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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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Consistency of a test - convergence of quantile

I have given statistical model $((0,1)^n, \mathcal{B}(0,1)^n,\mathcal{P}_n)$, where $\mathcal{P}_n=\{ P_{\theta}^{\otimes n} \ |\ \theta \in (0, \infty) \}$ and each $P_{\theta}$ has density function $...
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Unbiased and consistent estimator with positive sampling variance as n approaches infinity? (Aronow & Miller) [duplicate]

In Aronow & Miller, "Foundations of Agnostic Statistics", the authors write on p105: [A]lthough unbiased estimators are not necessarily consistent, any unbiased estimator $\widehat{\...
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For an ideal Kalman filter, I have that the NEES test passes but NIS test does not?

Sorry if this is more of a debugging question, but I have been stuck on this supposedly simple NIS test for a very long while. If anyone knows any sources which cover the theory or implementation of ...
Minecraft dirt block's user avatar
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Granular regression with repeating dependent variable within group

I am estimating a standard OLS regression model where the unit of observation is inventor-firm-year level. The dependent variable I am interested in is patent count (a measure of inventor productivity)...
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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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What is the difference between unbiasedness, consistency and efficiency of estimators? How are these interrelated among themselves? [duplicate]

!Efficiency(https://stackoverflow.com/20240427_193105.jpg). Given snapshot of the book states that among the class of consistent estimators, in general, more than one consistent estimator of a ...
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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 ...
DarkenExcalibur's user avatar
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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 ...
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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 ...
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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 ...
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What's the relationship between "bias-variance tradeoff" and "consistent model selection"?

I'm very confused about the relationship between "bias-variance tradeoff" and "consistent model selection". Based on my current interpretation, the ultimate goal of taking care of ...
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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 ...
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Are sample quantiles consistent with population quantiles?

The Wikipedia page about quantiles describes two approaches to the definition of quantiles: population quantiles, and sample quantiles. The section on sample quantiles lists nine different flavors of ...
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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 ...
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Bias vs consistency in instrumental variable estimation

So in Mostly Harmless Econometrics, page 154, they analyse the bias of instrumental variables: They consider the case of one endogenous variable $x$, multiple instruments $Z$, and $\eta$ is the ...
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Mathematical Step for consistency

Let me state my problem from the beginning: Let $i$ be an index representing countries ($i = {1,2,\ldots,N }$), and $t$ represent time, denoted as available data for country $i$ ($t = {1,2,\ldots,T_i }...
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Difference between consistent and unbiased estimator [duplicate]

I have a problem where I have to think of an example to explain a practical example of consistency and unbiased. The example I thought of is the sample mean. Consistency is when the estimator (sample ...
stats_noob's user avatar
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Is convergence in probability implied by consistency of an estimator?

Every definition of consistency I see mentions something convergence in probability-like in its explanation. From Wikipedia's definition of consistent estimators: having the property that as the ...
Estimate the estimators's user avatar
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1 answer
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Does increasing number of observations lead to the decreasing of Mean Square Error of consistent estimators?

I know that not all weakly consistent estimators exhibit MSE-consistency : https://stats.stackexchange.com/a/610835/397467. Anyway, does increasing the sample size leads to a reduction in their mean ...
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Strong consistency of kernel density estimator

I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise: $\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
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Calculating MLE under restriction on coefficients

Consider the following simple linear regression model: $y_i = a + b \cdot x_i + \epsilon_i \space\space\space\space\space\space\space where \space i= 1, 2, 3, \cdots , n$ here $\epsilon_i \space's$ ...
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Proof of Variable Selection Consistency for LASSO in Zhao & Yu, 2006

I'm going through the proof of Proposition 1 in Zhao & Yu, 2006 (https://www.jmlr.org/papers/volume7/zhao06a/zhao06a.pdf), titled On Model Selection Consistency of LASSO. The proof is in Appendix ...
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Questions about efficiency of estimators with longitudinal data

I'm modelling data with repeated observations; I'm reading up on options and pitfalls, and have a few questions. Coefficient estimates are still unbiased and consistent in the presence of ...
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Why are lags of the dependent variable a no-no in traditional random effects models?

