# UMVUE of $\cos\theta$ when $X_i\sim U(0,\theta)$

$$X\sim U(0,\theta)$$. To find the umvue of $$\cos\theta$$ is it enough to find the umvue of theta and substitute for it. Umvue of $$\theta$$ being $$(n+1)X_{(n)}/n$$, is the answer $$\cos (n+1)X_{(n)}/n$$?

• There is no invariance property of UMVUE. – StubbornAtom Apr 22 at 4:35

Assuming you have a sample of $$n$$ observations.

The density of the complete sufficient statistic $$X_{(n)}$$ is

$$f_{X_{(n)}}(t)=\frac{nt^{n-1}}{\theta^n}\mathbf1_{0

Any function of $$X_{(n)}$$ that is unbiased for $$\cos\theta$$ will be UMVUE of $$\cos\theta$$. Let $$g(\cdot)$$ be that function.

Set up the equation

$$E_{\theta}\left[g(X_{(n)})\right]=\cos\theta\quad,\,\forall\,\theta>0$$

That is, $$\int_0^\theta g(t)t^{n-1}\,dt=\frac{\theta^n\cos\theta}{n}$$

Differentiating both sides of the last equation wrt $$\theta$$, one can solve for $$g(\cdot)$$.