Questions tagged [efficiency]
A measure of the quality of a statistical estimator.
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How to show mathematically that random effects are more efficient than fixed effects?
I have read in several places now that random effects estimators are more efficient than fixed effects estimators, in particular here
I’ve searched this site and Google and couldn’t find this result. ...
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Estimating ratio of regression coefficients
What is the best method of estimating a ratio of regression coefficients $\beta_1/\beta_2$ under the usual assumptions / in practice? I have two relatively well approximated signals $X_1, X_2$ and ...
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Questions about efficiency of estimators with longitudinal data
I'm modelling data with repeated observations; I'm reading up on options and pitfalls, and have a few questions.
Coefficient estimates are still unbiased and consistent in the presence of ...
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Asymptotic efficiency of IQR
I was wondering about the asymptotic efficiency of the Interquartile Range (IQR) in the Gaussian case. I have calculated it empirically using a Monte Carlo estimator, and it appears to be equal to ...
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Is the smooth transformation of an asymptotically efficient estimator still asymptotically efficient?
Suppose I have an estimator $\widehat{\theta}$ that is asymptotically efficient (for example, it could be the MLE of the mean $\mu$ and variance $\sigma^2$ of $Normal(\mu,\sigma^2)$). Suppose I also ...
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Adjusting confidence interval of estimator by efficiency
Summary: If we have an unbiased MLE $\widehat{\sigma_1}$ of an exponential distribution parameter, and the confidence intervals for its estimates are given by the $\chi^2$ distribution; and we find ...
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misspecified models coverage and efficiency
Suppose I have model $M$ generating data $Y=\beta_0+\beta_1X+\beta_2Z+\beta_3W$ with all $\beta$'s known. Instead of using model $M$, I used misspecified models $M':Y=\beta'_0+\beta'_1X+\beta'_2Z$, $M'...
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Compare ARE of mean and median
I have followed some lectures on the internet on asymptotic relative efficiency. When I had calculated the relative efficiency $ARE(median, mean)$ than my result was $ARE(median, mean)=\frac{2}{\pi}\...
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Counterexamples: Minimax but not efficient, efficient but not minimax
Are there examples of estimators that are minimax, but not efficient? Perhaps, to be more concrete, any of the following:
(strong) Estimator sequence for each $n$ that is minimax but does not match ...
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Is there any advantage to PPS sampling if, in the end, you sample the same number of your sub quantity?
PPS sampling (Probability-proportional-to-size sampling), it seems to me, is based on the scenario where you are limited to sampling a fixed number, $n$, units of observation containing some varying ...
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Many crossed-random effects increase multicollinearity, decreases efficiency or consistency or introduces bias?
Many crossed-random effects increases multicollinearity, decreases efficiency or consistency or introduces bias?
What are the trade-offs of using many random-effects?
user318514
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Derivation of the formula for the asymptotic relative efficiency of two estimators with different estimands
Background
In their book, Huber & Ronchetti (pp. 2-3) compare the efficiency of the mean absolute deviation $d_n$ with the standard deviation $s_n$ with the following formula:
$$
\operatorname{ARE}...
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How does one test the efficiency and completeness of an estimator using monte-carlo simulation?
How does one test the efficiency and completeness of an estimator using monte-carlo simulation?
In particular,
I want to use-montecarlo simuation to answer. Maybe the better question is how does one ...
user318514
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Efficiency of sample mean
Since the variance of the sample mean estimator decreases with $n$
$$
Var(\bar{X}) = \frac{\sigma^2}{n}
$$
does that mean that it becomes more efficient as the sample size grows?
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Comparing efficiency between estimators
Suppose that $\hat \theta_1, \hat \theta_2$ are two estimators of $\theta$. Furthermore, assume that
\begin{align}
\sqrt{n}(\hat \theta_1-\theta)\overset{d}{\to}N(0,V_1)\\
\sqrt{n}(\hat \theta_2-\...
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What does asymptotic efficiency mean?
I read some comparison articles, and always find "asymptotic efficiency," "asymptotically less efficient," and "asymptotically normal."
I am really confused about the ...
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Monte Carlo in R simulation for Efficiency [closed]
An exercise displayed in the image below shows example of finding the efficiency of an estimator. I am trying to replicate this example in R using monte carlo. Y1,Y2,Y3 are random samples of normal ...
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unbiased estimator and efficiency
can someone plese clarify a doubt for me?
