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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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24 views

statsmodels logistic regression with binned variables has large coefficients and standard error for some variables

I'm fitting a logistic regression (binary) using Python's statsmodels, and here's a snippet of summary from the model: I have noticed that the large coefficients ...
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0answers
12 views

glmer model convergence question

We are working with a longitudinal dataset, with three variables: WAIP, BPSRRI and group. WAIP and BPSRRI are measured repeatedly for 10 times and group refers to the group assignment of our subjects ...
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12 views

Cannot seem to converge beyond a loss of 3 on an object detector being trained on YOLO

Data The you only look once YOLO algorithm is used for object detection. I have scoured the internet for resources on how to tackle this problem, but there seems to be a lot of resources that point ...
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8 views

How does network structure (model complexity) affects covergence speed?

I trained Bi-GRU and HAN (Hierarchical Attention Networks) on my own datasets, and found HAN converges faster than Bi-GRU, within less number of epochs. What would be the reason for this? I guess ...
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1answer
106 views

Convergence of the Matérn covariance function to the squared exponential

The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ...
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1answer
196 views

Central limit theorem (CLT) writing

Is there a reason why we are used to write the CLT as $\sqrt{n}(\overline{X}_n-\mu)\stackrel{d}{\rightarrow}N(0,\sigma^2)$ and not as $\overline{X}_n\stackrel{d}{\rightarrow}N(\mu, \frac{\sigma^2}{n})$...
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2answers
41 views

Rate of convergence of sum of two random variables

Let $X_n$ and $Y_n$ be random variables such that $X_n=o_p(1)$, $Y_n=o_p(1)$, $X_n - Y_n = o_p(1)$. Is the following correct? $o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$
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1answer
58 views

Does convergence in distribution imply asymptotic stationarity?

Let ${\bf \tilde{x}}_1, {\bf \tilde{x}}_2, \ldots$ be a (possibly non-stationary) stochastic sequence of $d$-dimensional random vectors that converges in distribution. Does it immediately follow that ...
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1answer
23 views

Nested model failing to converge - how to make decisions about random intercept only model?

I have two models: modela <- lmer(perception~1+self+actual+(1|id/rid),data=data) modelb <- lmer(perception~1+self+actual+(1+self+actual|id/rid),data=data) ...
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1answer
46 views

Bayesian consistency in compact uncountable parameter space

Let $p(y_i \mid \theta)$ be the likelihood we are using of a single data point, $p(\theta)$ be the prior, and $f(y_i)$ the true distribution of the data. Also, let $\theta_0$ be the parameter that ...
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1answer
56 views

convergence in distribution?

I have a question. Let $X_n$ converge to $X$ in distribution, on the other hand, $Y_n$ converges to $Y$. What can we obtain about convergence of division of $X_n/Y_n$ in distribution? Does it ...
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1answer
52 views

convergence of an algorithm [closed]

I want to know when we speak about the convergence of an algorithm, what are the conditions that we should check. For example, I was looking for the convergence of the policy iteration algorithm in ...
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1answer
85 views

Convergence in probability of $\frac{1}{n}\sum_{i=1}^n X_i^2$ when $X_i$'s are i.i.d $N(0,1)$

Question: My approach: And after this I am stuck..How do I put the modulus over here and how do I determine the appropriate value of "k" ? (here k signifies the value of convergence in probability ...
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1answer
40 views

Can you perform a likelihood ratio test on two linear mixed effects models with different optimizers in lme4?

I ran into an error with my full (but not simple/null) model, so I had to use a different optimizer to avoid the fitting problems. Can I still do an LRT test using those models?
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1answer
18 views

jackknife estimator with central limit theorem

Let $\hat{\theta}_n$ be an estimator of the parameter $\theta$ from the sample $\Omega_n$ of $n$ observations, satisfying that $\sqrt{n} (\hat{\theta}_n-\theta) \overset{d}{\longrightarrow} \mathcal{N}...
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0answers
16 views

Convergence radius of random power series

I have a problem with getting how I should interpret the random power series. I am given $X_n$ that are i.i.d random variables. Further the random power series, $\sum_{n=0}^{\infty} X_{n}z^{n}$ ...
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0answers
29 views

CLT and convergence of Variance

I am looking at a problem where the sum of the individual $X_i$ is $S_n=X_1+\dotsm+X_n$. The probability is given as, $P(X_i=i)=P(X_i=-i)=\frac{i^{-\alpha}}{4}$ and $P(X_i=0)=1-\frac{i^{-\alpha}}{2}$. ...
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80 views

policy iteration convergence

There is a question here for 2014 about the convergence of policy iteration algorithm with two answers > Question However, it is not clear for me how we change the value functions after one policy ...
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0answers
24 views

