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.
247
questions
0
votes
0answers
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
$...
1
vote
0answers
18 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 ...
2
votes
0answers
17 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:
"...
1
vote
1answer
27 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 ...
0
votes
0answers
17 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
150 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 $n$...
1
vote
0answers
14 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 ...
1
vote
0answers
22 views
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{\...
0
votes
1answer
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 ...
1
vote
0answers
10 views
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 ...
0
votes
0answers
10 views
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 ...
0
votes
0answers
18 views
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]$...
1
vote
0answers
26 views
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 ...
1
vote
1answer
49 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 ...
2
votes
1answer
69 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 arenāt allowed in GAM.
0
votes
1answer
53 views
the approximation of the variance of MLE (Cramer-Rai Lower Bound)
This is in In Casella's Statistical Inferenceļ¼page 473, the approximation of the variance of MLE (Cramer-Rao Lower Bound). I really confused with the conclusion:
$Var_{\hat{\theta}}h(\hat{\theta})$ ...
3
votes
1answer
85 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$$
Data is given for a ...
13
votes
1answer
442 views
OLS as approximation for non-linear function
Assume a non-linear regression model
\begin{align}
\mathbb E[y \lvert x] &= m(x,\theta) \\
y &= m(x,\theta) + \varepsilon,
\end{align}
with $\varepsilon := y - m(x,\theta)$...
1
vote
0answers
14 views
Consistent estimation of discrete probabilities given probabilistic observations
Suppose $X$ and $Y$ are Bernoulli distributed.
When we have observed data, it's clear we can estimate marginal and joint probabilities from counting occurrences. For example, suppose we have the ...
1
vote
1answer
43 views
Question regarding cointegration and superconsistency
I am reading this PDF:
https://warwick.ac.uk/fac/soc/economics/staff/gboero/personal/hand2_cointeg.pdf
where on pages 4 and 5 it says that if the residuals are stationary, the OLS regression is ...
4
votes
1answer
66 views
Non-Linear regression and variance misspecification
Given a non-linear regression model for cross-section data
$$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 ...
1
vote
0answers
8 views
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 ...
2
votes
1answer
35 views
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. ...
0
votes
1answer
71 views
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 ...
7
votes
1answer
240 views
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 ...
0
votes
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 ...
0
votes
0answers
11 views
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 ...
1
vote
1answer
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 ...
2
votes
0answers
31 views
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 $...
1
vote
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 ...
3
votes
2answers
57 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 ...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
0answers
28 views
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?
1
vote
0answers
96 views
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$$
Thus I obtain the two estimators $\hat{\...
3
votes
2answers
94 views
Consistency of MLE of $\alpha$ when pdf is $f(x;\alpha,\beta)=\frac{\alpha x^{\alpha-1}}{\beta^\alpha}1_{0<x<\beta}$
I have a sample of size $n$ from the following distribution:
$$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{\...
2
votes
0answers
22 views
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 ...
4
votes
2answers
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,...
0
votes
0answers
16 views
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, $...
0
votes
0answers
16 views
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:
...
1
vote
0answers
27 views
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 ...
10
votes
1answer
245 views
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 ...
0
votes
0answers
21 views
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 , \...
0
votes
0answers
39 views
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 ...
16
votes
4answers
2k views
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 $\forall \epsilon>0$
$$\lim_{n\...
2
votes
2answers
134 views
Proof of (weak) consistency for an unbiased estimator
I want to prove a theorem stating:
An unbiased estimator $\hat{\theta}$ of the unknown parameter $\theta$ is consistent if $V(\hat{\theta}_n$) $\to0$ for ${n\to\infty}$.
I've tried using the ...
1
vote
1answer
104 views
Show that $nX_{(1)}$ is not consistent
Consider a random sample from exponential distribution with mean $\frac{1}{\theta}$. I have to prove that $nX_{(1)}$ is not consistent for $\frac{1}{\theta}$ . A sufficient condition for consistency ...
3
votes
1answer
91 views
Consistent estimator for conditional expectation
Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d.
$X_i$ has finite support. Let $x$ be a point in the support of $X_i$.
Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite.
Is it ...
0
votes
1answer
51 views
$P(\hat{\theta}\neq \theta) \rightarrow 0$ as the sample size increases implies $\hat{\theta}= \theta+o_p(1)$?
While I think it is reasonable, I cannot show this result.
Suppose $\hat{\theta}$ is an estimator of $\theta$ and $P(\hat{\theta}\neq \theta) \rightarrow 0$ as the sample size increases, that is, $P(\...
0
votes
1answer
61 views
Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$
Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that
$$T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$$
MY ATTEMPT
To begin with, let us notice that
\begin{...
