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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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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Finding a consistent sequence of estimators such that $\lim_{n\to\infty} E_\theta[(W_n-\theta)^2]\ne 0$

There are many ways to check if a sequence of estimators is consistent. By definition, a sequence of estimators $W_n = W_n(X_1,X_2,\ldots,X_n)$ is a consistent sequence of estimators of the parameter ...
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Retrieve random effects from an instrumented RE model

I am writing my term paper and I feel a demand to retrieve the individual effects from a Random Effects Instrumented Model (due to the usage of lags of regressors as the instruments, the panel is ...
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understanding the proof that the average of sum of i.i.d cauchy is not a consistent estimator of location parameter

Consider $X_1, X_2, ... ,X_n \sim_{i.i.d} Cauchy(\theta), \bar{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$ To prove that it is inconsistant, consider the characteristic function of $X_i$ and $\bar{X}$, which ...
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Differences Between the Central Limit Theorem and Consistency

I have recently finished studying the central limit theorem and the idea of consistency. I am still a little fuzzy about them, so I was wondering what are some key similarities and differences of the ...
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convergence rate of sample covariance matrix

I have a question about deriving the rate of convergence of sample covariance matrix. For the sake of simplicity, we can assume that our sample $\{ X_i\}_{i=1}^{n}$ is i.i.d. (I known we can relax ...
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Efficiency and consistency of an estimator

I recently encountered the following question, which I am struggling with for two days: Let $X_1, \ldots, X_n$ be an i.i.d. sample from a $\text{Geometric}(1/(1 + \theta))$ distribution with pmf $$f(...
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Multiple roots of the likelihood equations vs. consistency

I'm trying to understand the implications of the Huzurbazar-Chanda theorem in finite samples. The result basically says that of all roots of the likelihood equations, one and only one tends in ...
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Identifiable but has no consistent estimator

Let $P_\theta$ denote the distribution of the random variable $X$. The distribution depends on the parameter $\theta$ that lies in some parameter space $\Theta$. Consider a function $f(\theta)$ of $\...
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Checking for consistency in pairs or triplets of distributions?

Background I am involved in a very large project studying plants. We have millions of plants, and for each plant, two or three different research groups are going to measure three properties $X$, $Y$, ...
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Consistency under serial correlation and endogenous instrumental variables

We are estimating the following model: $$ \text { y}_{i, t}=\text {y}_{i, t-1} \alpha+x_{i t}^{\prime} \beta+\theta_{t}+c_{i}+u_{i t} $$ where $x_{i t}$ are vectors of stricly exogenous variables but ...
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What is an initial consistent estimator and how do I find one?

When maximizing a likelihood function $L(\psi)$, the gradient-based optimization procedure is generally $$ \tag{5.1} \hat{\psi}_{r+1} = \hat{\psi}_{r} + \left| I^{*}(\hat{\psi}_{r}) \right|^{-1} D \...
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How to check the consistency of OLS estimator in macroeconomic models

Problem: We have a model $$C_t = a + b Y_t + e_t$$ and $$ Y_t = C_t + I_t$$ It's known that $Cov(I, e)$ is zero. A student estimates the following model: $$C_t = a + b Y_t + e_t$$ Are the estimators $\...
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Are neural networks consistent estimators?

In a nutshell Has consistency been studied for any of the 'typical' deep learning models used in practice, i.e. neural networks with multiple hidden layers, trained with stochastic gradient descent? ...
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Invariance Property of MLE and Consistency

I am posting this because I want to make sure I understood the concept of invariance property of consistent estimators and Maximum Likelihood Estimators. The invariance property of consistent ...
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Will ratio of two consistent estimator be consistent in general

If I have two statistic say $T_1$ and $T_2$ both of which are consistent for $\theta_1$ and $\theta_2$, then will the ratio $\frac{T_1}{T_2}$ be also consistent for $\frac{\theta_1}{\theta_2}$ in ...
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Does this Monte Carlo method evidence the consistency of an estimator?

