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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25 views

Consistent estimator of $p^2$

$(X_1, X_2,...,X_n)$ is a random sample of size $n$ from $Bernoulli(p)$ distribution. $S_n=\sum_{i=1}^nX_i$. I have to check whether $\frac{S_n(S_n-1)}{n(n-1)}$ is a consistent estimator for $p^2$. $...
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Is the estimator 0.5X1 + 0.5(n-1)^(-1) * the sum from i=2 to n of Xi an unbiased estimator? Is it consistent?

Let {Xi} from i=1 to n be an i.i.d. sample from a distribution f. I suspect this is unbiased, but is it consistent? I'm not sure how to approach it as I think the variance converges to 0, but won't it ...
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Asymptotic distribution after replacing quantities by consisent estimators

Suppose that we wish to estimate $T(\theta_1,\theta_2)$, a continuous function of several parameters. Suppose that we know the asymptotic distribution when $\theta_1$ is replaced by an estimator $\hat{...
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87 views

Basic Questions about regression formula, sampling variability, and 'identification'

lets say I run the simple regression, $y_i = \beta_o + \beta_1x_i + \epsilon_i$.. Assume $cov(\epsilon,x)$=0 This yields the formula people write in terms of covariances for the slope parameter: $\hat{...
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$\sqrt{n}$-consistent and strongly consistent

Can you show that $\bar{X}$ is a $\sqrt{n}$-consistent and strongly consistent for $\mu$? Where $X_1, X_2,..., X_n$ be iid from $P\in{\wp}$ and $\mu$, mean of $P$ is assumed to be finite.
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46 views

Theoretical justification of Parametric bootstrap?

I've been reading about bootstrap, and while it's relatively easy to find theoretical results (consistency and higher-order correctness) for the nonparametric bootstrap (e.g., Asymptotic Statistics by ...
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Citation: Sample mean as consistent and unbiased estimator of the expected value

A reviewer asked for a citation that the sample mean is a consistent and unbiased estimator of the expected value and therefore converges towards the expected value. I know I can easily do the ...
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Advantages of using panel data structure when estimating causal effects consistently

What are the main advantages of using panel data structure compared to the pure cross-sectional data when we want to estimate causal effect of some phenomenon consistently? I know what the general ...
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Heteroskedasticity leads to inconsistent estimate in log-linear model

My question concerns the following paper. Silva, J. M., & Tenreyro, S. (2006). The Log of Gravity. Review of Economics and Statistics, 88(4), 641-658. doi:10.1162/rest.88.4.641 To summarize, ...
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Variance estimator using mixture of scaled and unscaled data

Given two datasets: $X_1, \dots, X_n \sim N(1, \sigma^2)$ and $X_{n+1}, \dots, X_N \sim N(1, 2\sigma^2)$ My proposed estimator for $\sigma^2$ is simply a scaled combination of both classical ...
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If $\theta_{n, 1}, \dots ,\theta_{n, m} \stackrel{p}{\rightarrow} \theta$, does $m^{-1}\sum_{i}\theta_{n, i}$ converge in probability to $\theta$?

Question Details If $\theta_{n, i} \stackrel{p}{\rightarrow} \theta$ for $i = 1, \dots ,m$, where $m$ is fixed, then does this imply $$\frac{1}{m}\sum_{i = 1}^{m}\theta_{n, i} \stackrel{p}{\rightarrow}...
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For regression: Are clustered standard errors(say specified correctly) only consistent, or both unbiased and consistent estimators?

Basically are clustering standard errors only an asymptotic argument or does it possess finite sample properties as well?
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Proving the consistency of this OLS estimator for $\hat\beta_1$?

So in this particular linear regression model we are given that $\beta_0=0$. The goal is to find the estimator, $\hat\beta_1$, and show that it is consistent. I managed to find $\hat\beta_1$ as ...
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Regression problem with “error in variables”

Suppose that there is a deterministic relation $y_t=ax_t$ where $x_t,y_t$ are real sequences or real functions and $a$ a constant. But only $X_t=x_t+e_t$ and $Y_t+u_t$ can be observed, with $e_t, u_t$ ...
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Calculating consistent estimators

Let $X_1, X_2,\dots$ be $iid$ random variables with density $f(x|p), 0<p<1$ being the unknown parameter. Suppose that there exists an unbiased estimator T of $p$ based on sample size 1, i.e. $E(...
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Consistent estimate of an interval

Suppose I have an interval $[a,b]$, where $a$ is known and $b$ is unknown. Suppose I have a consistent estimator for $b$ denoted as $\widehat{b}$ so that $\widehat{b}=b+o_{p}(1)$. My question: is the ...
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1answer
35 views

In a weighted least squares regression, can we use the weight as a control variable?

I have found Weighted Least Squares with Endogenous Weights but the answers primarily tackle the question of when $w_i$ correlates with $\epsilon_i$. I would like to ask if we use $w_i$ as a control ...
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Using Kendall's W or Kendall's Tau?

So I have a dataset of 5,000 observations annotated by two respondents each, which comes out to 10,000 unique annotations. In these 10,000 unique annotations, I had 600 unique respondents, with the ...
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1answer
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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 Conditional Expectation Function in ...
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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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1answer
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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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1answer
59 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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1answer
108 views

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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1answer
87 views

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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1answer
90 views

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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1answer
470 views

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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1answer
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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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1answer
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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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1answer
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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
31 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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1answer
84 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
32 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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2answers
83 views

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
144 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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