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Questions tagged [estimators]

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

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1answer
514 views

Looking for unbiased estimators for Poisson probabilities

I am looking for unbiased estimators for Poisson probabilities. That is, some estimator $\hat{g}(k)$ such that $ E( \hat{g}(k) ) = \text{Poisson}(k|\lambda) $ I discovered one in this old paper: ...
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128 views

How does one calculate Fisher-consistency factor for Rousseeuw and Croux's $S_n$ for empirical distribution?

In "Alternatives to the Median Absolute Deviation" (Rousseeuw and Croux, J. Amer. Statistical Assoc, 88(424), 1993, pp.1273–1283), the authors described an estimator of SD better than median absolute ...
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4k views

constant $\times$ distribution

I know that if $U\sim\chi^2(k)$ then $aU\sim \Gamma(k/2,2a)$ for $a>0$. But i read about the estimator and its distribution $$\hat{\sigma}_k^2=\frac{1}{2k}\sum_{i=1}^k (X_{2i}-X_{2i-1})^2=\frac{2\...
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15 views

Expected value of the parameters in linear regression (trying to understand which part is constant and which isn't and why)

I'm studying Linear Regression and trying to proof/demonstrate some properties of the parameters. When I started working with the expected value of the slope, I got confused with something. I actually ...
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2answers
195 views

What does the variance of an estimator for a regression parameter mean?

I may be asking dumb or non-sensical question, but what does the variance of an estimator for a regression parameter (e.g. $\beta_{0}, \beta_{1}$) mean? How does it even have variance? Isn't it a ...
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1answer
1k views

Variance of Estimator (uniform distribution)

In my script for statistical signals, I have some troubles to get the same result for the variance of an estimator $T$. Here is the example: Given the observations $X_1, \dots , X_N$ of a uniquely ...
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1answer
134 views

estimators with singular covariance matrix

Suppose I have 2 vectors of random variables $\boldsymbol\theta_1 \in \mathbb{R^n}$ and $\boldsymbol\theta_2 \in \mathbb{R^m}$ with asymptotic covariance $\Sigma_1$ and $\Sigma_2$ respectively. I want ...
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12 views

Why is having low variance important in offline policy evaluation of reinforcement learning?

Intuitively, I understand that having an unbiased estimate of a policy is important because being biased just means that our estimate is distant from the truth value. However, I don't understand ...
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26 views

Which is the better estimator for standard deviation?

Let $X_i \sim^{\textrm{iid}} N(\mu, \sigma^2)$. If I have measured $n$ values of $\textrm{std}(X_i)$ as $\sigma_1,\cdots,\sigma_n$, then what is the better estimator for $\sigma$: $$\hat{\sigma}_1 = ...
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5 views

How to estimate an activity based on self reported activity in a social network

I help run a social network for rock climbers who self report what they have climbed on a specific day, and optionally who they climbed it with. These climbers who self report are a perfect lower ...
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25 views

Comparison of GMM and ML estimators for regression with correlated errors

Consider a linear model with normally distributed, autocorrelated errors \begin{aligned} y&=X\beta+\varepsilon \\ \varepsilon&\sim N(0,\sigma^2_{\varepsilon}) \text{ and autocorrelated.} \end{...
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2answers
260 views

Sample mean is always an optimal estimator of the mean?

Suppose we have $T_i,i=1..n$ i.i.d. with unknown distribution and we want to estimate $E[T]$. Note that in this setting we are not estimating E[T] as a parameter of a parameter-dependent family of ...
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10 views

Are there differentiable estimators for Entropy?

I have recently came across a paper on estimation of Information theoretic measure such as Entropy, Mutual Information and divergence, using a Mean Nearest Neighbor approach. Since, the estimator is ...
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1answer
497 views

Why do we divide by the degree of freedom?

This might be trivial and vague question, but I still don't understand why when creating test statistics or estimators we always divide by the degree of freedom. Just to give examples of what I'm ...
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1answer
13 views

Principled way comparing and evaluating learned features/variables of estimators?

Does anyone know of any principled ways to compare and evaluate estimators based on what features have they learned. Basically I am interested in showing what features have these estimators picked up ...
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26 views

Questions on BLUE and MMSE estimators for a linear observation model

I have this observation model $$ \begin{cases} Y_i = X + V_i, \quad i=1,\dots, p \\ V_i \perp V_j, \quad \forall i,j \\ V_i \sim (0, \sigma_i^2) \end{cases} $$ where $X$ is a scalar random variable $...
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23 views

Difference between “Within” estimator for $Y_\textit{diff} = Y_{t_i} - Y_{t_{i - 1}}$ vs “First Difference” estimator for $Y_{t_i}$

I am using a diff-in-diff strategy and have a few question. I am not getting equivalent coefficient estimates for the two following regressions on panel data: a panel regression with a fixed ...
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26 views

Are point estimators and set estimators the same thing?

My professor defined a point estimator as A mathematical rule that maps a random sample $\lbrace x_i \rbrace_{i=1}^n$ into a 'best guess' at the parameter $\theta$ I am confused about what ...
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33 views

Are a distribution's higher-order features harder to estimate?

In what sense, if any, are a distribution's higher-order features (e.g., moments, cumulants) harder to estimate than its lower-order features, for at least some distributional families? For example, ...
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1answer
37 views

“Ice skater” / “figure skating” / “ISU” method of discarding outliers

So I need a way of ruling out outliers and "the ice skater method" has been suggested. The person who suggested it has a good deal of experience of doing the task I am doing, so I am certainly ...
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47 views

How do I correctly estimate the size of a subpopulation using noisy observations?

