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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199 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 ...
2
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0answers
43 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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0answers
16 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
2
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1answer
39 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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0answers
75 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
79 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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1answer
41 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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0answers
34 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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0answers
20 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
66 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
43 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 ] + ...
1
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1answer
74 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 ...
3
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1answer
115 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 $$\int_\mathbb{R} f^2 \,\mathrm{d}x<\infty?$$ ...
2
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0answers
49 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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0answers
39 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
52 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
26 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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1answer
147 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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0answers
42 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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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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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
124 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
66 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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97 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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0answers
16 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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2answers
93 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 ...
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1answer
28 views

Given 50 IID normals, find the exact SE for the estimate of $\sigma^2$?

Given 50 I.I.D Normal distributions random variables $X_i$, the Maximum Likelihood estimator for $\sigma^2$ is $\hat{\sigma}^2$, as proven in my lecture notes. Find the EXACT SE. My Attempt: ...
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0answers
28 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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74 views

Identification of real functions

this my second question, so I'm still new... thanks in advance for any help! Basically, I'm looking for some references and tools to study the following problem. Consider the following function ...
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1answer
69 views

Finding the MLE of pareto dist., and trouble interpreting $\prod$ notation properly.

I am generally having trouble understanding how to use product notation when calculating Maximum Likelihood Estimators. The example bellow is from a random sample $X_1,...,X_n$. Find the MLE of ...
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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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1answer
27 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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42 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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0answers
75 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
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0answers
11 views

Estimating variance from the sequence

Suppose that we have $\{X_n\}\to X\sim N(0,\Omega)$ where $X_n$ can be obtained from observations. My problem is to estimate $\Omega$ consistently. If $var X_n$ converges to a "finite" matrix, then ...
4
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1answer
180 views

Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics

Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $M_n=\max\{X_1, \ldots, X_n\}$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda ...
1
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1answer
139 views

Find the Method Moment Estimator of parameter $\theta$

Find the MME of parameter $\theta$ in the distribution with the density $f(x,\theta)=(\theta +1)x^{-(\theta+2)}$, for $x>1$ and $\theta >0$. So far I think I have a basic understanding of the ...
1
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1answer
67 views

Is Sample Covariance Tied to a Specific Distribution

In many sources on data analysis, the author(s) talk about calculating covariance of the data, and the formula is given as such $$ \Sigma = cov(X) = E[(X-E[X])(X-E[X])^T]$$ This formulation is given ...
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0answers
84 views

How to simplify conditional probability of union of several events

I have an output binary scalar, $y∈B=[0,1]$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=[0,1]$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
0
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1answer
17 views

Likelihood function for only one trial

I have a trial $\mathbb X = (X_1,X_2,....X_n)$. $X_i$ has specified distribution with unknow parameter $\theta$. I want to find an estimator of this parameter. So I can use methods like Likelihood ...
0
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2answers
69 views

Asymtptotic limit of $e^x$ [closed]

I am looking for functions $A,B$ such that $$ A < e^x < B.$$ $A,B$ should be as close to $e^x$ as possible. I was trying to find something, but all I found was very distant. Can someone suggest ...
3
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1answer
70 views

How is the “cooking” done in surveys

In my country there's an official center undertaking surveys of voting intention every 4 months. However, they provide only "direct" voting intention, and the statistics obtained are usually pretty ...
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1answer
27 views

Estimator of a Random Variable

Given a random varable $Y$ where $$ f_Y(y) = \begin{cases}e^{-(y-k)} \quad x>k\\0\quad \text{otherwise}\end{cases} $$ Given $n$ observations of $Y$. Is the sample mean $\bar{Y}$ an unbiased ...
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2answers
112 views

Estimation theory: Finding MSE and Variance of an Estimator

Let $\hat{A}$ be an estimator of $A$ where $a<A<b$ and $\tilde{A}$ be another estimator such that$$\tilde{A} = \cases{a \text{ if } \hat A \leq a\\\hat A \text{ if } a< \hat A <b\\{b ...
3
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2answers
75 views

Calculating likelihood of event based on retrospective analysis

I have a simple dataset consisting of the dates/times at which certain medications were taken by a patient. By looking retrospectively I'd like to make a best guess estimate as to which medication ...
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0answers
24 views

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 ...
2
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3answers
88 views

More accurate estimation of mathematical constant $e$

Very often in books and also on Wikipedia we can find that: $$e \approx \left(1+\frac{1}{n}\right)^n$$ but I want more accurate estimation, it means instead using $\approx$ I wonder if I can use ...
0
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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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0answers
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 ...