CV.GLMNET warnings convergence

I get the warnings when I try to perform a cross validation with cv.glment for a Logistic penalized model ...
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1answer
80 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 ...
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1answer
59 views

Convergence of $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$ - central limit theorem

Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$, where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\...
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2answers
53 views

Prove convergence in distribution for n times the minimum of an unknown positive distribution

Let $Z_1, Z_2, ...$ be independent and identically distributed random variables with some density $f$. Suppose that $P(Z_i > 0) = 1$, and that $$ \lambda = \lim_{x\to 0} f(x) > 0$$ Let $X_n = ...
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0answers
13 views

Prove bi-directional relationship between convergence in distribution and convergence of probability mass functions

Let $X$ be a random variable that is positive and integer-valued. Let $X_1, X_2, ...$ also be random variables that are positive and integer-valued. Prove that $X_n$ converges in distribution to $X$ ...
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1answer
26 views

Can we conclude, with the strong law of large numbers, that $n$ random variables are independent? [closed]

Suppose we have a sequence of identically distributed random variables $X_1, \ldots, X_n$, and that we know $(X_1 + \ldots + X_n)/n$ converges almost surely to $\mu = E[X]$ as $n$ approaches infinity. ...
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37 views

How to select parameters for ADAM gradient descent

I am using ADAM for performing gradient descent. I am having difficulty in setting the learning rate, $\beta_1$ and $\beta_2$. Along with gradient descent, I am projecting the paramters on $L_1$ ball ...
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1answer
24 views

Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable

Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$. Suppose that $X_n$ has the support set $\{X, e^n\}$, and $P(X_n = X) = 1 - \frac{1}{n}$ $P(X_n = e^n) = \frac{1}...
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1answer
43 views

Does $X_{n}=o_{p}\left(Y_{n}\right)$ imply that $P\left(Y_{n}=0\right)=0$ for all $n$?

Suppose that $X_{n}=o_{p}\left(Y_{n}\right)$. Does this imply that $P\left(Y_{n}=0\right)=0$ for all n ? Or only, e.g., that $P(Y_n = 0)\rightarrow 0$? I guess it is a matter of definition. If $X_{n}=...
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16 views

Choose between ratio of estimators or estimator from the ratio of data

I have to estimate a function which is, say, the proportion of people under 20 in each point of a given territory. Let's call that $h(x,y)$ for $(x,y)$ in my territory. To get that, I have, for every ...
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0answers
15 views

Almost Sure Convergence and Subsamples

My actual question is in the last paragraph, but I will start with a basic example. In the book "A Course in Large Sample Theory" (Ferguson), they present the Strong Law of Large Numbers as the ...
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1answer
77 views

Limiting distribution of a ratio using Basu's theorem

Edit: there's seems to be a typo in original question. This is a past exam question that I'm trying to solve. Suppose that $X_1,\ldots, X_n$ are i.i.d. Uniform (0, $\theta$) random variables. Let $...
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1answer
43 views

Proving $X_n \rightarrow X$ in distribution implies $a+bX_n \rightarrow a+bX$ in distribution by definition

As stated in the title, I am trying to prove that if $X_n \Rightarrow X$ in distribution, then $a+bX_n \Rightarrow a+bX$ ( where $a,b\in\mathbb{R}$) in distribution using the definition as follows: $...
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Proving the convergence of k-means algorithm using fixed point theorem

I'm learning machine learning and functional analysis this semester. When I learn the k-means algorithm , it came to me that the stopping criterion is very similar to the fixed point theorem thought. ...
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17 views

Examples of convergence in distribution using CDF directly [duplicate]

I find very interesting the example that if we let $Q_n$ be the maximum of n i.i.d. with distribution $U[0,\alpha]$, then $n(Q_n - \alpha)$ converges to an exponential distribution. See e.g. here for ...
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3answers
121 views

Proof of Convergence in Distribution with unbounded moment

I posted the question here, but no one has provided an answer, so I am hoping I could get an answer here. Thanks very much! Prove that given $\{X_n\}$ being a sequence of iid r.v's with density $|x|^{...
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0answers
14 views

Different random weight initilization leading to different performances

I'm training a 3D U-Net on an EM dataset of a brain. The objective is to segment neurons in it. During the experiments, I've noticed, different random initialization of the network leads to different ...
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0answers
15 views

Proving new features theoretically help learning

I've been recently working with feature construction. I was wondering, whether how one would go about proving, that given a standard classification setting, a new feature can improve a classifier's ...
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0answers
24 views

How does the concept of consistency apply to the full bayesian posterior as opposed to a single estimate?

