Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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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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About the efficiency of MobileNet v1 architecture

The salient feature, which was said to make MobileNet v1 efficient in terms of computational complexity, is the usage of depthwise convolutions, which is in essence ...
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1answer
38 views

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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What is the statistical efficiency of the sample standard deviation?

I've been reading about measures of scale and data dispersion, and often come across the term statistical efficiency, which is used as a means of comparison (as well as breakdown points), of different ...
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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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2answers
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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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152 views

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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1answer
120 views

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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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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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 ...
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341 views

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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139 views

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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951 views

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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206 views

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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1answer
952 views

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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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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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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188 views

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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1answer
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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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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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1answer
276 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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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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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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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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Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

I want to prove the statement in the title using the following statement from Wikipedia: it was proved that efficient estimation is possible only in an exponential family, and only for the natural ...
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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
204 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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1answer
252 views

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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1answer
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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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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 ...
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1answer
848 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 ...