Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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1answer
727 views

efficiency of variance / standard deviation

Is there any literature on the efficiency/asymptotic efficiency of the variance estimators like sample variance? I can find enough analysis about efficiency of mean estimators but nothing on that of ...
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0answers
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Maximum Likelihood Efficiency

It is a well-known fact that ML estimates are not efficient in the class of consistent, asymptotically normal estimates due to the existence of superefficient estimates. The right way to put it is ...
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459 views

How general is the backfitting algorithm?

Hastie \& Tibshirani's original approach to fitting generalized additive models was the backfitting algorithm. For a model of the form $$ y = \alpha + \displaystyle\sum_k f_k(x_k) + \epsilon $$ ...
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Examples where method of moments can beat maximum likelihood in small samples?

Maximum likelihood estimators (MLE) are asymptotically efficient; we see the practical upshot in that they often do better than method of moments (MoM) estimates (when they differ), even at small ...
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1answer
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D-efficiency in choice-based conjoint analysis

How to calculate the D-efficiency of experimental designs in conjoint analysis? Specifically, how do you specify the $X$ and the number of $nBetas$ in this formula: $$ D_e=\frac{|X'X|^{1/nBetas}}{...
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1answer
492 views

Does the Hodges-Lehmann estimator perform better than trimmed/winsorized means?

I've been reading about the HL estimator, and a question came to mind. I could fairly easily create a mean-estimator where I trim or clip 29% of the data on either side and have a statistic with a ...
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1answer
1k views

Restricted OLS have less variance than OLS?

According to Gauss-Markov Theorem, ordinary least squares (OLS) is the best linear unbiased estimator (BLUE). How then can restricted OLS have less variance? Please tell me the reason.
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Is multiple imputation worth doing at all for data missing completely at random?

If data are missing completely at random (MCAR), then obviously multiple imputation (MI) won't serve its lauded function of lessening bias in your findings, since there's no bias to lessen. However, ...
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Why is the asymptotic relative efficiency of the Wilcoxon test $3/\pi$ compared to Student's t-test for normally distributed data?

It is well-known that the asymptotic relative efficiency (ARE) of the Wilcoxon signed rank test is $\frac{3}{\pi} \approx 0.955$ compared to Student's t-test, if the data are drawn from a normally ...
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0answers
266 views

Efficiency of the OLS estimator

I have this linear regression: $y_{i}=\beta_{0} + \beta_{1}x_{i} + u_{i}$ with $i=\{1..n\}$. Say $\hat{\beta}_{1}$ is the OLS estimator of $\beta_{1}$. $\hat{\beta}_{1}$ is BLUE since the Gauss ...
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Critera for “horse-racing” normally-distributed estimators?

I am seeking to compare how two estimators perform on a test set. These estimators are approximately normal -- they produce both point estimates and $\alpha$ intervals. One would like the point ...
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3answers
324 views

How to improve the efficiency of an estimator when the estimates significantly vary among subgroups?

In a clinical trial, patients enter the hospital (hence, the study) at different time points. We have observed that patients who respond to the primary therapy early survive longer than those who ...
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2answers
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Increase training performance of a neural network with low learning rate?

I am trying to train an Artificial Neural Network for classification. In the input layers, I have 402 neurons; the first 400 are binary, and the last two are floating points in the range -1 to 1. In ...
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667 views

Mincer-Zarnowitz for model combination, how to construct?

Lets say we have two competing models to forecast something. We can test with the Mincer-Zarnowitz regression if model 1 is unbiased and encompessing all the information of model 2: $y_i = \beta_0 + ...
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2answers
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Why $\sqrt{n}$ in the definition of asymptotic normality?

