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 ...
1
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0answers
22 views
Unbiasedness and consistency
Assume the simple regression model satisfying all Gauss-Markov assumptions.
Somebody suggests the estimator
Why may someone consider such an estimator?
Why will this estimator be consistent? Why ...
2
votes
1answer
44 views
Why is it important that estimators are unbiased and consistent?
I am clear on the definition of unbiasedness and consistency. But why are these the criteria we use to judge whether an estimator is a good one? There are other criteria, of course, like the variance ...
1
vote
0answers
23 views
How does the concept of consistency apply to the full bayesian posterior as opposed to a single estimate?
Towards the goal of making a bayesian statistical inference, I start by collecting $M$ independent and identically distributed data observations $D_i$. Then I take a Bayesian approach to learning the ...
2
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0answers
20 views
Asymptotic consistency and normality
I need help getting the following problem
Let $X_1,..,X_n$ be independent $N(\mu,1)$-distributed random
variables. Define $\hat{\theta_n}$ as the point of minimum of
$\sum_{i=1}^n(X_i-\theta)^4$...
0
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0answers
25 views
Yn = min(X1 … Xn), X1 … Xn ~iid with pdf e^-(x-θ), X > θ . Why is the cdf of Yn = 1-e^-n(x-θ)
Yn = min(X1 ... Xn)
X1 ... Xn ~iid with pdf e^-n(x-θ), X > θ
The answer to a problem I couldn't figure out states that the cdf of Yn =
1-e^-n(x-θ)
However, When I try to determine the cdf of Yn, I ...
4
votes
3answers
173 views
How is it that an ML estimator might not be unique or consistent?
Christian H Weiss says that:
In general, it is not clear if the ML estimators (uniquely) exist and if they are consistent.
Can someone explain what he means? Do we not generally know the shape of ...
0
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1answer
39 views
Consistency of the estimator of the variance of the error
In the classical linear regression model, the estimator of the variance of the regression error is $s^2 = \frac{e'e}{n-k} = \frac{u'Mu}{n-k}$ where u is the error vector, e is the residual vector, and ...
1
vote
1answer
32 views
instrumental variables property
How can I show that the instrumental variables (IV) estimator is consistent from this equation using the two stage least squares method?
Where does this equation come from?
1
vote
1answer
35 views
Role of random sample assumption in consistency of OLS estimator
I guess in part what this all amounts to is what does the assumption {(x_i,y_i) : i=1,2,...,n} being i.i.d. imply about the i.i.d-ness of functions of it?
I am confused because for example I have ...
0
votes
0answers
39 views
MLE Asymptotics of Two independent samples
Given two independent samples, parameterized by the same parameter yet with different distributions (for example, Exponential(lambda) and Gamma(lambda, 2)), under what conditions is the parametric MLE ...
0
votes
1answer
11 views
Cronbach alpha and item selection
I am a relative novice when it comes to statistical analysis, so forgive me if my question is unclear, or simply stupid. I am also not a matemathician, as I work in social science research, but I have ...
0
votes
1answer
51 views
Does “random effect” really exist in real data when we use random/mixed effect model? [closed]
If I understand correctly, here is a standard case when we need the mixed effect model:
We are interested in studying the how drugs influence human health conditions, so we collected information ...
2
votes
2answers
90 views
$\sqrt{n}$-consistency of M-estimator based on plug-in estimator
Note: This is a follow-up on a previous question that was concerned about consistency, but this time seeking $\sqrt{n}$-consistency.
Suppose we estimate a quantity $\theta_0$ by the $\tilde{\theta} = ...
6
votes
1answer
107 views
Consistency of M-estimator based on plug-in estimator?
Suppose we estimate a quantity $\theta_0$ by the $\tilde{\theta} = \hat{\theta}(\eta)$ that solves the estimating equation
$$S_n(\tilde{\theta}, \eta_0) = 0$$
where $\eta_0$ is a nuisance ...
0
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0answers
19 views
Bayesian asymptotics
I'm reading about Bayesian asymptotics, but I'm getting confused by the different definitions and notations used here and there.
Could anybody explain to me what is the difference between
...
3
votes
1answer
35 views
Consistency and rates of convergence
Suppose that I have two statistics that are known to be consistent , e.g :
$ S_{n} ^2 $ (biased sample variance about sample mean) and $ S_{n-1}^2$ (bessel-corrected sample variance, that is unbiased)....
0
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0answers
12 views
Seeking a measure of multiple rater consistency for cateorical ratings with sparse data
I have multiple raters and sparse data. The raters can decide how many attributes apply to a person. I have many empty cells due to many attributes and few attributes actually chosen. What tools ...
2
votes
1answer
54 views
Most relaxed assumptions to get consistency of linear regression?
