# Questions tagged [slutsky-theorem]

The tag has no usage guidance.

21 questions
Filter by
Sorted by
Tagged with
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 $E(Y_{2i}|Z_{i})=\mu_{2}+\beta Z_{i}$ ...
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 ...
52 views

If $X_n\overset{p}{\rightarrow}0$, and $Y_n\overset{d}{\rightarrow}Z\sim Normal$, does $X_nY_n\overset{p}{\rightarrow}0$? According to Slutsky theorem, I can directly get $X_nY_n\overset{d}{\... 0answers 82 views ### Convergence in distribution and Slutsky's theorem It is known that from the CLT, if$X_i \stackrel{\text{iid}}{\sim} F$for some distribution$Fwith finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\... 1answer 108 views ### product of asymptotic standard normal distribution Suppose Z_n\xrightarrow{d} Z \sim N(0,I_p), why Z_n^TZ_n\xrightarrow{d}\chi^2_p? I encounter this problem when we get the asymptotic distribution of the maximum likelihood estimator (MLE). Suppose ... 2answers 127 views ### asymptotic normality for MLE Suppose under suitable assumptions,$$[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)|... 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, ... 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} ... 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 ... 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 ... 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 ... 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 ... 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 ... 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}... 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}... 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} $$where X_1, \cdots, X_n \overset{iid}{\sim} \operatorname{Exp}(1) I know it converges in ... 1answer 289 views ### Proving the delta method I am interested in proving the delta method, where we show that$$\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)... 2answers 345 views ### Intuition about Central limit theorem SupposeX_{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}$. From the Central ... 0answers 262 views ### Proving convergence to normal distribution using the delta method The following is a question and instructor provided solution from a recent exam I took. Question: Prove that$\sqrt{n}T_n$converges in distribution to N(0, 4).$T_n$=$(\bar{X}^2 - \hat{\sigma}^2)/\...
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 ...
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 ...