This post says: Lagged versions of the dependent variable are a no-no in traditional random effects models. The problem is that they are correlated with the random intercept and produce inconsistent ...
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Consistency of the pooled standard deviation estimate

Suppose that $X_{ik}\sim\mathcal N(0,\sigma^2)$ for $k = 1,2,\dots, n_i$ are independent and identically distributed for each $i \in\{ 1,2\}$. Note that I assume equal means ($0$) and variances ($\...
Syd Amerikaner's user avatar
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1 answer
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Unbiased and constistency of OLS

For the linear regression $y_t = Bx_t+e_t$ where we have the assumptions: $E(e_t)=0$, $E(e_t^2) = \sigma^2$, $E(e_t e_s)= 0$ for $s\neq t $ ...
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Conditions needed for the convergence of Bayesian posterior distribution to point mass (posterior consistency)?

The following 2 theorems (from Bayesian Data Analytics 3rd edition by Gellman, appendix B) show proofs for why Bayesian posteriors converge to a point mass around θ0. Where θ0 is the true parameter ...
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Example consistency of functions

I know that as long as the population moments exist and the data are identically distributed the raw sample moments, the central sample moments and the sample quantiles are consistent. This concept of ...
Marie's user avatar
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Proving consistent/inconsistency of a fusion of KF estimates

I have a distributed fusion scenario with a single target where two sensor nodes $i,j$ estimate the true state $\mathbf{x}$ using a local Kalman filter. The (linear, Gaussian) measurement errors of ...
Nikhil Sharma's user avatar
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6 answers
402 views

How to create optimal cut-off scores for a test placing students into different courses

Edit: Shared my solution as an answer here Our goal is to determine optimal cut-off test scores for course placement. The course placement has already been manually assigned to each test-taker. The ...
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2 answers
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What does the likelihood function converge to when sample size is infinite?

Let $\mathcal{L}(\theta\mid x_1,\ldots,x_n)$ be the likelihood function of parameters $\theta$ given i.i.d. samples $x_i$ with $i=1,\ldots,n$. I know that under some regularity conditions the $\theta$ ...
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Network meta-analysis with placebo only comparison

I have a network meta-analysis where all treatment are compared to placebo. Obviously this is not properly a disconected network but consistency cannot be assessed. Is it problematic?
Jason Shourick's user avatar
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1 answer
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Kolmogorov Smirnov Test Consistency

I was reading that the Kolmogorov Smirnov 2 sample test is consistent, that is Probability of rejection under $H_1$ is 1 for sample size going to infinity. Say we have 2 random variables X and Y. K-S ...
Andrew741's user avatar
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1 answer
182 views

What are the minimum conditions needed for the consistency of OLS estimator in the following linear regression model?

Suppose $Y_i=X_i'\beta+\epsilon_i$ with $E(\epsilon_i|X_i)=0$. Consider the usual OLS estimator for $\beta$ using a random sample $\{X_i,Y_i\}_{i=1}^n$: $\widehat{\beta}=(\frac{1}{n}\sum_{i=1}^nX_iX_i'...
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6 votes
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What 's the $(\Omega,\mathcal{F},P_{\theta} )$ those $T_{n}$ defined on?

Definition (Consistency) Let $T_1,T_2,\cdots,T_{n},\cdots$ be a sequence of estimators for the parameter $g(\theta)$ where $T_{n}=T_{n}(X_1,X_2,\cdots,X_{n})$ is a function of $X_{1},X_{2},\cdots,X_{n}...
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If I have T observations from a binomial(n,p), how to consistently estimate n and p?