A sequence of estimators $U_n$ for a parameter $\theta$ is asymptotically normal if $\sqrt{n}(U_n - \theta) \to N(0,v)$. (source) We then call $v$ the asymptotic variance of $U_n$. If this variance is ...
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Extra variable in regression model increases variance of parameter

Suppose my true Model is: $$y = Xb + u \tag 1$$ But I am estimating: $$y = Xb + Zd + u \tag 2$$ I can get the estimate of $b$ from $(2)$ by using $Mz$ operator as: $$\hat{b} = (X'MzX)^{-1}(...
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1answer
49 views

Mean estimation under known variance heterogeneity

We observe $X_1,\ldots,X_N$ and consider following model: $$ X_i = \theta + w_i\epsilon,\quad \epsilon \sim N(0, 1). $$ Based on above model, we want to estimate $\theta$ given $X_1,\ldots,X_N$ and $...
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1answer
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Why the Covariance between efficient and inefficient estimator is equal to variance of the efficient one in Hausman test?

I tried to prove Hausman test for two estimators efficient and inefficient one. Then I encountered with the statement for variance of two estimators saying that: Covariance between the efficient ...
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20 views

Test for informational efficiency

I need to compare one benchmark models regarding their "informational efficiency" against three other models. From my search I can now say, that this usually refers to the question how well a model is ...
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1answer
491 views

Efficiency of GLS over OLS when regressors are not fixed

Suppose we have a regression model $$ y=X\beta+u,\quad E(u)=0,\quad E(uu')=\Sigma. $$ Let $\hat\beta$ and $\bar\beta$ respectively denote the OLS and GLS estimator. Then, when $X$ is fixed (or when $X$...
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342 views

Creating separate model for each level of a factor using caret in R

I have a large data frame in R with 51 variables/predictors one of which is a factor variable that represents the user (6 in my data) that those observations belong to and I want to predict type. <...
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1answer
786 views

Relative efficiency: mean deviation vs standard deviation

I have difficulties following a seemingly elementary claim from Tukey (1960): It is well known that, in large samples, the relative efficiency as a measure of scale of the mean deviation compared ...
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How does CoDi compares to the deep network?

I like the idea of CoDi model which uses a von Neumann neighborhood method which has four types of cells. How this models compares to the deep network in terms of efficiency? Are there any scenarios ...
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0answers
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Estimator's Efficiency vs. Consistency

I know the definition of both (I think), but they seem so equal at the same time. Any clarification? I know that as the sample size goes to infinity; the estimator converges to the population ...
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Is the efficiency of biased estimators judged according to the the mean squared error criterion of optimality?

I understand that the Cramér–Rao bound relates to achieving the lowest possible mean squared error amongst unbiased estimators. Is the same standard used to judge biased estimators? Why/why not?
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1answer
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OLS Regression : Efficiency of the estimator of the variance of the residuals under the assumption of normality

My question is probably already answered somewhere but I did not find it. In the standard linear regression model under the assumption that residuals are normally distributed, we have a result ...
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1answer
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Efficient estimators and CRLB

An estimator is efficient if it reaches the Cramér-Rao Lower Bound and since it is efficient, it is also the UMVU estimator of the parametric function $\tau(\theta)$. But Cramér-Rao inequality and the ...
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1answer
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How do we know if using IV model is more efficient than OLS?

What are we looking for when we want to determine which model is most efficient? In my course slide they often discuss which model is most efficient, but I don't really know what they are looking and ...
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1answer
35 views

How to define user efficiency time

I'm making a user study. Users had to perform the same task 10 times. I've recorded the completion times of each task. Now I need to calculate the efficiency time. As you can imagine the user, when ...
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0answers
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Asymptotics of the MLE: a different flavor of proof? [Reference request]

I'm currently trying to understand more about the properties of the maximum likelihood estimator. It's known that, in the large data-limit, the MLE becomes an unbiased estimator with almost Gaussian ...
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What is the statistical efficiency of L-moments?

In particular I am interested in the scale estimator. Hopefully it is much better than that of IQR.
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404 views

Comparing the efficiency of estimators

I've taken the following explanation from Wikipedia page on efficiency. If $T_1$ and $T_2$ are estimators for the parameter $\theta$, then $T_1$ is said to ''dominate'' $T_2$ if: Its mean squared ...
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1answer
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Relative efficiency of Wilcoxon signed rank in small samples

I have seen in published literature (and posted on here) that the asymptotic relative efficiency of the Wilcoxon signed rank test is at least 0.864 when compared to the t test. I have also heard that ...
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1answer
108 views

Particle Filter Inefficiency

As I understand it, Particle Filters are a Monte Carlo method to narrow down a search space and find a posterior through a survival-of-the-fittest type method. The particular application of Particle ...
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1answer
1k views

Biased and Efficient estimators

Is unbiasedness a necessary condition for an estimator to be efficient? For example, if $\hat {\theta}= \frac{\sum_i^n X_i}{3}$, I assume $\hat {\theta}$ can't be efficient in a Cramer-Rao lower ...
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0answers
352 views

Summary of estimator properties (consistency, bias, sufficiency, etc.)