The Statistic Let $\hat{\theta}(x)$ be a statistical estimator of population parameter $\theta$ whose exact distribution is unknown. Let $\hat{\theta}_n(x)$ be the estimator calculate from a sample of ...
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Checking the consistency of the statistic

Let $X_1, X_2,...,X_n$ be n random samples from $N(\theta, \theta^2)$. We have a statistic $T = \sum Xi^2$. I need to prove the consistency of $\frac{T}{2n}$ for estimating $\theta^2$. I know that ...
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Gauss-Markov and Asymptotic Properties

Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $\sigma^2 I$) can imply "consistency" and "asymptotic normality" of the OLS estimator? ...
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Doubt in the Invariance Property of Consistent Estimators

Let $X_1,X_2,X_3,..,X_n,X_{n+1}$ be random samples from $N(\mu,1)$. Let us define $\bar {X}_n = \frac{\sum X_i}{n}$ and $T = \frac{1}{2}(\bar {X}_n + X_{n+1})$. It is required to test whether $T$ is ...
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Intuitively Assert Consistency

Consider a model $$\mathbf y=\beta_0+\mathbf x_1\beta_1+\mathbf x_2\beta_2+\mathbf e$$ and assume $\mathbb E[\mathbf e\mid\mathbf X]=0$. Under this scenario, the OLS estimator is a consistent ...
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Consistency of a simple Bayes classifier

Let $p(x,y)$ be the joint distribution of random variables $X$ and $Y$ where $$ \begin{aligned} Y&\sim \operatorname{Bernoulli}(\pi),\\ X\mid Y=y&\sim N(\mu_y,\sigma_y^2). \end{aligned} $$ Let ...
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How much would an endogenous variable bias the regression coefficient of other covariates

I came across an expression for inconsistency in regression coefficient of the exogenous variable, when an endogenous variable is included. I didn't quite get how to go from the auxiliary equation to ...
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Can I use interclass correlation with only two rankers?

I was wondering if using an interclass-correlation coefficient with only two rankers is appropriate. I can't find anything on the internet. Cheers!
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How to measure internal consistency of 1 person over 3 conditions

I have conducted a survey in which people rate 4 images out of 5, rank 4 images on a scale 1-4 and choose only one image when displayed 2. The results were normalised, so each image for each method ...
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Adapting OLS to a parametric regression coefficient

Consider the following linear model $$ Y_i=X_{i1}\beta_1+\eta_i X_{i2}\beta_2+\epsilon_i $$ Let $\beta\equiv (\beta_1,\beta_2)$ and $X_i\equiv (X_{i1}, X_{i2})$. Assume that [A1] We have an i.i.d. ...
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ML estimator of $\theta>0$ is $\hat{\theta}_n=\frac{\sum_{i=1}^nX_i^2}{n}$ and $I(\theta)=\frac{1}{\theta^2}$. Show $\hat{\theta}_n$ is consistent

In this problem we have that $I(\theta)=\frac{1}{\theta^2}$ is the Fisher information for a general probability density function $f(x;\theta)$ and $X_1,..., X_n$ are IID random variables from this ...
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Mean-square convergence of maximum likelihood estimators: Examples?

From what I've gleaned from the literature, Cràmer, in his 1947 monograph Methods of Mathematical Statistics, proved convergence in probability of an MLE under certain regularity conditions. ...
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Evaluating rating consistency between an algorithm and a group of experts

Let's have a look on this example: $N$ experts rate the taste of $M$ cakes by a score from 0 (awful) to 10 (best cake in the world). Since experts are rare and quite expensive, a prototype machine was ...
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NLS more consistent than GMM

I'm trying a simple code to test whether NLS has better performance than GMM. The R code looks like this ...
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Question about consistent estimators and asymptotic distributions

Lets say you have an estimator that is consistent, and you do not have any information on the asymptotic distribution, what can you do with such an estimator? Also when using aysmptotic distribtuions, ...
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General recipe for finding unbiased or consistent estimator? [closed]

I am wondering whether there is a general recipe for finding unbiased and consistent estimators of some non-random quantity. For concreteness, I will discuss only discrete probability distributions ...
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consistency of weight matrix

Take the model $Y = X'\beta + e$ with $\mathbb{E} [Ze] = 0$. Let $\tilde{e}_i = Y_i - X'_i \tilde{\beta}$ where $\tilde{\beta}$ is consistent for $\beta$ (e.g. a GMM estimator with some weight matrix)....
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Measurement error in dependent variable leading to inconsistent estimates

Suppose the true model is: Y_i = \alpha + \beta X_i + \epsilon_i Suppose there is a measurement error v in the dependent variable. If v and \epsilon_i are not correlated, is that enough to say our ...
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Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples?

