Questions tagged [estimators]

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

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0answers
26 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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22 views

Skewed outcome variable, sem model: is it a problem? [on hold]

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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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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70 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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42 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
25 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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14 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
45 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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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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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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1answer
27 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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(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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30 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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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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22 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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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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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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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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113 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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1answer
36 views

Problems to obtain the maximum likelihood estimator

I'm trying to calcul the MLE from a Beta-Binomial distribution. However, I'm having problems defining the estimator. I'm using the following function: $$ \widehat{\ell\,}(\theta\,;x)=\sum_{i=1}^n ...
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23 views

For the German Tank problem, why do we assume symmetrical samples?

I have read that for the German Tank problem, one assumption to make is the following: 'By symmetry, one would suppose that the number of unobserved labels above X(n) should be about equal to the ...
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30 views

What are biased and inefficient estimators?

I’m studying statistics from Schaum’s Outline, which gives the following: ...
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Estimation of variance of mean of Bernoulli distribution, if sample is degenerated [duplicate]

If $X_1,X_2,…,X_n∼Bernoulli(p)$ Variance of the average of $X$ is $Var[S_x/n]=\frac{p(1−p)}{n}$ But if we have sample, where all $X$ are equal, $\hat{p}=1$ (or zero), and estimation of var of ...
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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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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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1answer
68 views

Finding bias of $\hat\theta=\max\{x_1,\ldots,x_k\}$ where $x_i$'s are discrete uniform

I am working through some textbook problems and came across a problem I am having difficulty with. The problem asks to give the bias of a point estimate, namely for a given set of data $X = \{x_1, x_2,...
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1answer
32 views

Use of Weighting Matrix (GMM)

While conducting estimation via the Generalised Method of Moments, or GMM, I understand that we need to minimise the following expression: $Q_n(\theta)=g_n(\theta)'W_ng_n(\theta)$ Where $g_n(\theta)$...
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Is this equation true for any joint probability distribution (used in orthogonality principle proof for estimators)?

Given two random variables $x, y$, is it true that $p(x - \hat{x}|y) = p(x|y) - \hat{x}$ where $\hat{x}$ is a known constant I came across this in this kalman filter derivation, Corollary 3.2.1, ...
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44 views

Derivation of the distribution of $\hat{\phi}=[\hat{\phi}_1, \cdots, \hat{\phi}_p]$ in AR(p) models

Background Consider the following AR($p$) model: $$ \dot{X_t} = \phi_1 \dot X_{t-1} + \phi_2 \dot X_{t-2} + \cdots + \phi_p \dot X_{t-p} + \epsilon $$ where $\dot{X} := X - \mu = X - \mathbb{E}(X)$...
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1answer
27 views

Can any unbiased estimator be changed into a consistent estimator when estimating functions of the mean [closed]

For an i.i.d sequence of Random Variables $X_1, \dots, X_n$, each with mean $\mu = \mathbb E[X]$, the goal is to estimate some continuous function $f$ evaluated at the mean, $f[\mathbb E[X]]$. If ...
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1answer
47 views

Kaplan-Meier method or estimator?

This is a really basic question. I say Kaplan-Meier estimator and always tell people to say estimator and not method or methodology. However, I've seen a few places where statisticians also say Kaplan-...
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27 views

Hopefully a quick semantics question (Maximum Likelihood Estimator)

I was working through a portion of this paper, when I came across something that seemed odd to me. In Appendix E (pg24), equation (E4), the following line pops up: $\widehat{D} = \frac{\widehat{\...
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1answer
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“Appropriate conditions” for method of moments estimator to exist, be consistent, and asymptotically normal?

My statistics text has the following theorem, and alludes to "appropriate conditions on the model", but never specifies what those conditions are. What conditions are necessary? Let $\hat{\theta}_n$...
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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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1answer
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Corroborating a differnce in differences identification strategy

I read in Mostly harmless econometrics that a good way of testing whether a difference in differences is a good identification strategy is running this equation: where the first sums are post-...
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1answer
34 views

Rao-Blackwell for Minimum-Variance Unbiased Estimator

Let $X$ be an observation from a distribution with probability mass function:$f(x;\theta) = \left(\frac{\theta}{2}\right)^{|x|}(1-\theta)^{1-|x|}I_{\{-1,0,1\}}(x), \, \theta \in (0,1).$ Use Rao-...
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Within estimator and between estimator?

I have understood that the within estimator is for fixed effects model. Can I say that the between estimator can only be used for a random effects model? (in reference to panel data)
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1answer
196 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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1answer
39 views

Standard error of sample variance

We know that an unbiased estimator of the variance is: $$ \hat{\sigma}^2_{unbiased} = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2$$ I was wondering, does it have the smallest possible standard error? ...
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1answer
52 views

Normal distributed random sample: find the least variance from the set of all unbiased estimators of $\theta$

Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample from $X\sim\mathcal{N}(0,\sigma^{2})$. (a) Find the least variance from the set of all unbiased estimators of $\sigma^{2}$. (b) Find a sufficient ...
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Parameter Estimation using hazard function

Following the notation in this paper [ref], assume we have a random sample of $x_1, \dots, x_n$ of a distribution with PDF $f(x)=f(x,\theta)$ and CDF $F(X)=F(x,\theta)$ and we wish to estimate $\...
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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 ...
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Manipulation of asymptotic bounds for distance between estimators

Suppose I know some asymptotic bounds: $$\mathbb{E}(|D(a,\hat{a})|) \lesssim O(n^{-1/2}),$$ where $D$ is some distance between probability measures, and $a$ is a probability measure while $\hat{a}$ ...
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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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1answer
32 views

How do I find bias and variance of estimators of a binomial distribution?

A product-lot arrives in two containers with respectively 300 and 700 units in each container. We examine 30 units in the first container and find that 𝑋1 of them is defective. We check 70 units in ...
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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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2answers
29 views

Expectation on estimator for Poisson distribution

I'm reading through the textbook "All of Statistics" and one of the problems gives the following estimator for the lambda parameter of the Poisson distribution: $\hat{\lambda} = \frac{\sum_{i=1}^n ...
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Least square estimate for post-stratification sampling

I figured out via the normal linear regression method that Beta0 hat = ybar - Beta1 hat xbar. But I am not sure how to find out the least square estimate for Chat. Is anyone able to help me? Thanks!! ...
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1answer
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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 ...