4
votes
Accepted
Error in Derivation for Control Variate Variance?
Your derivation is correct; the two expressions for $\text{Cov}(f,h)$ referred to in the comments are the same:
$$\mathbb{E}[f(h-\mathbb{E}h)] = \mathbb{E}[fh]-\mathbb{E}[f\mathbb{E}h] \\
= \mathbb{E}...
- 41.1k
3
votes
Accepted
How is this minimum variance worked out for this importance sampling estimator?
An intuitive explanation is that we want $q$ to be large whenever either $p$ or $|f|$ is large. Otherwise, our estimate of $E_p[f]$ might have a lot of error, since we're "missing out" on sampling the ...
- 26.5k
2
votes
How is this minimum variance worked out for this importance sampling estimator?
An easiest and intuitive answer [in addition to the earlier one that is completely to the point!] is that, when $f$ is a positive function, the variance of the resulting optimum is$$\text{var}[\hat ...
- 108k
2
votes
The UMVUE of ratio of parameters for two uniform distributions,
First, one only need look at sufficient statistics:
Second, one need find an unbiased estimator based on a sufficient statistic:
Last, one can call for a completeness argument.
- 108k
2
votes
Accepted
Proving an Estimator of the sample variance to be MVUE
I'm pretty sure the question was actually asking about the Normal case, but the general case is interesting (if unhelpful).
The statement is true, under various much weaker conditions. In order to ...
- 46.8k
1
vote
Accepted
Minimizing variance of sequence of independent but not identically distributed random variable
Since $X_i$s are independent of each other,
$$
\begin{align}
Var{\sum_{i=1}^n w_i X_i} &= \sum_{i=1}^n w_i^2 Var X_i \\
&=\sum_{i=1}^n iw_i^2
\end{align}
$$
Apply Cauchy's inequality,
$$
\sum ...
- 43
1
vote
Accepted
Experimental Design: Choose Data Points to Minimize Quadratic Term Variance in Multiple Regression
First, due to the conditions on $k$ values, we know that there are $nk_1$ points at $-1$, $nk_2$ points at $0$ and $nk_3$ points at $1$. For each such point, there's a respective row in $X$: Either $(...
- 4,037
1
vote
Rao-Blackwell for Minimum-Variance Unbiased Estimator
Hint: Write out the sampling distribution for the statistic $|X|$. It is an extremely simple and well-known distributional form. Once you have found this, finding the MVUE should not present much ...
- 133k
1
vote
Accepted
Showing that the minimum-variance estimator is the OLS estimator
Because $\sum_{i=1}^n a_i = 0$, $\sum_{i=1}^n a_i x_i =\sum_{i=1}^n a_i x_i -\bar x \sum_{i=1}^n a_i = \sum_{i=1}^n a_i (x_i-\bar x)$. So (4) can be written as
$$V(\tilde{\beta}) = \frac{\sigma^2 \...
- 7,561
1
vote
Accepted
How to measure how "good" or accurate a probability distribution is? Entropy, variance or what?
By definition, the notion of entropy is the one that gives you the amount of knowledge in a probability distribution (from Shannon information theory).
I would say that the variance is only one part ...
1
vote
Accepted
Good parameter estimates vs good computed moment estimates
You have data from some distribution family $f(y; \theta)$ and some estimator $\hat{\theta}$ of $\theta$ with "good properties". But you are interested in some function of $\theta$, say $g(\theta)$ (...
- 82.8k
1
vote
A Proof of Tukey's Inequality
See https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-19/issue-1/Approximate-Weights/10.1214/aoms/1177730297.full?tab=ArticleLinkCited or the Casella&Berger's "...
- 639
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