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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8 views

Truncated CEF of normally distributed RV. Is sample analogue a consistent estimator of the 'population' truncated CEF?

If I have a random variable that is normally distributed, and truncated such that I only see $y$ if $y\geq 0$, and I want to do some calculations with the truncated CEF in my sample, where it equals $...
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Is maximum-a-posteriori estimation consistent?

I am wondering if Maximum-a-Posteriori (MAP) estimates are consistent in the frequentist sense. When I am searching for this, usually what pops up is posterior consistency, for example in the sense ...
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Strong consistency in quantum estimation problem

I'm reading the paper: Strong consistency and asymptotic efficiency for adaptive quantum estimation problems by Akio Fujiwara. In this paper, describes the next adaptive scheme of estimation: "...
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Consider this estimator of a cumulative distribution function. Can you tell me if it is consistent? What about the assymptotic distribution?

This is an estimator of a cdf of F(x) of a iid random sample x1, x2, ..., xn of observations. My question is if for a given value of x, this estimator is consistent. And how can you derive the ...
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Is sample variance of identical but correlated variables a consistent estimator for true variance?

We know that sample mean and sample variance for iid random variables is a consistent estimator for true mean and true variance, but how about if the random variables follows same distribution but ...
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Are inconsistent estimators ever preferable? A twist

The thread "Are inconsistent estimators ever preferable?" and @whuber's answer in it shows that there exists an inconsistent estimator that can outperform a reasonable consistent one for all finite $n$...
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Weakest possible assumptions to guarantee consistency in linear regression estimators?

For a linear regression model where $Y = \beta_0 + \beta_1 X + \epsilon$ and $E[\epsilon|X=x]=0$, $Var[\epsilon|X=x]=\sigma^2$ for any $x$. For a sample $(X_1,Y_1),\dots,(X_n,Y_n)$, I'm wondering if ...
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Asymptotics of Marginal Likelihood

I'm working with Bayes factors, and I want to develop some intuition for the result $$ \frac{m_1(\mathbf{X})}{p_n(\mathbf{X}|\hat\theta_n)}\xrightarrow{p}\frac{\pi_1(\theta_0)\sqrt{2\pi}}{\sqrt{\...
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55 views

Weibull's MLE consistency and asymptotic normality

Let X = $(X_1, \dots, X_n)$ be a sample from Weibull distribution $W(\alpha, \beta)$ with fixed and known $\alpha$. Find MLE of parametric function $g(\beta) = \beta^{\alpha}$. Check if bias is equal ...
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M estimation for independent but not identical random variables

I am looking for advanced theories for M estimation. Suppose $X_1,\dots,X_n$ comes from some parametric family. They are independent but not identical with one common parameter $\theta$ and one own ...
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Instrumental variables, do we only need cov(z,e)=0 for exogeneity, and not E[e|z]=0?

Looking at the formula for instrumental variables estimator, we have : B_iv = B + (z'x)^-1(z'e) and then taking probability limits it becomes evident that we need cov(e,z)=0 and cov(z,x)=/=0.. but ...
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Question about consistency and unbiasedness of least square estimators in linear regression

For a random variable $X$ and $Y$, the $MSE$ is defined as $MSE(b_0, b_1) = E((Y - b_0-b_1 X)^2)$ and is minimized when $b_1 = \beta_1 =\frac{Cov(X,Y)}{Var(X)}$ and $b_0 = \beta_0 = E[Y]-\beta_1 E[X]$...
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What is an example of a weakly consistent but not strongly consistent estimator?

I just can't think of any example. I am using definitions: weakly consistent: $\forall \varepsilon > 0 \lim_{n\rightarrow \infty} P(|\hat{\theta}_n - \theta| \geq \varepsilon) = 0$; strongly ...
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Can bias of an estimate be decreased by increasing sample size?

I understand that in case of consistent estimates, larger the sample size, there's a higher probability that the estimate converges to true value of parameter. Now, using the sufficient condition of ...
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Are GAM suitable for inference?

Are the estimators unbiased efficient and consistent? Or is GAM better for classification and prediction than non additive models? Interaction terms aren’t allowed in GAM.
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the approximation of the variance of MLE (Cramer-Rai Lower Bound)

This is in In Casella's Statistical Inference,page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion: $Var_{\hat{\theta}}h(\hat{\theta})$ ...
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Two step regression using group effects and DAG

Consider the following model $$y_i = \sigma_{c(i)} + \mathbf x_i^\top\beta + u^y_i $$ $$\sigma_{c} = z_c\lambda + \eta_c$$ where for all $i$ $$\mathbb E[u^y_i \lvert x_i] = 0$$ Data is given for a ...
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OLS as approximation for non-linear function

Assume a non-linear regression model \begin{align} \mathbb E[y \lvert x] &= m(x,\theta) \\ y &= m(x,\theta) + \varepsilon, \end{align} with $\varepsilon := y - m(x,\theta)$...
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Consistent estimation of discrete probabilities given probabilistic observations

Suppose $X$ and $Y$ are Bernoulli distributed. When we have observed data, it's clear we can estimate marginal and joint probabilities from counting occurrences. For example, suppose we have the ...
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Question regarding cointegration and superconsistency

I am reading this PDF: https://warwick.ac.uk/fac/soc/economics/staff/gboero/personal/hand2_cointeg.pdf where on pages 4 and 5 it says that if the residuals are stationary, the OLS regression is ...
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Non-Linear regression and variance misspecification

Given a non-linear regression model for cross-section data $$y_i = f(x_i,\theta_0) + \epsilon_i,$$ where it is assumed that $\mathbb E[y_i\lvert x_i] = f(x_i,\theta_0)$, I understand that it is a ...
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Effect of heteroskedasticity in hierarchical (non-)linear models

Unlike linear models estimated via OLS where heteroskedasticity lead to inconsistency of the variance estimator but not the coefficient estimates, heteroskedasticity causes inconsistency of both ...
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Alternative to Cronbach's alpha if $n$ is too small for McDonald's omega?

I have a very high Cronbach's Alpha of $0.966$ (and composite reliability of $0.969$). My questionnaire has a 1-factor-structure, which makes it compatible with Cronbach's Alpha, but it has 22 items. ...
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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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1answer
26 views

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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72 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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1answer
26 views

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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1answer
120 views

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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126 views

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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156 views

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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104 views

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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51 views

$P(\hat{\theta}\neq \theta) \rightarrow 0$ 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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61 views

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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