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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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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1answer
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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
$$
\...
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2answers
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What is difference between $\hat{X}_n \overset{p}{\to} \bar{x}$ and $(\hat{X}_n - \bar{x}) = o_p(1)$?
Let $\{\hat{X}_n\}$ be a sequence of estimators that converges in probability to the constant $\bar{x}$, which I take to mean that, for any $\epsilon > 0$, $\lim \limits_{n \to \infty} \Pr(|\hat{X}...
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2answers
77 views
How to find a good estimator for $\lambda$ in exponential distibution?
I have an Exponential distribution with $\lambda$ as a parameter.
How can I find a good estimator for lambda?
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1answer
259 views
How to prove $s^2$ is a consistent estimator of $\sigma^2$?
I am trying to prove that $s^2=\frac{1}{n-1}\sum^{n}_{i=1}(X_i-\bar{X})^2$ is a consistent estimator of $\sigma^2$ (variance), meaning that as the sample size $n$ approaches $\infty$ , $\text{var}(s^2)...
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1answer
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How do you prove that an IV estimator is (in general) inconsistent if the first stage regression does not include a constant?
Given an equation
$$
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
$$
is the true data generating ...
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Consistency and asymptotic unbiasedness?
I understand the differences between the two concepts, but they look similar so I was searching for some theorems which tie them. I found that a sufficient condition for an estimator $T_n$ to be ...
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1answer
34 views
Calculate the consistency of an Estimator
I need to determine whether the following estimator $T$ is asymptotically unbiased and consistent for an i.i.d. sample of Gaussian distributions with $X_{i} \sim N(\mu, \sigma)$:
\begin{equation*}
T = ...
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118 views
Why does MLE tend to normal distribution
We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$).
If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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MLE and consistency and efficiency of MLE
Here is my understanding of Likelihood function, maximum likelihood estimator (MLE) and consistency and efficiency of MLE.
(Notes: Comments are not main parts of this post and can be skipped. Only ...
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0answers
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Intercept in a dynamic panel model
I have been taught that including fixed/random effects in a dynamic panel model yields inconsistent estimates when using OLS and hence motivates the usage of other estimation methods.
However, does ...
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votes
1answer
408 views
Consistent estimator - consistent with what exactly?
Lets assume, that the real DGP (real world data) is generated from the model:
$$y_i = \beta_0 + \beta_1x_{1i} + \beta_2x_{2i} + \varepsilon_i$$
Lets further assume, that $x_1$ and $x_2$ are correlated....
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1answer
32 views
For $X_1, \ldots, X_n \overset{iid}{\sim} N(\mu,\sigma^2)$, is $\frac{1}{n+\delta}\sum_{i=1}^n X_i$, for $\delta>0$, consistent for $\mu$?
For $X_1, \ldots, X_n \overset{iid}{\sim} N(\mu,\sigma^2)$, suppose we define an estimator for $\mu$ as
$$
\theta_n = \frac{1}{n+\delta}\sum_{i=1}^n X_i
$$
for some $\delta>0$. Intuitively it ...
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1answer
43 views
CLT with inconsistent estimator
So I have the OLS estimator that is inconsistent due to the mean independence assumption being violated. I'm asked whether $\sqrt{n}(\hat{\beta}-\beta)$ converges when the sample size $n$ goes to ...
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votes
1answer
55 views
Prove that the MLE exists almost surely and is consistent
I need to show that given an i.i.d sample $X_1,\dots X_n$ arising from the model:
$$\{f(x,\theta)=\theta x^{\theta-1}exp\{-x^{\theta}\},x>0,\theta\in (0,\infty)\}$$
that the MLE exists with ...
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votes
1answer
34 views
How to test if two methods of measurement are consistent?
I am measuring (the same) property of of 20 different objects. To measure this property I have two methods at my disposal. Which statistical test should I use to determine whether these two methods ...
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1answer
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What is the justification for using the sample mean in confidence intervals?
In the set-up for the classical CLT we have that
$$\frac{\sqrt{n}}{\sigma}(\bar{X}_n-\mu)\to^d N(0,1)$$
as $n\to \infty$, which gives rise to the $1-\alpha$ asymptotic confidence interval formula for $...
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votes
0answers
36 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 ...
1
vote
1answer
36 views
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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1answer
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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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1answer
43 views
$\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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1answer
48 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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1answer
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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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0answers
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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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2answers
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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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2answers
168 views
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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0answers
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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
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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
27 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 Conditional Expectation Function in ...
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39 views
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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0answers
20 views
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
30 views
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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29 views
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 ...
5
votes
2answers
207 views
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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19 views
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{\...
