In my script for statistical signals, I have some troubles to get the same result for the variance of an estimator $T$.
Here is the example:
Given the observations $X_1, \dots , X_N$ of a uniquely distributed random variable
$$ X: \Omega \rightarrow [0,\theta] $$
with $[0, \theta] \subset \mathbb{R}$ such that the CDF $F_X (\xi) = \frac{\xi}{\theta}$ and the PDF $f_X(xi) = \frac{1}{\theta}$ if $0 \leq \xi \leq \theta$.
To estimate the upper bound $\theta$ of the uniform distribution, the expected value $\mathbb{E}[X] = \frac{\theta}{2}$ is used which is the mean of this partictular uniform distribution. Then the estimator $T$ is given as:
$$ T = 2 \cdot \underbrace{\frac{1}{N} \sum_{i = 1}^N X_i}_{Average}: \quad x_1, \dots , x_N \rightarrow \hat{\theta} $$
Now the expected value for this estimator is calculated as:
$$ \mathbb{E}[T(X_1, \dots , X_N)] = \mathbb{E}[\frac{2}{N}\sum_{i = 1}^{N}X_i] = \frac{2}{N}\sum_{i = 1}^{N}\mathbb{E}[X_i] = \frac{2}{N} \cdot N \cdot \frac{\theta}{2} = \theta $$
which makes this estimator unbiased since the expected value is exactly the wanted parameter $\theta$.
Finally the Variance $Var[T]$ is just given as:
$$ Var[T] = \frac{\theta^2}{3N} $$
However, I don't know how to obtain this result.
I tried two approaches to get the same result for the variance:
In my first approach, using the definition of the variance, depending on the expected value of the estimator, there seems to be an error in my calculations or something that I miss.
The second approach gives me the same result but I don't really understand the properties/rules for the variance I used.
First approach:
According to the definition the variance is:
$$ Var[T(X_1, \dots , X_N)] = \mathbb{E}[(T-\mathbb{E}[T])^2] = \mathbb{E}[T^2] - \mathbb{E}[T]^2 $$
Now in order to get the variance, $\mathbb{E}[T^2]$ and $\mathbb{E}[T]^2$ are needed. $\mathbb{E}[T]^2 = \theta^2$ is already available (see above). In my calculations of $\mathbb{E}[T^2]$ seems to be an error:
$$ \mathbb{E}[T^2] = \mathbb{E}\left[\frac{4}{N^2}\left(\sum_{i = 1}^N X_i\right)^2\right] $$
Because of my assumed independence of $X_i, X_j$ this equation results in (where I am really not sure if this step is correct):
$$ \mathbb{E}[T^2] = \frac{4}{N^2}\sum_{i = 1}^N \mathbb{E}[X_i^2] $$
Now using functions of random variables $E[g(x)] = \int_{\mathbb{R}}{g(x)f_x (x)dx}$, where $g(x) = x^2$, the previous equation is:
$$ \mathbb{E}[T^2] = \frac{4}{N^2}N \int_{0}^{\theta}{x^2 \frac{1}{\theta} dx} = \frac{4}{N} \frac{\theta^3}{3} \frac{1}{\theta} = \frac{4}{N} \frac{\theta^2}{3} $$
However, this obviously leads to:
$$ Var[T] = E[T^2] - E[T]^2 = \frac{4}{N} \frac{\theta^2}{3} - \theta^2 = \frac{\theta^2 (4 - N3)}{N 3} $$
which is wrong. Can you please tell me where I made a mistake?
Second approach using properties of variances (correct result):
$$ Var[T] = Var[\frac{2}{N} \sum_{i = 1}^N X_i] = \frac{4}{N^2} Var[\sum_{i = 1}^N X_i] = \frac{4}{N^2} \sum_{i = 1}^N Var[X_i] = \frac{4}{N^2} N \frac{\theta^2}{12} = \frac{\theta^2}{3} $$
where I used the variance of the uniform distribution $Var[X_i] = \frac{\theta^2}{12}$ and the following rules for variance:
$$ Var[\alpha X + \beta] = \alpha^2 Var[X] $$
and
$$ Var[\sum_{i = 1}^N X_i] = \sum_{i = 1}^N Var[X_i] + \sum_{i \neq j} Cov[X_i,X_j] $$
For the second rule I assumed independence for the statistics $X_i$ which leads to $\sum_{i \neq j} Cov[X_i,X_j] = 0$. Though I am not sure if this is right but it lead to the correct result.
Could you please tell me how to derive these rules?
I would be glad to get the variance using my first approach with the formulas I mostly understand and not the second approach where I have no clue where these rules of the variance come from.
