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Questions tagged [convergence]

Convergence generally means that a sequence of a certain sample quantity approaches a constant as the sample size tends to infinity. Convergence is also a property of an iterative algorithm to stabilize on some aim value.

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Checking model after convergence issues with glmer logistic regression

I'm running a logistic regression model where subject ID is nested within litter ID, using glmer. The equation is as follows: glmer(outcome ~ group * time + previousStatus + (1|litter/pup), data, ...
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Ascertaining sub-probabilities using re-sampling of the data. Multivariate convergence?

I am trying to compute a payout for an online game that has the following information: 40% chance of a loss with value -2 10% chance of a loss with value -1 remaining 50% chance with four potential ...
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Convergence rate of Nadaraya–Watson estimator in Holder Space

I'm currently learning non-parametric regression using some online public materials. Specifically, consider the model $$ y_{i} = f_{0}(x_{i}) + \epsilon_{i} $$ where $x_{i}\in \mathcal{X} \subset \...
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Proof of Strong consistency of Beta posterior distribution

Suppose that we have random variable $X_{1}, X_{2}, ..., X_{n} \sim^{iid} \text{Bernoulli}(p_{0})$ with $p_{0}$ true unknown probability in $[0,1]$. Now, I want to implement Bayesian machinery to ...
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non-positive-definite Hessian matrix/non-convergence problem with glmmTMB

I've got a dataset that has temperature (21c or 29c), inoculation (mock (m),single inoculations (c or r), or coinoculation (rc)), and age group (y or o). I am trying to model the interactive effects ...
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Convergence of MLE for non-IID data

Consider calculating optimal model parameter $\theta$ using MLE for the following 2 cases: Data generating process is independent but non identical: $L(y;\theta) = \prod_{i=1}^{n} f_{i}(y_i;\theta)$....
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If the variance converges to zero, when do we have almost sure convergence

We have that $\mathbf{E}(X_n)=c$ where c is a positive constant and $\lim_{n \rightarrow \infty} \mathtt{Var}(X_n) =0$. Then $$ X_n \rightarrow c \quad \mbox{in probability as} \quad n \rightarrow \...
GCru's user avatar
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Do convergence rates for (convex) gradient descent apply when domain is (convex) subset of reals?

I have a convex multi-variate optimization problem where each variable lies on the domain $[x, \infty)$ for some positive number $x$. I know the problem has a unique finite solution in the domain, ...
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The sum of $O_p$ --$ O_p \left(s^2\frac{\log d}{n}+s\sqrt{\frac{\log d}{n}} \right) $

I read papers in the area of inference for high-dimensional graphical models and these papers always state the convergence rate of the estimator. Using $O_p$ is a good choice. Maybe I made some ...
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Can we have $|\overline{X}_n - E[X_1]| > \varepsilon$ infinitely often?

Let's say I have some random variables, $X_1, X_2, ...$ which are identically distributed with finite expected value, $E[X_1]$ and say they satisfy the requirements of the law of large numbers. Let $$\...
roundsquare's user avatar
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SVRG vs full gradient descent

Stochastic gradient descent allows us to avoid the computation of full gradients at the expense of introducing a noise floor to convergence. To decrease this noise floor, SGD requires a decrease in ...
hegash's user avatar
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Doubt regarding limiting distribution on Vasicek model

I was reading an article from Vasicek where he's concerned about deriving the limiting loss probability distribution on a credit risk model with 2 factors. I am here presenting a somewhat different ...
Chaos's user avatar
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Convergence of a Bayesian classifier

Background Let $y_k$ be a noisy measurement at time $k$ and let $\{p_{k-1}(i)\}_{i=1}^n$ be (a discrete) prior probability distribution. Using Bayes rule, one can update the prior in function of $y_k$ ...
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How to approximate the point a sequence is converging to?

I have created a poker solver as part of my Master's Thesis. This solver uses Counterfactual Regret Minimization (CFR) to compute a Nash Equilibrium of Hold'em or Omaha Poker. The solver uses existing ...
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Does the mean of the maxima of a set of distributions converge?

This question is related to a recent one I posted. In that question I ask what statistic might best represent the central tendency of the true discrete distribution of a property for a sample for ...
Buck Thorn's user avatar
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Almost sure convergence in Bayesian setting

Lets say I have a probability space with random variables X1,X2,.... These random variables have a parameter Θ. Given Θ, X1,X2,... are iid. This implies that conditional on Θ, the sample mean ...
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How to prove that support converges to probability?

In the body literature of Association Rule Mining (apriori algorithm is one of them) there's a lot of information about te usage of many metrics, whithin them 'support'. Support is defined as the ...
Oscar Flores's user avatar
2 votes
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Can I add two independent results from the central limit theorems?

