Questions tagged [estimators]

A rule for calculating an estimate of a given quantity based on observed data [Wikipedia].

169 questions with no upvoted or accepted answers
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985 views

Why don't asymptotically consistent estimators have zero variance at infinity?

I know that the statement in question is wrong because estimators cannot have asymptotic variances that are lower than the Cramer-Rao bound. However, if asymptotic consistence means that an estimator ...
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1k views

What are the main different/alternative correlation estimators?

I'm looking to learn about the main/popular alternatives when it comes to estimating correlations that I've missed in the following list. The best answer will provide a reference (can be Wikipedia), a ...
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49 views

Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
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55 views

Rank forecasting from uneven samples

Say we have a bag of chocolate balls. There are $I$ different unique colors. Our bag has an unknown number of balls for each color. We want to get a sense of which color people like the most and ...
4
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557 views

MSE of an individual coefficient from ridge or lasso vs. OLS

Consider a multiple regression model $$ y = X\beta + \varepsilon $$ with $K$ regressors in $X$. If the model is correctly specified, the OLS estimator $\hat\beta_{OLS}=(X'X)^{-1}X'y$ will be the ...
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61 views

Estimating the mean with least median of squares

I have a set of real numbers $x_{1}...x_{n}$ and would like to estimate their mean (let's call it y) so that $median(x_{i}-y)^{2}$ is minimal. Is there an algorithm and a correctness proof for ...
4
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68 views

Forming an unbiased estimator of the maximum of several parameters, given independent estimators of each parameter?

Say I have $K$ independent normals, $X_i \sim \mathcal{N}(\mu_i, \sigma_i)$ for $i = 1,...,K$. How can I form an unbiased estimator of $\max_i \mu_i$ using $X_i$'s?
3
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1answer
42 views

Mean Squared Error as quantifier of the Bias-Variance tradeoff

I have acquired the impression that many of the people doing statistical work, will prefer a biased estimator $\hat b$ to an unbiased one $\hat \beta$, if the former has lower Mean Squared Error. This ...
3
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73 views

Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
3
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0answers
43 views

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 ...
3
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78 views

Can an asymptotically efficient estimator be biased?

In "Theory of point estimation" by Lehmann and Casella (1998) there is the following definition: It is also said that So terms of the asymptotically normal sequence of estimators can be ...
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48 views

Find unbiased estimators for $\lambda$ and $\lambda^2$.

For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$. Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
3
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162 views

Paired comparison of instruments using different measurement samples

I have instruments A, B, C and D - I'm in search of the best one. The problem: For illustrative purposes, let's use an example of evaluating the best among the instruments measuring difference in ...
3
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72 views

How to derive an estimator for the parameter of a continuous uniform distribution

$X_1, X_2,\dots.,X_n$ are i.i.d. random variates drawn from a continuous uniform distribution over $[0,\theta].$ The sufficient statistic is denoted $\max$. I want an estimator $e$ of $\theta$ that ...
3
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69 views

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?
3
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65 views

Fixed parameter estimates of parent factors in a nested design

Summary: What is happening with parameter estimates of factors that are the 'parents' of nested factors? Data: My analysis involves testing the effect of different parameter settings for automatic ...
3
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72 views

Bayesian inference with the wrong distribution

When an observation $x$ is generated by $P(x|\theta)$ for a parameter $\theta$ the Bayesian optimal estimator for the value of $\theta$ is $\hat\theta_{BEST}=\mathbb{E}[\theta|x]=\frac{1}{P(x)}\int d\...
3
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81 views

Trying to understand an example where unbiased estimators don't exist

I am new to statistics especially in the topic of estimators and sufficient statistic. I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". ...
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250 views

Quantile regression analyzing the conditional quantiles of one of the regressors?

Given response $Y_t$ and predictor $X_t$, we can use OLS to analyze the conditional mean; $E[Y_t | X]$. Quantile regression can be used to analyze the conditional quantile function; $Q(Y_t(\tau)|X)$. ...
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29 views

Parameter estimation by averaging over all high-likelihood possibilities?