What are the most relaxed assumptions to get consistency of the linear regression estimates with $p$ variables?
The most basic assumptions that I know are in White (1984):
1) The model is correct
2)...
7
votes
2answers
717 views
why does unbiasedness not imply consistency
I'm reading deep learning by Ian Goodfellow et al. It introduces bias as
$$Bias(\theta)=E(\hat\theta)-\theta$$
where $\hat\theta$ and $\theta$ are the estimated parameter and the underlying real ...
3
votes
1answer
122 views
Parameter estimation and model selection consistencies
In a highly cited paper by Zhao (2006) it is stated that (Section 2)
An estimate which is consistent in terms of parameter estimation does not necessarily consistently selects the correct model (or ...
3
votes
1answer
155 views
Consistency of lasso
I would appreciate help in understanding the following theorem from Knight and Fu (2002) paper:
Consider linear regression model of the form
$$Y_i = \beta_0 + x_i'\beta + \varepsilon_i,$$
where $\...
0
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0answers
36 views
Big Oh Pee / Little Oh Pee & Matrix Calculus
It's clear from Big Oh Pee / Little Oh Pee calculus that for $A_n$, $B_n$ scalars, we have
$A_n = Op(a_n), B_n = op(b_n) \implies A_n B_n = op(a_n b_n)$
However, does the same applies to matrices? ...
1
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0answers
34 views
Are the Inverses of two asymptotically equivalent matrices themselves asymptotically equivalent
Suppose $M_n = P_n + op(1)$.
Is it the case that $M_n^{-1} = P_n^{-1} + op(1)$, if both $M_n^{-1}$ and $P_n^{-1}$ exist with probability going to 1 as $n$ increases? Can the Continuous Mapping ...
1
vote
0answers
31 views
Consistency of the risk estimator in the L1 regularized logistic regression
Prof. Ryan Tibshirani in CMU explained in his class notes how to prove the consistency of the risk estimator in the L1 regularized linear regression, but how to prove the consistency of the risk ...
6
votes
1answer
37 views
Is $R^2_{adjusted}$ both unbiased and consistent under the alternative in simple regression?
Consider a simple regression model
$$
y_i = \beta_0 + \beta_1 x_i + \varepsilon_i.
$$
and suppose it is the correct model for the data. As far as I know, $R^2_{adjusted}$ is an unbiased estimator of ...
1
vote
1answer
102 views
Showing Bayes Estimator is Consistent
$X_1, X_2, ... , X_n$ are iid $N(0, \theta)$ random variables with $\theta$ in $(0, \infty)$. With the prior distribution $\pi(\theta)$=$\frac{4e^\frac{-2}{\theta}}{\theta^3}$, I calculated the ...
0
votes
0answers
18 views
Model selection consistency of Dantzig selector
Is it known that Dantzig selector of Candes and Tao: https://arxiv.org/abs/math/0506081 has model selection consistency, i.e, with high probability aporoaching 1 the model will select true features,(...
5
votes
0answers
42 views
Do shrinkage estimators solve the Neyman-Scott paradox?
I read the following SE question: What problem do shrinkage methods solve? And I wondered if shrinkage estimators provide a consistent estimator of the sample variance in a "mixed-effects" model using ...
0
votes
2answers
123 views
Showing that estimator is consistent
Let $\hat{\theta}_n= -\frac{n}{\sum_{i=1}^n \log(X_i)}$, where $X_i$ are i.i.d. samples from distribution with pdf $\theta x^{\theta-1}$ for $x \in (0,1)$. How to prove that $\hat{\theta}_n$ is ...
0
votes
0answers
56 views
Proving Consistency of 'theta' for an exponential distribution
$X_1, X_2, \dots X_n$ constitute a random sample of size $n$ from an exponential distribution:
a) show that $\bar{X}$ is a consistent estimator of the parameter $\theta$
b) with reference to part a,...
0
votes
0answers
80 views
Asymptotic distribution of the first order statistics and consistency
$ X_{i} $'s are i.i.d and $ X_{1},...,X_{n} $ ~ $ f(x;a,\theta )=\dfrac{1}{\theta}e^{-\dfrac{(x-a)}{\theta}}I_{[a,\infty)}(x)$
I found that MLE of $ a $ is $\widehat{a}=X_{(1)} $ and MLE of $\theta$ ...
0
votes
0answers
40 views
How to prove $Var_f[\tilde{\mu}] \approx Var_p[\hat{\mu}](1+Var_f[W])$ is consistent estimator?
Here it's given a method of drawing an estimator for a variance in the context of importance sampling.
http://www.stat.uchicago.edu/techreports/tr348.pdf
My question is, how would one show
$$Var_f[\...
