# Questions tagged [estimation-theory]

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66
questions

**-2**

votes

**0**answers

28 views

### Is kernel density estimator a linear transformation?

I am reading the book Nonparametric econometrics, I am thinking since the kernel density estimator is given as
$$\hat{f}(x)=\frac
{1}{nh}\sum_{i=1}^nK\left(\frac{X_i-x}{h}\right),$$
where $K(\cdot)$ ...

**-1**

votes

**1**answer

111 views

### Sufficient conditions on $ a_i,b_i$ for $a_1\phi(n)+b_1, \cdots, a_k\phi(n)+b_k$ to be simultaneously prime infinitely often?

I am really interested in sufficient conditions on $a_i, b_i$ guaranteeing that the linear forms $a_1\phi(n)+b_1,\dots, a_k\phi(n)+b_k$ become simultaneously prime for infinitely many positive ...

**0**

votes

**0**answers

107 views

### continuity with respect to the diffusion coefficient of the solution of a semilinear parabolic equation

Let $\Omega \subset
%TCIMACRO{\U{211d} }
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},n\geq 1$ be an open bounded subset has a boundary $\Gamma $ of class $%
C^{2}$, $Q=\Omega \times \left( 0,T\...

**2**

votes

**0**answers

53 views

### Extended Kalman Filter and its State Transition Matrix

Sorry for what might be a long post, I want to give background.
Initially I had regular Kalman filter, and the state model was defined by Newtonian kinematics, with initial position 0 and speed of 2. ...

**0**

votes

**0**answers

21 views

### The existence of maximum likelihood estimator of matrix-variate distribution

For a matrix-variate Gaussian distribution, the negative log marginal likelihood is
\begin{equation}\label{matrixLikelihood}
\mathcal{L} = \frac{nd}{2}\ln(2\pi) + \frac{d}{2}\ln \det(K') + \frac{n}{...

**1**

vote

**0**answers

54 views

### Distances between up and down crosses in Gaussian Processes

Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...

**-2**

votes

**2**answers

133 views

### Lower bound of q pochhammer symbol [closed]

How one could prove, that q pochhammer symbol $(1,1/n) = \prod_{k = 1}^{\infty}(1-\frac{1}{n^k}) \geq 1 - \frac{1}{n-1}$

**1**

vote

**1**answer

59 views

### Error metric for joint estimation of mean and variance

Background:
Let $\mu:\mathbb{R}^n\to\mathbb{R}$ and $\sigma:\mathbb{R}^n\to\mathbb{R}_+$ be two unknown functions, and consider a stochastic model of the form
$$
\mathbb{E}[Y|\mathbf{x}] = \mu(\...

**5**

votes

**1**answer

77 views

### Estimating the size of the remainder in a random partition

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with ...

**-2**

votes

**1**answer

77 views

### Existence or impossibility of Gaussian factory

Gaussian factory problem: given an iid sequence $x_i \sim \mathcal{N}(\mu,\sigma^2)$, $i=1,2,\dots$, with $\mu$ and $\sigma^2$ both unknown, construct a realization $y \sim \mathcal{N}(0,1)$.

**0**

votes

**1**answer

45 views

### Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...

**3**

votes

**1**answer

72 views

### Optimal linear measurement operator

Let $x\in R^n$ be an unknown vector. Suppose I am allowed to choose any $A\in R^{m\times n}$, under the constraint that each row of $A$ has $\ell_2$ norm at most $1$. Then I carry out a "measurement", ...

**1**

vote

**0**answers

306 views

### Functional Taylor expansion for differential entropy

Consider an continuous distribution $F$ with density $f$. The (differential) Shannon entropy of $f$ is
$h(f)=-\int f(x)\log f(x) dx$.
In the literature of differential entropy estimation, ...

**2**

votes

**1**answer

144 views

### Finding a similarities and differences of sent of matrices

Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,

**2**

votes

**1**answer

360 views

### Distribution of ratio between complex Gaussian and Chi-square R.V.s

What would be the distribution (p.d.f.) of the following ratio?
$$z = \frac{x_{1}}{|x_{1}|^2 + |x_{2}|^2 + ... + |x_{M}|^2}$$
where $x_{i} \sim \mathcal{CN}(0,a), \forall i$ and $a > 1$. As can ...

**3**

votes

**1**answer

150 views

### Proving bounds on analytic functions using only the Taylor expansion

I wonder if there is a general method for obtaining bounds on an analytic function using only its Taylor expansion (not using its special properties such as satisfying a good differential equation, ...

**3**

votes

**1**answer

101 views

### maximum likelihood estimation of X is better than that of f(X)?

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) ...

**2**

votes

**2**answers

358 views

### An alternative proof of Bayesian Cramer-Rao

My question is:
Are there an alternative proof of Cramer-Rao lower bound that does not use
Cauchy-Swartz inequality?
Let me outline the classical proof and explain why I am interested in this ...

**1**

vote

**0**answers

76 views

### Extended Kalman filter for initial values estimation

I try to make extended Kalman filter for estimation of initial values of small celestial body. I have:
$(x_1^0, x_2^0, x_3^0, v_1^0, v_2^0, v_3^0) = (x^0, v^0)$ -- inaccurate initial values.
$z = ...

