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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4 views

Estimate time it takes the minimum mean cycle cancelling algorithm to converge

This particular algorithm solves the circulation problem, equivalent to the minimum-capacitated flow. My question rather than only from this particular algorithm, but for combinatorial solutions in ...
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29 views

Estimators of the binomial distribution

This is a follow-up of this previous question and elaborates on the answer I received there. $\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}\def\a{\alpha}X_1,X_2,\dots, X_n$ are a random sample from ...
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129 views

Polluting an image with Gaussian anisotropic noise and estimate the covariance matrix

Assume that we have a $d\times d$ grey-scale image represented as a vector $$ \mathbf{x}=(x_1,\ldots,x_D)^T\in[0,255]^D, $$ where $D=d\times d$. We would like to import some noise concerning the ...
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1answer
25 views

$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
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7 views

Asymptotic coincidence of the MAP and MMSE estimator

In many works, simulations show that as number of samples increases, the mean-square-error (MSE) of the MAP estimator attains the minimum MSE. Where can I find a theoretical proof to these empirical ...
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1answer
28 views

Does consistent estimators have in-variance property?

If $(T_n)$ is a sequence of consistent estimators of a parameter $\theta$ ( i.e. for every $ \epsilon >0$ , $\lim_{n \to \infty} P [ \space |T_n -\theta|< \epsilon ]=1$ ) , then is it true that ...
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1answer
36 views

Maximum likelihood estimate vs likelihood ratio tests?

Can someone explain to me the intuition behind why we need likelihood ratio tests. From my understanding, they make use of maximum likelihood estimators over different parameters space and they are a ...
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2answers
40 views

Biased MLE estimate of mean (expectation)

Please give an example of p.m.f. or p.d.f. , the maximum likely-hood estimate of whose mean (expectation) is a biased estimator . Thanks
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1answer
300 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
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26 views

Working with the sum of two independent random variables, and estimating a parameter

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
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1answer
211 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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21 views

Estimation of real estate

I'm working on a project on estimating the price value of real estate. First I have collected a lot of data (500 000 instances) with details such as postal code, number of bedrooms, build year, ...
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24 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
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1answer
26 views

estimation problem for two-parameter weibull distribution

Suppose the two-parameter Weibull distribution is given by the pdf $$ f(x;a,b) = \left(\frac{x}{a}\right)^b\frac{b}{a}\exp\left\{-\left(\frac{x}{a}\right)^b\right\}, $$ where $x,a,b>0$. I am ...
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23 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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1answer
48 views

Finding an efficient estimator for $\theta$ in $U[0, \theta]$ in terms of the sample maximum

This question appeared in a past exam paper, in the form: Let $X = (X_1\dotsc X_n)\in\mathbb{R}^n$ be an i.i.d. sample from $U[0, \theta], \theta>0$ Apply Rao-Blackwell's theorem to the unbiased ...
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1answer
24 views

How to compute MAP estimate of y?

Suppose that a scalar random variable y is of the form $y=z+v$, where the pdf of $v$ is $p_{v}(t)=\frac{t}{2}$ on the interval $[0,2]$, and the pdf of $z$ is $p_{z}(t)=2t$on the interval $[0,1]$. Both ...
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0answers
14 views

Optimal combination of multiple estimates of a random variable

For the following estimation problem: y = hx + n, x is the sent data, y is the observation (received data), h is a scaling factor (known), n is an AWGN random variable with zero mean ...
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1answer
29 views

Shapiro-Wilk test

I am trying to determine if a given sample comes from a Normal distribution. For that purpose I want to perform a Shapiro-Wilk test in the way stated on wikipedia. My concern comes with the vector ...
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3answers
4k views

Prove the sample variance is an unbiased estimator

I'm trying to prove that the sample variance is an unbiased estimator. I know that I need to find the expected value of the sample variance estimator $$\sum_i\frac{(M_i - \bar{M})^2}{n-1}$$ but I get ...
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2answers
109 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
206 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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33 views

paramter estimation (maximum likelihood) of a mixture density

I have this mixture distribution $f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma) $ where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution. I now have random ...
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1answer
35 views

Bayes - dual estimation of parameter value and parameter growth

I am trying to find an bayesian approach to the following problem: Image a bucket with 100 white balls and an unknown number of red balls During each year, one can take a sample with replacement of ...
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15 views

OLS estimators in stationary process

Given a stationary process xt=a+b*t+et with et a white noise, how can I find the OLS estimators for a and b? Cheers
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0answers
60 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
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1answer
75 views

Order related to Empirical distribution function and Normal distribution

Let $X_1,\dots,X_n$ are i.i.d with distribution function $F$. Let $\hat F_n$ be its empirical distribution function, i.e., $$ \hat F_n(x)=\frac1n\sum_{i=1}^n1_{\{X_\le x\}}(x) $$ where $1_A(x)$ is the ...
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0answers
21 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?
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26 views

Convergence rate of an estimator

Say we are interested in estimating some unknown real scalar parameter $\alpha$ using data. Suppose the estimator $\widehat \alpha_N$ of $\alpha$ using the data is consistent. I want to know what it ...
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27 views

compute maximum likelihood estimator and confidence interval

Let $X$ have a poisson distribution with parameter ${\lambda}$. Find the maximum likelihood estimator of ${\alpha}=P(X=0)$. In a sample of size 100 from a poisson distribution, it is found that the ...
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16 views

Show that E(g(T-p)) < E(g(S-P) for any convex function g if T and S are estimators of p

The more detailed question. I'm kinda having some trouble starting out with answering this question. My initial approach would be to g(x)= x^2 since that is a convex function and find the expected ...
2
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1answer
60 views

How to compute the unique MLE from an Exponential Family of Distributions?

