3
votes
0answers
27 views

Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
1
vote
1answer
44 views

convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
1
vote
1answer
19 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
0
votes
1answer
31 views

stats - limiting distribution of $X_i$

Suppose $P(X_{n}=i) = \frac{n+i}{3n+3},$ for $i = 0, 1, 2$. Find the limiting distribution of $X_{n}$. Is the cdf $F_{x_{n}}$ like: " if x = 0, $F_{x_{n}} = \frac{n}{3n+3}$; if x = 1, $F_{x_{n}} = ...
0
votes
2answers
29 views

stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
0
votes
1answer
48 views

stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
1
vote
0answers
26 views

Multivariate convergence in distribution

Assume $X_i$ are iid with mean 0 and variance $\sigma^2$ and $E(X^3_i) =0$. Define $\bar{X}$ and $S^2 = \frac{\sum(X_i^2)}{n}-\bar{X}^2$ Prove that convergence in Distribution of $$ \sqrt(n) ...
1
vote
0answers
45 views

Convergence in Probability of an estimator

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
1
vote
0answers
25 views

Convergence rate of an estimator

Say we are interested in estimating some unknown real scalar parameter $\alpha$ using data. Suppose the estimator $\widehat \alpha_N$ of $\alpha$ using the data is consistent. I want to know what it ...
0
votes
0answers
33 views

Why is Expectation Maximization algorithm guaranteed to converge to minimum, even local?

I have read a couple of explanations of EM algorithm (e.g. from Bishop's Pattern Recognition and Machine Learning and from Roger and Gerolami First Course on Machine Learning). The derivation of EM is ...
0
votes
0answers
91 views

Convergence of empirical quantiles

Let $X^1,\dots,X^n$ be a sample of (not necessarily iid) random variables and denote $$F^n(x)=\frac{1}{n}\sum_{i=1}^n \mathbf 1_{X^i\leq x}$$ the empirical distribution function. Suppose that we know ...
4
votes
2answers
68 views

Convergence of random increment of two bins

If bins A and B are initialized to 1, and then I continually increment one of them by 1 with a probability proportional to their values, would A/B ever converge? To demonstrate ...
1
vote
1answer
159 views

Shifted exponential limiting distribution

Assume we have random variables $X_1,..,X_n$ which have the pdf $f(\theta):=\exp(-(x-\theta))$ for $x>\theta$ and zero elsewhere. Let $X_{(1)} =\min(X_1,...,X_n)$ then I want to find the pdf of ...
0
votes
0answers
59 views

Asymptotic distribution/properties of ratio of sample mean and (biased) sample variance

Define: $$Q = \frac{2\overline{x}}{\left(s_x^*\right)^2} +1 $$ where: $\overline{x} = \frac{1}{T} \sum_{t=1}^T x_t$ $\left(s_x^*\right)^2 = \left(\frac{T-1}{T}\right)s_x^2$ where $s_x^2 = ...
0
votes
0answers
64 views

Delta Method : Ambiguity in the proof by K. Knight

I am posting this message because I read the proof of the Delta Method (theorem 3.4) in K. KNIGHT, Mathematical Statistics, Chapman \& Hall CRC, 2000, p120-121. and there is something that ...
0
votes
0answers
62 views

Weak and strong consistency of estimators application needed

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
2
votes
1answer
36 views

Find a sequence $a_n$ such that $X_{(n)} - a_n$ converges in distribution, where $X_{(n)} = \max(X_i), i=1:n$

Let $X_1,X_2,...,X_n$ be iid and exponential(1). Define $X_{(n)} = \max(X_i), i=1:n$. What is a sequence $a_n$ such that $X_{(n)} - a_n$ converges in distribution? I think it would also converge to ...
0
votes
0answers
80 views

Limiting distribution of standard Negative Binomial Random Variate.

Let $X_{\alpha}\sim NB(\alpha,p)$ $$P(X=k)=\binom{\alpha+k-1}{k}p^k(1-p)^{\alpha};\quad x=0,1,\ldots$$ $\alpha>0$ and $0<p<1$ let $$W_{\alpha}=\frac{X_{\alpha}-\mathbb ...
2
votes
1answer
217 views

Is the MLE strongly consistent and asymptotically efficient for exponential families?

It is known that the Maximum Likelihood Estimator (MLE) is strongly consistent and asymptotically efficient under certain regularity conditions. By strongly consistent I mean that $\hat{\theta}_{MLE} ...
1
vote
2answers
150 views

Does convergence in probability not imply convergence in distribution for Least Squares estimators?

I have a question relating to convergence in probability and distribution for least squares estimators. Frequently, I see in textbooks that $\hat{\beta} \rightarrow^p b$. Where $b$ is the population ...
1
vote
0answers
88 views

Convergence almost surely via Simple linear regression

Let $X$ anf $Y$ be two random variables with finite second moments ($EX^2 \leftarrow\infty$ , $EY^2 \leftarrow\infty$) and are related by $Y=aX+b+\epsilon$ when $a,b \in \mathbb R$ and $\epsilon$ is a ...
1
vote
0answers
59 views

Convergence in distribution of a quadratic form

If $Q_n=X_nM_nX_n=\sum_{i,j=1}^n X_i m_{nij}X_j$, $X_n=(X_1,...,X_n)$ where $X_j$ are iid random variables and $M_n=(m_{nij})$ is a symmetric matrix with extending rownumber in $n\to\infty$. Iam ...
1
vote
1answer
58 views

Help with understanding a derivation in a book

I'm reading a book on time series analysis and I came across with a derivation I cannot understand. The following picture is a part from my book and I have highlighted with red color the part I'm ...
0
votes
1answer
239 views

Does sample mean converge in distribution to population mean?

