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Let B > 0 and let X1 , X2 , … , Xn be a random...

Let B > 0 and let X1 , X2 , … , Xn be a random sample from the distribution with probability density function.

f( x ; B ) = β/ (1 +x)^ (B+1), x > 0, zero otherwise.

(i) Obtain the maximum likelihood estimator for B, β ˆ .

(ii) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 = 2.0, x 5 = 4.0. Obtain the maximum likelihood estimate for B, β ˆ

(iii) Obtain a method of moments estimator for B, β .

(iv) Suppose n = 5, and x 1 = 0.3, x 2 = 0.4, x 3 = 1.0, x 4 = 2.0, x 5 = 4.0. Obtain a method of moments estimate for B, β ~ .

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