Questions tagged [slutsky-theorem]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
2
votes
0answers
32 views

Slutsky's theorem applied to a sample mean conditional on a Bernoulli variable?

Let $(Z_{i},Y_{2i})$, $i=1,2,\ldots,N$ be iid random vectors, where $Y_{2i}$ is the outcome vector and $Z_{i}\sim\operatorname{Bernoulli}(\delta)$. Assume that $E(Y_{2i}|Z_{i})=\mu_{2}+\beta Z_{i}$ ...
1
vote
1answer
84 views

Is the limit in probablity of an inverse matrix equal to the inverse of the limit in probability of the matrix?

Suppose $X_n$ is a random matrix, which converges in probability to a matrix of constants, $Y$. It seems intuitive that therefore $X_n^{-1} \xrightarrow{p} Y^{-1}$ - so the limit in probability of an ...
2
votes
1answer
52 views

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$?

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$? According to Slutsky theorem, I can directly get $X_nY_n\overset{d}{\...
0
votes
0answers
82 views

Convergence in distribution and Slutsky's theorem

It is known that from the CLT, if $X_i \stackrel{\text{iid}}{\sim} F$ for some distribution $F$ with finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\...
1
vote
1answer
108 views

product of asymptotic standard normal distribution

Suppose $Z_n\xrightarrow{d} Z \sim N(0,I_p)$, why $Z_n^TZ_n\xrightarrow{d}\chi^2_p$? I encounter this problem when we get the asymptotic distribution of the maximum likelihood estimator (MLE). Suppose ...
3
votes
2answers
127 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
46 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
49 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} ...
2
votes
1answer
350 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
550 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
108 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
527 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
176 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
684 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
73 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
150 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
289 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
345 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
262 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
4k 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 ...