By Barry C. Arnold

The idea that of conditional specification of distributions isn't new yet, other than in common households, it has now not been good constructed within the literature. Computational problems definitely hindered or discouraged advancements during this path. even though, such roadblocks are of dimished value at the present time. Questions of compatibility of conditional and marginal requirements of distributions are of primary significance in modeling eventualities. versions with conditionals in exponential households are rather tractable and supply worthy versions in a huge number of settings.

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3. 6. w1 6 –6 –3 0 w2 12 –6 –6 0 ⎛ 1/3 B=⎝ 0 1/3 w3 12 –6 –3 0 v1 30 –18 –12 1 ⎞ 2/3 0 1/3 2/3 ⎠ . 37) We cannot resolve this case using cross-product ratio matrices (there are no positive 2 × 2 submatrices). 2 may be used without difficulty. 38) ⎟ τ = ⎜ ⎟ , τ ≥ 0, ⎜ 1/3 −1/3 ⎟ ⎜ 0 ⎜0⎟ ⎟ ⎜ ⎜ ⎟ 0 1/6 ⎟ ⎜ −1/6 ⎜0⎟ ⎟ ⎜ ⎜ ⎟ 0 0 ⎟ ⎜ 0 ⎜0⎟ ⎠ ⎝ ⎝ ⎠ 0 −1/3 1/3 0 1 1 1 1 which by removing redundant equations can be written as: ⎛ ⎛ ⎞ ⎞ 1/6 0 −1/6 0 0 ⎟ ⎜ 1/3 −1/6 ⎜0⎟ ⎝ ⎠ τ = ⎝ ⎠ , τ ≥ 0. 3. 26), the conditional probability matrices A and B are incompatible.

In addition, they suggests alternative ways in which most nearly compatible distributions can be defined in incompatible cases. A related concept of ǫ-compatibility arises naturally in the discussion of incompatible cases. 2 Review and Extensions of Compatibility Results We are interested in discrete random variables X and Y with possible values x1 , x2 , . , xI and y1 , y2 , . . , yJ , respectively. A candidate conditional model for the joint distribution of (X, Y ) consists of I × J matrices A and B with nonnegative elements in which A has columns which sum to 1 and B has rows which sum to 1.

117) λi = 1 i=1 λi ≥ 0, i = 1, 2, 3, 4, τ1 ≥ 0. 117) that the minimum value of ǫ is obtained for π1 = 0, λ1 = 1, λi = 0, i = 1, leading to 9/214, as found before. The corresponding values of the X-marginal are τ1 = 42/107 and τ2 = 65/107. 118) 61 61 , which for the optimum case ǫ = 9/214 leads to P1 = 1 107 7 39 35 26 , P2 = 1 214 23 69 61 61 . Method 3. Solving the linear programming problem, we get the same ✷ solution for τ and ǫ above. We can use the concepts of ǫ-compatibility to give us yet more definitions of a most nearly compatible matrix P for a given pair A, B.

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