# Questions tagged [estimation-theory]

The estimation-theory tag has no usage guidance.

102
questions

1
vote

0
answers

47
views

### Estimate for the gradient of solutions in an elliptic differential equation in a Sobolev space

Let $\Omega$ be a bounded or unbounded domain in $\mathbf R^{3}$ with a smooth boundary $S$ and a normal vector given by $n$. Now, we consider the following second-order elliptic problem with Neumann ...

1
vote

0
answers

46
views

### Parameter estimation of a Taylor expansion

Let $a,b$ two real numbers, $\theta$ a real parameter and suppose that you have an analytic function of the form:
$$
f_\theta(x)\triangleq \sum_{k\in\mathbb{N}}a_k(\theta)x^k \quad\forall x\in[a,b],
$$...

0
votes

1
answer

59
views

### Derivative of log-likelihood function for Gaussian distribution with parameterized variance

Suppose we have a parameter $\theta \in R^{n}$ that defines some noisy observation $z=\mu(\theta)+\eta, z\in R^{m}$ where the noise follows a Gaussian distribution whose covariance is a function of ...

7
votes

1
answer

203
views

### Singular Fisher information matrix and existence of unbiased estimators

I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...

0
votes

0
answers

51
views

### Estimation of Nonzero Coefficients of Binary Cylotomic Polynomials

I am reading Fouvry's paper https://msp.org/ant/2013/7-5/ant-v7-n5-p07-p.pdf . I am still confused on section 4.2 why $P\leq x^{\frac{20}{9}\gamma -\frac{2}{3}}\mathcal{L}^{-16}$ leads to estimate in ...

2
votes

0
answers

77
views

### A complex problem involving densities (likelihood functions) and optimization

Consider the following autoregressive process with normal errors:
\begin{equation}\label{7YlUV4i8nuO}\tag{I}
y_t = \phi y_{t-1}+ u_t, \quad u_t \overset{iid}{\sim} N(0,\sigma^2)
\end{equation}
We ...

4
votes

2
answers

197
views

### Generalization of van der Corput's estimate on oscillatory integrals

Question: Given exponents $0<\alpha<\beta$ and an interval
$[a,b]\subset(0,\infty)$ are there constants $C,d>0$ such that for any
$\lambda_1,\lambda_2\in\mathbb{R}$,
$$\left|\int_a^be(\...

1
vote

0
answers

51
views

### Functional approximation with derivatives

I am trying to solve a functional approximation problem.
Consider a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm x}$ and ...

2
votes

0
answers

87
views

### Comparing the truncated $\ell^{1}$-norm of polynomial coefficients with the supremum norm on the unit disc

Let $p=a_{0}+a_{1}z+\ldots+a_{n}z^{n}$ be a polynomial. Consider the following truncated $\ell^{1}$-seminorm of the coefficients of $p$:
$$\|p\|_{\ell^{1},\text{trun.}}:=\sum_{k=1}^{n}|a_{k}|=\|p-a_{0}...

2
votes

1
answer

98
views

### DKW inequality for $L^1$-norm

Suppose that $X,X_1,X_2,X_3\dots$ is a sequence of $\mathbb{P}$-i.i.d. random variables supported in the interval $[0,1]$. Let $F$ be the cumulative distribution of $X$, i.e. $F(x):=\mathbb{P}[X \le x]...

3
votes

0
answers

60
views

### Is the Kalman Filter computationally optimal for Kalman filtering?

Kalman filtering is known to be a recursive process that minimizes mean square error in linear problems.
My question is: has anybody shown that this algorithm is computationally optimal, i.e. that you ...

3
votes

1
answer

267
views

### A problem of using Schauder estimate in the paper of Yau's proof of calabi conjecture

[This question is looking at the paper
Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002/...

0
votes

0
answers

18
views

### Estimatives for elliptic systems involving the laplacian

Considering the problem
\begin{equation}
\left\{
\begin{array}[c]{11}
\Delta(\Delta \chi -\chi) = 0 & \text{in } \Omega, \\
\Delta \chi -\chi = h_2 - h_1, & \text{on } \partial\Omega \\
\end{...

