Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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Difficulty carrying out Automatic Portmanteau Test on R

I'm trying to replicate part of the empirical analysis of the following paper: https://www.tandfonline.com/doi/full/10.1080/23322039.2020.1719574#...
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For what (symmetric) distributions is sample mean a more efficient estimator than sample median?

I have labored under the belief that the sample median is more robust measure of central tendency than the sample mean, since it ignores outliers. I was therefore surprised to learn (in the answer to ...
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Asymptotic relative efficiency of median vs mean for Student t distribution

I am looking for a theoretical expression for the ARE of mean and median for student-t distribution against sample size (degree of freedom), exactly the blue curve shown in John Cook's blog. Can ...
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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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Is $X_{(1)} + X_{(n)}$ a good estimator for $\theta$?

Problem 8.7 From Van der Vaart's Asymptotic Statistics: Given a sample of size $n$ from the uniform distribution on $[0,\theta]$, the maximum $X_{(n)}$ of the observations is biased downwards. ...
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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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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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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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178 views

Multinomial logit model with alternative-independent coefficients

Having a multinomial logit model $U_{ij}=x'_{i}\theta_{j} + \epsilon_{ij}$ for 3 alternatives, I checked the efficiency of the estimators of this model through 1000 Monte-Carlo iterations, and ...
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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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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, ...
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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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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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ARMA Estimation Efficiency of the Mean

Usually in books related to ARMA Time Series, it is assumed that the series is 0 mean. If not, the recommended standard procedure is to simply subtract the sample mean from the series and continue ...
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Seasonal adjustment a la Hyndman

Rob J. Hyndman has two posts here and here on forecasting weekly series. He suggests using a regression with ARIMA errors, $$ y_t = a + \sum_{k=1}^K (\alpha_k \ \text{sin}(2\pi k t/m)+\beta_k \ \...
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Compare efficiency of several linear combinations of random variables

I have the following linear combinations of (non-independent) random variables: $Y1=X1+X2$ and $Y2=X1+X2+X3$ With the following general formula, I can calculate the variance for combinations $Var(Y)=...
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Efficiency of Bayesian Estimator

In the Wikipedia article for Minimum Mean Square Error, the Bayesian estimator is referred as "asymptotically efficient" using the similar arguments of Fisher information from frequentist statistics. ...
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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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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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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 ...
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Does C-efficiency exist? If so, how is it defined?

When using an algorithm to construct an optimal experimental design one has a measurement of optimality to find an optimal design with this optimality-criterion. For any arbitrary chosen design a D- ...
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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?
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What is the difference between BLUE and MVUE?

What is the difference between a Best Linear Unbiased Estimator (BLUE) and a Minimum Variance Unbiased Estimator (MVUE)? I know that "best" mean efficient, but that is also what "minimum variance" ...
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Are there efficient estimators for the variance of an exponential family?

Let us consider the Gaussian model $\mathcal{N}(\mu,\sigma^2)$, where both $\mu$ and $\sigma$ are unknown. I have learnt that (for example, from Amari's information geometry book) the exponential ...
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243 views

Efficient Estimator from Insufficient Statistic

Suppose that I have a statistic $T(X)$, and I know for sure that it is not sufficient to estimate a parameter $\theta$. Is it still possible to have an estimator $\hat\theta(T(X))$ that is efficient (...
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Distribution of $\bar{X^2} $ when $X\sim N \left( \theta, \sigma^2 \right) $

How can I derive the distribution of $$\bar{X^2}\quad \text{when}\quad X\sim N \left( \theta, \sigma^2 \right) $$ The context of this question is an exercise requiring me to show that $\bar{X^2}- \...
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Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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QR decomposition computational efficiency

I am struggling to find a reference for this: In terms of big Oh notation does anyone know of any expressions for the computational time taken by commonly used algorithms for QR decompositions?
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991 views

The concept of efficiency

I have some problems in understanding the concept of efficiency as related to an estimator. My sources (Mukhopadhyay, 2000 and Casella, Berger, 2002) do not treat this argument as I expected since ...
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Speeding up in R [closed]

for(i in 1 : n){ y[i] <- length(X[X >= X[i]]) } This is my code in R (partially given) to obtain the number of X's greater than or equal to each Xi, where X is a vector of values. When I run ...
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A Primer on Estimator Efficiency?