I'm reading introduction to mathematical statistics by R. Hogg, et al. I have some trouble to understand a limiting distribution. Let $X_1,\cdots,X_{n_1}$ be iid random variables from $Bernoulli(p_1)$ ...
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Sum of asymptotically independent random variables - Convergence

Let $\theta_N=\frac{1}{N}\sum_{i=1}^N \pi_i\cdot g_i$ where $0<\pi_i<1$ and $0<g_i<1/\pi_i$ such that $\theta_N\overset{N\rightarrow \infty}{\rightarrow}\theta$. If $X_i\sim Ber(\pi_i)$, I ...
Pierfrancesco Alaimo Di Loro's user avatar
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Almost sure convergence using exponential tail bound

I have a question about a theorem in the following set of lecture notes 'A Gentle Introduction to Empirical Process theory' (http://www.stat.columbia.edu/~bodhi/Talks/Emp-Proc-Lecture-Notes.pdf). In ...
Stan's user avatar
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Which test do I use for checking chain convergence on an mcmc glmm with a factorial response variable?

I am running an mcmc glmm (mcmc package in R) with the following structure: continuous response variable + continuous response variable + factorial response variable ~ all of my covariates+etc. This ...
Juliette's user avatar
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2 answers
523 views

Asymptotic unbiasedness + asymptotic zero variance = consistency?

Here, Ben shows that an unbiased estimator $\hat\theta$ of a parameter $\theta$ that has an asymptotic variance of zero converges in probability to $\theta$. That is, $\hat\theta$ is a consistent ...
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Chain of convergence of random variables

Let $\{X_i\}_{i=1}^{\infty}$ and $\{Y_i\}_{i=1}^{\infty}$ be two sequences of random variables. If $P(X_i\neq Y_i \text{ i.o.})=0$ and $n^{-1/2}\sum_{i=1}^n Y_i \to Z $ in distribution, can I conclude ...
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Consistency of IV Estimator

I have a quick question about the proof of the consistency of the IV estimator. I following the Davidson and MacKinnon (1st ed.) text where, as one of their assumptions, they state the following ...
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Understanding Asymptotic Relative Efficiency and how to compute it

I am learning about asymptotic relative efficiency (ARE) in class, and I am trying to understand exactly how to compute the ARE. From my understanding, asymptotic relative efficiency refers to ...
Harry Lofi's user avatar
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In a mixed effects model, can leaving out the intercept-slope correlation parameter inflate type I error?

I am considering leaving out the intercept-slope correlation parameter in a mixed effects model to avoid convergence issues (i.e., in nlme::lme, ...
Evan's user avatar
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1 answer
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Stochastic boundedness in consistency proof

I'm reading Knight and Fu (2000), Asymptotics for Lasso-Type Estimators and I don't understand why (6) and (7) imply consistency in Theorem 1 (copied and pasted below). I'm familiar with the standard ...
Giacomo's user avatar
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Almost sure probability in convergence, versus 0 probability in reality

I'm having some issues with what it means for a probability distribution to converge. Consider this example problem: for a fair dice thrown $6x$ times, for an integer $x$, define $y$ as the ...
Taw's user avatar
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How can we compare the "performance" of different Markov chain Monte Carlo algorithms?

How can we judge the performance a Markov chain Monte Carlo (MCMC) algorithm? I guess we could consider one of the following: The variance of $X_t$ for a given $t\in I$; The asymptotic variance of $(...
0xbadf00d's user avatar
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Ergodicity-definition for general statistic

I'm struggling with the definition of ergodicity within time series. Consider a time series denoted as $X = (X_i)_{i\in\mathbb{Z}}$, where each $X_i$ represents a random vector defined on the same ...
Albert Paradek's user avatar
2 votes
1 answer
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Weak Law of Large Numbers: Conditional Expectations in Random Subsequences

Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
Albert Paradek's user avatar
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Prove the convergence of the sample PACF

I'm looking for a proof that the sample PACF converges in probability to the PACF in probability as T goes to infinity when m = p. Moreover, I am looking for a proof that when m > p, that the ...
Jerry Qu's user avatar
3 votes
1 answer
30 views

convergence and closeness of two estimators

Suppose $\hat{\theta}_n$ is the estimator obtained by solving $g_n(\theta) = 0$ $\tilde{\theta}_n$ is the estimator obtained by solving $h_n(\theta) = 0$ If $g_n(\theta) = h_n(\theta) + o_p(n^{\alpha})...
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Model failed to converge (gamma model, self-paced reading data)

I'm trying to run a Gamma analysis in a self-paced reading data. However, the model successively fails to converge. I've seen some answers here trying to solve this problem for other people, but none ...
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1 answer
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Assumptions needed for consistency of plug-in estimator

