Question

Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1 : θ-3. Does your MP decision rule depend on the value of θ in Hi? What does this suggest?

Answer 1

$(a)~MLE~of~\theta=X_{(n)}.\\ The~likelihood~function:~L(\theta)=a^n\theta^{-a}\left(\prod_{i=1}^nx_i \right )^{a-1}\\ \sup_{\theta\leq \theta_0}L(\theta)=a^n\theta_0^{-a}\left(\prod_{i=1}^nx_i \right )^{a-1}\\ \sup_{\theta}L(\theta)=a^nx_{(n)}^{-a}\left(\prod_{i=1}^nx_i \right )^{a-1}\\ Likelihood~ratio=\Lambda=\frac{\sup_{\theta\leq \theta_0}L(\theta)}{\sup_{\theta}L(\theta)}=\left(\frac{x_{(n)}}{\theta_0} \right )^a\\ We~reject~H_0~at~level~\alpha~if\\ \Lambda<k\\ or,~X_{(n)}<c\\ where,~c~is~such~that\\ P(X_{(n)}<c|\theta_0)=\alpha~or,~P(X_1<c,X_2<c,...,X_n<c|\theta_0)=\alpha\\ or,~\prod_{i=1}^nP(X_i<c|\theta_0)=\alpha~since~X_i's~are~independent\\a\theta_0^{-a}\int_0^cx^{a-1}dx=\alpha~or,~c^a=\theta_0^a\alpha~or,~c=\theta_0\alpha^{\frac{1}{a}}.\\ Hence~Level~\alpha~rejection~region:~\{(X_1,...,X_n):X_{(n)}<\theta_0\alpha^{\frac{1}{a}}\}\\$

$(b)~Power~function=P(X_{(n)}<\theta_0\alpha^{\frac{1}{a}}|\theta)\\ =P(X_1<\theta_0\alpha^{\frac{1}{a}},X_2<\theta_0\alpha^{\frac{1}{a}},...,X_n<\theta_0\alpha^{\frac{1}{a}}|\theta)\\ =\prod_{i=1}^nP(X_i<\theta_0\alpha^{\frac{1}{a}}|\theta),~since~X_i's~are~independent\\ =\prod_{i=1}^n\left(\int_0^{\theta_0\alpha^{\frac{1}{a}}}a\theta^{-a}x^{a-1}dx \right )=\prod_{i=1}^n\left(\alpha\left(\frac{\theta_0}{\theta} \right )^a \right )=\alpha^n\left(\frac{\theta_0}{\theta} \right )^{an} =\beta(\theta),~say\\$

$(c)~f(x_1,x_2,...,x_n|H_1:\theta=\theta_1)>kf(x_1,x_2,...,x_n|H_0:\theta=\theta_0)\\ or,~a^n\theta_1^{-an}\left(\prod_{i=1}^nx_i \right )^{a-1}C(X_{(n)},\theta_1)>k a^n\theta_0^{-an}\left(\prod_{i=1}^nx_i \right )^{a-1}C(X_{(n)},\theta_0)\\ C(X_{(n)},\theta)=1~if~X_{(n)}<\theta.\\ =0~otherwise\\ If~X_{(n)}<c<\theta_0<\theta_1~then\\~f(x_1,x_2,...,x_n|H_1:\theta=\theta_1)>kf(x_1,x_2,...,x_n|H_0:\theta=\theta_0)\\ Hence~the~level~\alpha~MP~crititical~region~is~X_{(n)}<c~where\\ P(X_{(n)}<c|H_0)=\alpha~or,~c=\theta_0\alpha^{\frac{1}{a}}.\\ Therefore~for~\theta_0~the~MP~critical~region~is\\ \{(X_1,X_2,...,X_n):X_{(n)}<5\alpha^{\frac{1}{a}}\}.\\ We~see~that~it~does~not~depend~on~\theta~in~H_1.~Hence~it~is~also~level~\alpha~UMP~critical\\ region.$