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.

learn more… | top users | synonyms

4
votes
0answers
238 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
3
votes
0answers
47 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
3
votes
0answers
112 views

Estimate number of distinct items

I have a large array of $n$ integers, some of which may be repeated, and I want to estimate how many distinct integers are in the array. Say the number of distinct integers is $N$. I can sample with ...
2
votes
0answers
29 views

Simulation Velocity of a harmonic oscillator system

I am write a simulation for get true Velocity of a harmonic oscillator system as Where P=[p1 p2;p2 p3] can find using Rung-Kutta Integration method with P(0)=[1 0; 0 1] This is code to find p Now, ...
2
votes
0answers
17 views

Risk in density estimation: grasping the definition

When generalizing estimators to an entire function what is the space in which we perform the integral to obtain the expected value (with respect to this function)? For example, when estimating ...
2
votes
0answers
143 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 ...
2
votes
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 ...
2
votes
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 ...
2
votes
0answers
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 ...
2
votes
0answers
53 views

Unbiased estimator with conditional expectation.

Suppose that $X$ has a binomial distribution with parameter $N=1$ and $p=1/2$. Y, which is independent of $X$, has a normal distribution with mean $\mu$ and variance 1. Consider the estimator $\mu$ of ...
2
votes
0answers
44 views

What are the properties of median-unbiased estimators?

On Wikipedia it says that " A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." How to prove this? Note that I asked on Cross ...
2
votes
0answers
193 views

Fisher Information and minimum variance estimators

I am trying to understand what can be proved about minimum variance estimators. I have changed the question to make it more specific. Let us assume we have some finite set $S$ of elements and we just ...
2
votes
0answers
95 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
1
vote
0answers
17 views

Help in understanding derivation of density function and expectation maximization

I am unable to understand how the density function is derived in this paper Semiblind System Identification with Symbolic Chaotic Sequences The Authors have ...
1
vote
0answers
22 views

asymptotic unbiasedness of weibull mle

It's known that the MLEs of the two-parameter Weibull distribution scale and shape parameters are not available in a closed form. It is, however, known that they do exist, are unique, and moreover, ...
1
vote
0answers
19 views

Estimation of Linear Projection

Given a linear system: $Y=AX+W$ Where: $X$ is the input signal of size $N \times K$ $Y$ is the output signal of size $M \times K$ $A$ is a projection of size $M\times N$; with $M >> N$ $W$ ...
1
vote
0answers
16 views

Unbiased estimator for maximum

Assume $n$ independent random variables with unknown distributions $\{X_1,X_2,...,X_n\}$. Multiple "samples" or observations for each of these variables are given (not necessarily with the same ...
1
vote
0answers
35 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 $Y_1, Y_2, ...$ ...
1
vote
0answers
74 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} + ...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
71 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
1
vote
0answers
22 views

Making sense of an equation

I'm wanting to impliment a formula but I'm having issues understanding some of the components that make it up. The premise of the equation is to use a modified version of Kalman filter that estimates ...
1
vote
0answers
23 views

A question on Stochastic Approximation

I have just started learning stochastic approximation methods, so the question I'm going to ask may be a trivial one in this field, but I need to know this seriousely. I know, that if $g(x,\xi)$ is a ...
1
vote
0answers
23 views

ML Estimation for number of animals in a park. Hypothesis Testing.

A park of area $S=10 000 km^2$ was surveyed for bears, and out of $n$ disjoint regions of equal area $s=1km^2$, there were $n_k$ regions with $k=0,1,....,N$ bears. On each of these regions, the amount ...
1
vote
0answers
22 views

Is there a way to estimate the range of fitting coefficients from only the data?

Considering an approximation $f$ for a set of $N$ data points $(x,y)$ using, for example, $M$ radial basis functions at arbitrary sites in the domain $f_i = \sum_{j=1} ^M c_j\phi(||x_i-x_j||)$ where ...
1
vote
0answers
80 views

Hypothesis testing problem of Normal distributions.

Consider the following Hypothesis Testing problem: Hypothesis $H_0$ : $X \sim N(\mu_0, \sigma_0)$. Mean $\mu_0$ is known but only upper and lower bounds on $\sigma_0$ are known. Hypothesis $H_1$ : ...
1
vote
0answers
74 views

Estimating the number of observations from a set of samples

I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs. All of the hidden inputs are driven by an experimenter ...
1
vote
0answers
71 views

Showing that statistic is unbiased

Let $X $ be observed data. Let $\hat{\theta}(X)$ be an unbiased estimate of $\theta$ and let T be a sucient statistic for $\theta$. Define the new estimator $\hat\theta^{*}$ of $\theta$, $$ ...
1
vote
0answers
49 views

Estimate the size of a set given random sub sets.

