Questions tagged [consistency]

Refers generally to a property of a statistical procedure to go to the "right" place as the sample size tends to infinity, primarily referring to estimators converging to the true parameter value as the sample sizes diverges. Use also for Fisher consistency, the property that an estimator when applied to the complete population gives the right answer.

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When the sample mean converges to the population mean, does the probability that the sample mean is equal to the population mean tend to 0?

Let $y_1, y_2, \ldots , y_N$ be arbitrary real numbers and consider an infinite triangular array of real elements, where each row is indexed by $N \in \mathbb{N}$: $$ \begin{matrix} y_{1,1} \\ y_{2,1}...
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Does a misspecified model always have lower likelihood value than the correct model?

Suppose the true dgp is $$ x_i \sim d_1(\theta_1), \quad i=1,\ldots,N $$ where $d_1$ is some probability distribution with parameter(s) $\theta_1$, but I wrongly assume $$ x_i \sim d_2(\theta_2). $$ ...
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Are the assumptions and implications for ordinary least squares listed relevant, comprehensive or too-relaxed for generalized linear models? [closed]

The following diagrams show which assumptions are required to get which implications in the finite and asymptotic scenarios. I have no idea whether GLMs share these assumptions or whether GLMs have ...
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Does one need consistent estimate of "$S$" in ordinary least squares for unbiased, consistent fixed-effects level-one estimate and correct inference? [closed]

What is a complete list of the usual assumptions for linear regression? says that one obtains a Consistent Estimate of $S$ when the regressors have finite fourth moments. $$ \hat{S}=\frac{1}{n} \sum_{...
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Is instrumental variables estimator applicable when a covariate is Spearman $\rho>0$ but not Pearson correlated with the residuals?

Is instrumental variables estimator applicable when a covariate is Spearman but not Pearson correlated with the residuals? Does Spearman non-zero correlation with residuals imply loss of consistency?
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Do covariates correlated with residuals in generalized linear models make estimates not consistent or other problems?

Do covariates correlated with residuals in generalized linear models make estimates not consistent or make other problems? Economists raise an issue about endogenous variables in OLS and do some 2SLS ...
-1 votes
0 answers
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Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators?

Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators? Are sufficient statistics consistent, efficient, complete and unbiased, ...
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28 views

Are generalized estimating equations estimates affected by endogenous covariates (covariates correlated with model residuals)?

Are generalized estimating equations consistent or still okay with endogenous variables? The authors of GEE say here that their estimates remain consistent without any qualifications about endogeneity ...
2 votes
1 answer
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For the logit model, how is the maximum likelihood and minimum distance estimator related?

Suppose I have data $\{y_i,x_i\}_{i=1}^N$, where $x_i\in\{s_1,...,s_K\}$ and follows a discrete uniform distribution. For each realized $x_i$, $y_i$ is generated by the logit model, i.e., $Pr(Y_i=1|...
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Does a Heteroskedasticity and Autocorrelation Consistent Estimator for generalized linear (mixed/non-mixed) models exist?

Does a Heteroskedasticity and Autocorrelation Consistent Estimator for generalized linear models exist? That would make GEEs outdated unless no-free lunch theorem suggests otherwise. I am only aware ...
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Are the classical moments consistently estimated from a single realization drawn from a given PSD?

Given a sequence $\{x_k\}_{k=-N}^{N}$ having power spectral density $S(f)$, we know that that "single realization PSD" $$ \frac{\Delta t^2}{T} \left| \sum_{k=-N}^{N} x_n \exp(-2\pi i f n \...
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Eckart–Young–Mirsky theorem for $n \gg m$

It has been proven that the best reconstruction error in the $k$ rank matrix estimation problem in terms of Frobenius or $L2$ norm is given by the $k$-truncated SVD as shown here. I've read in ...
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Using time as a variable to calculate consistency

How can I use time as a factor in calculating consistency/reliability. Given a simplified scenario : I am trying to rank researcher based on how often they publish a paper by giving them a score. ...
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2 answers
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Cronbach’s alpha not equal to ICC 3,k

I am calculating ICC over 16 items rated by 41 raters, using the ICC function from the psych package. The output is: ...
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1 vote
1 answer
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Internal consistency of correlation [duplicate]

I have this question below , and I am unable to understand what is internal consistency, can anyone please tell the concept , I have read its wiki page but I couldn't understand how to solve a ...
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1 answer
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Can I say that $T_n$ is a consistent estimator of $\theta$ by Monte Carlo simulation under this setting?