I am refereeing a chemistry paper. The authors are trying to interpret some experimental data by comparison with numerical simulations. They have run many simulations using different combinations of ...
2
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24 views

variance estimator for a symmetrical two-sides censored normal distribution

Suppose to draw a sample of $n$ observations from $X \sim \mathcal{N}(0,\sigma)$, with observations outside the interval $(-c,+c)$ censored; $c$ is known and one can conveniently set $c=1$, for ...
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48 views

Best way to estimate the probabilities of a random variable

I have some confusion about estimating the probability of a particular value of a random variable. For simplicity, consider the case of a coin and the random variable being $X = \{H,T\}$, where $T$ is ...
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19 views

Prove that bias of MLE for Weibull process

In the case of fixed observation interval $[0,T]$ of a Weibull process, I learned that the MLE of the shape parameter $\beta$ and MLE of the scale parameter $\alpha$ are as follow: $$\frac{1}{\hat{\...
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85 views

Is there a UMVUE for arbitrary distribution with density and variance?

Let F be the family of all distributions with probability density and finite variance, and $X_1, ..., X_n$ be random samples from F. Does UMVUE for variance exists for this situation?
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32 views

Conceptual questions on efficient estimators for MA model

I am trying to estimate parameters of a MA(p) system where p is the order. E.g., $$y[n] = \sum_{i=1}^p {\theta}_i u[n-i] + e[n] = \mathbf{\theta}^T\mathbf{u}[n] + ...
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1answer
35 views

Statistical theory proof intuition (UMVU estimators)

I've been working through this problem in Theoretical Statistics by Keener, but could not solve it. I looked up the answer and I do understand why it's correct, but I don't understand what intuition ...
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33 views

When you see an item twice for the first time during k random choices, the total number of items is ~ k² / 2

Let $X_1, X_2, ..., X_k, ...$ be a sequence of i.i.d. random variables with $X_i \sim \mathcal{U}\{1, 2, ..., n\}$ (discrete uniform distribution). The parameter $n$ is unknown. Let's say the first ...
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34 views

How to get an estimator from this heuristic argument?

In the answer of this question, a practical solution was suggested for: Let $X_1, X_2, ..., X_k$ be a sequence of i.i.d. random variables with $X_i \sim \mathcal{U}\{1, 2, ..., n\}$ (discrete ...
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206 views

Comepare two Theil-Sen estimators for significance

Since an ANCOVA was not possible with my data, I moved on to use Mann-Kenndall as well as Theil-Sen estimators to analyze my data. I have three datasets. For two of them Sen's slopes were found to be ...
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57 views

Comparing methods of aggregating standard deviations

Background Problem One of the economic/finance software systems that I am using is adopting the following approach to aggregate standard deviations: Let $R_t$ be the time series of interest. The ...
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82 views

Can we improve on the sample mean as an estimator of the true mean of a Pareto distribution, 1 < α < 2?

Suppose I have a sample drawn from a population which is approximately distributed i.i.d. according to the Pareto distribution for values of x greater than X*. Suppose, moreover, that the tail index 1 ...
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732 views

Variance of an unbiased estimator is 0 when the sample size goes to infinity

So I would like a proof for the following but I can't seem to do it myself. I have a random variable $X$ and I draw $n$ samples($\{X_1, \ldots, X_n\}$) from it and I have $$ Z_n = \frac{\sum_{i = ...
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61 views

Is M-estimation valid only for regression models?