Assume $X,Z$ are random variables and let $x_0$ be a fixed number. I want to estimate $A =\mathbb{E}_{X,Z}[\frac{X}{P(X=x_0|Z)}]$. If $P(X=x_0|Z=z)$ is known for all $z$ we can apply the LLN and ...
James's user avatar
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1 answer
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Show convergence in distribution to a bivariate normal vector

Let $Y_1,...,Y_n$ be iid exponential random variables with mean $\theta>0$. Let $$ \hat\alpha_n:= \frac{1}{n} \sum_{i=1}^n Y_i \quad \text{and} \quad \hat\beta_n:=\sqrt{\frac{1}{n} \sum_{i=1}^n (...
Hepdrey's user avatar
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2 votes
3 answers
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Example of non-consistency of M-estimators in case of pointwise converging criterion functions

When one wants to establish consistency of an M-estimator $\widehat{\theta}_n$, one typically requires uniform convergence of the criterion function $\theta \mapsto M_n(\theta)$. That is, one requires ...
Stan's user avatar
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Expected standard errors depend on the underlying distribution in regression

I am investigating the effect of distributions of predictive variables ($x$) on their standard errors in regression. I thus programmed a little simulation to see how the standard errors behave. I used ...
POC's user avatar
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3 votes
1 answer
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Error in mixed-models. Which to detect? Collinearity? Singularity in backsolve at level 0, block 1

Firstly, I would like to admit that even though it is not the first time I am working with linear mixed models, the mathematical foundations escape me. I am running a linear mixed-effects model using ...
Javier Hernando's user avatar
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lme4 convergence warning in specific subset of data

Our main analysis consists of univariate logistic mixed models using lme4’s glmer to check for an association between plasma ...
JED HK's user avatar
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Model converges when using orthogonal polynomials but fails to converge when using raw polynomials

I am fitting linear mixed effect models with random slopes (lmer4 package), and I recently attempted to add a quadratic term due to some theory behind (the quadratic ends up also in the random slopes, ...
LiamV's user avatar
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2 votes
0 answers
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Almost sure convergence of $\frac{2}{n(n-1)}\sum\limits_{1 \leq i < j \leq n} X_i X_j$

I'm trying to prove that: Given a sequence $(X_n)_{n \geq 1}$ of independent and identically distributed random variables, $E(X_i^2) < +\infty$ for all $i \geq 1$, then $$\frac{2}{n(n-1)}\sum\...
Ta Thanh Dinh's user avatar
5 votes
1 answer
510 views

Does the Law of Large Numbers work better for some Distributions? [closed]

Here are two popular principles in Statistics: 1) Law of Large Numbers: If $X$ is a random variable with a probability density function $f(x)$ and an expected value $E[X] = \mu$. If we take a sample ...
Uk rain troll's user avatar
5 votes
1 answer
232 views

Convergence rate of a nonparametric estimator

Optimal rate of convergence for a nonparametric estimator is well-known. This rate is derived for when we don't anything about functional form (expect perhaps degree of smoothness). Suppose we know ...
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1 answer
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Convergence of squared sample average [closed]

If $y_{1}, y_{2}, . . . , y_{N}$form a sample of independent standard-normally distributed random variables and $\bar{y}$ is the sample average. Is it correct to say that $$\bar{y}^2 \overset{p}{\...
user407052's user avatar
1 vote
3 answers
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Asymptotic normality implies consistency

I'm trying without success to solve the following exercise in my econometric textbook: Show that $\sqrt{N}\left(\widehat{\beta_1} - \beta_1 \right) \xrightarrow{d} \mathcal{N}(0,a^2)$, where $a^2$ is ...
Residual Claimant 's user avatar
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Sample size in simulation and stopping criteria

I want to estimate the average of a random variable by simulation. Also, I want to estimate a proportion by simulation. I know that there are formulas to calculate the minimum sample size so that the ...
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When is a function of an ergodic stationary process itself ergodic stationary?

I am working with a function which has the form $f(X_1, \dots, X_n)$, where $\\{X_n\\}$ is an ergodic stationary process. Theorem 5.6 in "A first course in stochastic processes" by Karlin &...
Kristan's user avatar
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Demonstration of Convergence in Probability of the Average Prediction Error for a Consistent Machine Learning Algorithm

I'm quite new to this topic, but I've set myself the task of understanding how to demonstrate that the average of prediction errors in the sample for a machine learning algorithm, which consistently ...
Tomás Rubio's user avatar
2 votes
0 answers
31 views

Stochastic order symbols - Intuition [closed]

I apologize for the following set of questions, which may seem trivial. Let $X_n$ denote a sequence of random variables. Then, $X_n = o_p(1)$ means that $\lim_{{n \to \infty}} X_n = 0$. I can use ...
Dimitru's user avatar
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