Questions tagged [estimator]

For questions about estimators; an estimator is an approximations of a parameter that is using the avaiable data. There could be multiple estimators for a particular parameter.

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9 views

Prove an estimator is biased

I want to prove that an estimator in biased. For a big n, like $10^{13}$ we define a distribution of values $x_i \in [0,1]$ for $i \in [n]$. We want to estimate with some error $\epsilon$ the value ...
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1answer
16 views

Unbiased estimator for $p^2$. Bernoulli distribution.

Let $X_{1},...,X_{n}$ be a random sample from Bernoulli (p), find an unbiased estimator for $p^{2}$. I think It's the same estimator for $\mathrm{Bin}(n,p)$ so: $${V}(X) = np(1-p) = np - np^2 \\ p^2 ...
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Error term covariance of two-stage estimators

I am looking into properties of two-stage estimators (2SLS). My setting is as follows: 1) $y_1 = y_2\beta+\epsilon$ 2) $y_2 = z \pi_2 + \eta$. Equation 1 represents the second-stage estimation, ...
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1answer
17 views

Is sample minimum an unbiased estimator for population mean?

Given $\mu$ as the population mean and $X_{(1)}$ as the lowest value of a sample extracted from this population, I want to know if $X_{(1)}$ is an unbiased estimator for $\mu$, i.e., if $E(X_{(1)}) = \...
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22 views

Bayesian estimator under 0-1 loss

If you were given the posterior distribution $P(\theta |X)$, as well as $P(X)$, and under no further assumptions, how could the Bayesian estimator be found under the 0-1 loss. I understand that I ...
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1answer
17 views

Find the relative efficiency of estimators [closed]

Let $X_1, X_2, \ldots , X_n$ be a random sample from $\operatorname{Unif}(0, \theta).$ Let $X(1) = \min(X_1, X_2, \ldots , X_n)$ and $X(n) = \max(X_1, X_2, \ldots , X_n)$. Consider the following two ...
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2answers
37 views

PDF of unbiased estimator

Given two samples $\left\{x[0], x[1]\right\}$ which are independently observed from a $\mathcal{N}(0, \sigma^2)$ distribution. The estimator, $$\hat{\sigma^2} = \frac{1}{2}(x^2[0] + x^2[1])$$ is ...
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26 views

Checking if estimator $\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}$ is unbiased

Let $P_\theta(X=x) = \left(\frac{\theta}{2}\right)^2(1-\theta)^{1-x^2}$ for $x=-1,0,1$ Let $\hat{\theta}=\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}$ be an estimator of $\theta$. Is it unbiased? So the ...
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Calculate the estimator and its standard deviation by Parametric Bootstrap

The third quantile of a distribution function F is the point q such that F(q) = 0.75. Note the q (α; λ) (Gamma distribution). Then, the quantile would be estimated by ^q = q (^α; ^λ). We do ...
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2answers
29 views

MLE for a regression with alpha = 0

Consider the model $Y_i = \beta x_i + \varepsilon_i$ where $i = 1,\ldots, n$. We know that $\varepsilon_1,\ldots, \varepsilon_n$ is iid sequence of random variables from $N(0,\sigma^2)$ and $x_i, i =...
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33 views

Finding the expectation of $\hat \theta$, if $\theta = \max(X_1, …, X_n)$ [duplicate]

I am trying to make $\hat \theta$ an unbiased estimator for a uniform distribution on $(0, \theta)$. So I am given the fact that $\hat \theta$ is the MLE for $\theta$, but I need to show that if I ...
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33 views

Limit at $\theta \rightarrow 0$ is showing that $\frac{1}{n}\sum^n_{i=1}X^2_i$. is not the MLE?

