UMP unbiased test

Question

I was wondering if any of you folks could help me with this statistics problem. Here is the problem : set T an exhaustive statistic which $T \sim \Gamma(n,1)$. We need to find a UMP unbiased test for $H_0 : \tau = \tau_0 \ vs \ H_1 : \tau \neq \tau_0$ . I know that the form of reject zone must be $\mathcal{R} = \{T \notin [q_1, q_2]\}$ and I found two equations relating $q_1$ and $q_2$. I have : $$ F(q_2) - F(q_1) = 1 -\alpha $$ and $$ q_1^ne^{-q_1} = q_2^ne^{-q_2} $$ \tau is the parameter of the density $$f(x)=\frac{\tau}{x^{\tau +1}}$$ and I know that $$T= \tau \sum_{i=1}^{n} \log(X_i) \sim \Gamma(n,1)$$ I need to find a single equation and that solve it numerically. Thank you for your help

Answer 1

Assuming $X_1,X_2,\ldots,X_n$ are i.i.d with pdf

$$f(x)=\frac{\tau}{x^{\tau +1}}\mathbf1_{x>1} \,,\quad \tau>0$$

I suggest an alternative way to go about this.

A UMPU test as you said is of the form

$$\phi(t)=\begin{cases}1&,\text{ if }t<q_1 \text{ or }t>q_2 \\ 0 &,\text{ otherwise }\end{cases}$$

For a size $\alpha$ test, $(q_1,q_2)$ are such that $E_{H_0} \phi(T)=\alpha$ and $\operatorname{Cov}_{H_0}(T,\phi(T))=0$.

Note that $2T\sim \chi^2_{2n}$ under $H_0$. The two restrictions above then reduce to

$$P(q_1'<\chi^2_{2n}<q_2')=1-\alpha $$

and

\begin{align} \operatorname{Cov}_{H_0}(T,\phi(T))=0 &\iff E_{H_0}(T(1-\phi(T))=E_{H_0}(T)E_{H_0}(1-\phi(T)) \\& \iff E_{H_0}\left[2T(1-\phi(T))\right]=(1-\alpha)E_{H_0}(2T) \\& \iff \int_{q_1'}^{q_2'} \frac{e^{-x/2}x^n}{2^{n+1}\Gamma(n+1)}\,dx=1-\alpha \\& \iff P(q_1'<\chi^2_{2n+2}<q_2')=1-\alpha \end{align}

with $$(q_1',q_2')=(2q_1,2q_2)$$

So for $\alpha=0.05$ (say), if $G_{2n}$ is the cdf of $\chi^2_{2n}$ distribution, you have

$$G_{2n}(q_2')-G_{2n}(q_1')=0.95 \tag{1}$$

and $$G_{2n+2}(q_2')-G_{2n+2}(q_1')=0.95 \tag{2}$$

Now for example, we can write $0.95=0.975-0.025$, so one choice of $(q_1',q_2')$ from $(1)$ would be that pair which satisfies $G_{2n}(q_2')=0.975$ and $G_{2n}(q_1')=0.025$. Similarly, with the decomposition $0.95=0.995-0.045$, you have another choice of $(q_1',q_2')$. Take four or five of these choices and choose that pair which satisfies $(2)$ the closest. If this works out, you have an approximate solution to $(q_1',q_2')$.

You can also try the same procedure with the two equations you derived involving $q_1,q_2$.