Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component.

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11
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2answers
690 views

You see a route 14 bus on the moon. What is the most likely number of bus routes on the moon?

This question was asked on a forum and while many argued that the answer is 14 (since the probability of you seeing bus 14 is maximum in this case), I argued against it that they were working ...
1
vote
1answer
252 views

How to find parameters from an equation

the question could be "stupid" but i don't know if it is feasible or not, please don't kill me :) EDIT WITH NEW FORMULAS! I have an equation like this: (unfortunatly in my first Q&A i cannot ...
5
votes
1answer
188 views

Finding the joint distribution of $X_{1:n}$ and $\overline{X}$

I need to show that, given a random sample of independent variables $X_1, ... , X_n$, each following a distribution EXP($\theta$,$\eta$), that is, ...
4
votes
1answer
141 views

Showing that $X_{1:n}$ is sufficient for $\eta$, by factorization

I'm asked to show that $X_{1:n}$ (the minimum order statistic) is sufficient for $\eta$, in the case of a random sample $(X_1, ... , X_n)$ where $X_i\sim EXP(1,\eta$) (this is the two-parameter ...
2
votes
2answers
158 views

Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given pdf?

Let $X$ be a continuous random variable with pdf, $$f(x;\theta)=\frac {\theta^m.x^{m-1}e^{-\theta x}} {(m-1)!} ; x\geq0, \theta>0$$ Is $\frac{m-1}{x}$ an unbiased estimator of $\theta$ for given ...
4
votes
1answer
146 views

MLE of the mean of a heteroscedastic Gaussian time series

Suppose we observe $Y_i\sim \mathcal{N}(\theta_0 + \theta_1 x_i, \sigma_i^2)$, with $x_i$ and $\sigma_i^2$ known for all $i = 1,\ldots,n$ and $Y_1,\ldots,Y_n$ independent. Assume $\theta_0$ is ...
-1
votes
1answer
363 views

Intuitive explanation of Fisher Information and Cramer-Rao bound [closed]

I am not comfortable with Fisher information, what it measures and how and how is it helpful. Also it's relationship with the Cramer-Rao bound is not apparent to me. Can someone please give an ...
7
votes
1answer
2k views

Minimum variance unbiased estimator for scale parameter of a certain gamma distribution

Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with p.d.f., $$f(x;\theta)=\theta^2xe^{-x\theta} ; 0<x<\infty, \theta>0$$ Obtain minimum variance unbiased estimator of ...
2
votes
2answers
784 views

Some expectation values for a Gamma distribution

Assuming I have a Gamma distributed random Variable $x \sim Gamma( \alpha, \beta )$. Now I like to have the following two expectation values (integrals): $E \left[ x \ln x \right]$ $E \left[ \ln ...
4
votes
2answers
6k views

Difference between logarithm of an expectation value and expectation value of a logarithm

Assuming I have a always positive random variable $X$, $X \in \mathbb{R}$, $X > 0$. Then I am now interested in the difference between the following two expectation values: $E \left[ \ln X ...
5
votes
3answers
269 views

Estimating population size

Let's suppose there are $n$ real numbers $a_0 < ... < a_n$ uniformly selected from interval [0, 1). If one knows $k$ numbers on consecutive positions $a_i < ... < a_{i+k-1}$ how good is ...
1
vote
3answers
1k views

How to estimate failure probability from count until first failure?

What would be the formula to estimate the rate of failure of some test as a percentage chance of failure from the number of runs of the test until the first failure was seen? For example, considering ...
1
vote
1answer
132 views

Likelihood function estimating two different means

I completely understand how to find the likelihood functions of simple pdfs. However, how would you attempt to find the likelihood function of a pdf with a negative exponential function with two ...