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2
votes
0answers
28 views

MLE and CRLB with mismatched likelihoods

Suppose that I can do a Karhunen-Loeve expansion of a log-likelihood function $p(\bf{x};\theta)$ into N terms and that these accounts for a fraction $1-\delta$ of the total energy. Now consider ...
0
votes
0answers
33 views

Kalman filter in estimating state space model

When using Kalman filter to estimate parameters in state space model,it is assumed that the variance of error in observation equation and state equation is known. However, in applications, the ...
3
votes
0answers
66 views

Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...
1
vote
0answers
33 views

Equivalence of Graphical model selection algorithms

Suppose, a jointly Gaussian random vector is denoted by $X \in \mathbb{R}^{p}$ and $X$ has a distribution given by $\mathcal{N}(\mu,\Sigma)$. It is known that estimating the graphical model that ...
0
votes
0answers
28 views

non-coherent estimation problem

I have the following signals $$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...
1
vote
0answers
78 views

Distribution of a signal covariance matrix

A common estimation problem in signal processing assumes the following signal model \begin{equation} \mathbf{r} = \sum_{i=1}^{Q}\alpha_i\mathbf{s}\left(w_i\right)+\mathbf{n} \end{equation} where ...
2
votes
1answer
67 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...
0
votes
0answers
25 views

Computing covariance matrix from patchy data

Take a linear Gaussian model given by the following normal distributions: $\mathcal{N}(z_n | z_{n-1}, A), \ \ \ \ \ \ \ \ \mathcal{N}(x_n | z_{n}, B) $ Let variable $z$ be a hidden variable and $x$ ...
2
votes
1answer
68 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 $\theta$ ...
1
vote
3answers
138 views

How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$. Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...
2
votes
2answers
106 views

estimating variance of dependent normal distributed data

Let $X_{ij}$ with $1\leq i<j\leq n$ (that are $X_{12},\dots, X_{1n},\dots,X_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that $ \text{corr}(X_{ij},X_{rs})=\rho ...
2
votes
2answers
76 views

Markov-type functions

I'd like to have some informations about Markov-type functions (or Cauchy-type): \[ f(z)=\int_{\Gamma} \frac{\mathrm{d}\gamma(\xi)}{\xi-z}.\] $\gamma$ is a positive measure with compact support ...
1
vote
0answers
63 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
3
votes
1answer
154 views

Equivalent method for maximum likelihood estimation of covariance parameters

My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function: $$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
0
votes
1answer
67 views

About the suboptimality of linear estimators

Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$. ...
1
vote
0answers
97 views

Moments of Matrix Gamma distribution

Matrix gamma distribution (defined for example in http://en.wikipedia.org/wiki/Matrix_gamma_distribution) is one way to generalize Wishart distribution. In our course work that distribution was used ...
1
vote
0answers
114 views

Shrinkage (or Stein's phenomenon) in low dimensions, discrete contexts

I am trying to understand shrinkage, or the Stein phenomenon. As someone without a statistics background, the focus in most introductory presentations on normal distributions and squared error loss ...
4
votes
0answers
167 views

Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...
0
votes
1answer
105 views

Signal model classification between two possbile candidates

How to decide the most possible signal model between two model candidates besed on the received signal vector? Assume the received signal vector is $y$, the possible signal model candidates could be: ...
2
votes
0answers
156 views

Worst-case error and Cramer-Rao Lower Bound - is there any mathematical relation between them?

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
491 views

Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form, $J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$ where $N$ is the number of ...
0
votes
1answer
272 views

Estimation of Temporal Correlation of Signal

I have a signal and i'd like to estimate its temporal correlation. My limited understanding is i should compute the PSD by estimation using a parametric model such as AR. However, i'm not quite ...
1
vote
1answer
199 views

Is an unbiased estimator with arbitrarily small variance necessarily consistent?

Given an unbiased estimator $\hat \theta_n$ of a parameter $\theta$, if the estimator has small variance (approaching $0$ as $n\to\infty$), it seems reasonable to expect that the estimator is ...
0
votes
2answers
438 views

Estimating a sum

Sorry for the vague title but I couldn't find a better one. I want to compute the sum $S = \frac{1}{N}\sum_{i=1}^N c_i x_i$ where $c_i$s are known positive constants. The problem is that computing ...
0
votes
0answers
77 views

Estimation of X in Gaussian noise

Given $\textbf{x}=[x_1 x_2 ... x_n]^T$ where $\textbf{x} \in \{ 0, a_1, a_2, a_3\}^n, a_i \in \mathbb{C}$ and $\textbf{z} = \{z_1 z_2,...,z_n \}$ where $z_i \textbf{~} N(0,\sigma^2)$ is a Complex ...
0
votes
0answers
128 views

Why does the OLS estimator simplify as follows for the single regressor case?

I was reading in "A Guide to Econometrics" that given $Y = X \beta + \epsilon$, the variance covariance matrix of $\beta^\text{OLS}$ is given by $\sigma^2 (X' X)^{-1}$ where $\sigma^2$ is the variance ...
2
votes
2answers
4k views

Maximum likelihood estimator for Power-law with Exponential cutoff

Hi, for fitting empirical data to power-law I am aware of the work by Clauset et al. (http://arxiv.org/abs/0706.1062) and how to use maximum likelihood estimation. There exists also a simple maximum ...
3
votes
0answers
684 views

Interpolating Wavelet Coefficients

Hi! I was instructed via reddit that this place would be the best place to post this question. Fingers cross you can help... Ive been writing some code to get rid of noise "spikes" in a signal. I'm ...
2
votes
1answer
397 views

Why doesn't Stein effect happen for multinomial distributions?

(Medeen, et all, 1998)" show that Maximum Likelihood estimate is admissible for multinomial distribution under squared error. On other hand, James and Stein showed that arithmetic average is not an ...
6
votes
0answers
1k views

Using Fisher Information to bound KL divergence

Is it possible to use Fisher Information at p to get a useful upper bound on KL(q,p)? KL(q,p) is known as Kullback-Liebler divergence and is defined for discrete distributions over k outcomes as ...
1
vote
1answer
267 views

Is there a text on estimation theory online?

Where can I find graduate level, thorough, parameter estimation/ estimation theory material on the web?