Is M-estimation valid only for regression models or does it's working hold good for robust estimation of parameters in other statistical models? I understand that M-estimators are asymptotically ...
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226 views

Derivation of OLS estimator

Dear mathematicians/statisticians, I have a question regarding the derivation of the OLS estimators. In one explanation I have found the following step: However, I don't understand why it's -b_1 ...
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79 views

linearization of an estiamtor

Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional ...
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48 views

Poisson counting process parameter

Two quick questions: What's the maximum likelihood estimator of the parameter of an homogeneous Poisson counting process? To estimate $\lambda$ I'm currently using number of events/total time, $N(t)/\...
2
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148 views

Non-Measure Theoretic Argument for Var(X) = 0 iff X is constant (X continuous RV)

I am studying out of DeGroot and Schervish trying to carefully understand the math of prob/stats. In ch 4.3 on variance, they state the theorem that given X a RV whose mean and var exist, then Var(X) =...
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189 views

Estimator of Bessel function?

I am trying to estimate the parameters of the modified Bessel function of the first kind for integer order case. $I_n(wt) = \sum\limits_{m=0}^\infty \frac{1}{m!(m+n)!}(\frac{wt}{2})^{2m+n}$ In ...
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0answers
70 views

Transforming an estimator to overestimate

I have an estimator $\mu^*$ of a the mean $\mu$ of a certain distribution that I obtained using a variational technique (basically just establishing a bound on $\mu$ and finding a trial function that ...
2
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360 views

What do people call the groups into which quantiles divide the population?

Is there a correct technical name for the group of observations between two quantiles? For example, if you have the values of the four cut-points that divide a population into five groups of equal ...
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51 views

Minimax estimators in Bayesian analysis

I got recently introduced to minimax methods in statistical decision theory. Is there an analogue in Bayesian analysis and some resources related to this?
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1answer
25 views

Expected value without complete sample space

The book way: Suppose, we have a bag with 8 balls numbered 1-8, we want to estimate the population parameter mean. we note down the entire sample space. (1,1)(1,2).. (8,8) calculate mean of each ...
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26 views

cross-covariance estimation and variance reduction

Let $X,Y$ be two vector variables and $$ \mathrm{Cov}(X,Y) = \mathbb{E}[(X-\mathbb{E}X)(Y-\mathbb{E}Y)^T] $$ their cross-covariance (but I think we could just pretend that's the covariance between two ...
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31 views

Sample mean lognormal variables

Suppose you've got $x_1, ..., x_n$ independant realisation drawn from a $LogNormal(\mu, \sigma^2)$. Could someone explain me why $exp(\mu + 0.5*\sigma^2)$ $\neq$ $\frac{1}{n}(x_1 + ... + x_n)$ ? Here ...
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1answer
42 views

How does the TraMiner Package Calculate Standard Error Using Weighted Data?

The TraMiner Package includes an option to include sampling weights in the analysis. However, I haven't found any discussion in the package documentation (or associated user manual) of how standard ...
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0answers
22 views

How I can sketch the proof of consistency of only one beta in multiple regression?

Now assume you additionally obtained data on average parental incomes (PI) and the ethnic composition (EC) of the pupils in school. You regress the score on STR PI EC and a constant. State the ...
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1answer
23 views

Definition of curvature

Kay (Fundamentals of Statistical Signal Processing) defines the curvature of a log-likelihood function to be the "negative of the second derivative of the logarithm of the likelihood function at its ...
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56 views

Spherical error variance in OLS estimation of AR($p$)

Consider the linear model $\boldsymbol y=\boldsymbol X\beta+\boldsymbol\varepsilon$. One of the assumptions for the OLS estimator is the spherical error variance assumption which states that $\...
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0answers
26 views

Existence of Huber M-estimators

I am working on a paper about optimization using the Huber's Loss function, which is defined as: \begin{equation} \psi(x)=\begin{cases} \frac{x^2}{2\gamma},& \text{if } \lvert x\rvert\leq\gamma\\ ...
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0answers
124 views

Consistency vs. Asymptotic Efficiency of estimator

I'm thinking about the relationship between an asymptotically efficient estimator and a consistent estimator, and I'd like to make sure that my thinking is correct. An estimator is asymptotically ...