Let $\mathbf X = (X_1, . . . , X_n)$ consist of independent and identically Normal $N(0, θ)$ random variables, with mean $0$ and variance $θ > 0.$ The density of $f_{\theta}(\mathbf x)$ is equal ...
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1answer
47 views

How to show that this statistic is complete

Suppose that $S, {f_θ : θ ∈ Θ})$ is a statistical model, corresponding to an observed random vector $\mathbf X = (X_1, . . . , X_n).$ Let $\theta_1(\mathbf X)$ and $\theta_2(\mathbf X)$ be unbiased ...
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11 views

How to show the first order condition of LAD estimator fo median regression is of order $o_{p}(\frac{1}{\sqrt{n}})$?

Given the model $y_{i}=x_{i}^{\prime}\beta_{0}+\epsilon_{i}$ with $E[sgn(\epsilon_{i})|x_{i}]=0$ and $E[sgn(\epsilon_{i}-c)|x_{i}] \neq 0$ if $c=c(x_{i}) \neq 0$, where $sgn(\epsilon)=1-2 \cdot 1(\...
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1answer
33 views

Variance of Area and Average Estimators in Monte Carlo Estimation of Pi

I know of 2 Monte Carlo estimators of $\pi$. Rick Wicklin discusses these 2 methods here. https://blogs.sas.com/content/iml/2016/03/14/monte-carlo-estimates-of-pi.html 1) The area method throws ...
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1answer
35 views

$X \sim U(0, \theta^*]$, how to find the MLE? [closed]

Given $X \sim U(0, \theta ^*]$. How can I show that $\frac{1}{12}max_{1 \leq i \leq n}X_i^2$ is an MLE of $Var(X)$?
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30 views

Given an MLE $\hat{\theta}$, how to show $f(\hat{\theta})$ is also an MLE?

Given an injective function $f: \Theta \rightarrow \mathbb{R}$ and an MLE $\hat{\theta}$ of $\theta ^*$, how do I prove that $f(\hat{\theta})$ is an MLE of $f(\theta^*)$. I know that because $f$ is ...
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1answer
48 views

What is the Least Squares Estimator

A simple question. When people talk about "the least square estimator", what is this estimator? Is it an unbiased estimator of the slope of the regression line? In a paper I'm reading, Let's Take ...
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1answer
68 views

Understanding the unbiased estimator

I am having hard time understanding what an estimator actually is ( I miss the intuition ). The definition ( for unbiased estimator) is as follows: $T$ is unbiased for the parameter $\theta$ if $E[T]...
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1answer
132 views

Unbiased Estimator of the Standard Error of another Estimator

Suppose that $Y_1,...,Y_n$ is an IID sample from a uniform $U(\theta, 1)$ distribution. The method of moments estimator for $\theta$ is $\tilde \theta=2\bar Y-1$. The standard error of $\tilde \...
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1answer
138 views

If best unbiased estimator exists then it's maximum likelihood estimator?

Our teacher proved in class that if the best unbiased estimator exists, then it is an MLE using a theorem that if $\hat{\theta}-\theta$ is proportional to the score of $\theta$ with probability $1$, ...
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81 views

Optimal unbiased estimator

I have the sample $X_1,...,X_n$ of i.i.d. from $U(\theta - 1/2; \theta +1/2)$. It is well known that $T = (X_{(1)}; X_{(n)})$ is a sufficient but not complete statistic, because $X_{(n)}-X_{(1)} - (n-...
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60 views

Point Estimator.

When the point estimator under consideration has a pdf , the $P[T=\tau(\theta)]=0 $ , where $\tau(.)$ is some function of parameter $\theta$ and $T$ is an estimator of $\tau(\theta)$. But I did ...
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1answer
662 views

Unbiased estimator

Let $X_1, X_2, \dots,X_n$ be the random samples from $$f(x,\theta) = \frac{2x}{\theta^2}, \quad 0 < x < \theta, \; \theta > 0.$$ Find the uniformly minimum variance unbiased estimator of $\...
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2k views

Unbiased estimator questions

If $X_1,X_2,\ldots,X_n$ are i.i.d. $\mathrm{B}(1,p)$, find the best unbiased estimator of $p^n$. Attempt: Use indicator functions to show every observation has mean equal to 1 so this is the same as ...