Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . Give the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ,σ, and the corresponding quantile zα of the standard normal distribution. Find the ML estimate of g(θ). Derive the bias of g(θˆ) as an estimator of g(θ) and use it to provide a bias-corrected estimator g ̃ of g(θ). Derive the standard deviation of g ̃ and the consequent standard error of g ̃.

Give the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ,σ, and the corresponding quantile zα of the standard normal distribution.

the moment estimator of θ;• the MLE of θ

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T ,

Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T , without the need to differentiate the log likelihood function with respect to θ.

LetX = (X1, . . . , Xn)Tdenote a random sample of size n on a discrete random variable X which hasprobability function f (x; θ), where θ is an unknown parameter.Suppose we wish to estimate f (k; θ), the probability that X is equal to agiven arbitrary (non-negative) integer k. We shall subsequently write f (k; θ)as the function g(θ), suppressing its dependence on the value k.Let U (X1) be a statistic defined as:U (X1) ={1, X1 = k,0, otherwise.(i) On letting T (X1, ..., Xn) denote a sufficient statistic for θ, show thatW (T ) = E{U (X1) | T (X1, ..., Xn)}is a statistic; that is, explain why it does not depend on θ.(ii) Show that W (T ) is an unbiased estimator of g(θ).(iii) Without assuming the result of any theorem, show that the variance ofW (T ) will be less than the variance of U unless U is constant given Twhere in this case W will have the same variance as U .(iv) Explain why W (T ) is the UMVU estimator of g(θ) in the case whereT is a complete, minimal sufficient statistic for θ.