Questions tagged [slutsky-theorem]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
2answers
57 views

asymptotic normality for MLE

Suppose under suitable assumptions, $$[I(\theta_0)]^{1/2}(\hat{\theta} - \theta) \xrightarrow{d} N(0, I_p),$$ where $\hat{\theta}$ is maximum likelihood estimator of $\theta$. $I(\theta_0) = I(\theta)|...
0
votes
0answers
16 views

Convergence in distribution of random sequence plus sequence of numbers

Let $X_n$ be a sequence of random variables and $a_n$ be a sequence of real numbers. Slutsky's Theorem says that if $Y_n$ is another random sequence and $Y_n\overset{d}{\to} Y, X_n\overset{d}{\to}X$, ...
2
votes
1answer
43 views

Convergence Question

Suppose that $X_m \in \mathbb{R}^d$ and $W_m \in \mathbb{R}^{d \times d}$ be a sequence of random variables such that the following asymptotic statements are true \begin{equation*} \begin{aligned} ...
1
vote
1answer
95 views

CLT + Slutsky for the t-test

If we want to test a mean and are lucky enough to know the population variance, we can use a z-test. Even if our population is not normal, for a sufficiently large sample, we can appeal to the central ...
1
vote
0answers
168 views

How to calculate confidence interval and p-value for percent change of treatment relative to control?

I'm analyzing the result of an experiment where the dependent variable is a count variable (# of purchases), and the unit of observation is an individual. The way I'm calculating the treatment effect $...
0
votes
0answers
57 views

convergence in distribution when parameters converges almost surely

let $X_n$ be sequence of random variables with the associated distribution $N(\mu_n,\beta_n+E)$. That is a sequence of normally distributed random variables with changing mean and variance. $\beta_n$ ...
8
votes
3answers
292 views

When does $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y$ imply $X_n+Y_n\stackrel{d}{\rightarrow}X+Y$?

The question: $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y \stackrel{?}{\implies} X_n+Y_n\stackrel{d}{\rightarrow}X+Y$ I know that this does not hold in general; Slutsky's ...
1
vote
1answer
123 views

Proving a remainder term converges to 0 in probability

So we have these definitions: σ̂^2_1= (1/n)∑(Xi−μ)^2 σ̂^2_2= (1/n)∑(Xi−Xbar)^2 I have shown that n^0.5(σ̂^2_2−σ^2)= n^0.5(σ̂^2_1−σ^2)- n^0.5(Xbar-μ)^2 I am trying to show that the remainder term ...
1
vote
1answer
293 views

Slutsky's theorem applied to Bernoulli random variables

Suppose $X_n$ is a sequence of $n$ Bernoulli random variables with unknown $p$, and I try to get a confidence interval for $p$. Using central limit theorem, I've got \begin{align*} \frac{\bar X_n - p}...
2
votes
2answers
64 views

Converges in distribution

Let, $X_1, X_2, \ldots, X_n$ be i.i.d. RVs with mean $0$ variance $1$ and finite fourth order raw moment. Find the limiting distribution of $Z_{n}=\frac{\sqrt{n}(X_{1}X_{2}+X_{3}X_{4}+\cdots+X_{2n-1}...
3
votes
1answer
99 views

Distribution convergence of $n\Big(\log(n)\Big) X_{\min} /X_{\max}$

I'm figuring out the asymptotic distribution of the following. $$ n\Big(\log(n)\Big) X_{\min} /X_{\max} $$ where $X_1, \cdots, X_n \overset{iid}{\sim} \operatorname{Exp}(1)$ I know it converges in ...
3
votes
1answer
230 views

Proving the delta method

I am interested in proving the delta method, where we show that $$\sqrt{n}(g(Y_n) - g(\theta)) \overset{\text{Dist}}{\to} \text{N}(0, \sigma^2 g'(\theta)^2).$$ We use Taylor expansion where $$g(Y_n)...
2
votes
2answers
267 views

Intuition about Central limit theorem

Suppose $X_{1}, X_{2}, ....$ is a random sample from a probability distribution with mean $\mu < \infty$ and $\sigma^{2} < \infty$. Let $\bar{X}_{n} = \frac{\sum X_{i}}{n}$. From the Central ...
1
vote
0answers
222 views

Proving convergence to normal distribution using the delta method

The following is a question and instructor provided solution from a recent exam I took. Question: Prove that $\sqrt{n}T_n$ converges in distribution to N(0, 4). $T_n$=$(\bar{X}^2 - \hat{\sigma}^2)/\...
2
votes
2answers
2k views

How does Slutsky's theorem extends when two random variables converge to two constants?

The Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random element $X$ and $Y_n$ converges in ...
12
votes
2answers
2k views

Is Slutsky's theorem still valid when two sequences both converge to a non-degenerate random variable?

I am confused about some details about Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random ...