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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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$\sqrt{n}$-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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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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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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,...
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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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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[\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mu$...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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 ...