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 2 ) f(x 1 , y 2 ) f(x 2 , y 1 ) whenever x 1 x 2 and y 1 y 2 Then, the conditional distribution of Y (X) stochastically increases as X (Y) increases a For the L L model...

From the definition it follows that a joint distribution of two discrete variables is positively ...

Lehmann (1966) defined (X, Y) to be positively likelihood-ratio dependent if their joint density satisfies f(x 1

Question:

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 µ_i > µ_h, show that the conditional distribution of Y is stochastically higher in row i than in row h. Explain why µ₁ = ... = µ_I is equivalent to the equality of the I conditional distributions within rows.

Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...

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