Given a iid random samples $X\sim N(\theta,1)$, we have a unknown parameter $\theta$ and its estimator $T_n=T_n(X_1,\dots,X_n)$. If we have strictly proved that $T_n$ is a consistent estimator, can ...
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1 vote
1 answer
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Checking consistency of answers within groups (e.g. household) in R

I got some survey data, in which the respondents answered various questions about environmental conditions on a 1-5 Likert scale. The respondents are also assigend to a household by an household ID. ...
1 vote
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Does a linear regression assume that the (unconditional) predictor data is i.i.d?

Say I have a linear, cross sectional relationship - $y_{i}=x_{i}b+e_{i}$. Where $E(e_{i}|X_{j})=0$ for all relevant $i,j$. Given this, one can prove that the OLS estimator is unbiased. However, ...
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Confusion about consistency of time series model parameter

Can someone clear this confusion. Lets say I have a time series model: $$X_t \text{ follows Poisson}(\lambda_t)$$ $$\lambda_t=a*X_{t-1}+b*\lambda_{t-1}$$ Then I find estimators for $a$ and $b$ called: ...
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root n consistency of parameter in mixture distribution

We have iid observations $\{X_i\}_{i=1}^n$ from CDF $\theta G + (1- \theta)H$ where $\theta \in (0,1)$ is unknown. Find a $\sqrt{n}$ consistent estimator for $\theta$ using the observations. Note that ...
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Testing Data Consistency and its effect on Multilevel Modeling Multivariate Inference

I have a MLM model looking at the effect of demographics of a few cities on a region wide outcome variable as follows: RegionalProgress = β0j + β1j * Demographics + u0j + e0ij The data used in this ...
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Using IV when regressor is not endogenous

Suppose I have a single regressor model and the regressor itself is uncorrelated with the error term. If I were to use IV estimation to estimate the coefficient, would the estimate be incorrect, and ...
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Weighted average estimator - unbiased and consistent

Take an estimator that produces a weighted average of all n observations in an i.i.d sample from a population with mean $\mu$ and variance $\sigma^2$. I.e.: $$ \bar{x}_w = \sum_{i=1}^{n} w_ix_i$$ ...
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What is the variance of $s^2$?

I am trying to calculate the variance of $s^2=\frac{1}{n-1}\sum (x_i-\bar x)^2$. So what I want to find is $ Var(s^2)$. I have seen different posts, but many of them seem to make the assumption that ...
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Proportion data: Logistic with MLE vs. OLS with logit-transformed response

This is an expansion of @Beethoven_90's comment on this question. Suppose I have proportion data $Y_i$ computed from a binomial; $Y_i = \frac{S_i}{N_i}$ where $S_i \sim Bin(N_i, p_i)$ and $p_i$ is the ...
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Finding the consistency of an estimator?

Suppose Y1, Y2,...,Yn is a random sample from the exponential pdf, fY(y; λ) = λe^(-λy), y> 0. a. Show that λn = Y1 is not consistent for λ. b. Show that λn = sum of Yi, from i=1 to n, is not ...
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Showing the sample mean estimator is consistent

I need to show that the sample mean estimator $(\sum x_i)/n$ calculated over the first n samples of $x_1,x_2,...$ iid of infinity size is consistent. Now it's a little bit difficult for me to ...
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Random effects tobit if random effects are non-normal distributed

I have the following (panel tobit) data-generating process: $y_{i, t}^* = a + bx_{i,t} + u_{i} + e_{i,t}$ $y_{i,t} = y_{i, t}^*$ if $y_{i, t}^* \geq 0$ and $y_{i,t} = 0$ otherwise. where $i = 1,..., N;...
1 vote
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Sample skewness consistency

Is sample skewness a consistent estimator of (moment coefficient) skewness? I have ran some experiments and it seems this is not the case, but I cannot find any definite answer online. Here is an ...
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Why when the number of data increase the consistency can’t guarantee that the bias induced by the estimator diminishes

Consistency ensures that the bias induced by the estimator decreases as the number of data examples increases. However, the converse is not true asymptotically, an unbiased estimator does not imply ...
1 vote
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Marginal Distribution of a Stochastic Block Model?

Let us say I have a Stochastic Block Model i.e. a random vector $X^{(n)} = [x_1,...,x_n]$ where $P(x_i = c) = P_{c}$, where $c \in \{1,...,u\}$, and a random simple undirected graph represented by $Y^{...
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1 vote
1 answer
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consistency of maximum likelihood estimator

For population with n size and following density function $$f(y, a)= (1/6a^4)y^3e^{-y/a}$$ For that, I have found the maximum likelihood estimator of a which is $\hat{a}= \bar{y}/4$ I have also shon ...
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Consistency for the estimator in a mixture of objective function

Current we have two discrepancy functions $f_1(x_1,x_2,y_1,y_2)$ and $f_2(x_1,y_1)$. $f_1$ reaches minimum when $x_1=y_1$, $x_2=y_2$; $f_2$ reaches minimum when $x_1=y_1$. We consider an objective ...
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Statistical test for questionnaire with various components

I was asked to analyze a research questionnaire that consists of a Likert scale, a checklist, and questions with yes, no, and don't know as response, as part of the pilot testing. I was struggling ...
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Consistency, efficiency and the asymptotic distribution of estimators. Is my solution right?

