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The background of the problem is as follows:

Suppose $X_1,...,X_n \sim U(a,b)$ independently where $a$ and $b$ are unknown parameters and $a < b$. Let $\hat\tau$ be the MLE of $\tau$, where $\tau = \int x dF(x)$.

How to analytically find the standard error of $\hat\tau$?

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  • $\begingroup$ Since $\tau=E(X_1)=\frac{a+b}{2}$, so $\hat\tau=\frac{\hat a+\hat b}{2}$ where $(\hat a,\hat b)$ is MLE of $(a,b)$. So looks like you have to find the variance of $\hat\tau$ for starters. $\endgroup$ – StubbornAtom Jan 2 at 7:06
  • $\begingroup$ As the sample space depends on the parameters a,b. The regularity condition does not meet. I have difficulty finding the Fisher information matrix for (a, b). $\endgroup$ – yalex314 Jan 2 at 7:38
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    $\begingroup$ Why would you need the Fisher information to derive the distribution of $\hat\tau$, which is a linear transform of $(X_{(1)},X_{(n)})$? $\endgroup$ – Xi'an Jan 2 at 8:16
  • $\begingroup$ Do you mean to directly take a variance of $\hat\tau$ and convert it to a linear combination of $\hat a$, $\hat b$? But in this case, I am not sure if $\hat a$ and $\hat b$ are independent. $\endgroup$ – yalex314 Jan 2 at 9:46
  • $\begingroup$ @yalex314 Yes, $\hat a=X_{(1)}$ and $\hat b=X_{(n)}$ are definitely dependent. But that is not a problem to find variance of $\hat \tau$ since we know the distribution of $(\hat a,\hat b)$. $\endgroup$ – StubbornAtom Jan 2 at 14:11
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As mentioned in comments, $\tau=E(X_1)=\frac{a+b}{2}$.

If $(\hat a,\hat b)$ is the MLE of $(a,b)$, then by invariance property MLE of $\tau$ is $$\hat\tau=\frac{\hat a+\hat b}{2}$$

Verify that $\hat a=X_{(1)}$ and $\hat b=X_{(n)}$, the minimum and maximum of the sample observations respectively.

You are looking for the standard error (standard deviation) of the statistic $\hat\tau$. So equivalently find the variance of $\hat\tau$ and take the positive square root.

$$\operatorname{Var}(\hat\tau)=\frac{1}{4}\left[\operatorname{Var} X_{(1)}+\operatorname{Var}X_{(n)}\right]+\frac{1}{2}\operatorname{Cov}(X_{(1)},X_{(n)})$$

The distributions of $X_{(1)}$ and $X_{(n)}$ are straightforward to derive, from which you can find the mean and variance of each. For the covariance, refer to the joint distribution (also easy to derive) of $(X_{(1)},X_{(n)})$:

$$g(x,y)=n(n-1)(F(y)-F(x))^{n-2}f(x)f(y)\,\mathbf1_{x<y}$$

, where $F$ and $f$ is the population cdf and pdf respectively.


In a different approach, you can derive the sampling distribution of $\hat\tau$ (sometimes called the sample mid-range) directly and find the standard error. Here is a relevant thread.

You might find it easier to define $Y_i=\frac{X_i-a}{b-a}\sim U(0,1)$ and then work with the $Y_i$'s in both methods.

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