Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity.

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Convergence of multivariate ECDF

Gilvenko-Cantelli assures uniform a.s. convergence of univariate ECDF. My questions are: Are there similar assurances for multivariate ECDF? How is the rate of convergence dependent on the ...
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14 views

Convergence Rate for Multivariate Kernel Density Estimators

I know that univariate kernel density functions converge uniformly a.s. to the true distribution. Is this also true for multivariate kdf? Is there a theorem that gives the rate of convergence in the ...
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12 views

Convergence of Estimation

The issue is, we can have a bunch of regression models to do prediction for the test set. For a specific testing data, some perform good and some do bad. How can I use existing method or design a ...
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1answer
22 views

On the stopping criterion of coordinate descent method for linear SVM with $\ell_1$-regularization

I am trying to implement the coordinate descent method to solve the dual of linear SVM problem, but blocked at the stopping criterion. The dual of linear SVM problem is: $$\min f(\mathbf{x}) = ...
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14 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
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32 views

Convergence of a matrix product

Let $A=o_{a.s.}(1)$; $A:k\times k$ matrix and $Vu=O_p(1)$; $V:k\times k$; $u: k\times 1$. Specifically, $Vu$ converges in distribution to $\mathcal N(0,I_k)$. Can we show that $VAu=o_p(1)$ or ...
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22 views

Any fancy application of convergence (in probability, law, CLT, etc)?

As part of the inference course in an applied stats masters degree, we've to prepare a talk about convergence (see, for example, Lehmann 1999 Chapter 2). We'll be explaining to other students some ...
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39 views

A lot of iterations before converging - GLMM vs GEE question

I'm wondering if there is any implication if a model takes 100+ iterations to converge. Can I still trust the results? I'm running cumulative logit with random intercept in proc glimmix in SAS. ...
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27 views

Is Heckman two-step overly sensitive to dummy variables?

I am doing a Heckman probit in Stata and whenever I try to include region dummy variables (which in normal probit are all significant), the model either doesn't converge, or does converge with rho ~ ...
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150 views

Convergence in Distribution\CLT

Given that $N = n$, the conditional distr. of $Y$ is $\chi ^2(2n)$. $N$ has marginal distr. of Poisson($\theta$), $\theta$ is a positive constant. Show that, as $\theta \rightarrow \infty$, $\space ...
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10 views

Geometric Mean of Uniform random variables convergence [duplicate]

I am doing some independent study in asymptotic statistics and point estimation and am aware that you can get from log transformations of uniform random variables (exponential) all the way up to ...
5
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88 views

Meaning of a convergence warning in glmer

I am using the glmer function from the lme4 package in R, and I'm using the bobyqa optimizer ...
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32 views

Asymptotic convergence involving EDF

Please help me proving this: Suppose that $Y_1,\ldots,Y_n$ are i.i.d. nonnegative RV's with CDF $F$ and $E(Y_i)=\mu<\infty$. Let $y_1,\ldots,y_n$ be a realization from which an EDF $\hat{F}$ is ...
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1answer
143 views

Does log likelihood in GLM have guaranteed convergence to global maxima?

My questions is: are generalized linear models (GLMs) guaranteed to converge to a global maximum, and if so why? Furthermore, what constraints are there one link function to insure convexity? My ...
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16 views

exact definition of rate of convergence

Can anyone concisely clarify within the framework of parametric regressions and sample size $N$ what is the rate of convergence? Increasing the exponent $k$ in $N^k$ (or $N^{-k}$) will increase or ...
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91 views

Estimating variance from sequence of random variable

Given $X\sim N(0,\Omega)$. Suppose that we can construct a sequence $\{X_n\}$ based on the observation such that $\{X_n\}\to X$ in distribution. My problem is to estimate $\Omega$ consistently using ...
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1answer
75 views

prove that this expression converges in probability to zero

Apparently $T^{-3/2} \sum\limits_{t=1}^T{y_{t-1}}u_t$ converges by law to $0.5\times T^{-1/2}\sigma^2 (X-1)$, where $X$ is a $\chi^2(1)$ random variable. $u_t$ is white noise and $y_t$ is an AR(1) ...
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1answer
71 views

Sample mean of random walk

I would like to find out if the sample mean $T^{-1}\sum\limits_{t=1}^T{y_t}$ of the simple random walk $y_t=y_{t-1}+u_t$ with $u_t \sim i.i.d$ $N(0,1)$ diverges or converges? I am looking for a ...
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24 views

