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How can I find the limiting distribution of $Z_n=\sqrt{n}\frac{X_1X_2+X_3X_4+\cdots+X_{2n-1}X_{2n}}{X_1^2+\cdots+X_{2n}^2}$?

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $E(X_i) = 0$, $\text{Var}(X_i) = 1$, and $E(X_i^4) < \infty$. We are tasked with finding the limiting distribution of  Z_n = \sqrt{n} \frac{...
• 64k

Overfitting the (non-nested) cross validation set

We may do model selection OR hyper-parameter tuning using non-nested cross validation. Model selection vs Hyper Parameter tuning. Suppose we are doing model selection. Then we train on k-1 folds and ...
• 295

What is the distribution of $X_i-\bar{X}$ when $X_1,...,X_n \sim \text{IID N}(\mu,\sigma)$?

Here is the multivariate extension using the centring matrix While there are other answers here that solve your problem, you might be interested to know that this problem is closely related to the ...
• 129k

• 103
Accepted

Rao-Blackwell Theorem

Let $(\Omega_X, \mathscr F_X,\mathfrak M:=\{P_\theta\mid \theta\in\Theta\})$ be a statistical experiment and $\mathfrak M$ be dominated. Consider the collection of unbiased estimators of $\theta$ as ...
• 9,427
Day5 (last) Let $A,B$ be respective the min and max fish in hand. Let $t_5$ be our threshold for rerolling the minimum fish on day5. Let $E_{5}[{A,B}]$ be the value of having the option to reroll when ...