Questions tagged [efficiency]

A measure of the quality of a statistical 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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Forecast efficiency: why no correlation between errors and available information?

(Applied Economic Forecasting using Time Series methods; Ghysels, Marcellino, 2018), in the chapter about forecast evaluation, relates efficiency as "the efficient use of the available information". ...
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Sterling vs Combustion Engines [closed]

This may be difficult to answer but, theoretically, if you had a car with a sterling engine that produces say approximately 300 horsepower how much of any specific heat source would need to be used if ...
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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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dummy variables and interaction terms: when the number of observations matters

Some authors conducted a univariate cross-section regression using 26 observations: $$Y_i = a + b \cdot X_i + e_i$$ Some others did the same regression and added another one where $X$ is split ...
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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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54 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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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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29 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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Derivative of Time-Transformed Stochastic Process

Given a continuous time stochastic process X(t), we can define the functional transformation, $$f(X)(t)=(X(t))^2−2X(t)$$ and evaluate the Hadamard derivative. Given a transformation on the real ...
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183 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 https://en.wikipedia.org/wiki/Minimum_mean_square_error the Bayesian estimator is referred as Asymptotically Efficient using the similar ...
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Proof: comparison between the statistical efficiency of two MLEs

Suppose that we have a simple random sample of size $n$. Based on this sample, we construct two log-likelihood functions. The first one is, \begin{eqnarray*} l_1 = \sum_{i=1}^r{\bigg[\ln f_\beta(y_i|...
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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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263 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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108 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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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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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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143 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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473 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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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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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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210 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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163 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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184 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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255 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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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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259 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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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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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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787 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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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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939 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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369 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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364 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
364 views

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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318 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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596 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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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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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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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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3answers
774 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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235 views

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 ...