Let $$ f(x;\theta)=\frac{1}{\pi} \frac{e^{\theta x}\cos(\theta \pi/2)}{\cosh(x)}, x\in{\mathbb{R}} $$ be a family of densities and which is clearly exponential family. Then what is the Maximum ...
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2answers
39 views

MAP Estimator with Laplacian Noise

I need to calculate the MAP estimator of $ x $ in the following case: $$ \left [ \begin{matrix} {y}_{1}\\ {y}_{2} \end{matrix} \right ] = \left [ \begin{matrix} x\\ x \end{matrix} \right ] + ...
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1answer
60 views

Conditional expectation and Rao Blackwell

Consider a family of densitites $f(x,\theta)=\frac{\exp(-\sqrt{x})}{\theta}$. Let $X_1$ be a single observation from this family. I have shown that $\sqrt{X_1}/2$ is an unbiased estimator. Now ...
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1answer
85 views

Is a probability density function necessarily a $L^2$ function?

If a nonnegative continuous real valued function $f$ is integrable over $\mathbb{R}$ with $$\int_\mathbb{R} f\,\mathrm{d}x = 1,$$ does it hold true $$\int_\mathbb{R} f^2 \,\mathrm{d}x<\infty?$$ ...
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41 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
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22 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
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2answers
49 views

how to estimate the phase parameter of a complex function

There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form: $$f(t)=Ae^{-iBt}$$ with unknown ...
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1answer
94 views

statistics inequality

Let $\theta$ be a discrete pararmeter and $\gamma_{n}$ be an estimator. Prove that for any $c>0$ we have that $$\text{E}[(\gamma_n-\theta)^2] \ge\Pr[|\gamma_n-\theta|>c]\cdot c^2$$
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1answer
141 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
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1answer
27 views

Mean Square Estimate problem

I have to find $\textbf{s}_{MS}$ given $\textbf{r} = h\textbf{s}+\textbf{n}$ where $h$ is a Bernoulli random variable with $Pr(h=1)=Pr(h=0) = 1/2$ and $\textbf{s}$ and $\textbf{n}$ are independent ...
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30 views

Predicting future outcomes from samples when sample sizes and distributions are not controlled and vary

I'm very stale in my statistics and am trying to calculate my confidence around a certain mean outcome from an investment firm (I'll use lay person terms so that I am not assuming any particular type ...
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26 views

ML estimate of sum of guassian variables?

consider the sum $z=x_{1}+...+x_{k}$, where the scalar variables $x_{i}$ are statistically independent and Gaussian, each having the same mean $0$ and variance $\sigma^2_{x}.$ how can I construct the ...
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1answer
73 views

log likelihood function of a cauchy distribution

What is the log likelihood function of a random varible x with cauchy distribution (0,1)? I've tried to work it out. I think its $\log (1+x)^2$. Is that correct?
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1answer
54 views

When is a minimum distance decoder also a maximum likelihood decoder?

It is well known that if we have a binary symmetric channel with crossover probability $\epsilon<0.5$ and we send a word $x$ through it, the most likely word is the one with minimum hamming ...
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93 views

Second derivative wrt complex parameter

I'm facing an estimation problem and I need to calculate the Cramer-Rao Lower Bound of an estimator. So I have 2 unknown parameters: the amplitude of the signal $A$ and its direction of arrival $u$. ...
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0answers
16 views

Can we predict the positions of zero entries in a sparse vector given a model

I am wondering if we can predict the positions of the zero entries in a $n$-dimensional sparse vector $x$ given the linear model: $y=Ax$ where $y\in\mathbb{R}^m$ and the matrix ...
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15 views

Looking for hints on how to prove the convergence of this iterative estimator! [duplicate]

Let $X_n$ be a Poisson process with mean $\lambda^*$. The following sequence estimates the parameter of the Poisson process: $ X_{n+1} = \hat{\lambda}_{n+1} + ...
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1answer
22 views

Literature study for Optimal Estimation Theory

It seems Optimal Estimation/Control Theory requires a lot more than undergraduate maths. Any good book that would help me get started? I have so far referred the following books but found them quite ...
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2answers
81 views

Maximum likelihood estimator?

I am looking at some questions from Mods 2010 and I can't figure this one out. I think my problem is technical... We have a sample (L1,R1), ...,(Ln,Rn) with Lj and Rj normally distributed independent ...