2
votes
2answers
105 views
Which is the relation between population/probability space/sampling?
I am trying to understand the relation between population/probability space/sampling. My arguments are divided in 3 sub-questions which trace my attempt to link in a logical way the three concepts. I ...
2
votes
1answer
28 views
Why using ratio-consistent estimator instead of consistent
I'm reading articles in high dimensional data. I didn't saw the definition of ratio-consistent estimator in any other field. But as in Chen an Qin's article to showing a estimator is consistent we ...
1
vote
1answer
66 views
Unbiased + Variance Vanishes = Convergence in Probability?
I am reading a proof of the consistency of sample variance estimator.
http://webpages.cs.luc.edu/~jdg/w3teaching/stat_305/sp11/consistentsamplevariance.pdf
On ...
2
votes
0answers
76 views
How to prove consistency of quantile regression estimators?
I am a bit confused. The assumptions that have to be fulfilled so that OLS estimators are consistent (and efficient) are fairly straightforward.
I am currently trying to prove consistency of quantile ...
5
votes
2answers
347 views
Is the sample mean always an unbiased estimator of the expected value?
Given a random variable $x$ with a well-defined expected value $\mu$, is the mean of the set of samples $\{x_1,\ \cdots,\ x_n\}$, which we'll call $\widehat{\mu}$, always an unbiased estimator of $\mu$...
3
votes
0answers
28 views
Are REML estimators of variance-covariance parameters consistent?
I am trying to locate some reference papers about consistency of REML estimators in linear mixed effects models. My understanding is that in linear model scenario, REML will produce the exact same ...
1
vote
1answer
463 views
Assumptions of MLE
I am currently reading up on Maximum Likelihood Estimation in Studies in Econometric Method. When describing the requirements for MLE to be consistent, they described it as the following:
A number ...
0
votes
0answers
20 views
is it true that existence of a consistent estimator does not imply identification?
A parameter in a statistical model is identified if there is a bijection between values of that parameter, and the set of probability distributions (over observable variables) that are consistent with ...
1
vote
0answers
110 views
Prove that (θ_n ) ̂ is a consistent estimator of θ
I am supposed to show the consistency of the estimator θ ̂ given that θ ̂_n is a sequence of estimators of θ∈R such that a_n (θ ̂_n-θ) -^d→N(0,σ^2) for some sequence of positive real numbers a_n→∞ as ...
2
votes
1answer
47 views
Reference book for practice problems on Inference
I was wondering if there is any book which has loads of problems on statistical inference.
Desired topics are
Unbiasedness
Consistency
Sufficiency
Completeness
Rao Blackwell Theorem
etc.
0
votes
1answer
251 views
General approach to proving the consistency of an estimator
I am studying statistical inference. I want to know what should be the general strategy for proving the consistency of an estimator.
In most problems whenever I prove consistency, I usually see ...
0
votes
0answers
36 views
Unsure of which intraclass correlation model to use to compare consistency of measures across bouts
I am comparing heart rate (HR) values across repeated bouts of exercise. I have 20 participants who each performed 4 bouts of exercise at the same intensity. I want to test whether the HR values were ...
2
votes
1answer
209 views
Proof of posterior consistency
Given the model
$x_1,...,x_n$ ~ $Poi(\theta)$ (iid),
and the prior
$\theta$~$ Gamma(a,b)$.
How do we prove the consistency of posterior distribution?
Thanks in advance.
Modification:
The definition ...
0
votes
1answer
336 views
Consistent estimator, that is not MSE consistent
I'm trying to understand the concept of consistency in point estimation. Could somebody give me an illustrative example of a (simply) consistent estimator that is not MSE consistent?
Just a recall of ...
0
votes
0answers
14 views
Why functional specification is important in RDD setting?
I'm reading Clark(2009) paper regarding the impact of education reform on school performance. The rule is that the school initiated a democratic vote at first and if 50% of students vote in favor of ...
0
votes
0answers
129 views
How to measure consistency of measurement over time
Let's say I have multiple wine experts who rank set of wines over time.
To put it formally, I have n experts, m wines in the set and c measurements.
Measurement is a matrix with n rows and m columns, ...
0
votes
2answers
43 views
Probability of error for consistent classifier (Random Forests)
I'm going through the paper "Consistency of Random Forests and Other Averaging Classifiers" and I'm stuck on a seemingly simple part of the proof for Proposition 1.
Here's the relevant part:
What I'...
2
votes
0answers
89 views
How to apply the Two stage least square
Given the arma (1, 1) model: $y_t=ρy_{t−1}+v_t+θv_{t-1}$
I found that OLS estimate for ρ is not consistent and that $y_{t-2}$ can be an instrument for $y_{t-1}$, can anyone help me know how to perform ...
