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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{...
Robert Long's user avatar
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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 ...
user2338823's user avatar
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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 ...
Ben's user avatar
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Asymptotically Normally Distributed

This is answered, but here's another explanation which might shed some more light. What is an estimator $\hat{\theta}_n$? It's just a function you calculate from the sample, i.e. $\hat{\theta}_n = h(...
Abhishek Divekar's user avatar
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Find expected value using CDF

Alternative derivation of $EX = \int_0^\infty \left(1-F_X(x)\right)\mathrm d x$ (for a positive r.v. $X$): For any $x \ge 0$ one has that $x=\int_0^x 1 \mathrm dt = \int_0^\infty \mathbf I_{t\le x}\...
Flo's user avatar
  • 103
2 votes

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 ...
User1865345's user avatar
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0 votes

expected value of a fishing strategy

The approach works the same as the dice problem, but now you work with the lightest fish instead of a single fish. You try to improve the expected value of the lightest fish and by doing that you ...
Sextus Empiricus's user avatar
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expected value of a fishing strategy

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 ...
enryuxbt's user avatar

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