Question: Consider n independentobservationsfroma N(, 2) distribution. (a) Focusingon , findthelikelihoodfunction.Showthatthelog-likelihoodfunctionisa concave,parabolicfunctionof . FindtheMLestimator . (b) Considering 2 to beknown,findtheinformationusing(i)equation(4.3),(ii)equation (4.4). Usethemtoshowthatthelarge-samplevarianceof is 2~n. Explainwhy

Consider n independentobservationsfroma N(μ, σ2) distribution.

(a) Focusingon μ, findthelikelihoodfunction.Showthatthelog-likelihoodfunctionisa concave,parabolicfunctionof μ. FindtheMLestimator ˆμ.

(b) Considering σ2 to beknown,findtheinformationusing(i)equation(4.3),(ii)equation

(4.4). Usethemtoshowthatthelarge-samplevarianceof ˆμ is σ2~n. Explainwhy ˆμ is the minimumvarianceunbiasedestimator.

(c) ShowthattheMLestimator ˆσ of σ is

¼

[Σi(Yi − Y )2]~n. Showthatthe large-sample varianceof ˆσ is σ2~2n.

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