# Questions tagged [efficiency]

A measure of the quality of a statistical estimator.

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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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### 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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### 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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### 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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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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### Are there efficient estimators for the variance of an exponential family?

Let us consider the Gaussian model $\mathcal{N}(\mu,\sigma^2)$, where both $\mu$ and $\sigma$ are unknown. I have learnt that (for example, from Amari's information geometry book) the exponential ...
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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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### A little confusion on Profile likelihood

As we all know, profile likelihood is an effective method for the estimation of conditional parametric model. But I still don't know exactly why it works. Profile likelihood was thoroughly studied by ...
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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 $... 0answers 68 views ### FGLS vs OLS tradeoff Hi, Can someone tell me what is the tradeoff between using OLS and FGLS? I know the conditions under which FGLS is more efficient than OLS. But what about robustness? It is usually said that the ... 0answers 601 views ### What is the difference between BLUE and MVUE? What is the difference between a Best Linear Unbiased Estimator (BLUE) and a Minimum Variance Unbiased Estimator (MVUE)? I know that "best" mean efficient, but that is also what "minimum variance" ... 0answers 277 views ### Efficiency of the OLS estimator I have this linear regression:$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 ... 0answers 109 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$Mzoperator as: $$\hat{b} = (X'MzX)^{-1}(... 0answers 71 views ### Is there a generalization of trimean to n-th order (central) moments? I think trimean is the cat's meow. Is there a generalization of this idea to n-th order (central) moments? Basically I live in a world where the pain of outliers vastly exceeds the pain of ... 0answers 171 views ### Quantifying the loss of efficiency incurred by a pseudo-likelihood estimator relative to the true maximum likelihood estimator I'm working on a couple of problems where I use pseudo-likelihood (a.k.a. composite likelihood) as an estimation method since full maximum likelihood is not feasible. To give some background without ... 0answers 63 views ### Deriving C-R inequality from H-C-R bound As mentioned in the title, I want to derive the Cramer-Rao Lower bound from the Hammersly-Chapman-Robbins lower bound for the variance of a statistic T. The statement for the H-C-R lower bound is ... 0answers 56 views ### Is the \sigma estimator more efficient than the \mu estimator? Are my empirical findings correct? How to get the same result analytically? I studied the efficiency of the mean and standard dev estimators:$$\mu_n=\sum \frac {x_i} {n}\space\space\space\space\... 0answers 32 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? 0answers 80 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 ... 0answers 151 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 ... 0answers 35 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 ... 0answers 63 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{... 0answers 190 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 ... 0answers 64 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 ... 0answers 56 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 ... 0answers 690 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 + ... 0answers 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 ... 0answers 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. <... 0answers 21 views ### 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 ... 0answers 2k views ### 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 ... 0answers 57 views ### 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 ... 0answers 424 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 ... 0answers 353 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. ... 0answers 240 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 ... 0answers 525 views ### 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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Before posting this question, I searched CV for a similar question that could helped me, but I didn't find one. So, I'm sorry if this has already been asked before. So, I'm self-studying from Keith ...
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### Examining the relative efficiency of the trimmed mean

Using the following reference A survey of sampling from contaminated distributions, I am trying to investigate the relative efficiency (RE) for the mean vs the trimmed mean, given the following ...
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### Asymptotic relative efficiency of median vs mean for Student t distribution

I am looking for a theoretical expression for the ARE of mean and median for student-t distribution against sample size (degree of freedom), exactly the blue curve shown in John Cook's blog. Can ...
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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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### 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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