Is it true that an estimator will always asymptotically be consistent if it is biased in finite samples? I feel like it is true but not sure exactly how to prove that...
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OLS estimator is consistent if the smallest eigenvalue of $X^TX$ goes to infinity as $n\to\infty$

I want to show that if $\lambda_{min}(X^T X)$ (i.e., the smallest eigenvalue of $X^TX$) goes to infinity as $n\to\infty$, then $\hat{\beta}$ is a consistent estimator of $\beta$. My approach is the ...
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Consistency of an estimator [closed]

I have an estimator for the coefficients of the model $$ y=X\beta+\varepsilon $$ with $y_{n\times1}$, $X_{n\times p}$, $\beta_{p\times1}$, $\varepsilon_{n\times1}$. The estimator is in the form $$ \...
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What is difference between $\hat{X}_n \overset{p}{\to} \bar{x}$ and $(\hat{X}_n - \bar{x}) = o_p(1)$?

Let $\{\hat{X}_n\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, which I take to mean that, for any $\epsilon > 0$, $\lim \limits_{n \to \infty} \Pr(|\hat{X}...
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How to find a good estimator for $\lambda$ in exponential distibution?

I have an Exponential distribution with $\lambda$ as a parameter. How can I find a good estimator for lambda?
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How to prove $s^2$ is a consistent estimator of $\sigma^2$?

I am trying to prove that $s^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2$ is a consistent estimator of $\sigma^2$ (variance), meaning that as the sample size $n$ approaches $\infty$ , $\text{var}(s^2)...
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How do you prove that an IV estimator is (in general) inconsistent if the first stage regression does not include a constant?

Given an equation $$ Y = \alpha + \beta X + u $$ where $X$ is an endogenous variable and $Z$ is a valid instrument for $X$. Then suppose that $$ X = \gamma + \pi Z + v $$ is the true data generating ...
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Consistency and asymptotic unbiasedness?

I understand the differences between the two concepts, but they look similar so I was searching for some theorems which tie them. I found that a sufficient condition for an estimator $T_n$ to be ...
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Calculate the consistency of an Estimator

I need to determine whether the following estimator $T$ is asymptotically unbiased and consistent for an i.i.d. sample of Gaussian distributions with $X_{i} \sim N(\mu, \sigma)$: \begin{equation*} T = ...
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Why does MLE tend to normal distribution

We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$). If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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MLE and consistency and efficiency of MLE

Here is my understanding of Likelihood function, maximum likelihood estimator (MLE) and consistency and efficiency of MLE. (Notes: Comments are not main parts of this post and can be skipped. Only ...
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Intercept in a dynamic panel model

I have been taught that including fixed/random effects in a dynamic panel model yields inconsistent estimates when using OLS and hence motivates the usage of other estimation methods. However, does ...
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Consistent estimator - consistent with what exactly?

Lets assume, that the real DGP (real world data) is generated from the model: $$y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \varepsilon_i$$ Lets further assume, that $x_1$ and $x_2$ are correlated....
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For $X_1, \ldots, X_n \overset{iid}{\sim} N(\mu,\sigma^2)$, is $\frac{1}{n+\delta}\sum_{i=1}^n X_i$, for $\delta>0$, consistent for $\mu$?

For $X_1, \ldots, X_n \overset{iid}{\sim} N(\mu,\sigma^2)$, suppose we define an estimator for $\mu$ as $$ \theta_n = \frac{1}{n+\delta}\sum_{i=1}^n X_i $$ for some $\delta>0$. Intuitively it ...
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CLT with inconsistent estimator

So I have the OLS estimator that is inconsistent due to the mean independence assumption being violated. I'm asked whether $\sqrt{n}(\hat{\beta}-\beta)$ converges when the sample size $n$ goes to ...

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