Can someone recommend a text with derivations of classical estimator efficiency results? I'm particularly interested in likelihood and pseudo-likelihood estimators for multi-variate discrete models
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Is this the only way to determine if a parameter can be estimated efficiently?

I am tasked with determining if a particular parameter can be estimated efficiently. Given that an efficient estimator is an unbiased estimator which achieves the Cramer-Rao lower-bound, is the only ...
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Using Mean Cumulative Function (Nelson-Aalen) to assess drug efficacy

I have a large medical retrospective longitudinal dataset of electronic health records. An individual is identified by an ID. A medical event or drug prescription event is identified using a code and ...
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How to control for severe medical cases using survival analysis and Cox regression?

I have a longitudinal medical record dataset. My cohort is made up of patients with a particular disease. There are no members of this cohort without this disease. Disease indications are denoted by a ...
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alternative to IQR [duplicate]

I am currently making a short literature study of robust and efficient estimators. Some very well known are the median absolute deviation (MAD) and the interquartile range (IQR). However they both ...
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Proof that sequential GARCH fitting is not efficient?

I've read that authors like Tsay (as well as several other researchers) use a sequential method for fitting a ARCH-type model. This means first estimating the conditional mean model (ARMA-type) and ...
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Examining the relative efficiency of the trimmed mean

Using the following reference A survey of sampling from contaminated distributions, I am trying to investigate the relative efficiency (RE) for the mean vs the trimmed mean, given the following ...
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Good Reading/Bibliography

Before posting this question, I searched CV for a similar question that could helped me, but I didn't find one. So, I'm sorry if this has already been asked before. So, I'm self-studying from Keith ...
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Is an inadmissible estimator necessarily dominated by some admissible estimator

Basic example: $X$ has a $p$-variate iid standard Normal distribution; the sample mean is not admissible if $p>2$ and is dominated by the Stein shrinkage estimator. However, the Stein shrinkage ...
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Proving First Difference is more efficient than OLS

I am trying to prove that the First Difference method is asymptotically more efficient than OLS when the error term follows a random walk. Assume the following model $$\begin{align}y_i&= \mathbf{...
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Efficiency of the command pmvnorm in R

Let $X_1,\,X_2,\ldots,\,X_n$ be $n$ independent random variables, where $X_i\sim\text{N}(\mu_i,\,\sigma_i^2)$, for all $i\in\{1,\ldots,\,n\}$. Consider the $n$-dimensional random vector $$\boldsymbol{...
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458 views

Efficient estimator for the mean of a Gamma distribution

Let $X_1$,$X_2$,...,$X_n$ be i.i.d. according to Gamma($\alpha$,$\beta$). Denote the mean by $\mu := E[X_i] = \alpha/\beta$. Can you find an unbiased and efficient estimator for $\mu$? MLE gives ...
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1answer
367 views

Why does asymptotic efficiency require normality?

Greene (Econometric Analysis), states the defintiion of Asymptotic Efficiency. My question is, why does this definition contain a reference to normal distribution? We already know by the central ...
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1answer
164 views

Using parallel tempering, is it possible to swap too often?

In parallel tempering I have replicas of the Markov chain I'm studying evolving at different temperatures, and intermittently I swap the replicas on a nearest-neighbouring temperatures basis. Between ...
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Efficient estimator in OLS and statistical power in t-test

Could please anybody help me to understand the difference between those two concepts? I know that statistical power refers to $1-\beta$, the type II error and that efficiency of estimator is related ...
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If $\operatorname{Var}\left(\epsilon_i\right) = h\left(X\right) \neq \sigma^2$, what can we know about $\operatorname{Var}\left(\hat{\beta}\right)$?

This question uses the derivations found here. The short version Consider a regression model. If the error variance is a known function of the data (rather than a constant), under what conditions ...
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How to speed up training of a Neural Network?

I'm writing a thesis where I developed a script that generates NN and precalculates weights and biases to reduce a required number of epochs when I train a network. In my work, using examples I ...
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Is it valid to calculate relative efficiency of the mean and median on data sampled from an asymmetric distribution?

Let the relative efficiency between two estimators $T_1$ and $T_2$ of a parameter $\theta$ be defined as: $$e(T_1,T_2) = \frac{E[(T_2-\theta)^2]}{E[(T_1 - \theta)^2]}$$ This definition indicates ...