Let $X_1, X_2,...,X_n$ be $iid$ continuous uniform $\mathcal{U} (0,\theta)$ and let $T=Max(X_i)$ Show that the family of distributions of T is complete.
Step I: Find the CDF (using independence property, and identical property): $$F_T(t)=P(X_{(n)}<t)=\prod_{i=1}^{i=n} P(X_i<t)=[P(X<t)]^n=\left(\frac{t}{\theta}\right)^n$$
Step II: Differentiate to get the PDF $$f_T(t)=\frac{d}{dx}F(t)=\frac{nt^{n-1}}{\theta^n}$$
Step III: Completeness requires showing $$E\left(g(T)\right)=\int_{0}^{\theta} g(T)\frac{nt^{n-1}}{\theta^n} dt=0 \implies \int_{0}^{\theta} g(T)t^{n-1}=0 \implies g(T)=0$$
The above is true since $n\in \mathbb{N} \implies x^{n-1}$ is polynomial or $x=1$, and hence it is a function with non-zero area between it and the $x$ axis. For the integral to be zero, g(T) must be zero.
I am concerned about two specific issues:
- Does T is a complete distribution mean T is a complete statistic.
- Is the last step regarding the integral valid.
Note: Could I have done this faster by simplying observing that $$\forall i, X_i>0$$ And hence the expectation will be zero only if $g(T)=0$
