# Tagged Questions

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.

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### Cronbach's alpha for one scale but different types of items

I developed a questionnaire on art expertise. There are 3 rather subjective questions("I know much about art","I know more than the average citizen","I am able to identify an artwork by a popular ...
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### Is E(u|x)=0 is a required condition for estimator consistency?

For OLS parameter estimates to be consistent it must be the case that E(u|x)=0. Is it true? E(u|x)=0 is a required condition for unbiasedness. But as far as I understand, unbiasedness does not ...
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### Show that a given estimator is biased and consistent

Let the model be $\log(W) = a + bX + U$ where $E(U) = 0$. We are allowed to assume that $\operatorname{Cov}(X,U) = 0$, and want to show $e^{xb^\text{ols}}$-1 is a consistent estimator for $e^{xb}$-1 ...
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### Why don't asymptotically consistent estimators have zero variance at infinity?

I know that the statement in question is wrong because estimators cannot have asymptotic variances that are lower than the Cramer-Rao bound. However, if asymptotic consistence means that an estimator ...
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### how to show the following consistency?

It is well-known that maximum likelihood estimate for a mixture model, with the mixture distributions known, and the estimation is done for the mixture coefficients is consistent (I think) -- the ML ...
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### when is an estimator consistent?

Say there are parameters $\theta$ such that $\theta_i > 0$ and $\sum_i \theta_i = 1$ and a model such as $p(x) = \sum_{i=1}^n \theta_i p_i(x)$ where $p_i(x)$ are fixed and defined over a domain of ...
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### How to show that the mean is (weakly) consistent

How can I show that the mean is weakly consistent? Is weakly consistent the same as consistent?
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### Is there a statistical application that requires strong consistency?

I was wondering if someone knows or if there exists an application in statistics in which strong consistency of an estimator is required instead of weak consistency. That is, strong consistency is ...
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### Sample variance Fisher consistency

I am reading conflicting references regarding the Fisher consistency of the sample variance. $$s^2 = \frac{\sum_i^n (x_i - \bar{x})^2}{n}$$ Could anyone explain me how to proof whether $s^2$ is ...
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### $a_n$ consistency and other consistency

From Jun Shao's Mathematical Statistics Definition 2.10 (Consistency of point estimators). Let $X = (X_1 , ..., X_n)$ be a sample from $P ∈ \mathcal P$ and $T_n(X)$ be a point estimator of $θ$ ...
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### Understanding and interpreting consistency of OLS

Many econometrics textbooks (e.g. Wooldridge, "Econometric analysis...") simply write something similar to: "If the population model is $y = xB + u$ and (1) $\text{Cov}(X,U) = 0$; (2) $X'X$ is full ...
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### Consistent estimator with var=0

I'm trying to think of an example of a consistent estimator with var=0 but I'm having a hard time. I pretty sure it exists but I'm unable to find one. Any help\hints would be much appreciated.
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### How to test inconsistency / within-subject variance across time?

I have this dataset where I have DV measured over 5 time points for every subject (over about 30 subjects). If I graph the values of individual subjects over time and look at them, there are no ...
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### counterexample for the sufficient condition required for consistency

We know that if an estimator is an unbiased estimator of theta and if its variance tends to 0 as n tends to infinity then it is a consistent estimator for theta. But this is a sufficient and not a ...
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### Consistency of sample quantile

I'm new to statistics and recently I learnt about non-parametric estimation. On the estimation of empirical quantiles, many notes talk about the asymptotic behavior of sample quantiles. The proof is ...
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### Is optimizing α for partitions of a scale a sensible method for factor analysis? (was: relation between consistency and unidimensionality)

Cluelessness Disclaimer I'm a statistics noob so if at all possible, please don't stone me. Also write slowly and in simple terms. I'm wondering what the relation between the internal ...
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### Assessing the consistency of an estimator of the average path length of a large network

I have found the following in a network-science article and would like clarification of what the authors are claiming: We observe that the root mean square of the difference between the empirical ...
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### Conditions in law of large numbers

The (Strong) Law of large numbers states that $\frac{1}{N}\sum_{k=1}^N h(X_k) \rightarrow \mathbb{E}\left[h(X)\right]$ a.s in $\mu$ as $N\rightarrow \infty$. but I can't find any conditions on ...
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### Inference on original population based on sample - estimating sampling probability

In my research I am working on some data (let’s call it sales data), however we almost have no information on the original sample. My goal is to come up with possible explanations on how it could be ...
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### Asymptotic normality of MLE in exponential with higher-power x

Given the distribution: $f(x;\theta) = \frac{3}{\theta}x^2e^{-x^3/\theta}$ if $x>0$ the MLE for $\theta$ is $\frac{1}{n}\sum_{i=1}^n x_i^3$. It's an unbiased estimator with variance $\theta^2/n$. ...
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### Consistency of estimator

Regression: Wage=b0+b1collegegrad, where collegegrad is a dummy variable. Suppose you want to estimate the wage ratio between college graduates and non-college graduates. Is the estimator theta=b0/b1 ...
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### Measures of internal consistency of survey items

I have a question on a survey that asks students how useful a feature of the software was that they used for learning. I then have three questions that assess in what way the feature was useful. The ...
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### Are inconsistent estimators ever preferable?

Consistency is obviously a natural and important property estimators, but are there situations where it may be better to use an inconsistent estimator rather than a consistent one? More specifically, ...
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### What is the difference between a consistent estimator and an unbiased estimator?

I'm really surprised that nobody appears to have asked this already... When discussing estimators, two terms frequently used are "consistent" and "unbiased". My question is simple: what's the ...
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### Consistent estimator for probability density function at a given point

I am trying to conduct a simulation study on a variation of the kernel density estimator. In that experiment, there is a parameter in one of my formulas that involves the value of the unknown pdf at a ...
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### Convex cost functions and Fisher consistency

In Section 4.2 of the paper by Boucheron et al., the authors argue that the minimizer $f^*$ of the cost functional $$A(f) = \mathbb{E}\{\phi(-Yf(X))\}$$ is such that the classifier $g^*$, constructed ...
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### Dynamic consistency and multilevel models using lmer

I've been using nlme and more recently lmer to fit multi-level models of time course data using orthogonal polynomials. My ...
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### How does one derive the Fisher-consistency correction factor for Rousseeuw and Croux's $S_n$ (and $Q_n$)?

In "Alternatives to the Median Absolute Deviation" (Rousseeuw and Croux, J. Amer. Statistical Assoc, 88(424), 1993, pp.1273–1283) and a few other papers from the same authors published in ...
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### How to show that an estimator is consistent?

Is it to show that MSE = 0 as $n\rightarrow\infty$? I also read in my notes something about plim. How do I find plim and use it to show that the estimator is consistent?