Suppose I have a dataset $\{S_t\}_{t=1}^T$, where $S_t\overset{i.i.d.}{\sim}Binomial(n,p)$, how to consistently estimate $n$ and $p$ using this dataset? It would be great if you could provide a method ...
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I need to prove that $\hat\theta=\max\{X_1,...,X_n\}$ is a mean square consistent estimator for $\theta$

Let $X_1,...,X_n$ a i.i.d from a population with distribution $U[0,\theta]$, i.e., $f_{X_i}(x)=\frac{1}{\theta}g_{[0,\theta]}(x)$, for $i=1, \ldots, n$ where \begin{align} g_{[0,\theta]}(x) = \begin{...
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Show that the MLE of $\alpha$ is consistent by definition

Suppose that $(X_1,\dots, X_n)$ is an iid random sample from $X\sim f(x;\alpha, \beta)$ and $$ f(x;\alpha, \beta)=\frac{\alpha x^{\alpha-1}}{\beta^{\alpha}}, \, 0<x\le \beta, \alpha>0, \beta>...
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root-n consistency of penalized M-estimator with fixed p

I am trying to find conditions for the root-n consistency of a generic L1-penalized M-estimator in a fixed p setting. I was able to find those for L1-penalized likelihood and regression (Fan, Li (2001)...
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1 answer
166 views

Posterior consistency for scale-mixture shrinkage priors in low dimension?

Consider the model [1] $$y_n=X_n\beta_n+\epsilon_n$$ $$\beta_i|\sigma^2,v_i \sim \mathcal{N}(0,\sigma^2 v_i), i=1,\ldots,p$$ $$v_i \sim \beta^\prime(a,b)$$ $$\sigma^2 \sim \mathcal{IG}(c,d)$$ where $\...
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Why is posterior consistency research focuses only high-dimensionality?

I have notice that most literature (especially recently) about posterior consistency as $n\rightarrow \infty$ only focuses on areas of high dimensionality i.e. on $p_n\rightarrow \infty$ as $n\...
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Consistent or inconsistent estimator

If $\hat{\theta}_n$ is an estimator for the parameter $\theta$, then the two sufficient conditions to ensure consistency of $\hat{\theta}_n$ are: Bias($\hat{\theta}_n)\to 0$ and Var$(\hat{\theta}_n)\...
user380598's user avatar
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Trouble understanding expected value (does it assume infinite sample size?) and bias vs consistent

This might be a dumb question.. but I was wondering if someone can help me out with the concept expectation. This question started from trying to understand bias vs consistent. So when we roll a dice, ...
chunguc1004's user avatar
4 votes
1 answer
689 views

(Why) is a logistic regression maximum likelihood estimator consistent?

A nice property of maximum likelihood estimators is that, while they can be biased, they are consistent for $iid$ observations. In a logistic regression, unless the conditional distributions all have ...
Dave's user avatar
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Proving estimator consistency

I have the following estimator: $\hat{\sigma}^2_N = \frac{1}{h^2}\sum\limits_{i=1}^{N}x^2_i$, where $x_i \sim i.i.d. \; \mathcal{N}(\mu\frac{h}{N}, \sigma^2\frac{h}{N})$. We can show that $E[\hat{\...
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are the marginals of boosted trees consistent if we can assume unconfoundedness?

given outcome $y$ and data $X$ with data generating process $y = f(X)+\epsilon$ where $\epsilon$ independent of $X$ and gradient boosted trees as the algorithm approximating $f$, does $\partial \hat{y}...
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*When* are mixed models with a lagged dependent variable inconsistent/biased?

Suppose panel data where multiple observations are made of units over time. Regressing a dependent variable measured at each time point on lag of the dependent variable and a unit-specific intercept (...
socialscientist's user avatar
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Proof of consistency of OLS estimator under Heteroskedasticity

$\DeclareMathOperator{\pl}{\operatorname{plim}}$ Consider a general linear regression model with heteroskedastic errors $$ \boldsymbol{y}=\boldsymbol{X}\boldsymbol{\beta}+\boldsymbol{u} \quad \text{...
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3 votes
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What is cor.smooth(R) : Matrix was not positive definite warning Cronbach alpha in Psych?

I'm getting the warning In cor.smooth(R) : Matrix was not positive definite, smoothing was done, but what is it in this case? Can I get away with that? code: <...
Larissa Cury's user avatar
3 votes
1 answer
88 views

Consistency of a simple estimator for $y_i = \beta_1 x_i + u_i$

Let $y_i = \beta_1 x_i + u_i$ for $i=1,2,..,n$. If I define $$\hat \beta_1 = \frac{y_1 + y_n}{x_1 + x_n}$$ then whether my $\hat \beta_1$ will be consistent or not in this setup? For my estimator to ...
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