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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Consistency of variance estimator in OLS [duplicate]
Given the model,
$$
y_i = x_i'\beta + \epsilon_i \quad \epsilon_i \sim N(0, \sigma^2)
\quad iid \quad \forall i = 1, ..,n
$$
how can I prove that the estimator of the variance $\hat{\sigma}^2 = \...
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8 views
Low internal consistency
I am currently analysing the data for a survey I have completed. In doing so, one question is quite simple (nominal) and require a typical “I do/I do not” and “I do not/I do” response from the ...
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1answer
45 views
How to measure the consistency of improvement on different conditions?
I want to measure whether the speed improvement of method 1 over method 2 is consistent on different conditions. Below are two examples of the speedup values of method 1 over method 2 on 5 conditions.
...
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1answer
47 views
Effect of adding more sample data on Maximum Likelihood estimator [closed]
I have samples $\{x_1, x_2, x_3, \dots , x_n\}$ of a random variable $X$. I compute Maximum Likelihood Estimator $\hat{\theta}_n$ using the sample data.
Now, if I collect one more sample $x_{n+1}$ ...
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Given a simple logit panel data model, the MLE estimators are inconsistent
Consider a binary choice model, $P \left( y _ { i t } = 1 | x _ { i t } , \alpha _ { i } \right) = F \left( x _ { i t } \beta _ { 0 } + \alpha _ { i } \right)$, $$F ( z ) = \frac { e ^ { z } } { 1 + e ...
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11 views
Is LinUCB Hannan consistent?
I'm going through the textbook Bandit Algorithms, by Lattimore and Szepesvari (http://downloads.tor-lattimore.com/banditbook/book.pdf).
It describes regret bounds for the LinUCB algorithm of the form:...
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1answer
52 views
Is MLE intrinsically connected to logs?
My mathematical exploration led me the following claim:
Claim: MLE is fundamentally connected to logs (and KL divergence, which also uses logs). It’s not correct to say log shows up simply to make ...
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1answer
50 views
Consistency in uniform distribution
I know what consistency is but in options C and D both U and V are given whose covariance is quite difficult to find.
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41 views
OLS assumptions: prediction vs inference
Taking an OLS model (actually, is it a "model" or an "estimator"?) as an example, there are several assumptions (such as strict exogeneity and spherical errors) which are important for the consistency ...
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18 views
Consistency metric explanation
I am trying to understand a bit more about the consistency metric (to understand how consistency-based subset evaluation works).
I find on this paper the following equation :
$$\text{Consistency}_s =...
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1answer
62 views
Invariance property
I am a bit confused regarding what exactly is the invariance property of sufficient estimators, consistent estimators and maximum likelihood estimators.
As far as I know,
Invariance property of ...
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1answer
79 views
OLS estimator for regression without intercept [duplicate]
Consider a linear regression model: $Y_i = \beta_1 A_i + \beta_2 B_i + u_i$ where all variables are assumed to have mean 0, and $A_{i}$ is distributed independently of both $B_{i}$ and $u_{i}$, but $...
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20 views
Are Poisson Regressions with Serial Correlation Biased or Inconsistent? (No Fixed Effects)
Let's say I've got panel data where a count outcome $y$ and continuous independent variable $x$ observed each time period $t=(1,2,...T)$ for each individual $i$. I am interested in how $x_{it}$ ...
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31 views
Bayesian analysis of multilevel model with lagged dependent variable
Currently, I am constructed a bayesian multilevel model to analyze a panel data set which now basically looks like the following: $y_{ijt} = \beta_{0ij} + X\beta + \epsilon_{ijt}$. So, now only a ...
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43 views
Are vanishing bias and variance enough for pointwise consistency for KDE-based estimation?
Question:
Is the condition that asymptotic bias and asymptotic variance goes to zero for infinite samples sufficient to guarantee the pointwise consistency of an estimator based on plug-in kernel ...
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1answer
50 views
Bayesian consistency in compact uncountable parameter space
Let $p(y_i \mid \theta)$ be the likelihood we are using of a single data point, $p(\theta)$ be the prior, and $f(y_i)$ the true distribution of the data. Also, let $\theta_0$ be the parameter that ...
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25 views
Gaussian process for machine learning consistent property explanation
I am currently reading Gaussian process for machine learning book from Christopher Williams, and I encounter a note on function-space view where consistency property is explained, what I am having ...
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32 views
consistency of an estimator not based on total sample size
How do I show the consistency of an estimator of a parameter, say $\mu$, that is not based on the sample size $n$ but a function of $n_{i}$'s where $\sum_{i=1}^{K}n_{i}=n$ ? Consider for example the ...
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35 views
Consistency of estimators
For $1\leq i\leq K$, I have an estimator of $\mu_{i}$ given by $\hat{\mu}_{i}=\frac{1}{K}\sum_{j\neq i=1}^{K}\frac{Y_{ij}}{n_{ij}}$, where $Y_{ij}\sim N(n_{ij}(\mu_{i}-\mu_{j}),\sigma^{2}n_{ij})$. ...
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14 views
Fisher information under different noise models
Directly lifted from Wikipedia: Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable $X$ carries about an unknown ...
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25 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 ...
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1answer
155 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 ...
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26 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 ...
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30 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
$\sum_{i=1}^n(X_i-\theta)^4$...
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3answers
257 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 ...
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1answer
65 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 ...
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1answer
39 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?
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1answer
57 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 ...
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0answers
48 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 ...
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1answer
16 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 ...
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1answer
56 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 ...
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2answers
109 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} = ...
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1answer
123 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 ...
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0answers
23 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
...
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1answer
71 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)....
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0answers
15 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 ...
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1answer
67 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)...
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2answers
1k 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
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1answer
128 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 ...
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1answer
258 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 $\...
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0answers
62 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? ...
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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 ...
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0answers
37 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 ...
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1answer
52 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 ...
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1answer
191 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 ...
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0answers
28 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,(...
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0answers
48 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 ...
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2answers
376 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 ...
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0answers
96 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 $\theta$ ...
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2answers
112 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 ...
