Question

3. Suppose that r,...,rn are a random sample from a B(a, B) distribution: Fin + β) r"-i(1-1)3-1. 3) Here E[X] a/(a + β) and E(X -(a + 1)o/{(a + β + 1)(a + β)). (a) Show that the method of moments, using the first two moments, gives the equations 0 (b) Determine the method of moments estimator of α and β based on the first two moments. Note: You can use the result of (a) even if you were unable to show it.

Answer 1

$(a)~According~to~method~of~moment:\\ m_1^\prime=\frac{\alpha}{\alpha+\beta}\Rightarrow (\alpha+\beta)m_1^\prime=\alpha~or,\alpha(1-m_1^\prime)-\beta m_1^\prime=0.........(1)\\ m_2^\prime=\frac{(\alpha+1)\alpha}{(\alpha+\beta+1)(\alpha+\beta)}\Rightarrow (\alpha+\beta+1)m_2^\prime=(\alpha+1) m_1^\prime\\ or,~\alpha(m_2^\prime-m_1^\prime)+\beta m_2^\prime=m_1^\prime-m_2^\prime.....(2)\\ (b)~(1)\times(m_2^\prime-m_1^\prime)-(2)\times (1-m_1^\prime),~we~get\\ -\beta\left(m_1^\prime(m_2^\prime-m_1^\prime)+m_2^\prime(1-m_1^\prime) \right )=-(1-m_1^\prime)(m_1^\prime-m_2^\prime)\\ or,~\hat{\beta}=\frac{(1-m_1^\prime)(m_1^\prime-m_2^\prime)}{m_2^\prime-m_1^{\prime 2}}......(3)\\$

$(1)\times m_2^\prime+(2)\times m_1^\prime,~we~get\\ \alpha(1-m_1^\prime)m_2^\prime+\alpha(m_2^\prime-m_1^\prime)m_1^\prime=m_1^\prime(m_1^\prime-m_2^\prime)\\ \alpha(m_2^\prime-m_1^{\prime 2})=m_1^\prime(m_1^\prime-m_2^\prime)\\ \hat{\alpha}=\frac{m_1^\prime(m_1^\prime-m_2^\prime)}{(m_2^\prime-m_1^{\prime 2})}.......(4)$