0
votes

0
answers

61
views

### Maximum likelihood estimator for power law with negative exponent

Background
I have data that roughly follows a power law with a negative exponent (up to a point; also, the parameters of the "fit" were just guesstimated by eye as a demonstration):
Now I ...

1
vote

0
answers

32
views

### Correlating two matrices $A,B$ with stochastic dependency structure imposed by cross-validation

Consider a labelled data set
$$D = \{(x_1, y_1),...,(x_n, y_n)\} $$
on which we want to evaluate a machine learning algorithm using $k$-fold cross validation with $m$ different random seeds. This ...

0
votes

0
answers

126
views

### How to estimate sums over arithmetic progressions?

For $x>1$
$$
N(x)=\sum_{0<n<x \\n \equiv 1 \pmod 4\\ n\text{ squarefree}} 1
$$
How to estimate $N(x)$'s order? (Like $N(x) \sim Ax$)
Furthermore, for $n=p_1p_2\cdots p_v$, define $\alpha (n)=...

4
votes

1
answer

180
views

### What journal(s) do you recommend for submitting a paper on a topic that spans information theory and estimation theory?

I've written a paper that a) demonstrates an equivalence between conditional complexity $K$($Y$|$X$) in information theory and the random component of an effect size estimate $r_{xy}$, and then b) ...

1
vote

1
answer

91
views

### Estimating the average of two gaussians' mean with minimal squared error

This is a follow-up to my previous question.
Assume that $X\sim \mathcal N(\mu_1,\sigma_1^2)$ and $Y\sim \mathcal N(\mu_2,\sigma_2^2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$....

1
vote

1
answer

454
views

### Estimating the average of two gaussians' mean

Assume that $X\sim \mathcal N(\sigma_1,\mu_1)$ and $Y\sim \mathcal N(\sigma_2,\mu_2)$.
I want to estimate $\frac{\mu_1+\mu_2}{2}$ after observing $X,Y$.
In my setting, $\sigma_1,\sigma_2$ are known ...

2
votes

0
answers

208
views

### Sobolev (Triebel-Lizorkin) norm estimate for $F \circ u - F \circ v$

Let $F \in C^1(\mathbb R^d;\mathbb R)$ be such that $F(0) = 0$ and
$$|F'(\tau v + (1 - \tau)w)| \leq \mu(\tau)(G(v) + G(w))$$
for some $\mu \in L^1([0,1])$ and some non-negative $G \in C^0(\mathbb R^d;...

4
votes

1
answer

328
views

### Showing that $\sum_{n=0}^\infty (4n+1)q^{\left (\frac{4n+1}{2}\right)^2} - \sum_{n=1}^\infty (4n-1)q^{\left (\frac{4n-1}{2}\right)^2} \geq 0.1$

Recently I came along the following problem concerning a lower bound on the difference of two series:
I want to show that for every $q \in [e^{-2},e^{-\frac{1}{2}}]$ we have
$$
f(q) := \sum_{n=0}^\...

1
vote

0
answers

77
views

### Calculating the mean squared error for an estimate of a large sum

Consider the set of all Boolean function $f: \{0, 1\}^{n} \rightarrow \{-1, 1\}$. Now, let's pick a function uniformly at random from this set. Let $F$ be the random variable corresponding to the ...

2
votes

0
answers

57
views

### Distribution of unbiased estimator of covariance matrix with missing values

Initial setup
Assuming $X_1, ..., X_n \in \mathbb{R}^m$ are iid, sampled from $\mathcal{N}(\mu, V)$, one can define the estimators for the sample mean $\hat{\mu} = \frac{1}{n} := X^T 1_n$, and sample ...

2
votes

0
answers

97
views

### L1 error of estimators

I came across the following problem and I have no clue how to approach it. I am looking for help with directions or references.
Consider the $\alpha$-stable distribution with unknown true mean $\mu$, ...

0
votes

1
answer

113
views

### How to detect, track and map a Markov chain

You are receiving a time series whose elements belong to a finite set. Assume the time series is distributed as a Discrete-Time Markov Chain. You receive one element at each time step.
For each time ...

