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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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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How to find a consistency index for a binary variable?

I am working on a project where there are 2 variables to monitor, say sales (actual) and projected sales (target). The sales ...
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Concentration inequalities for local smoothers (Nadaraya Watson)

Let $m(X)=E(Y|X)$ be a regression function with random design and let $\hat{m}_h(x)$ \begin{equation} \hat{m}_h(x)=\frac{\sum_{i=1}^n K_h\left(x-x_i\right) y_i}{\sum_{i=1}^n K_h\left(x-x_i\right)} \...
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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 }...
Maximilian's user avatar
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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 ...
Uk rain troll'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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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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Metric for run-to-run consistency of time series data

If I run $n$ samples of a physical experiment, I expect to see roughly similar time vs. position plots but with slight variations run-to-run. What are good statistical metrics to quantify the ...
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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 ($\...
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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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Can you use inter-rater reliability to measure reliability of a whole measure with items measuring different things? (Krippendorff's Alpha or similar)

Let's say that you have one case study, where 30 raters are asked to apply a measure that has 5 items. Can you compute a single Krippendorff's Alpha (kalpha) value across all 5 items (that are the ...
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How to decompose the scaled estimation error of the debiased machine learning estimator? [Chernozhukov et al 2018]

In page C4 of Chernozhukov et al (2018) the authors introduce a debiased ML estimator $\check{\theta}_{0}$ for scalar parameter $\theta_{0}$: $$ Y = D\theta_{0} + g_{0}(X) + U, \quad \mathbb{E}[U|X,D] ...
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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 ...
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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 ...
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6 answers
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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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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 ...
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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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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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Showing an estimator is consistent for a parameter

Suppose $X_1, ..., X_n$ i.i.d.~ $\mathscr{N}(0, \theta^2)$ with unknown $\theta>0$. I have an estimator $\hat{\theta}_n=\sqrt{\frac{\pi}{2}}\frac{1}{n}\Sigma_{n}^{i=1}|X_i|$ for $\theta$. How do I ...
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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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2 votes
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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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Can I delete a dimension of a scale if this dimension reach a very low internal consistency when doing psychological network analysis

I am trying to do a psychological network analysis, and I aim to do the analysis on dimension-level (summing all the items belonging to this dimension). However, I've found that one dimension of the ...
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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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Are the Grenander conditions on the explanatory variable to ensure OLS has consistent treatment effect estimates applicable to GLM/GLMM?

Are the Grenander conditions on the explanatory variable to ensure OLS has consistent treatment effect estimates applicable to GLM/GLMM where you have count,binary data? If you don't know what ...
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7 votes
1 answer
165 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)\...
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Methods describe the temporal consistency of kernel density data

I am working on a spatial time series analysis project. The task is to study the spatial distribution of point features (e.g., crime events, traffic accidents) over time. I aim to find the places with ...
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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, ...
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1 answer
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(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 (...
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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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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
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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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For Instrumental Variables, Why can we show that it is unbiased by taking E(Y|X,Z)?

My understanding is that instrumentals variables regressions estimator is consistent, but not unbiased, for identifying the causal effect of a variable x and y. I understand that for an instrument,z, ...
Steve's user avatar
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Is there a natural equivalent of consistency for the case of prediction?

Consistency is generally understood to be a pretty basic requirement for a decent estimator of a model parameter or other population quantity. If your estimator isn't consistent, then it won't ...
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