I've read about various properties of estimators, but I'm wondering if there's some source with a summary (maybe a list, table, or graphic) of the properties for different kinds of estimators. ...
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1answer
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Can someone define efficiency?

There are lots of questions about efficiency in CrossValidated, but I am no a sophisticated statistician. All I want is a simple layman's definition of efficiency. Thank you :)
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Is OLS Asymptotically Efficient Under Heteroscedasticity

I know that OLS is unbiased but not efficient under heteroscedasticity in a linear regression setting. In Wikipedia http://en.wikipedia.org/wiki/Minimum_mean_square_error The MMSE estimator is ...
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1answer
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Estimating VAR by GLS versus OLS: efficiency

Suppose I have a VAR model with different regressors in different equations (this could be due to restricting some coefficients of a full VAR($p$) model to zero or having some different exogenous ...
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0answers
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Relation between asymptotic relative efficiency for tests and estimators

The asymptotic relative efficiency for unbiased estimators is the limit of the ratio of the variances as the $n\rightarrow \infty$. Is there a relation to asymptotic relative efficiency according to ...
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215 views

Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient

Consider when $X_1, ..., X_n \sim $ Bernoulli($p$). We want to estimate $p(1-p)$. Suppose $\hat{p}=\frac{1}{n}\sum_{i=1}^nX_i$. Prove that the MLE $\hat{p}(1-\hat{p})$ is a asymptotically efficient ...
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Correctness of a proof for Hodges' estimator

We know the following is Hodges' estimator: $$ \delta_n = \begin{cases} \bar{X}_n & |X_n| \geq n^{-1/4} \\ a\bar{X}_n & |X_n| < n^{-1/4} \\ \end{cases} $$ where $X_1, ..., X_n \sim N(\...
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1answer
188 views

Asymptotic variance of GMM with efficient instrument

This question emerged from reading Wooldridge's Econometric Analysis of Cross Section and Panel Data, second edition, section 14.4.3, where the asymptotic distribution of the GMM (Generalized Method ...
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2answers
585 views

efficiency - bias trade-off

Under which conditions would a researcher choose optimally when there is a trade-off between the variance and bias of an estimator? I hope this question is not too broad... Any help would be ...
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2answers
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Efficiency of beta estimates with heteroscedasticity

I need something clarified and that is when you have non-constant variance, estimates won't be biased but will be a problem when it comes to the S.E. formulas and efficiency. Therefore OLS estimates ...
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0answers
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Is there a generalization of trimean to $n$-th order (central) moments?

I think trimean is the cat's meow. Is there a generalization of this idea to $n$-th order (central) moments? Basically I live in a world where the pain of outliers vastly exceeds the pain of ...
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2answers
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Is MLE more efficient than Moment method?

I have got some small data sets (about 8 to 11 data points for each set), following Normal distribution. I would like to find out the 95% confidence interval of the 0.005 and 0.995 percentile of each ...
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2answers
447 views

How to compute efficiency?

Suppose instead of maximizing likelihood I maximize some other function g. Like likelihood, this function decomposes over x's (ie, g({x1,x2})=g({x1})g({x2}), and "maximum-g" estimator is consistent. ...
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quantile regression analysis [duplicate]

What is quantile regression? How can I use quantile regression to analyse technical efficiency in rubber farmers? I also want to use the production frontier. The response (y) variable is production....
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3answers
2k views

Efficiency and the number of regressors?

An econometrician told me that I shouldn't keep adding new variables to the model even if I have reason to believe they're relevant to the response variable, as it "reduces the efficiency of the other ...