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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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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Unbiasedness and consistency of OLS in an AR(1) model with AR(1) residuals [duplicate]

consider equation 1 : , Now let , where the error component is iid with mean 0 and constant variance, and Is the OLS estimator of the coefficients in equation 1 unbiased and consistent under this ...
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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;...
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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 ...
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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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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 ...
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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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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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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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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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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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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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Consistency condition for Wiener's process

Currently, I am working in order to prove the statement written below. Maybe somebody has any ideas/hints to prove this statement? How can one show that the Wiener process (standard Brownian motion) ...
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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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Testing bias and consistency for a parameter given variance less than infinity

I proceeded to find the expectation of the estimator to check for bias. Since Therefore and hence biased. An estimator is consistent if MSE tends to 0 as n tends to infinity but I do not know how ...
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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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Weak orthogonality, consistency, and unbiasedness of the OLS estimator

My question is based on this question. Suppose we assume the sample is iid (so time series data is out) and $E[e_i X_i ] = 0$ but we're not sure about $E[e_i \mid X_i]$ = 0. Can you provide a ...
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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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Consistency of OLS when no intercept

Suppose I have a model $y_i = \beta_0 + \beta_1 x_i + e_i$ but instead I estimate $y_i = \beta_1 x_i + u_i$ using OLS. That is, I ignore the intercept. Working out the algebra, based on this post, we ...
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Consistency of $f( x| \theta ) = \exp(-(x- \theta ))$

Prove that the second smallest observation in a random sample of size n from following pdf is consistent estimator of $ \theta $ $$ f( x| \theta ) = \exp(-(x- \theta )) , \qquad x > \theta $$ ...
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Does the OLS estimator in simple linear regression converge a.s.?

Consider the following model. Assume $(x_i, u_i)$ is sequence of independent identically distributed random vectors in $\mathbf{R}^{d+1}:$ $x_i$ are $\mathbf{R}^d$-value random vectors, which will ...
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Asymptotic Normality and Consistency

I have difficulties understanding the concept of asymptotic normality and consistency. Take an estimator of a parameter which is consistent and asymptotically normally distributed. Because it is ...
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Understanding the proof for consistency of the OLS estimator

In my econometrics lecture we discussed the consistency of the OLS estimator ($\hat{\beta} \overset{P}{\rightarrow}\beta)$ and I don't understand why it holds that $(X'X)^{-1}=\Big(\frac{X'X}{N}\Big)^{...
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How do we say that two estimators are consistent with each other?

I have two estimators that return the results as $x_1 \pm \sigma_1$ and $x_2 \pm \sigma_2$. How do I tell if these two estimators are statistically consistent with each other? I heard that if $x_1$ ...
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Can you calculate a Bayesian version of the Smallest Detectable Difference?

Apparently, there are quite a few synonymous designations for the statistic in question: Minimal Detectable Difference, Smallest Worthwhile Effect, Minimal Clinically Important Difference, etc. Let's ...
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Consistency of slope given by SLR through the origin

I'm reading a post about the consistency of coefficients of SLR models: Consistency of estimators in simple linear regression Now I'm wondering whether there will be some similar conditions for SLR ...
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Consistency of maximum likelihood for not-identically distributed observations

Let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of positive real numbers and let $$ A=\left\{f:\mathbb{Z}_+\times\mathbb{R}_+:\,\sum_{k=0}^\infty f(k,\mu)=1,\;\sum_{k=0}^\infty k f(k,\mu)=\mu\;\forall\...
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Proving consistency of OLS estimator in an unfamiliar setting

I'm stuck on two questions, both of which relate to proving the (in)consistency of the standard OLS estimator. I'm given the following regression model, where $Y_i$ denotes the outcome measure for the ...
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Does Normalized Estimation Error Squared (NEES) and Normalized Innovation Error Squared (NIS) Only Apply to Kalman Filtering?

Regarding the NEES and NIS metrics, do they only apply to Kalman Filtering? Or can I use them for any estimator that outputs a prediction and has a covariance matrix? I have never seen NEES applied to ...
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1 vote
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Simple mean or kernel estimator?

Let $X$ be a continuous random variable with support $\mathcal{X}$ and density $f(x)$. Suppose I'm interested in constructing a consistent estimator of $E(X)$ using $n$ i.i.d. observations $(X_1,..., ...
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Consistency of OLS estimator with unobservable variable

Suppose I have the next model $Y_i = \alpha_i + \beta_i X_i + u_i$ where $u_i - N(0,\sigma^2)$ and $\alpha_i$ is unobservable, also $E(X_i \alpha_i) \neq 0$ My OLS estimator for $\beta_i$ is $\hat{\...
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Bootstrap consistency for maximum likelihood

I'm looking for references (textbook if possible) that treat the strong consistency of bootstrapped maximum likelihood (MLH) or more generally M-estimators. By strong consistency I mean that the ...
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OLS Consistency for Count Data?

Why would we choose Poisson / NB regression (GLM) over OLS for fitting count data? Is there a way to show that OLS estimator would lost it consistency and asymptotic normality for count data? I'm ...
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Linear probability model difference in difference?

I see frequently people opposed to linear probability models for binary outcomes, and I know there are related questions to this on this forum, but I want to make sure my understanding is correct. ...
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Homework Problem Consistancy of Estimator

We have to show: Let $\theta_0$ be a k-dim vector. Show, that the following statements are equivalent: (1) $\hat{\theta}_n$ is consistent for $\theta_0$. (2) For each component $i= 1,...,k$: $\hat{\...
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