Suppose that x1, . . . , xn are a random sample from a B(α, β)...

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Question

Suppose that x1, . . . , xn are a random sample from a B(α, β)...

Suppose that x1, . . . , xn are a random sample from a B(α, β) distribution:

f(x; α, β) = $\frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha) + \Gamma (\beta )}$ x^(α-1) (1-x)^(β-1)

Here E[X] = α/(α + β) and E[X^2 ] = ((α + 1)α)/{(α + β + 1)(α + β)}.

(a) Show that the method of moments, using the first two moments, gives the equations

0 = α(1 − m1 ) − βm1

m1 − m2 = α(m2 − m1 ) + βm2

(b) Determine the method of moments estimator of α and β based on the first two moments.

math Statistics-And-Probability

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Answer 1

Answer #1

$По + 3) ra-1 (1-2)3-1, 0 < x < 1 (a) f(z) = T(a)r(3) = 0 otherwise Γ(a + β) , , Γ(a + r)「(3) 1)dr 5-1 , Then for r = 1, E(X) for r = 2, E(X2)-「(a + β) × По +2)「(3) ala +1)$

$Now, using method of moment, (a+3)(a+i) + 1) (_)o+3+1 mi(β+1) or, m2(α+ β +1)-ml(a+1) (b) From (1), a 1-m From (2) and (3), we get, m1 - m2- m1 1 mi(m2 - m) +m2(1 mi) 1- m1 (1 - (mi-m2) m2 nm m1 m2-m From (3), a111-m2) m2-mm Hence MMEs of α and β are respectively : m1(m1 - m2)(1- m)(m1 m2) an m2-mm m2-mm$

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Answer 2

Suppose that x1, . . . , xn are a random sample from a B(α, β)...

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