To preface this question: please comment on whether my description of the problem is missing details or the problem itself is not well-posed as I'm seeing a lot of views but no activity. I am trying ...
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1answer
161 views

regression - $b1$ and $b2$ are normally distributed

Let's say fitted regression line is $y_i=b_1+b_2x_i$.That is, $b_1$ and $b_2$ are estimators. The text book says that if $e_i$ (residual) is normally distributed, then $b_1$ and $b_2$ are also ...
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1answer
155 views

The definitions of estimator and estimate

This example demonstrates the difference between a theoretical observation and a realized observation. A theoretical observation is a random variable with a probability distribution, while its ...
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51 views

“Magical” variance reduction problem

I recently came across this toy problem: You have two sticks of unknown lengths $a>b$ and a measuring device with constant variance $1$ that you can only use twice. How can you construct ...
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1answer
41 views

Skewed outcome variable, sem model: is it a problem?

My outcome variable is really skewed, and I want to include it in a SEM model (I am using lavaan - R). It is measured with a 7-points Likert scale (agreement) and consists of 5 items. If the model ...
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1answer
113 views

Why is the log derivative estimator considered of large variance?

It's mentioned in the paper Variational Bayesian Inference with Stochastic Search that, the variance of the following approximation may be very large, but I didn't quite understand why this is so. It ...
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1answer
66 views

Help to understand this. Expected value of $S^\alpha$ in Gaussian distribution [closed]

Lets $X_1,\cdots,X_n$ be simple random sample from $\mathcal{N}(\mu,\sigma)$. $\overline{x}$ is sample mean. Let $$S^2=\begin{cases}\sum_{i=1}^n (x_i-\mu)^2, \mathrm{ where\ } \mu \mathrm{\ is\ ...
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31 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 ...
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59 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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119 views

Why consider the variance rather than the entropy of estimators?

It is a rather common thing to be concerned with the variance of an estimator. For instance, confidence intervals for the mean can be constructed based on the standard error. Often, however, we look ...
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81 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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48 views

quantifying asymmetry on a sphere

I have a scalar quantity that is distributed on a sphere. I would like to quantify the asymmetry in this scalar field. is there any standard method to do this? Let's say that the function on the ...
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1answer
26 views

L1 distance between categorical distribution and any arbitrary estimator?

Given an unknown categorical distribution $p$ over $k$ categories, and any arbitrary estimator of this distribution vector $q$ constructed from $n$ i.i.d samples, can anyone point me to some results ...
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16 views

Taking into account the variance of an estimated population size to construct confidence intervals for count statistics

I had originally posted this on the Math Stack Exchange website, but was justifiably recommended to explore this site instead. When given confidence intervals that are developed for proportions under ...
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25 views

Is sample mean square is a consistent estimator of population variance?

If x ~Normal with population mean u and population variance (sigma)^2 Then is sample mean square which is asymptotically unbiased estimator for population variance then is it also consistent ...
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1answer
47 views

Proof for how the drift estimator, for a random walk with drift, is unbiased?

Random walk with drift formula is: (Yt = α + Yt-1 + εt ) How do I go about checking that the drift estimator α-hat is unbiased.. which is proving that E(α-hat) = α? Is this something I would need ...
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1answer
28 views

Unbiased estimator when only the magnitude can be measured

given an random variable $x$, which is drawn from a normal distribution $f(x| \mu_x, \sigma_x) = \frac{1}{\sqrt{2 \pi \sigma_x^2}} \exp\left(-\frac{(x-\mu_x)^2}{2 \sigma_x^2}\right)$. We are drawing $...
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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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1answer
33 views

Policy evaluation in contextual bandit setting

I am currently reading a paper whose links is (Exploration Scavenging) http://delivery.acm.org/10.1145/1400000/1390223/p528-langford.pdf?ip=128.135.98.49&id=1390223&acc=ACTIVE%20SERVICE&...
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(contextual bandit problem) What does 'identical draw' mean here?

I am currently reading a paper (Learning from Logged Implicit Exploration Data) whose link is below. https://arxiv.org/pdf/1003.0120.pdf The paper supposes we have a set of possibly deterministic ...
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15 views

Bias of Pearson correlation estimator of two Bernoulli variables

Crossposting link: https://math.stackexchange.com/questions/3312349/bias-of-pearson-correlation-estimator-of-two-bernoulli-variables Suppose we have two correlated Bernoulli random variables, $X_j$ ...
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1answer
27 views

quick questions about a contextual bandit problem

I am currently reading the paper "Learning from Logged Implicit Exploration Data" https://arxiv.org/pdf/1003.0120.pdf. But I believe the questions I have can be answered without reading the whole ...
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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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1answer
717 views

Huber M-Estimator calculation

I found out that we can calculate some estimator depends on the objective function. Where if we want to minimize the least square $\sum (x_i - \theta)^2$ the best estimator is the mean. And if we want ...
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1answer
53 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 ...
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23 views

Parameter estimation in bivariate linear models

I'd like to simulate data from a bivariate normal distribution to a regression problem. In other words, let $X = (X_1, X_2)$, where $X_1$ and $X_2$ be two matrices $n \times 1$. $X_1\sim N(3, 2)$ and $...
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27 views

Asymptotic Mean Squared Error of Maximum Likelihood estimator

I want to show that $n$ times mean squared error for the maximum likelihood estimator converges to the inverse of Fisher information, where $n$ is the number of samples. But The standard proofs of ...
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32 views

Why does Agresti define the natural exponential family this way?

I am reading through Agresti's Categorical Data Analysis. And in section 4.1.1 he defines the natural exponential family as: $f(y_i;\theta_i) = a(\theta_i)b(y_i)*exp(y_iQ(\theta_i))$ Why is this ...
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36 views

Process for finding UMVU estimator

I've been working on a problem from Theoretical Statistics: Topics for a Core Course by Keener. I spent a few hours on it making very little progress before caving and looking up the solution. I don't ...