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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Prove that an estimator is UMVU under the usual “assumptions of regularity”

I'm asked to prove that some estimator is UMVU under the usual assumptions of regularity. I'm not sure what is meant with 'usual assumptions of regularity'. Do they mean with this that I can just ...
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1answer
103 views

Maximum likelihood estimator of $P(X < y)$ for fixed $y$

I'm having a problem understanding the following question. Given the following density function $f_X(x; \theta) = (\theta +1)x^\theta$ on $0<x<1$, find the maximum likelihood estimator for ...
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22 views

Estimate the population mean when random selection is not possible

Consider I have a jar with marbles labeled 0 and 1 in it. They're not well mixed so the possibility of obtaining a sample sized 1000 with mean 0.6 and another sample sized 1000 with mean 0.4 is not so ...
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51 views

Computing an estimator for a piecewise distribution?

Suppose I have a random variable $X$ that follows a distribution with a piecewise function $f(x|\theta)$. What is the correct way to compute an estimator $\theta$ for this function? Should the ...
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1answer
19 views

Estimating Attendence

I am trying to estimate attendance at events. What I have is daily/weekly ticket sales numbers. The goal is to analyze the ticket sales each day and estimate where we might be at by the day of the ...
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43 views

Variance of a difference in estimated proportions with trivariate discrete distributions

Let a multivariate distribution be given by $P(Y,S_1,S_2)$, where all three variables are discrete, $Y$ is multivalued, $S_1=(0,1)$ and $S_2=(0,1)$, respectively, and all may be dependent. Define the ...
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1answer
24 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
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122 views

Maximum Likelihood

Find maximum likelihood estimator $\hat\theta$ of $f(x;\theta) = (1/2)\exp(-|x-\theta|)$, for $-\infty \leq x < \infty$ and $-\infty \leq x < \infty$. I am confused of how to deal with the ...
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1answer
250 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
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1answer
48 views

Suitability of skew normal for rating task and calculation

in an experiment, I ask participants to rate qualities on a continuous scale. I expect the results to be normal distributed and I am confident that assuming a normal works fairly well for most values. ...
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81 views

Markov Chain : Montonicity of Sample Mean

Let $\{X_n\}_{n\geq1}$ be an irreducible, ergodic Markov chain with discrete state-space $S$, transition probability matrix $P$ and steady state distribution $\pi = \{\pi_j\}_{j\in S}$. Let $f$ be a ...
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1answer
77 views

Bounds on least squares and weighted least squares estimator

I was wondering if I can get some help in getting bounds on the parameters estimated by least squares (LS) and weighted least squares (WLS) methods. Suppose our observation model is: $\mathbf{y} = ...
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1answer
226 views

Finding UMVUE from Lehmann-Sheffe Directly

I am having some trouble with an example from the book I am following. Let $X_1,X_2,...,X_n$ for $n>2$ be an iid set of $N(\mu,\sigma^2)$ random variables with, $\mu\in\mathbb{R}$ and ...
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42 views

How to calculate $Q_n$estimator

How to calculate $Q_n$estimator from this formula for sample $1\;3\; 6\;2\;7\;5$: $$Q_n=d\left\{|x_i-x_j|;\;\;i<j\right\}_{(k)}$$ where d is constant factor and $k={h \choose 2}\thickapprox ...
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1answer
103 views

Efficient method of approximating a distribution with Gaussian

Given a univariate uni-modal density function $f(x)$ (very hard to compute its cumulative distribution function (CDF) $F(x)$, not to mention its inverse CDF $F^{-1}(x)$), how to find the best ...
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1answer
87 views

Clasification of parameter estimation method

Consider that $P$ is the water pressure coming out from a valve A, therefore, the population is all the valve A pressure values. Let $P_{dif}$ be defined as the difference between the maximum and the ...
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187 views

Finding the efficiency of an unbiased estimator

I have a random sample drawn from a $N(\theta,\sigma^2)$ distribution with $\sigma^2$ known. I am trying to estimate $\theta$. I need to calculate the efficiency of the unbiased estimator, ...
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48 views

Identification of parameters problem

I always struggle to get the true essence of identification in econometrics. I know that we state that a parameter (say $\hat{\theta}$) can be identified if by simply looking at its (joint) ...
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2answers
181 views

how to calculate vehicle speed using mathematics and Image processing?

i am doing my project in image processing.using segmentation i have detected the moving part(i.e the car) in the video successfully. But now i want to calculate speed of vehicle. in the above ...
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1answer
366 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 ...
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1answer
227 views

CRLB to find UMVUE

In what situation can one obtain an estimator that reaches the Cramer-Rao lower bound, i.e. an efficient estimator? I know the rules for finding UMVUEs, and I know they are efficient if they reach ...
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1answer
339 views

How to estimate parameters of a uniform distribution?

I have information of the order in which students were classified in regard to their scores in a SAT test. I know the distribution of scores for each student is uniform with support [a,b]. I also know ...
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21 views

Estimation in Multiplicative and additive Noise problem

I have unsolved problems below for random process homework. Consider the problem of estimating a random vector $\underline{S}(u)$ from an observation $\underline{X}(u)$, where ...
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1answer
36 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
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1answer
38 views

When does Cramer-Rao Lower Bound fail to hold?

When does Cramer-Rao Lower Bound fail to hold? I have computed the bound, and still one of my estimators has less variance below the bound. Why does this happen? Should not this bound hold always?