Conditions for weak convergence of covariance matrix derivatives

I am considering a model that includes a covariance matrix which is a function of model's parameters: $$ \mathbf \Sigma = \Sigma(\mathbf \theta), $$ where $\mathbf \theta$ is $p$-dimensional, $\mathbf ...
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1answer
36 views

Show power divergence statistic converges to LRT statistic

This is a homework problem out of Agresti's Analysis of Categorical Data (1.34b) For counts $\{n_j\}$ the power divergence statistic for testing goodness of fit is $$ ...
8
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1answer
158 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 ...
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22 views

PROC KDE convergence problem

I'm using PROC KDE to fit a density to a large set of continuous data. I can't figure out how to get around the convergence issue below. Any suggestions? I could plot percentiles, but I prefer the ...
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27 views

Detecting convergence in Random walk

I am trying to detect convergence of a random walk on a graph. After doing some preliminary research, the Geweke convergence diagnostic seems to be most commonly used for this. This diagnostic calls ...
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88 views

Convergence in distribution (central limit theorem)

If $X_1, ... , X_n$ be iid exponential with mean $1/\lambda$. Let $S_n = X_1 + ... + X_n$. a) Show that $S_n$ is $\Gamma(n, 1/\lambda)$. Each $X_i$ is $\Gamma(1, 1/\lambda)$ by the ...
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1answer
218 views

Unable to fit repeated measures in R

I've been trying to fit a repeated measures model in R using lmer but I get an error message. I've used SAS in the past and I was able to fit a somewhat similar ...
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64 views

$X_{n}$ converges in probability where $X_{n}=\frac{Y_{n}}{n}$

Let $Y_{n}$ be a sequence of independent and identically distributed random variables and $X_{n}=\frac{Y_{n}}{n}$ . show that $X_{n}$ converges in probability. decide whether $X_{n}$ converges a.e or ...
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44 views

I want to show $Y_{n} \xrightarrow{ p} c$ ,then show that $f(Y_{n}) \xrightarrow{ p} f(c) $

Let $c \in \mathbb R$ . Let be a sequence of random variables . let $f:\mathbb R \to\mathbb R$ be continous function . if $Y_{n} \xrightarrow{ p} c$ ,then show that $f(Y_{n}) \xrightarrow{ p} f(c) $
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149 views

Find the limiting distribution of $\sqrt{n} \left(\sqrt{\bar{X}} -1 \right) $ if $\sqrt{n} \left( \bar{X}-1 \right) \to N(0,1)$

Find the limiting distribution of $\sqrt{n} \left(\sqrt{\bar{X}} -1 \right) $ if $\sqrt{n} \left( \bar{X}-1 \right) \to N(0,1)$. Can you please check my work below? In principle, the Delta method ...
4
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1answer
71 views

Limiting Distribution of $W_n=\frac{Z_n}{n^2}$ , $Z_n \sim \chi ^2 (n)$

My try ended in an awkward result. I thought it best to use the moment generating function (MGF) technique. We can derive the MGF of $W_n$ as follows: $$ E \left[ e^{tZ /n^2} \right]= ...
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1answer
38 views

Convergence in probability of a product of RVs

I have found the following very useful theorem and I would appreciate some help comprehending it fully. Theorem Let $\{X_n \} $ be a sequence of random variables bounded in probability and let $ ...
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1answer
112 views

Convergence in Probability

In the following question, I was able to compute part (i), and part (ii), but I am finding hard time to compute with part (iii). So far I have computed the MLE of $\theta$, which is ...
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2answers
46 views

Limiting pmf as $n \to \infty$

Consider the simple pmf: $$p_n (x)=\begin{cases} 1\quad x=2+1/n \\ 0\quad \text{elsewhere} \end{cases}$$ Then my book states that $\lim_{n\to \infty} p_n (x)=0$ for all values of $x$. Is that really ...
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1answer
26 views

determining the limiting distribution of a consistent estimator

Setting: $X_i$ ~ i.i.d $Uniform(\theta,2\theta)$, and say we have shown that $X_{(1)}$, the first ordered statistic, is a consistent estimator for $\theta$. Now we want to show that $$n \left( ...
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21 views

Variational distance between samples of the same distribution

Suppose that I have $n$ independent samples from a probability distribution. Then, I divide them into two separate sets both with cardinality $n/2$. I compute the pdf from the first and the second ...
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1answer
72 views

Law of Large numbers and central limit theorem

I have come across a bit of a dilemma in my exam revision, this is not a topic I am particularly strong with so help is most appreciated: Assume that $X_1,X_2,...$ is an i.i.d sequence, where ...
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1answer
65 views

What characteristics of the distribution of a test statistic can be inferred using a bootstrap?

