Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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45 views

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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2answers
46 views

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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53 views

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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1answer
76 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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1answer
66 views

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 ...
2
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1answer
70 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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0answers
54 views

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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46 views

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 ...
3
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1answer
61 views

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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0answers
31 views

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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29 views

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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1answer
562 views

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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0answers
52 views

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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0answers
16 views

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- ...
3
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1answer
132 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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2answers
115 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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0answers
410 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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0answers
120 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
241 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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1answer
45 views

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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0answers
67 views

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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0answers
127 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 ...
3
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1answer
41 views

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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0answers
32 views

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
240 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 (...
5
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1answer
180 views

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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0answers
166 views

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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58 views

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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0answers
55 views

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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1answer
366 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
163 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
183 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{...
2
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1answer
151 views

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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4answers
7k views

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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0answers
185 views

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
706 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
426 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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0answers
62 views

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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0answers
296 views

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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0answers
176 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 ...
2
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0answers
263 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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0answers
55 views

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 ...
4
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3answers
318 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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0answers
629 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 + ...
3
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1answer
453 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 ...
2
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0answers
104 views

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 $...
2
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1answer
1k views

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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0answers
19 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 ...
4
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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.