Context:
I am given a Probability Density Function, and the question wants me to show that $$X_{(n)}$$ is complete.
Given pdf:
$$f(x, \theta, \phi)=\frac{1}{\theta \phi}\left(\frac{x}{\theta}\right)^{\frac{1-\phi}{\phi}}, \quad 0 \leq x \leq \theta, \quad 0<\phi<1 $$
The Density of the statistic $$T=X_{(n)}$$ is given by
$$
f_{X_{(n)}}(x)=\frac{n}{\theta \phi}\left(\frac{x}{\theta}\right)^{n / \phi-1} \quad \text { for } 0 \leq x \leq \theta
$$otherwise zero.
So for the completeness I need to satisfy following conditions:
If $$\mathrm{E}_{\theta}(g(T))=0$$ for all θ then $$\mathbf{P}_{\theta}(g(T)=0)=1$$ for all θ.
My Expectation came out to be non-zero and in terms of n,ϕ, and x.
I don't know how to move further, any help is appreciated.
Note: ϕ is to be considered known.
