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Let X 1 …X n be a random sample from N (0 ; ө ).

Let be i.i.d Gamma random variables with unknown parameters  and  . Determine the ML estimator for  and .

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Let X 1 …X n be a random sample from N (0 ; ө ).

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  1. Let be i.i.d Gamma random variables with unknown parameters  and  . Determine the ML estimator for  and 

  2. In general the (log)-likelihood function can have more than one solution, or no solutions at all. Further, the (log)-likelihood function may not be even differentiable, or it can be extremely complicated to solve explicitly

  3. We wish to fit data (x1; y1),…,(xn; yn) with linear model Yj= 1 + βxj+ ejwith ej ~ N(0; σ2). • Find the maximum likelihood estimator for β • Either prove or disprove unbiasedness of the estimator you computed

  4. Let X1…Xn be a random sample from N(0; ө). • (a) Obtain an estimator of 1/өusing the method of moments. • (b) Find the likelihood function and use it to obtain the maximum likelihood estimator of 1/ө.

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