Let (X1, . . . , Xn) be a random sample of i.i.d. random variables with expected value $µ$ and variance $σ^2$
Consider the following estimator of $µ$:
$T_{n}(...
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Proving efficiency of GLS over OLS [duplicate]
I'm trying to prove that GLS is more efficient than OLS.
I found out it is, I need to show that:
$\text{Var}^{-1}({\beta}_{GLS})-\text{Var}^{-1}(\beta_{OLS})$ is the positive semi-definite matrix, ...
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Variable batch size for inputs of different length
We're fine-tuning a GPT-2 model (using the Adam optimizer) to some posts from a social network. The length of these posts varies quite dramatically, so while some are only two tokens long, others can ...
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Does the null space of the design matrix X have an interpretation in statistics?
I am going through the material I learned in my linear algebra classes and I am trying to view the material from a statistical perspective. This typically boils down to imagining what the theorems say ...
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Find asymptotic efficiency of MLE to UMVUE
Let $\{X_i\}_{i=1}^n $ be a sequence of i.i.d random variables with common pdf:
$$ f(x;a,\theta) =\theta a^\theta x^{-(\theta+1)} \boldsymbol 1_{(a,\infty)}(x) \, \,\text{; where } \theta, a > 0$$
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Efficiency of IV vs GMM
I am trying to understand how IV/just identified GMM and overidentified GMM compare when it comes to efficiency.
The way I understand it, we are able to identify the vector of coefficients in IV and ...
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Is the Hodges-Lehmann estimator 'optimal' for estimating the location parameter of Logistic distribution?
Is the Hodges-Lehmann estimator $\hat\theta_{HL}=\operatorname{median}\limits_{1\le i\le j\le n}\left\{\frac{X_i+X_j}{2}\right\}$ in some sense 'optimal' for estimating the location parameter $\theta$ ...
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Stratified sampling ratio with srswor
The variable X under study have rectangular distribution with interval (a, a+ d ) the interval is divided into k equal subintervals which form k equal strata of equals sizes . From each stratum ...
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Is sample mean an 'efficient' estimator for alpha stable distribution?
The question is just like the title. But...$\alpha$-stable distribution (for $\alpha\in (1,2)$) does not have the second moment, so the sample mean doesn't have variance well defined. Then for such a ...
Robert
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Relative Efficiency of Sample Variance vs Square of Difference
$\newcommand{\szdp}[1]{\left(#1\right)}
\newcommand{\szdb}[1]{\left[#1\right]}
\newcommand{\eff}{\operatorname{eff}}$
Problem Statement: Suppose that $Y_1, Y_2,\dots, Y_n$ is a random sample from a ...
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Is BIC asymptotically efficient for minimizing prediction error if the true model is being considered?
If a set of models is being compared using BIC and AIC, given the fact that the true model (the one which generated the data) is in this set (and given the other assumptions that guarantee BIC ...
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Does machine learning really need data-efficient algorithms?
Deep learning methods are often said to be very data-inefficient, requiring 100-1000 examples per class, where a human needs 1-2 to reach comparable classification accuracy.
However, modern datasets ...
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Solved: Relative Efficiency of Average versus Maximum Order Statistic on a Uniform Distribution
$\newcommand{\szdp}[1]{\!\left(#1\right)}\newcommand{\eff}{\operatorname{eff}}$
Problem Statement: Let $Y_1, Y_2, \dots, Y_n$ denote a random sample from the uniform
distribution on the interval $(\...
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Asymptotic efficiency of estimators of autoregressive models
Are OLS or MLE estimators of autoregressive model asymptotically efficient if errors are i.i.d?
Consider the case of an AR(1) model $$x_t=\alpha x_{t-1} + \epsilon_t$$
with $\epsilon_t$ ~ $i.i.d. N(0,\...
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Data-efficiency vs sample-efficiency
I noticed that in the context of RL, people call the ability to learn from little data "sample-efficiency". However, in the context of supervised learning, it is called "data-efficiency&...
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An efficient estimator can be biased? [duplicate]
Can someone explain to me whether an efficient estimator can be biased? What I've read so far seems to be always about efficiency being a property being alluded to unbiased estimators but I was ...
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Efficient influence function in proportional hazards model
I was hoping someone could help me with this problem in the cox proportional hazards model.
I am given the following setup.
T is a non-negative random variable with continous distribution and hazard ...
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How can I find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\theta}_{n}^{(2)}}$ (in terms of $\tau_1$, $\tau_2$ and $\alpha$)?
