New answers tagged expected-value
2
votes
Accepted
Expectations of estimators
Jensen's Inequality for random variables is involved here.
If, say, $X$ is a random variable and a function $f$ is convex then
$$\mathbb E[f(X)] \geq f\left(\mathbb E[X]\right).$$
The reverse ...
- 60.8k
2
votes
Expectations of estimators
Specifically, why is the expected value of a ratio not equal to the ratio of expected values?
Suppose
$$
(X,Y) = \begin{cases} (1,1) \\
(1,2) \\
(2,1) \\ (2,2) \end{cases} \text{all with equal ...
- 11k
0
votes
Proof of $\mathbb{E} (|X-Y|) = 0 \implies X^2 = Y^2$
If you are allowed to use the standard fact that a measurable function vanishes almost everywhere iff the Lebesgue integral of its absolute value vanishes, then you can write
$0 = \mathbb E(|X-Y|) = \...
- 6,650
1
vote
Bayes Predictor for linear regression with square loss and expected value properties
For the squared error loss $l(\theta, a)=(\theta-a)^2$, the Bayes estimate of $\theta$ after $X=x$ is observed is given by the value of $a$ which minimises the expected squared error loss $\mathbb{E}[(...
- 51
2
votes
(More complete) proof the Fisher information is additive
\begin{align}
\mathbb E\left[\frac{\partial}{\partial\theta}\log f(X_1)
\frac{\partial}{\partial\theta}\log f(X_2)\right] &=
\mathbb E\left[\frac{\partial}{\partial\theta}\log f(X_1)\right]
\...
- 108k
4
votes
Accepted
(More complete) proof the Fisher information is additive
One has to utilize the regularity conditions which ensure that the family is stable meaning the gradient and hessian of the likelihood function are uniformly bounded in a nbd of $\theta$ by integrable ...
- 10.3k
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