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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18 views

Simultaneity and Bias of OLS [closed]

I have the following information $q_i^d = \alpha_0 + \alpha_1p_i + u_i$ and $q_i^s = \beta_0 + \beta_1p_i + v_i$ and I know that $q_i^d = q_i^s = q_i$. Additionally $E(u_i)=E(v_i)= 0$ and $Cov(u_i,...
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Consistency of ADL/ARDL/ARIMAX coefficients

Enders in Applied Econometric Time Series (4th edition, p.282) has following statement about consistency of coefficients in ARDL models: "For the coefficients of C(L) to be unbiased estimates of the ...
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Is the PCA estimator used in regression root-n-consistent?

Consider a sample of $n$ observations $(y_i, x_i)$, $i=1,\ldots,n$ and assume without loss of generality that the samples are centered. The true model is $y_i=x_i^t\beta+\epsilon$, and the OLS ...
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Error distributions and consistent and unbiased OLS

If OLS estimator is unbiased and consistent, what does it imply about the distribution of error terms? In linear regression model: $ y_i = \boldsymbol{x_i' \beta} + \epsilon_i $ if the OLS estimator ...
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OLS with asymmetrically distributed disturbances

My question is related to this post: [[here]1]1 parameters-for-a-regression-with-asymmetric-and-non-zero-dis However, I want to know if it's possible to get unbiased and consistent OLS estimator if ...
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58 views

Consistent estimator of conditional expectation, when conditioning on binary variable

Suppose I have a sequence of i.i.d. random variables $\{Y_i,X_i,Z_i\}_{i=1}^n$ and $Z_i$ is binary. Is the following a consistent estimator of $E(Y_i*X_i|Z_i=1)$ as $n\rightarrow \infty$? Under which ...
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Efficiency of OLS versus Quantile regression estimator

If I have a linear model $ y_i = x_i'\boldsymbol\beta + \epsilon_i $ and I assume that OLS estimator of $\boldsymbol\beta$ is unbiased and consistent and Least absolute deviation (LAD) estimator of $...
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Show that bias term involving an indicator function convergences to zero

Assume that we have $N$ observations of i.i.d. data $(Y_i,X_i)_{i=1}^{N}$. We want to learn the model given by $Y=f(X)+\epsilon$. We use the data to estimate $\hat{f}$ using any machine learning ...
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Consistency of estimators vs sample size

I understand that consistency of an estimator is large sample property, but does it make sense to talk about consistency in small samples as well? Can I say about the estimator that it is consistent ...
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Unbiasedness and consistency of OLS

Does unbiasedness of OLS in a linear regression model automatically imply consistency? Edit: I am asking specifically about the assumptions for unbiasedness and consistency of OLS. If the assumptions ...
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Consistency of least absolute deviation estimator (LAD) vs OLS

If we know that OLS estimator of beta in linear regression model is unbiased and consistent and we don't have any further assumptions (on errors or anything), will LAD estimator of beta be also ...
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Quantile regression estimator - conditions for consistency and efficiency

What are the conditions for consistency and efficiency of Quantile regression estimator (for example LAD) in a linear regression model?
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Estimator, Bias and asymptotic distribution

I have a model; $$y_i = \beta_1 + \frac{1}{\beta_2}x_i+\epsilon_i$$ To simplify I use OLS to regress on; $$y_i = \delta_1 + \delta_2 x_1 + \epsilon_i$$ Thus I obtain the two estimators $\hat{\...
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Consistency of MLE of $\alpha$ when pdf is $f(x;\alpha,\beta)=\frac{\alpha x^{\alpha-1}}{\beta^\alpha}1_{0<x<\beta}$

I have a sample of size $n$ from the following distribution: $$f(x;\alpha,\beta)=\frac{\alpha x^{\alpha-1}}{\beta^\alpha}1_{0<x<\beta}\quad,\,\alpha>0$$ I found that the MLEs are $$\hat{\...
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Lasso feature selection consistency: $\frac{\lambda_n}{n} \rightarrow 0, \frac{\lambda_n}{\sqrt{n}} \rightarrow \infty$ hold at the same time

