# Standard Error of a function of ML estimators

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 $$\hat\tau$$?

• 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. – StubbornAtom Jan 2 at 7:06
• 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). – yalex314 Jan 2 at 7:38
• Why would you need the Fisher information to derive the distribution of $\hat\tau$, which is a linear transform of $(X_{(1)},X_{(n)})$? – Xi'an Jan 2 at 8:16
• 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. – yalex314 Jan 2 at 9:46
• @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 $(\hat a,\hat b)$. – StubbornAtom Jan 2 at 14:11

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

, 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.