Towards the goal of making a bayesian statistical inference, I start by collecting $M$ independent and identically distributed data observations $D_i$. Then I take a Bayesian approach to learning the ...
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0answers
13 views

convergence issues moving from dirichlet to multinomial-dirichlet in JAGS (implemented in R)

I am modeling microbiome count data, using a multinomial dirichlet. The number of times I observe each microbial "species" depends on its fractional abundance within a microbial community, and the ...
2
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0answers
14 views

Theory of convergence for this case?

I hope you can guide me to the right place. I am estimating two equations, where each equation produces an output which is needed in the other equation as input. Something like this: $Y_t = AX_t + f(...
3
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1answer
106 views

If the Markov assumption is wrong, will a learner still converge to a stable policy?

I'm trying to figure out what guarantees can be made if a learner wrongly assumes a problem obeys the Markov transition property. Assume I have a problem defined by a partially observable Markov ...
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1answer
91 views

Using Markov's Inequality to show that $E(X_i) \rightarrow 0$ implies $P(X_i=0) \rightarrow 1$

I am trying to use Markov's inequality to show that for a sequence of positive random variables $X_1, X_2, ....$ with values in $N=\left\{0,1,2,...\right\}$ and $\lim_{i\rightarrow\infty}E[X_i]=0$, it ...
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0answers
39 views

Influence of RMSE versus MSE on Convergence of Gradient Descent

I am working in TensorFlow and was wondering if I choosing an MSE loss function would cause different convergence behaviour when compared to a RMSE loss function. The square root will influence the ...
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1answer
23 views

Help understanding a probability inequality

I'm working throught Wasserman's "All of Statistics" book. When proving convergence of random variables/distributions in chapter 5, he lists the following inequality: $$F_n(x) = \mathbb{P}(X_n\le x)=\...
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0answers
140 views

Showing that a Gamma distribution converges to a Normal distribution

Consider $G = \operatorname{Gamma}(p)$. As $p$ goes to $\infty$, the Gamma becomes more and more bell-shaped. How do I show that $\frac{G - p}{\sqrt{p}} \to Z \sim N(0,1)$ as $p \to \infty$? I ...
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0answers
18 views

Impulse response for general VAR lag-p model: when does it converge?

Consider the VAR lag-p model: $$Bx_t = \Gamma_0 + \sum_{i=1}^p\Gamma_i x_{t-i} + \epsilon_t,\quad x_t\in\Bbb R^n,\,\forall t\in\Bbb Z$$ Setting $B$ to be upper-triangular and $A_0:=B^{-1}\Gamma_0,\,...
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2answers
92 views

Convergence to a Uniform Distribution

$\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} $ Show that if $P(X_n = i/n)=1/n$ for every $i = 1,...,n$, then $X_n$ converges in distribution to a uniformly distributed random variable $X$. ...
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1answer
53 views

Convergence almost sure of sequence random variables with Bernoulli distribution

i stacked in one example. We have independent sequence of random variables $X_{n}$, where $X_{n}$ has Bernoulli distribution with parameter $\frac{1}{n}$, so $X_{n} \sim Bernoulli(\frac{1}{n})$ and ...
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0answers
24 views

Asymptotic consistency and normality

I need help getting the following problem Let $X_1,..,X_n$ be independent $N(\mu,1)$-distributed random variables. Define $\hat{\theta_n}$ as the point of minimum of $\sum_{i=1}^n(X_i-\theta)^4$...
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2answers
84 views

Show that if $E\psi(x-\theta)= 0 $ then $P(X< \theta) \leq p \leq P(X \leq \theta)$

Define $$\psi(x)=\begin{cases} 1-p & x < 0 \\ 0 & x=0 \\ -p & x> 0 \end{cases}$$. I have to show that if $$E\psi(x-\theta)= 0 $$ then $$P(X< \theta) \leq p \leq P(X \leq \theta)$$...
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1answer
50 views

ABC: Population Monte Carlo (PMC) convergence statistics?

I'm using the abcpmc code: Approximate Bayesian Computing (ABC) Population Monte Carlo (PMC) implementation based on Sequential Monte Carlo (SMC) with Particle Filtering techniques. described in ...