Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators?

Given that some statistics are estimators, are test-statistics consistent, efficient, complete and unbiased, estimators? Are sufficient statistics consistent, efficient, complete and unbiased, ...
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The GMM estimator is reliable under the given conditions

Consider the linear model $$y_t= x_t'\beta +\epsilon_t$$ for $t=1,...,T$. where $x_t= ( x_{1t} \ \ x_{2t} \ \ ... \ \ x_{kt}) $ and $\beta$ is $(k\times 1)$ vector of unknown coefficients. Given $z_t= ...
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Stratified randomization: adjustment or not?

In randomised trials, there's a mantra saying "analyse as you randomise". So if I deploy a stratified randomisation, i.e. randomise within strata of some baseline variable, then this would ...
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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 ...
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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 in statistic?

I reads some comparison articles, and always find " asymptotic efficiency", "asympototically less efficient", and "asympotoically normal". I really confused about the ...
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Calculating "relative efficiency" of a single estimator in two samples from different processes

I want to assess how well the multilevel AR(1) model performs on not-so-long time series wherein the AR(1) model assumptions are violated. I have three data generating processes (say, $P_1$, $P_2$, ...
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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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Relative efficiency of two estimators for the cardinality of a set from which we sample without replacement

I would like to check my understanding of the relative efficiency for the following estimators for the problem of estimating the cardinality of a set, from which you sample without replacement; i.e. ...
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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 ...
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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.
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1 vote
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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 ...
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In the case of lognormal distribution, when can the median be more efficiently estimated than the mean?

I understand, that in the case of normal distribution, the estimation of the mean (from samples) is more efficient (i.e. of less risk), than the estimation of the median. According e.g. to this post, ...
2 votes
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Efficiency of OLS versus Quantile regression estimator

If I have a linear model $ y_i = x_i'\boldsymbol\beta + \epsilon_i $ and I assume that OLS estimator of $\boldsymbol\beta$ is unbiased and consistent and Least absolute deviation (LAD) estimator of $...
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Quantile regression estimator - conditions for consistency and efficiency

What are the conditions for consistency and efficiency of Quantile regression estimator (for example LAD) in a linear regression model?
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When can't Cramer-Rao lower bound be reached?

The Cramer-Rao lower bound (CRLB) gives the minimum variance of an unbiased estimator. One sentence in the wiki page says "However, in some cases, no unbiased technique exists which achieves the bound....
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FGLS vs OLS tradeoff

Hi, Can someone tell me what is the tradeoff between using OLS and FGLS? I know the conditions under which FGLS is more efficient than OLS. But what about robustness? It is usually said that the ...
3 votes
1 answer
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What may be an inefficient estimator of the population mean?

If the sample mean is an efficient estimator of the population mean, what may be an example of an inefficient such estimator?