Question: Lehmann (1966) defined (X, Y) to be positively likelihood-ratio dependent if their joint density satisfies f(x 1 , y 1 ) f(x 2 , y
Lehmann (1966) defined (X, Y) to be positively likelihood-ratio dependent if their joint density satisfies f(x1, y1) f(x2, y2) ≥ f(x1, y2) f(x2, y1) whenever x1 < x2 and y1 < y2. Then, the conditional distribution of Y (X) stochastically increases as X (Y) increases.
a. For the L × L model, show that the conditional distributions of Y and X are stochastically ordered. What is its nature if β > 0?
b. In row effects model (9.8), if µi > µh, show that the conditional distribution of Y is stochastically higher in row i than in row h. Explain why µ1 = ... = µI is equivalent to the equality of the I conditional distributions within rows.
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