In some references regarding sign consistency of Lasso such as Zhao, Peng; Yu, Bin, On model selection consistency of Lasso, J. Mach. Learn. Res. 7, 2541-2563 (2006). ZBL1222.62008. there is a ...
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demeaning or differentiation in fixed-effect equation

I have panel data (with short time-dimension $T>2$) and I consider a simple model of the form: $$y_{i,t} = x_{i,t} \beta + c_i + \epsilon_{i,t}$$ where $E(x_{i,s}\epsilon_{i,t})=0$ and $\epsilon_{i,...
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Consistent estimator for $\mathbb{E}[\mathbb{E}[f(X_1,X_2)|X_1]^2]$

Suppose $(x_{i})_{i=1}^n$ are i.i.d. sample. I want to construct a consistent estimator of $\mu=\mathbb{E}[\mathbb{E}[f(X_1,X_2)|X_1]^2]$, where $f(x_1,x_2)\ne f(x_2,x_1)$. I use $X_1$ for random, $...
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Cronbachs alpha interpretation - what does var.r and med.r mean?

i have used the psych::alpha to find the cronbachs alpha values, but i cannot find any interpretation for some parts of the output, namely: ...
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Non-frequentist Consistency of Bayesian Estimates

this question seems weird. But I am just wondering if there exists any sorts of theories about 'non-frequentist consistency' of Bayesian methods? So far all the results I found are about the ...
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The No-Free-Lunch Theorem and K-NN consistency

In computational learning, The NFL theorem states that there is no universal learner. For every learning algorithm , there is a distribution that causes the learner output a hypotesis with a large ...
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Consistency under mild endogenity

Assume the usual linear model: $$Y_i = X_i\beta + \varepsilon_i, \quad 1\leq i \leq n$$ whit $E(\varepsilon_i)=0, Cov(\varepsilon_i, \varepsilon_j) = \sigma^2 \delta_{ij}$ and $Cov(X_i , \...
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Testing the consistency of two sets of measurements

Suppose I'm measuring a quantity $X$ twice, $x_{1}\pm\sigma_{1}$ and $x_{2}\pm\sigma_{2}$. $\sigma_{1}$ and $\sigma_{2}$ are in general not equal to one another. If I want to know if the two values ...
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Why do we need an estimator to be consistent?

I think, I have already understood the mathematical definition of a consistent estimator. Correct me if I'm wrong: $W_n$ is an consistent estimator for $\theta$ if $\forall \epsilon>0$ $$\lim_{n\...
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Proof of (weak) consistency for an unbiased estimator

I want to prove a theorem stating: An unbiased estimator $\hat{\theta}$ of the unknown parameter $\theta$ is consistent if $V(\hat{\theta}_n$) $\to0$ for ${n\to\infty}$. I've tried using the ...
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Show that $nX_{(1)}$ is not consistent

Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
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Consistent estimator for conditional expectation

Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d. $X_i$ has finite support. Let $x$ be a point in the support of $X_i$. Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite. Is it ...
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$P(\hat{\theta}\neq \theta) \rightarrow 0$ as the sample size increases implies $\hat{\theta}= \theta+o_p(1)$?

While I think it is reasonable, I cannot show this result. Suppose $\hat{\theta}$ is an estimator of $\theta$ and $P(\hat{\theta}\neq \theta) \rightarrow 0$ as the sample size increases, that is, $P(\...
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Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$

Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $$T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$$ MY ATTEMPT To begin with, let us notice that \begin{...
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Consistency test

I have data that consists of 7 characteristics for each participant. There are about 170 participants. The test was repeated in some time and the same 7 characteristics were acquired. I need to find ...
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Cronbach's alpha for different items?

