Questions tagged [efficiency]
A measure of the quality of a statistical estimator.
104
questions
3
votes
2answers
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 ...
2
votes
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{...
1
vote
0answers
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 ...
1
vote
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_{...
4
votes
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
votes
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.
1
vote
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\...
0
votes
0answers
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
votes
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, ...
2
votes
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 $...
1
vote
0answers
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?
9
votes
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....
2
votes
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 ...
0
votes
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
votes
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?
4
votes
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 ...
1
vote
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" ...
4
votes
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 ...
1
vote
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?
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
7
votes
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
votes
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.
...
4
votes
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 ...
1
vote
0answers
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{...
0
votes
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)=...
3
votes
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 ...
2
votes
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 ...
1
vote
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
$$\boldsymbol{...
2
votes
1answer
151 views
Efficient estimator in OLS and statistical power in t-test
Could please anybody help me to understand the difference between those two concepts? I know that statistical power refers to $1-\beta$, the type II error and that efficiency of estimator is related ...
3
votes
4answers
7k views
How to speed up training of a Neural Network?
I'm writing a thesis where I developed a script that generates NN and precalculates weights and biases to reduce a required number of epochs when I train a network. In my work, using examples I ...
1
vote
0answers
185 views
Is it valid to calculate relative efficiency of the mean and median on data sampled from an asymmetric distribution?
Let the relative efficiency between two estimators $T_1$ and $T_2$ of a parameter $\theta$ be defined as:
$$e(T_1,T_2) = \frac{E[(T_2-\theta)^2]}{E[(T_1 - \theta)^2]}$$
This definition indicates ...
1
vote
1answer
706 views
efficiency of variance / standard deviation
Is there any literature on the efficiency/asymptotic efficiency of the variance estimators like sample variance? I can find enough analysis about efficiency of mean estimators but nothing on that of ...
5
votes
0answers
426 views
How general is the backfitting algorithm?
Hastie \& Tibshirani's original approach to fitting generalized additive models was the backfitting algorithm. For a model of the form
$$
y = \alpha + \displaystyle\sum_k f_k(x_k) + \epsilon
$$
...
1
vote
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 ...
0
votes
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, ...
0
votes
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
votes
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 ...
1
vote
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
votes
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 ...
1
vote
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
votes
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
votes
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}(...
1
vote
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
votes
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}}{...
1
vote
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
votes
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.
