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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3
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1answer
205 views

MLE of Uniform on $(\theta, \theta +1)$ and consistency/bias

I see there were a few questions on SE about MLE of Uniform already but none of them helped me with this one: We are to compute MLE of $U(\theta, \theta +1)$ and check if it is biased and consistent. ...
3
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2answers
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 ...
2
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3answers
72 views

Unbiased estimator of variance

My question is why is the best and most commonly used estimator for the variance (in a Gaussian distribution) the sample variance with constant 1/n-1 when the sample variance with constant 1/n+1 ...
2
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2answers
29 views

Finding unbiased point estimate of population variance

Q.The contents of each of a random sample of 100 cans of a soft drink are measured. The results have a mean of 331.28 ml and a standard deviation of 2.97 ml. Show that an unbiased estimate of the ...
2
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1answer
36 views

Estimator problems

I am stuck with some parts of a problem in my textbook, and the solutions in my textbook do not seem to help me. The problem goes: Two independent observations $X_1$ and $X_2$ are made of continuous ...
2
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1answer
82 views

MVUE for a function of variance of Normal Distribution

Let $X_1, X_2, ..., X_n$ be a random sample from a $N(\theta_1,\theta_2)$ distribution. Find the uniformly minimum variance unbiased estimator of $3{\theta_2}^2$. Using factorization theorem, I found ...
2
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1answer
41 views

Asymptotic Properties of OLS estimators

This is an econometrics exercise in which we were asked to show some properties of the estimators for the model $$Y=\beta_0+\beta_1X+U$$ where we were told to assume that $X$ and $U$ are independent. ...
2
votes
1answer
148 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 $1$, ...
1
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2answers
54 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 ...
1
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1answer
92 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]...
1
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1answer
38 views

Can an Maximum Likelihood Estimator be 0 or undefined?

I was working on this MLE problem which I derived to be \begin{equation*} f(x;\theta) = \frac{1}{x\cdot \ln \theta} \end{equation*} where 1 < x < $\theta$ \begin{equation*} f(x_1,x_2,...
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2answers
46 views

$\hat{\theta} = \sqrt{\frac{3}{n}\sum_{i = 1}^{n} Y_i^2}$ is an unbiased estimator?

If $\hat{\theta} = \sqrt{\frac{3}{n}\sum_{i=1}^{n}Y_i^2}$ and $\{Y_i\}_{i=1}^n \sim U\lbrack 0, \theta \rbrack$ and they are iid, is $\hat{\theta}$ in unbiased estimator of $\theta$? Suppose I am ...
1
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1answer
40 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 $Var(X)$?
1
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1answer
689 views

Unbiased estimator [closed]

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 $\...
1
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1answer
34 views

Determine all $\overrightarrow{a}$ for which the estimator is an unbiased estimator for the variance

consider a random variable $X$ and stochastically independent repetitions $X_1,...,X_n$ of $X$. For each vector $\overrightarrow{a}=(a_1,...,a_n) \in \mathbb{R}^{n} \text{ with } a_i > 0 $ we ...
1
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1answer
19 views

Can L1 Distance Give Underestimates In a 2-D Rectangular Lattice?

I am working on this program that plays PacMan and involves calculating the Manhattan distance between the player and some enemies. Here's my problem, if we have an enemy straight ahead of the ...
1
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1answer
25 views

Find the MoM estimator of $\theta$ when $X_i \sim N(a_i\theta, 1)$

Let $X_1, ..., X_n$ be independent random variables on some probability space such that for each $i = 1, ..., n$ we have that $X_i \sim N(a_i\theta, 1)$, where $a_1,...,a_n$ are given constants ....
1
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2answers
31 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 =...
1
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1answer
56 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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0answers
22 views

What does symmetrised tuple mean?

I was reading the paper "Remarks on some nonparametric estimates of a density function" by Murray Rosenblatt (1956) and in one part he writes that an estimate $S(y;X_1,...,X_n)$ of the density ...
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0answers
18 views

On the pooled variance in the estimating difference of means

Suppose that we need to compare means of two sample spaces, say P, Q. Let $\mu$ and $\nu$ be the expectation value of $P$ and $Q$, respectively. Consider $m$ independent random samples from $P$ and $n$...
1
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1answer
42 views

What is the maximum likelihood estimate of $\theta$?

A random sample of size $7$ is drawn from a distribution with p.d.f $$f_{\theta}(x)=\frac{1+x^2}{3\theta(1+\theta^2)}, -2\theta \le x \le \theta,\;x>0 \;\text{and otherwise zero}$$ and the ...
1
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1answer
23 views

How to obtain unbiased estimator

Let $X_1, ...,X_n$ iid with distribution $f(x) = \theta x^{ \theta - 1}$, where $0 < x < 1$, $\theta > 0$. Let $Y_i = - \log (X_i)$. Then it can be shown that $Y_i \sim \text{ Exp} (\theta)$....
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0answers
25 views

Finding approximate variance of the logistic transform of an estimator for the survival function

So, let $\hat{S}(t)$ be the Kaplan–Meier estimator of an unknown survival function $S(t)$. Consider the logistic transform of this estimator: $$\hat{S}_{\text{logistic}}(t) = \log \left(\frac{\hat{S}(...
1
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0answers
30 views

Unbiased estimator of linear regression

I need to check if an estimator $\hat\beta = \frac{1}{n}\sum\limits_{i=1}^{n} \frac{Y_i-\bar{Y}}{X_i-\bar{X}}$ of regression $ Y_i = \alpha +\beta X_i + \epsilon_i, i = 1,...n $ is unbiased. My idea ...
1
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0answers
33 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 ...
1
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0answers
33 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 ...
1
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1answer
490 views