I am not sure about the results of some econometrics exercises from Hansen 2020. I would really appreciate if you could tell if my solutions look right. The model il $Y= X\beta + e$, $E[e|X]=0$ and $X ...
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Proving the MLE of N(θ,θ) is consistent?

I had successfully found the MLE to be (sqart(1+4S)-1)/2, where S = average sum of square of X1...Xn, but how do I actually prove it's consistency? I don't find finding the mean of the MLE easy and ...
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Observing tails in an i.I.d. data sample

Is there a result that says that in an I.I.d. data sample one shouldn’t observe tails “too soon”? I am trying to prove consistency of a certain MLE and want to exclude scenarios of the form: In $\...
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Clarification of line in proof of consistency theorem (Vapnik)

In Vapnik's Statistical Learning Theory (1998 edition) on pages 89-92, he proves a "key theorem of learning theory" that states the conditions for when: "the following two statements ...
1 vote
1 answer
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Would decision tree propagate error?

Given a regression dataset $X,Y$. Suppose there are two different decision tree (CART) $T_1, T_2$ fitted from it. Each using different feature encoding method. And we get two different tree. And there ...
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1 vote
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consistency of mle of double exponential distribution ( not advanced)

Let $y_i\sim DE(\mu, \sigma), $ $i=1,2,...,n, \ i.i.d.$ Where DE represents the double exponential distribution. The the MLE of \sigma is: $\hat\sigma = \frac{1}{n} \sum_{i=1}^{n}|y_i-med(y_i)|$, ...
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Consistency of Posterior

I have a conceptual question regarding the definition of the consistency of a Bayesian posterior. I found this definition on the web: Given the i.i.d. data $x_1,...,x_n$ and the model $f_\theta(x)$, ...
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1 vote
3 answers
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How to prove that, in finite sample, Bayesian posterior is more informative than the prior?

Suppose there is a space of possible models $\theta \in \Theta$, and that we can generate i.i.d. data $\{x_1, x_2,...\}$ from the true model. Asymptotically, the Schwartz Theorem shows that Bayesian ...
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1 vote
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Maximum likelihood estimation when the model is misspecified (and the true data generating process is a mixture model)

I'm interested in the properties of maximum likelihood estimators under a particular form of model misspecification: We observe data $\left\{X_i\right\}$ generated from a finite mixture model Let $\...
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1 vote
1 answer
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Find the expectation of an exponential distribution estimator

So we've got a sample data coming from exponential distribution with parameter $\lambda$, and we take an estimator $\lambda_n = \frac{n}{X_1+X_2+\cdots+X_n}$. I need to show that this is a biased and ...
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Consistent Estimator for the Dispersion of a GLM

I am trying to figure out the proof for consistency of the estimators for an exponential dispersion family. The proof is well covered in the paper "Consistency and Asymptotic Normality of the ...
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Sequence of probability measures is consistent

Denote $\varphi (y|x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-(y-x)^2}{2}}$. Show that the sequence of probabilit measures $\\$ $P_n(B) = \int...\int_B\varphi(x_1|0)\varphi(x_2|x_1)...\varphi(x_n|x_{n-1})d(...
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2 votes
2 answers
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Proof of consistency of Maximum Likelihood Estimator

I would appreciate some help comprehending a logical step in the proof below about the consistency of MLE. It comes directly from Hogg, McKean, Craig, Introduction to Mathematical Statistics, 6th ...
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2 votes
1 answer
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Fisher vs. Asymptotic Consistency - Example using a single observation as the population mean estimator

I am learning about Fisher Consistency and came across this section of a Wikipedia article (https://en.wikipedia.org/wiki/Fisher_consistency#Relationship_to_asymptotic_consistency_and_unbiasedness) ...
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2 votes
0 answers
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Types of consistency: practical difference [duplicate]

Consistency is usually a desired property for an estimator. We have the definition of consistency for an estimator $T_n$ for $\theta$, stating that it converges in probability to $\theta$, and the ...
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4 votes
3 answers
305 views

Is this an ARMA(2, 1) process?

I am puzzled by an equation, $$ y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + u_t + \varepsilon_t - \varepsilon_{t-1}, $$ where $u_t$ and $\varepsilon_t$ are independent white-noise processes. Is this an ...

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