Assuming there is a set $S$ that you are given subsets of, $s_1, s_2, ..., s_n$, estimate $|S|$ (and a confidence interval if possible) making as few assumptions as possible. I'm not going to quibble ...
1
vote
0answers
121 views

Worst-case error related to Cramer-Rao bound

I would like to understand the relation (if any) between the Cramer-Rao Lower Bound of estimation theory and the following simple definition of "reconstruction accuracy" which doesn't use any ...
1
vote
0answers
102 views

Maximum Likelihood Estimator of SNR for a Known Signal Superimposed in AWGN

I would like to evaluate the Maximum Likelihood Estimator for the SNR of a given signal: $ x(t) = as(t-\tau) + n(t) $ Under the following assumptions (This is the model of Radar Signal): The input ...
1
vote
0answers
29 views

MLE estimation of parameters, converting normalized observations to integers and back

I am fitting a model's parameters to grouped data by maximizing the likelihood equation: $L(\theta)=N!\prod_{i=1}^{G}\frac{p_i(\theta)^{n_i}}{n_i!}$ $\theta$ is the vector of parameters. $n_i$ is ...
0
votes
0answers
9 views

How to estimate variances for Kalman filter from real sensor measurements without underestimating process noise.

As the title says, I want to estimate the variances needed for a Kalman filter from real sensor measurements only. For example we can take a temperature sensor, but the solution shall be as ...
0
votes
0answers
21 views

Description length in model coding

In class, our professor posted the following: We will discretize $\theta$ (some model) into $1/\sqrt{n}$ distinct values. Intuitive argument: with N data points, our estimation error for $\hat ...
0
votes
0answers
12 views

How can I find the nonlinear and linear MS estimates of y in terms of x and the resulting MS errors?

If $y=x^3$, find the nonlinear and linear MS estimates of $y$ in terms of $x$ and the resulting MS errors? This is what I got for the nonlinear MS estimation: Since $e=E\{[y-C(x)]^2\}$, $C(x)=x^3$ and ...
0
votes
0answers
9 views

how can I find the resulting MS error with linear estimation?

The problem says: If $\eta_x=\eta_y=0, \sigma_x=\sigma_y=4$ and $\hat{y}=0.2x$ (linear estimate), find $E\{(y-\hat{y})^2\}$. I am doing this, based on Papulis formulas for homogeneous linear estimate: ...
0
votes
0answers
12 views

How Can I show that a=A in this linear MS estimation problem?

How can I show that if the constants A,B and a are such that $E\{[y-(Ax+B)]^2\}$ and $E {\{[(y-\eta_y)-a(x-\eta_x)]^2 \}}$ are minimum, then $a=A$. I am trying to use this: $e=e_m$ is minimum if ...
0
votes
0answers
13 views

Simulation Position of a harmonic oscillator system

I am write a simulation for get true Postion of a harmonic oscillator system as Now, I want to write matlab code to get the true postion z of the system. However, ...
0
votes
0answers
7 views

Sensitivity analysis of paramaters and input variables

I am trying to perform a sensitivity analysis of an optimization problem $f(x,\alpha)= \min_{ Q} {g(x,\alpha , Q)}$ where $x$ is an input variable for our function, and $\alpha $ is a parameter. ...
0
votes
0answers
19 views

Find the MLE of mu/sigma and its expected value?

The problem: $X_1, \ldots, X_n$ be random sample from a normal distribution $N(\mu, \sigma^2).$ Find the MLE of $\mu/\sigma^2$ and its expected value. My solution: Since $\hat{\mu} = \bar{x}$ and ...
0
votes
0answers
34 views

Second derivatives of MLE to multivariate normal distribution

I want to calculate the second derivatives to the MLE's $\hat{\mu}$ and $\hat{\Sigma}$ to confirm that the extremums indeed are maximums to the multivariate normal distribution $$ ...
0
votes
0answers
6 views

Find the spectral density of white noise $\omega$~(0,2)

I am find the spectral density of a white noise given by $\omega$ ~ (0,2). Could you help me to find it? Thank all.
0
votes
0answers
46 views

Determine a distribution of a gaussian stochastic at different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
0
votes
0answers
4 views

Least square estimator: $N( \beta x_i, \sigma^2)$

Let $ Y_1,...,Y_n$ be i.i.d $N(\beta x_i, \sigma^2) $ with known $ x_i's$. It is asked to find the Mean Squared Estimator for $\beta.$ I didn't understandmuch about this method of pbtaining an ...
0
votes
0answers
12 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 ...
0
votes
0answers
39 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 ...
0
votes
0answers
9 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 ...
0
votes
0answers
28 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 $\theta$ ...