Statistics | Part IB, 2001

Let X1,,XnX_{1}, \ldots, X_{n} be independent, identically distributed N(μ,σ2)N\left(\mu, \sigma^{2}\right) random variables, where μ\mu and σ2\sigma^{2} are unknown.

Derive the maximum likelihood estimators μ^,σ^2\widehat{\mu}, \widehat{\sigma}^{2} of μ,σ2\mu, \sigma^{2}, based on X1,,XnX_{1}, \ldots, X_{n}. Show that μ^\widehat{\mu} and σ^2\widehat{\sigma}^{2} are independent, and derive their distributions.

Suppose now it is intended to construct a "prediction interval" I(X1,,Xn)I\left(X_{1}, \ldots, X_{n}\right) for a future, independent, N(μ,σ2)N\left(\mu, \sigma^{2}\right) random variable X0X_{0}. We require

P{X0I(X1,,Xn)}=1αP\left\{X_{0} \in I\left(X_{1}, \ldots, X_{n}\right)\right\}=1-\alpha

with the probability over the joint distribution of X0,X1,,XnX_{0}, X_{1}, \ldots, X_{n}.


Iγ(X1,,Xn)=(μ^γσ^1+1n,μ^+γσ^1+1n)I_{\gamma}\left(X_{1}, \ldots, X_{n}\right)=\left(\widehat{\mu}-\gamma \widehat{\sigma} \sqrt{1+\frac{1}{n}}, \widehat{\mu}+\gamma \widehat{\sigma} \sqrt{1+\frac{1}{n}}\right)

By considering the distribution of (X0μ^)/(σ^n+1n1)\left(X_{0}-\widehat{\mu}\right) /\left(\widehat{\sigma} \sqrt{\frac{n+1}{n-1}}\right), find the value of γ\gamma for which P{X0Iγ(X1,,Xn)}=1α.P\left\{X_{0} \in I_{\gamma}\left(X_{1}, \ldots, X_{n}\right)\right\}=1-\alpha .

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