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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104 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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31 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
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39 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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70 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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45 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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38 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
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185 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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93 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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20 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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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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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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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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68 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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90 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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59 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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21 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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73 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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68 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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67 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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116 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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101 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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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 ...
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18 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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19 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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12 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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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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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

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 ...
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21 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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28 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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92 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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37 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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10 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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51 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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23 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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47 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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41 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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40 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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40 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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40 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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78 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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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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175 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, ...