$X_i$ is i.i.d. random variable. $\overline{X}=\frac{1}{n}\sum X_i$ Can we say that $\overline{X} \stackrel{d}{\longrightarrow } E(X)?$
0
votes
1answer
69 views

Convergence of probability densities

I appreciate if you can give me some guidance on how to approach this question: Suppose $f_n(x) \text{ and } g(x)$ are densities such that for all x, $f_n(x) \rightarrow g(x)$ as $n \rightarrow ...
2
votes
0answers
37 views

Convergence of $c_j = \sum_{k=1}^nx_k\cos(2\pi\frac{k-1}{n})j$

S. Kim, K, Umeno, and A. Hasegawa, Corrections of the NIST Statistical Test Suite for Randomness (available at http://arxiv.org/pdf/nlin/0401040.pdf) mention page 8-9 that: $c_j = ...
1
vote
1answer
30 views

Condition for Law of Large Numbers, Monte Carlo

In some lecture notes I am reading, there is the following; Consider $X_{1},...,X_{n}$, each with pdf $g$ (the instrumental distribution). Our aim is to estimate $E_{f}[h(X)]$ where $h(X)$ is some ...
1
vote
1answer
75 views

Stochastic variable equals indicator function?

An exercise in my statistics & probability theory course goes as follows: $\Omega = [0,1], \mathcal{B} = \mathcal{B}([0,1]), P$ the Lebesgue measure on $[0,1]$. We have the sequence of ...
0
votes
1answer
42 views

Central Limit Theorem with (linear) weights

I have a question about the CLT. Suppose we have the independently and identically distributed random variables $X_i$ with mean $\mu$ and variance $\sigma^2$. Then, by the Central Limit Theorem $$ ...
3
votes
1answer
103 views

Godambe estimating equation (Proof)

Let $Y_1, \ldots ,Y_n$ be iid with density $f(y;\theta)$. We assume that $\dfrac\partial{\partial\theta}\log f(y ; \theta)$ and $\dfrac{\partial^2}{\partial\theta^2}\log f(y ; \theta)$ exist for all ...
1
vote
1answer
61 views

Convergence of a running average

I'm trying to understand convergence in probability and have a specific problem; how do you show that the sample mean of n random variables $(X_k)$, each of which is the mean of $Y_{k-1}$ through ...
4
votes
1answer
380 views

Continuous Mapping Theorem (CMT) for a sequence of random vectors

I need help proving the Continuous Mapping Theorem (CMT) for random vectors. I'm currently reading Econometric Analysis for Cross Section and Panel Data by Jeffrey M. Wooldridge (Chapter 3, pp. 40 - ...
0
votes
0answers
116 views

When does distribution bootstrap mean converge to distribution sample mean?

Let $\bar{X}_n$ denote the sample mean of n iid random variables. Let $\bar{X}^*_n$ be the bootstrap sample mean. Does $\left|\mathbb{P}\left(n^{1/2}\bar{X}_n\leq ...
1
vote
0answers
68 views

Estimating the number of observations from a set of samples

I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs. All of the hidden inputs are driven by an experimenter ...
2
votes
0answers
65 views

Root Convergence rate of Iterative Scheme [closed]

I have an iterative sequence for optimizing an EM algorithm based loss function $L(X)$ with $t$ being the iteration number as: $X_t=ABX_{t-1}+CX_{t-1}+X_{t-1}$ where $A$ is a diagonal matrix, $B$ and ...
2
votes
2answers
259 views

An interesting problem about almost sure convergence

Assume that $X_1,X_2,\ldots$ are independent random variables (not necessarily of the same distribution). Assume that that $Var[X_n]>0$ for all $n$. Assume also that $$\sum_{n=0}^\infty ...
0
votes
1answer
183 views

Almost sure convergence of random variables

Assume that $X_n$ are independent (but not necessarily of the same distribution) and that $Var[X_n]>0$ for all $n$. We know that $$\frac{X_n-E[X_n]}{n}\to 0 \textrm{ almost surely as ...
1
vote
0answers
541 views

So is this, finally, the difference between convergence in probability and almost sure convergence?

I've been trying to come up with a intuitive, practical distinction between convergence in probability and convergence almost surely. Can someone please tell me if the following is correct? Let $X_n$ ...
0
votes
1answer
240 views

Incorrect use of Borel Cantelli?

I am confronted with the following argument which I think may not be right: Let $(X_n)$ be a sequence of independent random variables s.t. $$P[X_n = 1] = 1- P[X_n = 0 ] = \frac{1}{n}$$ in order ...
1
vote
1answer
814 views

Chi-square approximation to standard normal (0,1)

Supose that $S_n$ has a $\chi^2$ distribution with $n$ degrees of freedom. Show that $$ \mathbb{P}(S_n \le x) = f\left(\sqrt{2x}-\sqrt{2n}\right) $$ where $f(u)$ is the normal distribution. I tried ...
1
vote
1answer
98 views

Convergence of Bayes Error to zero

Given, $$ 0<p_{i} ,q_{i} <1 $$ define ${E}{}_{k}$ as, $$\begin{array}{l} {E_{1} =min(p_{1} ,q_{1} )+min(1-p_{1} ,1-q_{1} )} \\ {E_{2} =min(p_{1} p_{2} ,q_{1} q_{2} )+min(p_{1} (1-p_{2} ...