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$X\sim U(0,\theta)$. 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 $\cos (n+1)X_{(n)}/n$?

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    $\begingroup$ There is no invariance property of UMVUE. $\endgroup$ – StubbornAtom Apr 22 at 4:35
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Assuming you have a sample of $n$ observations.

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

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

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 $g(\cdot)$.

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