By B.L.S. Prakasa Rao
The matter of identifiability is simple to all statistical equipment and knowledge research, taking place in such different parts as Reliability idea, Survival research, and Econometrics, the place stochastic modeling is familiar. arithmetic facing identifiability in line with se is heavily relating to the so-called department of "characterization difficulties" in likelihood thought. This booklet brings jointly correct fabric on identifiability because it happens in those various fields.
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Additional info for Identifiability in Stochastic Models. Characterization of Probability Distributions
Example text
IDENTIFIABILITY 35 BY SUM AND MAX (OR MIN) Then Fo(z) = Ae """Η'^ζ) where A is constant so that i o ( o o ) = 1 and '<*> = W ' F M < l e > = ^ - ·1 2 )9 An analogous result holds characterizing probability measures by sum and minimum. The result is due to Kotlarski. and X2 be independent random variables and Let Χο,Χι Yi = X0 + Xi and Y2 = X0AX2. Let Mi(a) = EeaXi,i = 0 < a < αο,αο > 0 0,1. 130) Suppose that Mi(a) ^ M ) ( # o ) is a fixed constant. a n that the distribution functions Fi of Xi,i is finite for Further suppose = 0,1,2 satisfy the conditions Fi(x) < 1 for χ < ao, ao < oo and lim eaoZF^{z) z—>oc = 0.
156). Then 0*(r) = g # ( r ) ) . 157) ς#(*))· Hence « ( 0 ( r ) ) 9( ^ ( t ) ) - g ( # r M t ) ) , r, t € Ä. 158) 42 CHAPTER 2. IDENTIFIABILITY BASED ON FUNCTIONS In view of the properties (i) and (ii) of Q and Q*, it follows that q(u)q(v) — q(uv), u,ν G SQ. 159) By property (iii) of Q and Q*, this relation can be extended to all of S by analyticity and we have q(u)q{v) = q(uv), u,v G S . 161) S for some constant c. 162) seS. OO Suppose 0 ( s ) = y ^ p n 5 n and Q*{s) = y " j 7 * s n . 161), it follows that £(iV) = £iV* · c.
4 Identifiability by Maximum and Minimum Let Χο,Χι and X2 be independent random variables. Define Yx = XoAX! and Y2 = X0VX2 . 1 : Let Fi be the distribution function of Xi,i = 0,1,2. 88) F0(a + 0) = 0, F0(b - 0) = 1, F 0( x 0) = q and Fo is strictly increasing in (a, 6). Then the joint distribution of (Yi, Y2) uniquely determines the distributions FQ,FI and F2. Proof : For —oo < yi < y2 < oo, P{Yi>yi,Y2