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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16 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 , \...
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How to measure consistency/similarity between 2 or more sets? [on hold]

Say I have 2 sets of things that can't be ranked or anything. Each object in a set is unique but that object may or may not be in both sets. How to measure the consistency between the two sets? By ...
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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 ...
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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\...
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2answers
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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 ...
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65 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 ...
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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 ...
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How to measure “well-sampled”ness i.e. how well a distribution is sampled in a given set of data

Say we have samples $x_i$ from an unknown distribution $F(x)$. We want to know the number of samples $n$ such that we can say the distribution is "well sampled". Generally we need some sort of ...
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45 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(\...
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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{...
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Consistency test

I have data that consists of 7 characteristics for each participant. There are about 170 participants. The test was repeated in some time and the same 7 characteristics were acquired. I need to find ...
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Cronbach's alpha for different items?

Can I perform a Cronbach's alpha test when the questions or items are completely different to each other as opposed to having similar questions written differently as a consistency measure?
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Lagged dependent variables, bias and consistency

I am working through Christopher Dougherty's Introduction to Econometrics, and am struggling to fully grasp the consequences of lagged dependent variables in terms of bias and consistency. The key ...
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Consistent estimation and valid inference when performing regressions on data with differing levels of granularity

Imagine that a dataset has a combination of variables of differing levels of granularity (e.g. an international sample of firms containing both firm-level and country level information). There are $K$ ...
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Endogeneity and Consistency

So I learned about the endogeneity problem of linear regression in class today, where E[XU] and Cov[X,U] isn't equal to zero but some random constant c times a standard basis k-element vector with 1 ...
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Consistent estimator and distribution function

a general question: If the distribution function $F_n$ of some estimator $T_n$ suffices \lim_{n \rightarrow \infty} F_n(x) = 1 \text{ or } 0 \forall x}. Does that imply that $T_n$ is consistent? I ...
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first factor saturation vs. general factor saturation

a simple explanation between these two in the context of reliability analysis, specifically Cronbach's alpha.
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Can any unbiased estimator be changed into a consistent estimator when estimating functions of the mean [closed]

For an i.i.d sequence of Random Variables $X_1, \dots, X_n$, each with mean $\mu = \mathbb E[X]$, the goal is to estimate some continuous function $f$ evaluated at the mean, $f[\mathbb E[X]]$. If ...
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Chronbach's Alpha of Likert Item

My questionnaire is consists of ten 5-point scaled likert items. If the likert items have a good chronbach alpha, is it ok to take the average score of 10 items?
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Difference between Cronbach's alpha and Pearson's coefficient

I am creating an indicator for social development which includes variables of health, education and economy. Since I have many variables, I decided to remove some of them based on dominion expertise ...
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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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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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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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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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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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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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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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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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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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199 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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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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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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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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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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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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1answer
277 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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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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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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112 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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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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1answer
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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 ...