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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102 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 ...
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19 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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67 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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38 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 ...
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35 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 ...
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105 views

Cramer-Rao bound for $\chi^2$ distribution parameter estimates.

I've stuck in unpleasant problem with noncentral $\chi^2$ distribution. I work with random variables, distributed as $\chi^2_{\nu}(\lambda)$, where $\nu$ is the degree of freedom and $\lambda$ is ...
2
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171 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
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92 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 ...
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22 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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18 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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62 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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70 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 ...
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51 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 ...
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16 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 ...
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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 ...
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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 ...
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65 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 $H_1$ : ...
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66 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 ...
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66 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 $\theta$, $$ ...
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48 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 ...
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107 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 ...
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98 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 ...
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28 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 ...
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16 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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19 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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16 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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20 views

Sufficient statistic of a geometric rv

Can anyone help me prove the sufficient statistics of geo r.v. I am stuck and cant cancel out the thetaws . thanks.
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81 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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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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7 views

How to compare two different estimators

What is the correct to compare the 'performance' of an estimator? Suppose I have two estimators $\hat{\pmb \theta_1}$ and $\hat{\pmb \theta_2}$ for the same problem of estimating $\pmb \theta$. For ...
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26 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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9 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 ...
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27 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 ...
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21 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 ...
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21 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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41 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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40 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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35 views

maximum likelihood estimate

Two random variable X and Y have the following joint pdf: $P_{XY}(x, y) = \begin{cases} 10y,& \text{for } 0\leq y \leq x^2 \text{and} 0 \leq x \leq 1,\\ 0, & ...
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34 views

Maximization of The Likelihood Function of Vector Entries and Its Norm

I'd be happy for assistance with the maximization of the likelihood function of the following model. The Parameters Vector $ \mathbf{\Theta} = [{x}_{1}, {x}_{2}] $. The measurement vector is $ ...
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38 views

Assigning prior to $\gamma$ in composite power function $P(t) = max[\lambda t^{-\beta}, \gamma]$

I want to estimate the parameters $\lambda, \beta$ and $\gamma$ using a bayesian approach and an MCMC sampler. With the exception of $t$ all variables are random variables between $0$ and $1$. $t$ is ...
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18 views

Difference between two model fitting schemes

We have some experimental data, $x \mapsto \hat{f}$ and we're trying to fit a known model of the form $$f(x\ \left|\right.\ a_1, a_2, a_3, b_1, b_2, b_3) = a_1 F(b_1, x) + a_2 F(b_2, x) + a_3 F(b_3, ...
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49 views

What is a zero order estimate?

In a paper I am examining (Multiband signal processing by using nonuniform sampling and iterative updating of autocorrelation matrix by Modris Greitans), the author uses the term zero order estimate. ...
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76 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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25 views

ML Estimation, graphical method?

I have a problem in statistics that I don't quite know how to do: "Generate a 1000-element data sample from the Rayleigh distribution. Graph the log-likelihood function $\ln L(\alpha; \vec{x})$ as a ...
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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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25 views

Proportional-Integral Estimator Question with Agents and Graphs

My question concerns a collection of $n$ agents, with interconnections described by a graph $G = (V,E)$, with Laplacian $L$ and adjacency matrix $A$. The point behind the following dynamical-system is ...
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35 views

How to obtain confidence intervals for a sample of independent normally distributed random variables

Let $X_1$, ..., $X_n$ be independent observations with distribution $N(\mu,\sigma^2)$ where $\mu \in \mathbb{R}$ and $\sigma^2>0$ are both unkown. What is an easy way to derive confidence ...
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50 views

Variability in estimations over a non-ergodic/non-regular Markov process

Imagine we have a non-ergodic/non-regular Markov Process with with $n$ states. Among these $n$ states, there are $k$ absorbing states. For each of the $n-k$ non-absorbing states, it is not possible ...
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27 views

Shrinkage estimator's risk function

How do you compute the risk function under squared loss of an estimator of the form $\begin{align*} \hat{\mu}(x) &= \bar{x} + \left(1-\frac{k}{||x-\bar{x}||_2^2}\right)(x - \bar{x}) \end{align*}$ ...
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154 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, ...