**2**

votes

**0**answers

46 views

### Rate of $L_1$ loss in estmating density on $[0,1]$

Let $f$ be a density on $[0,1]$ and let $X_1,X_2,\ldots$ be $\textit{iid}$ $f$-distributed. Also, let $f_n$ denote the kernel density estimator, i.e.
$$f_n(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\...

**1**

vote

**1**answer

112 views

### Fisher information with vanishing probability

I am confused about the definition of the Fisher information and the case when probability is 0. Consider discrete set $\epsilon$ of possible measurement outcomes. Fisher information is defined as:
$$...

**1**

vote

**0**answers

84 views

### A different objective function in liner regression analysis

I'm an undergraduate student who is green in statistics. I have a problem in the chose of objective function when estimating the parameters.
Let $Y = \beta^TX + \epsilon $ be the standard liner ...

**1**

vote

**1**answer

115 views

### Reconstructing the number of distinct elements from a random projection

Assume we have an unknown sequence $x_1,\ldots, x_n\in \mathcal U$.
We get to observe the sequence $h(x_1),h(x_2),\ldots, h(x_n)$, where $h:\mathcal U\to \{1,\ldots, k\}$ is a random function such ...

**1**

vote

**0**answers

72 views

### sufficient statistics that are irrelevant

I'm designing a lecture on hypothesis testing and want to do an example on a certain matter, but I cannot come up with a good one.
If we should decide upon $H_0$ or $H_1$ given observed data sets ${\...

**1**

vote

**0**answers

43 views

### A question about the prediction error

I am reading about the prediction error estimation and I found the following:
Suppose we have ${\mathbf{Y}}=\mathbf{x}_0+ \epsilon$, where, $\epsilon$ is normally distributed as $\sim \mathcal{N}(0, \...

**2**

votes

**0**answers

64 views

### Calculate sample mean confidence interval of noisy logistical distribution

I have $n$ samples which follow a logistic distribution with unknown $u$ and $s$; it is affected by a Gaussian noise with 0 mean.
I would like to estimate its average $u$ with a confidence interval (...

**1**

vote

**0**answers

53 views

### Cramer Rao bound for relative estimation

I have an observed vector ${\bf y}$ from which I would like to estimate a parameter vector ${\bf c}$ (denote the estimate $\hat{{\bf c}}$).
A feature of our estimation problem is that the involved ...

**2**

votes

**0**answers

43 views

### Perturbation results for statistical estimators

Suppose I have a continuous random variable whose distribution $f$ is some parametric form (normal, exponential, etc.) that is known to me. If I draw many independent samples $x_i$ from $f$, I can ...

**3**

votes

**0**answers

73 views

### Uniform mean-square-error estimates

Consider a standard statistical estimation problem with iid real observations $\{X_i\}_{i=1}^N$. For a collection of real functions $\mathcal{F}$, I want to get an estimate of the uniform rate of ...

**5**

votes

**1**answer

158 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...

**1**

vote

**0**answers

58 views

### Robust weighted estimator of location

Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, ...

**1**

vote

**0**answers

34 views

### Bounding Hidden Markov model Bayesian filter error with inexact models

In context of a hidden Markov model, I am interested in bounding the error of a Bayesian filter when using inexact state transition and observation models.
Consider a hidden Markov model (HMM) with ...

**2**

votes

**0**answers

65 views

### Elliptic Equation with Wentzell boundary condition

I'm looking for a reference showing how to obtain a priori estimate for solutions to a linear second-order elliptic equation with Wentzell boundary condition in a bounded domain in $H^1$ space.
The ...

**2**

votes

**1**answer

209 views

### Literature question on the convergence rate of the empirical distribution

Assume that given $n$ i.i.d samples $(X_1, X_2, ..., X_n)$ drawn from $p_X$, an unknown probability mass function defined over a finite alphabet $\mathcal{X}$, one wants to estimate $p_X(x)$ for each $...

**2**

votes

**0**answers

165 views

### Maximum-likelihood estimation for univariate responses from multivariate data

I am new in the field of machine learning, so I hope I will be able to formulate my question in a clear way...
I have some data represented by vectors $\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n \...

**1**

vote

**1**answer

228 views

### KL divergence Inequality

I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading.
Consider two probability measures $P$ and $Q$ both ...

**2**

votes

**0**answers

132 views

### How to find moment condition for generalized method of moments?

Consider a scalar system with $2K$ outputs and $K+2$ unknowns:
$y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables.
To estimate $a_1$ and $...

**4**

votes

**1**answer

228 views

### Cramér-Rao bound for randomized estimator

As is well known, the Cramér-Rao bound (or information inequality) sets a lower bound on the variance of estimators of a parameter.
Consider the case when the parameter is a scalar, the estimator is ...

**2**

votes

**0**answers

46 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 ...

**4**

votes

**1**answer

331 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

**0**answers

44 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 ...

**1**

vote

**0**answers

223 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

**1**answer

82 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 ...

**2**

votes

**1**answer

81 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 $\theta$ ...

**2**

votes

**3**answers

257 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

**2**answers

171 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
...

**3**

votes

**2**answers

123 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

**0**answers

73 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

**1**answer

236 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

**1**answer

77 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 $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...