Uniformly minimum variance unbiased estimator

How to prove $ \overline{X}=\frac{1}{n}\sum_{i=1}^nX_i$ is the uniformly minimum variance unbiased estimator of $\mu$ when $X_i\sim N(\mu,\sigma^2),$ and $\sigma$ is known. Idea: Let $X=(X_1,X_2,...,...
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0answers
88 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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0answers
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 ...
0
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1answer
67 views

Calculate weighted estimated variance [duplicate]

Hello I have to calculate the estimated variance between 3 samples Here is an example ...
0
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1answer
31 views

Biasedness and iid observations

Suppose we have a sample of 100 observations, each observation detailing the weekly wage of an individual in a country. Would the sample mean(estimator) be biased? I know this can be deduced by ...
0
votes
1answer
168 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 \...
0
votes
1answer
28 views

Prove that $n \cdot\min\{T_1,…,T_n\}$ isn't allowable as an estimator of $\mu$

Let's suppose we have some electronic device which duration follows an Exponential distribution of unknown mean $\mu$. Some research team wants to estimate $\mu$ and uses a sample of $n$ devices to do ...
0
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1answer
40 views

minimum mean square error of $X_{1}$ given $S_{n}=\sum_{i=1}^{n}X_{i}$ when $p=0.5$ and $X_{1},X_{2},\ldots$ are bernoulli iid

I need help with this one: Let $X_{1},X_{2},\ldots$ be bernoulli i.i.d random variables with $$\forall i:\mathbb{P}\left\{ X_{i}=1\right\} =\mathbb{P}\left\{ X_{i}=0\right\} =\frac{1}{2}$$ define $S_{...
0
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1answer
13 views

Measure of error in smoothness of approximation of sphere

I'm meshing a sphere and am solving a physics problem on this. What I want to show is that the error in the model scales like$$ \varepsilon = \epsilon^p, $$ where $\epsilon$ is the "error" in the ...
0
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1answer
16 views

Question about Estimators and Notation

I was studying the notes for my statistics course and I was given the following two results: bias($\hat\theta$) = $\operatorname{E}({\hat\theta(X_{1},...,X_{n})})-\theta$ and MSE($\hat\theta$) = ...
0
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1answer
35 views

Is the following estimator biased or unbiased?

Let $X_1, ... ,X_n$ be i.i.d with a common density function: $f_x$, with finite mean $\mu$ and variance $ 0 < \sigma^2 < \infty$. Consider the estimator $\hat \mu_n = \frac{1}{n-1} \sum_{i=1}^{...
0
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1answer
58 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 ...
0
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1answer
44 views

Determine the value of $α$ for which the $MSE(T)$ is minimal.

Let $X_1$ be an estimator for the probability $θ$ of unauthorized access. Let $X_2$ be another estimator for $θ$. Assume that $X_1$ and $X_2$ are independent, unbiased estimators for $θ$. ...
0
votes
1answer
270 views

minimum variance estimator for $\mu^2/\sigma^2$

I'm trying to do the following exercise Let $(X_1,...,X_n)$ be a random sample from an $N(\mu, \sigma^2)$ distribution with $\mu$ and $\sigma$ unknown. Assuming $n>3$, find a minimum variance ...
0
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1answer
23 views

Unbiased estimator for median (lognormal distribution)

Assume that $Y \sim N(\mu,\sigma^2)$ $X = e^Y$ Then X is lognormal distributed with parameters $\sigma$ and $\mu$. I know that $E(X) = E(e^Y) = \eta e^{\frac{\sigma^2}{2}} $ The median in ...
0
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0answers
34 views

Determine all $\overrightarrow{a}$ for which $T_\overrightarrow{a}$ is an unbiased estimator for the variance $Var(X)$ of $X$.

consider a random variable $X$, whose mean value is known 𝜇 and stochastically independent repetitions $X_1,...,X_n$ of $X$. For each vector $\overrightarrow{a}=(a_1,...,a_n) \in \mathbb{R}_{n} \text{...
0
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0answers
9 views

conservative two-sided equal-tailed confidence interval

Consider a random sample $X_1,...,X_n$ from a Bernoulli distribution with unknown parameter $p$ that describes the probability that $X_i$ is equal to $1$. The maximum likelihood ($ML$) estimator for $...
0
votes
1answer
53 views

Mean square error of MLE

$\textbf{X}=(X_1,...,X_n)$ is a random sample from a shifted exponential distribution with common density $f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \...
0
votes
1answer
30 views

Is this Bayes estimator result correct

I am trying to see where I went wrong in this calculation of the Bayes estimator or if there is a hole in my understanding. We have a common discrete density $$f(x|\theta)=\frac{\theta^{x}e^{-\theta}}{...
0
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1answer
48 views

Determining an unbiased estimator

Say we have a shifted exponential distribution with common density $$f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \end{matrix}\right.$$ We have $\theta$ ...
0
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0answers
17 views

Deriving OLS estimator dependent on X being orthogonal

Let $X \in \mathbb{R}^{n \times d}$ be a predictor matrix with orthonormal columns and $y \in \mathbb{R}^n$ as output vector and vector (OLS estimator) $\beta_{LS} \in \mathbb{R}^d$ for estimating a ...
0
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1answer
26 views

how do i find the variance of an estimator?

If the Estimator was simply the sample mean $s=\frac{\sum{x}}{n}$ taken from a binomial distribution (a random example) how would i calculate the variance of this? I am trying to use the difference ...
0
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0answers
18 views

How to derive formula of variance of sample median?

I wonder how to derive the formula of variance of median of sample. I searched it for a day but couldn't find, so I'm asking help here. You may just notice me the link of it. I saw the result form is ...