By B.L.S. Prakasa Rao

The matter of identifiability is simple to all statistical tools and information research, taking place in such assorted components as Reliability idea, Survival research, and Econometrics, the place stochastic modeling is familiar. arithmetic facing identifiability in keeping with se is heavily on the topic of the so-called department of "characterization difficulties" in chance idea. This ebook brings jointly appropriate fabric on identifiability because it happens in those various fields.

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In all the above cases, the joint distribution of (£7, V) determines the unknown distributions of the random variables involved in their definition. The discussion given here is based on Kotlarski (1985). 11 Identifiability by Random Linear Forms Suppose X\,X2 and X3 are three independent real-valued random vari- ables. Let Yi,Y2>*3 be random variables independent of Xi,X2,X$ and independent among themselves with known distributions. Let wx = y i X i + y 2* 2 , W2 = YiXi + YsXs . 202) The question now is to find conditions under which the joint distribution of (W^i, W2) determines the distributions of Xi,X2,Xz.

94) implies that Fo(y) = F0{y) for - oo < y < x0 . 96) Similarly we can prove that Fo(y) = F0(y) f o r x 0< y < + o o . 98) completing the proof of the theorem. 2 (Explicit determination) : Given the joint distribution of (Fi, Y2), one can explicitly write down the distributions of Χ ο , Χ ι and X2. Let H(u,v) = P{Yl>u,Y2

0 ( α ) ι Κ / ? ) 6 5ο, (ii) Q is non-vanishing and one-to-one on So, and (iii) Q can be extended analytically from So to S. 152) i=i Then the joint distribution of (£/, V) uniquely determines the distributions of X, Y and N. 9. IDENTIFIABILITY BY RANDOM SUMS Proof : The characteristic function %(r, t) of (f7, V) is given by (r,t) X = E[eirU+itv ] = E{E[eirU+itV \N}} = = 0]P(N = 0) E[eirU+itv \N OO +^£[exp{w-(X1 + ···+ Χ*) n=l +it(Y! + ••• + YN)}\N=n]P(N = = n) P(N = 0) OO + ^ £ [ e x p O > ( X 1 + · · · + Xn) + + · · · + Yn)}]P(N = n) n=l (by the independence of Ν and X^Y^i > 1) oo = P(N = 0) + 5 > ( Γ ) Γ Μ « ) Γ Ρ ( ^ = η) = Q(