Setup: Let $p_n(x) = \mathbb{P}(S_n \leq x)$ and let $p_n^*(x) = \mathbb{P}^*(S_n^* - S_n \leq x)$, where $S_n$ is some zero-mean test statistic that can validly be bootstrapped, eg a sample mean, and ...
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1answer
75 views

Why do we examine the rate of convergence when we find a confidence interval?

I would like to help me with this. I don't understand why do we examine the rate of convergence. Also what do we mean by saying "the error is usually dominated by the variance, not the bias" ...
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44 views

How to show a series with Cauchy converges in distribution to a degenerate random variable? [closed]

I would like to show that $\sum \frac{X^m}{a_n k^2}$ from $k=1$ to $k=n$ converges in distribution to the degenerate random variable 0. We have that some constant $m > 0$, $X$ ~ $Cauchy$, or the ...
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1answer
75 views

Convergence in probability and inequality

Is it true that if $$ 0 \leq x_n \leq y_n \xrightarrow{p} 0,$$ then $$x_n \xrightarrow{p} 0?$$ If this is true, I would be very grateful if you could give me some references where I can find the ...
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1answer
82 views

Fractional polynomial model not converging in Stata

I am having trouble getting a fractional polynomial logistic model to converge using Stata 12. fracpoly, logistic: y x I have tried centering the dependent ...
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1answer
97 views

Deriving the asymptotic distribution of a particular equation

Question: Assume we have the following equation: $$\widehat{\Theta}(\rho) = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) \ \ \ \ ...
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1answer
110 views

Test the convergence of a neural network

I'm using Weka multilayer-perceptron classifier to do classifications. I want to know after exactly how many epochs the neural network converges(weights don't update any more). I'm using its java API, ...
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52 views

Convergence in Probability of the minimum

This is a homework question. I think I have the correct answer, but I am not sure. Also, the wording sounds very awkward. Is there a better way to show this (or better way to word this)? Let ...
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49 views

Dramatic changes in the results from glmm with slight changes in dataset

I am running a GLMM in R using the glmmPQL code from the MASS package. I have been getting a convergence error from the following code. ...
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1answer
74 views

Rate of convergence for SLLN

I am interested in writing a non-asymptotic rate of convergence for SLLN as a function of number of samples. From the literature I've read so far, CLT provides an asymptotic convergence rate of ...
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2answers
94 views

Proof of Convergence in Probability

Theorem Suppose $X_n \stackrel{P}\to X\quad \text{and}\quad Y_n \stackrel{P}\to Y $. Then $X_n+Y_n \stackrel{P}\to X+Y $. Proof Let $\epsilon >0 $ be given. Using the triangle inequality, we can ...
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1answer
51 views

Convergence in probability of minimum

This is a homework problem. Suppose we have a random sample $X_1,\ldots,X_n \overset{iid}{\sim} F$ with density $f(x) = 2(x-\theta)$ for $x\in (\theta,\theta+1)$. Let $X_{(1)} = ...
2
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2answers
109 views

Convergence in probability of reciprocal

This is a homework problem. If $X_n$ converges in probability to 1, show $X_n^{-1}$ converges in probability to 1. My attempt: $$\begin{align*} P(|X_n^{-1}-1| > \epsilon) &= P(|X_n^{-1}-X_n ...
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1answer
91 views

Convergence in distribution

For a statistic $T_n = \frac{1}{n} \sum_{i=1}^nY_i - \frac{1}{a}$. Prove directly (without CLT) that scaled and appropriately shifted version of $T_n$ converges in distribution to $N(0,1)$. [EDIT] ...
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2answers
115 views

Show $Y$ converges to $a$

Given: $f_{Y_{(1)}}(y) = nbe^{-nb(y-a)}$, where $b> 0$ and $y \geq a$. Show that as $n \rightarrow\infty$, $Y_{(1)}$ converges to $a$ in probability. I have calculated $E[Y_{(1)}] = \frac{1}{nb} ...