Questions tagged [asymptotic-covariance]

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Convergence in probability with double limits

Suppose you have a sequence of random variables $ \left\lbrace X_{i}\right\rbrace_{i=1,...,n}$ which converges in probability to a random variable $X$, shown by $ X_n \ \xrightarrow{p}\ X$ as n goes ...
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46 views

Variance of sample moments - clarification on Serfling (1980)

In "Serfling, R. J. (1980). Approximation theorems of mathematical statistics", we read In Theorem A, as one suspects, $k=1,2,...$, indicating the integer-moments, while $n$ is the sample ...
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35 views

Do these two random variables have the same asymptotic distribution?

Let $\{X_k\}$ be a sequence of dependent random variables with mean 0. Define $\bar{Y}_k = \frac{1}{\sqrt k}\sum_{i=1}^k X_i$. Let $\{W_k\}$ be a sequence of i.i.d. random variables with mean 1 and ...
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36 views

Linear regression with one generated regressor

Suppose I have the regression model: $Y_i=T^{\top}_{i}\beta_0+e_{i}$ with $E(e_i|X_i)=0$, where we have two regressors $X_i,\ E(D|X_{i})$ so that $T^{\top}_{i}=[X_i,\ E(D|X_{i})]$. $X_{i}$ is a ...
3
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143 views

variance of an autoregressive process

Let $\{x_t\}_{t\in\mathbb{N}}$ be a zero mean strictly stationary sequence of random variables and $c:\mathbb{N}\to\mathbb{R}$ the (auto)covariance function. If the process follows the AR(1) model $$...
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0answers
8 views

How to proof the asymptotic properties of the penalized spline estimator using asymptotic notations?

Please could someone proof how the Average Mean Squared Error of penalized spline estimator is given as \begin{eqnarray} AMSE(\hat{l} )=\...
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0answers
50 views

What is the sample form of the following variance

If $x_1, \ldots, x_n \in \mathbb{R}^p$ i.i.d. a specific distribution. denote: \begin{equation*} \mathbf{X} = \left( \begin{array}{c} x_1^{\top} \\ \cdots \\ x_n^{\top} ...
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1answer
95 views

OLS estimator of ARMA(1,1) process

When I solved the DGP in the picture, I got an ARMA(1,1) process with intercept term (1-a)*mu. To solve my problem I need the (X'X)^-1(X'Y) form equation of "mu hat". How can I derive the equation in ...
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0answers
2k views

Variance of evidence lower bound(ELBO) loss function

When using Bayesian optimisation in a neural network our loss function is equal to: Here the first term is the KL divergence between the approximate and true posteriors. The second term is the ...