Given that $n^{\alpha}[\hat{\theta}_{n}^{(1)}−\theta_0] \xrightarrow{L} \tau_1 H$ and $n^{\alpha}[\hat{\theta}_{n}^{(2)}−\theta_0] \xrightarrow{L} \tau_2 H$, find $ARE_{\hat{\theta}_{n}^{(1)},\hat{\...
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Are these statements about the maximum likelihood estimator and efficiency correct?
I'm trying to understand efficiency and its relation with maximum likelihood estimators so I need someone to confirm or correct these statements I deduced :
1/ If the maximum likelihood estimator ...
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Expected variance of biased estimators
Suppose that $\mu$ is an unknown $k$-vector, which one seeks to guess. You observe $n$ i.i.d. normal variates $x_i \sim \mathcal{N}\left(\mu,\Sigma\right)$, say stacked in the matrix $X$, then produce ...
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Using calculus of variations to deduce lower bound on Pitman efficiency (asymptotic relative efficiency) between $t$-test and sign test
Let $Y_1,...,Y_n$ be iid draws from a location family $\{f(\cdot - \theta) : \theta \in \mathbb{R}\}$. $f$ is a symmetric density w.r.t. the Lebesgue measure on $\mathbb{R}$ with finite variance. We ...
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Accuracy of MLE
Hi I have a question related to MLE.
Consider a simple linear regression model:
y = a + b*x + u
Here, if the regressor X has small variability, the OLE estimator could be inaccurate.
My question is ...
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Why does MLE tend to normal distribution
We have $X_1,\dots, X_n$ are iid (the distribution can be of any type, e.g. Bernoulli (p), normal ($\mu, \sigma^2$), Poisson ($\lambda$).
If we use MLE $\hat \theta$ to estimate any parameter $\theta$ ...
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MLE and consistency and efficiency of MLE
Here is my understanding of Likelihood function, maximum likelihood estimator (MLE) and consistency and efficiency of MLE.
(Notes: Comments are not main parts of this post and can be skipped. Only ...
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Vector-valued estimators, intuitively why $var(\widetilde{\beta})-var(\widehat{\beta})$ being p.s.d. means $\widehat{\beta}$ more efficient?
For two scalar unbiased estimators $\widehat{\alpha}$ and $\widetilde{\alpha}$, we know that if one has smaller variance, then we say it is more efficient, which intuitively means that this estimator ...
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In regression modeling, are there any caveats to always using robust standard errors?
Aside from efficiency issues, is there anything else to this?
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How can I find the asymptotic relative efficiency of two quantities, estimating $\sigma$?
Let $X_1,...,X_n$ be a random sample from $N(0,\sigma^2)$, where $\sigma>0$ is unknown. We try to estimate $\sigma$ using $T_1=\sqrt{\frac{\pi}{2}}\frac{1}{n}\sum^n_{i=1}|X_i|$ and $T_2=\sqrt{\frac{...
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Deriving C-R inequality from H-C-R bound
As mentioned in the title, I want to derive the Cramer-Rao Lower bound from the Hammersly-Chapman-Robbins lower bound for the variance of a statistic $T$.
The statement for the H-C-R lower bound is ...
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Does a balanced design have to be connected?
Under a general block design setup, consider $v$ treatments allocated to $b$ blocks. Let $r_i$ and $k_j$ be the replication number of the $i$th treatment and $j$ the block respectively. We define $n_{...
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Which is a better estimator, averaged functions vs. A function of an average?
Problem:
Assume that we want to estimate $f(\theta)$ with a pre-specified strictly increasing function $f$ and a parameter $\theta$.
Let $\hat{\theta}_1$ and $\hat{\theta}_2$ be unbiased estimators ...
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Are GAM suitable for inference?
Are the estimators unbiased efficient and consistent? Or is GAM better for classification and prediction than non additive models? Interaction terms aren’t allowed in GAM.
user271077
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Is the $\sigma$ estimator more efficient than the $\mu$ estimator?
Are my empirical findings correct? How to get the same result analytically?
I studied the efficiency of the mean and standard dev estimators:
$$\mu_n=\sum \frac {x_i} {n}\space\space\space\space\...
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Exponential family and efficient estimator
In my lecture notes there is the notion of efficiency related to the exponential family. More precisely, the lecturer stated that for an exponential family an efficient estimator always exists. How is ...