Can I perform a Cronbach's alpha test when the questions or items are completely different to each other as opposed to having similar questions written differently as a consistency measure?
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Lagged dependent variables, bias and consistency

I am working through Christopher Dougherty's Introduction to Econometrics, and am struggling to fully grasp the consequences of lagged dependent variables in terms of bias and consistency. The key ...
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Consistent estimation and valid inference when performing regressions on data with differing levels of granularity

Imagine that a dataset has a combination of variables of differing levels of granularity (e.g. an international sample of firms containing both firm-level and country level information). There are $K$ ...
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Endogeneity and Consistency

So I learned about the endogeneity problem of linear regression in class today, where E[XU] and Cov[X,U] isn't equal to zero but some random constant c times a standard basis k-element vector with 1 ...
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Consistent estimator and distribution function

a general question: If the distribution function $F_n$ of some estimator $T_n$ suffices \lim_{n \rightarrow \infty} F_n(x) = 1 \text{ or } 0 \forall x}. Does that imply that $T_n$ is consistent? I ...
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first factor saturation vs. general factor saturation

a simple explanation between these two in the context of reliability analysis, specifically Cronbach's alpha.
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Can any unbiased estimator be changed into a consistent estimator when estimating functions of the mean [closed]

For an i.i.d sequence of Random Variables $X_1, \dots, X_n$, each with mean $\mu = \mathbb E[X]$, the goal is to estimate some continuous function $f$ evaluated at the mean, $f[\mathbb E[X]]$. If ...
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Chronbach's Alpha of Likert Item

My questionnaire is consists of ten 5-point scaled likert items. If the likert items have a good chronbach alpha, is it ok to take the average score of 10 items?
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Difference between Cronbach's alpha and Pearson's coefficient

I am creating an indicator for social development which includes variables of health, education and economy. Since I have many variables, I decided to remove some of them based on dominion expertise ...
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Consistency of variance estimator in OLS [duplicate]

Given the model, $$ y_i = x_i'\beta + \epsilon_i \quad \epsilon_i \sim N(0, \sigma^2) \quad iid \quad \forall i = 1, ..,n $$ how can I prove that the estimator of the variance $\hat{\sigma}^2 = \...
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Low internal consistency

I am currently analysing the data for a survey I have completed. In doing so, one question is quite simple (nominal) and require a typical “I do/I do not” and “I do not/I do” response from the ...
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How to measure the consistency of improvement on different conditions?

I want to measure whether the speed improvement of method 1 over method 2 is consistent on different conditions. Below are two examples of the speedup values of method 1 over method 2 on 5 conditions. ...
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Effect of adding more sample data on Maximum Likelihood estimator [closed]

I have samples $\{x_1, x_2, x_3, \dots , x_n\}$ of a random variable $X$. I compute Maximum Likelihood Estimator $\hat{\theta}_n$ using the sample data. Now, if I collect one more sample $x_{n+1}$ ...
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Given a simple logit panel data model, the MLE estimators are inconsistent

Consider a binary choice model, $P \left( y _ { i t } = 1 | x _ { i t } , \alpha _ { i } \right) = F \left( x _ { i t } \beta _ { 0 } + \alpha _ { i } \right)$, $$F ( z ) = \frac { e ^ { z } } { 1 + e ...
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Is LinUCB Hannan consistent?

I'm going through the textbook Bandit Algorithms, by Lattimore and Szepesvari (http://downloads.tor-lattimore.com/banditbook/book.pdf). It describes regret bounds for the LinUCB algorithm of the form:...
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Is MLE intrinsically connected to logs?

My mathematical exploration led me the following claim: Claim: MLE is fundamentally connected to logs (and KL divergence, which also uses logs). It’s not correct to say log shows up simply to make ...
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Consistency in uniform distribution

I know what consistency is but in options C and D both U and V are given whose covariance is quite difficult to find.
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OLS assumptions: prediction vs inference

Taking an OLS model (actually, is it a "model" or an "estimator"?) as an example, there are several assumptions (such as strict exogeneity and spherical errors) which are important for the consistency ...
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Consistency metric explanation

I am trying to understand a bit more about the consistency metric (to understand how consistency-based subset evaluation works). I find on this paper the following equation : $$\text{Consistency}_s =...
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347 views

Invariance property

I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators. As far as I know, Invariance property of ...
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444 views

OLS estimator for regression without intercept [duplicate]

Consider a linear regression model: $Y_i = \beta_1 A_i + \beta_2 B_i + u_i$ where all variables are assumed to have mean 0, and $A_{i}$ is distributed independently of both $B_{i}$ and $u_{i}$, but $...