + B = + B = + B = + B = + B, (4.3) + A, (4.3) + A, (4.33) + A, (4.34) it can be easily verified that a, a 3, a and a 3 can all be identified. Let us only mention that in case (v) above, A = B = a = a 3 and A = B = a = a 3. In case of (i) through (iv) above, we can use (4.9) to identify a, a 3, a, a 3 uniquely. Now we consider (4.8). In this case, there are two possibilities: (A, B, A, B ) = (a, a 3, a, a 3 ) (4.35) or (A, B, A, B ) = (a 3, a, a 3, a ). (4.36) In case of (4.35), we have σ + a3 = σ + a3, which implies that σ3 + a3 = σ3 + a3 or a3 = a 3. The same is the conclusion when (4.36) occurs. First, note that in each of the five possibilities, (4.30) through (4.34), we can identify uniquely σ,, a, a 3, a, a 3. Thus in this case, if σ 3 is also identified, then a3 and a3 are also identified since σ + a 3 = σ 3 + a 3, + a 3 = σ 3 + a 3. (4.37)

14 658 A Mukherjea and M Elnaggar Next, when (4.8) occurs, but (4.34) does not, we have already seen in (4.37) that a3 = a 3. Suppose now that we have also identified + a σ3 3. Thus considering } r = min σ + A, σ + A, σ3 + A 3, + A 3, then A = a 3 and A = a 3, and if r < σ3 are also identified uniquely. it is clear that if r = σ3 A = a and A = a, and thus, a, a 3, a, a 3 + A 3, then Still the question remains: how do we identify the correlations ρ,ρ 3 and ρ 3 by knowing the nine parameters σ,σ,σ 3, a, a 3, a, a 3, a 3, a 3?Weusethe following method. Suppose we find that then r = + a, r = σ + σ σ ρ σ ( ρ ). (4.38) From Step, we know if a = 0, >0or< 0. By solving (4.38), we know ρ = ± (rσ )(r ) (4.39) rσ σ so that rσ σ ρ σ = [ rσ rσ rσ. (4.40) It follows that a = 0 ρ = σ, σ a < 0 we take the + signin(4.39), a > 0 we take the signin(4.39), Similarly, considering + a σ 3 = r, we get ρ 3; and we get ρ 3 from + a σ 3 = r. Finally, before we can complete the solution of our problem, we need to use equation (.3) and show how we can identify the parameters σ3, A 3 and B 3 in case the function f 3 in (.3) is given and a3 = a 3 in (.3). Let us define for a =±, s a = sup s } e st f 3 (t) = 0. (4.4) t a When a 3 = 0, it is clear that s a = σ3, for a = or. (4.4)

15 Identified minimum for the tri-variate normal 659 When a 3 > 0 and a 3 = a 3, then s a = σ3, for a = (4.43) and by Lemma 3., σ 3 < s a = σ 3 + a 3 for a = (4.44) + ρ Also, in the case when 0 < a 3 = a 3, by Lemma 3.0, a 3 = sup s } t est [ P(W 3 a 3 t, W 3 a 3 t) = 0 and by Lemma 3., (4.45) t t es t f 3 (t) = 0 (4.46) The same results (4.43) through (4.46) hold for 0 > a 3 = a 3, with replaced by and by. Now consider the last case when a 3 > 0 and a 3 = a 3. In this case, by Lemma 3.0, [ s = s = + a3 σ 3 (4.47) and π t t es t f 3 (t) =. (4.48) σ 3 a 3 In this case, (4.47) and (4.48) giveσ3 and a 3 uniquely in terms of s and the it on the left in (4.48). It is clear from the above results which of the following four cases hold: a 3 = 0, 0 < a 3 = a 3, a 3 = a 3 < 0 and a 3 = a 3 > 0; also, how in each case, we can identify σ3 and a 3 in terms of f 3 through a iting process. Finally, as we remarked earlier, when a3 may not equal a 3, we still need to identify σ3, and this can be done using the lemmas in 3 and following the method used above. See (4.4) through (4.48). In this case, σ,, a, a 3, a and a 3 are already identified, and thus, (4.37) tells us if a3 is greater than, less than or equal to a 3. In each of these cases, σ3 can be identified using (4.4) and our lemmas. We omit the details to avoid duplication. This completes the solution of the problem. Acknowledgement The authors are grateful to the referee for pointing out a gap in an argument and other useful comments. Questions or Comments, if any, should be ed to the first-named author (A.M.).

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