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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Use only the last sample as the posterior in MCMC

I am new to statistics. After an MCMC sampler warmed up, the posterior is better estimated as the mean of several samples. (e.g. related question: http://stats.stackexchange.com//questions/56077) ...
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1answer
28 views

Convergence of Gini index

Let $\theta(F)=2\int^1_0(t-q_F(t))dt$, where $\displaystyle q_F(t)=\frac{\int_0^tF^{-1}(s)ds}{\int_0^1F^{-1}(s)ds}$. For discrete distributions I'm assuming that $F^{-1}(s)=\inf\{x:F(x)=s\}$ (the ...
3
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22 views

X vs x Notation in law of large numbers

This may be a silly question, but I can't find a concise answer. I've been studying Convergence of Random Variables in Wasserman's All of Statistics, which starts out by explaining: $X_n$ is a ...
5
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70 views

Convergence in distribution of sum implies marginal convergence?

Let $X_n, X, Y$ be random variables such that $X_n + cY \stackrel{d}{\rightarrow} X + cY $ for every positive constant $c$. Prove that $X_n \stackrel{d}{\rightarrow} X$. I know if only we have joint ...
4
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92 views

Distribution function of maximum of n iid standard uniform random variables where n is poisson distributed

I am studying probability theory on my own and am trying to work the following problem in the book - Let $X_1, X_2, . . .$ be independent, $U(0, 1)$-distributed random variables, and let $Nm \in ...
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15 views

Let $f(x)$ be the density of $X$. What is $\lim_{n\rightarrow \infty}\mathrm{E}_X[nf(X+n)]$?

The expectation can be written as $\mathrm{E}_X[nf(X+n)] = \int_{-\infty}^\infty nf(x+n)f(x)\,\mathrm{d}x$. The expectation relies on the speed of tail $f(x+n)$ goes to $0$. I posted a related ...
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39 views

Reference request: local limit theorem for log-concave densities

The following is easy to prove and can't possibly be new. But I can't find it printed anywhere despite some effort. Can anyone tell me where it is published? Let $X_1,X_2,\ldots$ be a sequence of ...
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143 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
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29 views

What to Do When a Log-binomial Model's Convergence Fails

There are times when one might want to estimate a prevalence ratio or relative risk, in preference to an odds ratio, for data with binary outcomes - say, if the outcome in question isn't rare, so the ...
5
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71 views

Simulating Convergence in Probability to a constant

Asymptotic results cannot be proven by computer simulation, because they are statements involving the concept of infinity. But we should be able to obtain a sense that things do indeed march the way ...
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32 views

Logistic Regression doesn't converge if I use a particular baseline

Apologies if the question is too broad, then please close the thread. Or maybe this belongs in StackOverflow? I observed something strange and wonder if someone more educated can shed some light on ...
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36 views

Coefficients from a Dynamic Panel Data Model of Economic Growth

I am having difficulties with the interpretation of the regression results from estimating a growth regression in a dynamic panel data set-up, estimated using Stata. (I'm using difference GMM and ...
2
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22 views

Convergence of a sequence of random variables [duplicate]

Suppose $P(X=1)=P(X=-1)=1/2$ and define $$X_n=\begin{cases} X\ \text{with probability}\ 1-\frac{1}{n} \\ e^n\ \text{with probability}\ \frac{1}{n} \end{cases}$$ I then need to prove or disprove ...
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90 views

Limiting distribution of the first order statistic of a general distribution

Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i>0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)>0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting ...
3
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1answer
40 views

How to report results from analyses that did not converge

I am an analyst on a paper and, in writing up methods and results, noted that one of the proposed (logistic regression) models did not converge due to separation. I noted this in the results section ...
4
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66 views

Convergence in probability, $X_i$ IID with finite second moment

Let $\{X_i\}_{i\geq 1}$ be IID with finite second moment, and $$ Y_n = \frac{2}{n(n+1)}\sum_{i=1}^n \,i\cdot X_i \, , \qquad n\geq 1 \, . $$ Could you please tell me how can I show that $Y_n$ ...
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14 views

Uniform convergence of Monte Carlo approximation

Usually Monte Carlo method is used to compute integration. For example, let $g(x,\theta)$ be a continuous function about $x$ and $\theta$, $f(x \mid \theta)$ is a continuous pdf with parameter ...
2
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30 views

Difference between sample and sequence in Law of Large Numbers

I'm comparing the Chebychev (Weak) Law of Large Numbers (LLN) to the Kolmogorov (Strong) LLN in an econometric textbook and both definitions start off differently. The Chebychev LLN begins with If ...
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25 views

Checking convergence in MCMC with single chain

I have reed the Gelman-Rubin method for check the convergence in MCMC on $m\geq 2$ chain, but when I work with only one chain, what can i do to check the convergence? There is some method that work ...
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82 views

Gelman and Rubin convergence diagnostic, how to generalise to work with vectors?

The Gelman and Rubin diagnostic is used to check the convergence of multiple mcmc chains run in parallel. It compares the within-chain variance to the between-chain variance, the exposition is below: ...
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114 views

Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?

Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a ...
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55 views

Building an ARMA or GARCH estimation battery for models of increasing order (rugarch in r)

A loop should be build to fit ARMA and/or GARCH models of increasing order, say GARCH(0,1), GARCH(1,0), GARCH(1,1), GARCH(0,2) etc. The language is r, and I'm using the ...
2
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2answers
58 views

Convergence in distribution results not deriving from central limit theorem?

In the stats class I took, all the results I have encountered about the convergence in distribution of some random variables are in one way or another consequences of the Central Limit Theorem. Out ...
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36 views

MIMIC fail to converge

I am trying to fit a MIMIC model measuring poverty in a household sample. The model looks like this: (HUMCAP -> education job) (HOUSINGQUAL -> floor wall dwelling tv refrigerator radio electricity ...
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13 views

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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26 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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21 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 ...
0
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1answer
42 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}) = ...
2
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20 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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35 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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25 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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58 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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53 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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197 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 ...
2
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13 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 ...
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255 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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34 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 ...
8
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205 views

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

My questions are: Are generalized linear models (GLMs) guaranteed to converge to a global maximum? If so, why? Furthermore, what constraints are there on the link function to insure convexity? ...
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21 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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99 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 ...
2
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1answer
86 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) ...
4
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1answer
76 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
40 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 $$ ...
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201 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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39 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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31 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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104 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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343 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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65 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 ...