2
votes

0
answers

134
views

### 'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem.
Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...

2
votes

1
answer

125
views

### Spline Interpolation error of higher degree

It is well-known that the interpolation error of a cubic spline has at best order $O(h^4)$, which results from polynomials of degree $3$.
Can I assume that, if one uses polynomials of degree $p$ and ...

1
vote

0
answers

62
views

### Percentile interval Lemma

Let $\theta$ be a parameter and $\hat{\theta}$ the plug-in estimate, I need a proof of the following lemma, as given in [1], p. 173, in the form of a reference or of a direct argument:
Percentile ...

1
vote

0
answers

58
views

### Bootstrap-$t$ confidence intervals

I'm writing a dissertation about bootstrap methods and the main book I'm using is Efron, B., & Tibshirani, R.J. (1994), An Introduction to the Bootstrap (1st ed.), Chapman and Hall/CRC. Now I need ...

2
votes

1
answer

151
views

### Stability estimates on quotients of the form $ \frac{\prod_{j=1}^n a_j}{\prod_{j=1}^n b_j} $

Suppose that $a_j,b_j \in \mathbb C$ are complex numbers, $j=1,\dots,n$, with the property that $|a_j|,|b_j| \geq c > d >0$ where $c,d$ are positive real numbers. I'm interested in the stability ...

2
votes

0
answers

76
views

### The optimality of Kalman filtering

It is known that the Kalman filter estimates the state of the following system recursively.
$$x_{k+1}=Ax_k+w_k, \ \ w_k \sim \mathcal{N}(0,Q)$$
$$y_k=Cx_k+v_k, \ \ v_k \sim \mathcal{N}(0,W)$$
In the ...

3
votes

1
answer

120
views

### Design a random variable which has the maximal correlation with another random variable

$Y$ is a Gaussian distributed random variable with zero mean and known variance: $Y\sim N(0,\sigma_y)$. We measure $Y$ with a sensor, which is corrupted by white Gaussian noise: $Z=Y+V$; $V\sim N(0,\...

6
votes

3
answers

570
views

### How to estimate the integral involving the distance function

Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain with smooth boundary. Consider the following integral:
$$I(t)=\int_{\Omega}e^{-\frac{d^2(y,\partial\Omega)}{t}}{\rm d}y.$$
My problem is how ...

-1
votes

1
answer

142
views

### How to combine estimator with different variances?

Consider independent random variables $X_1,X_2,\ldots,$ that have the same expectation $\mathbb x=\mathbb E[X_1]=\mathbb E[X_2]=\ldots$
Further, assume that we know that $Var[X_i]=\sigma_i^2$.
In the ...

1
vote

0
answers

89
views

### How to retrieve back the input using Bussgang theorem?

If we have a non-linear function $f$, that is applied to input $x$,
we have then the output $y=f(x)$
Using Bussgang decomposition we can linearize this nonlinearity and express $y$ as
$y=Bx+ η$,
...

6
votes

1
answer

379
views

### Probability of complex eigenvalues

I find this is the best site to post this question, even though I considered cs.
It is a Monte Carlo experiment over the set of 10.000 n×n matrices.
If a single matrix eigenvalue is complex then ...

3
votes

0
answers

110
views

### Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe
$$
\tilde f = f \star h + n
$$
where $\star$ is convolution, $h$ is the point spread function (caused,...

3
votes

0
answers

171
views

### Proving the exponential decay of Green's function for the lattice $-\Delta+p$

The Green function $G(x,y) =G(x-y)$ of the discrete Klein-Gordon operator $-\Delta+p$ on $\mathbb{Z}^{d}$ is given by:
\begin{eqnarray}
G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}\frac{e^{...

1
vote

0
answers

49
views

### Estimation of parameters through multivariate Taylor expansion?

I do have a function $$f(t) = \prod\limits_{j=1}^{n} \left(1 + \sum\limits_{i=1}^{n} M_{i,j} t_i\right)^{-\alpha_{j}}$$ defined by parameters:
$M_{i,j} \in \mathbb{R}_{+}, \;\forall i \in 1,...,d,\; ...

-1
votes

1
answer

131
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 ...

2
votes

0
answers

320
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. ...

1
vote

0
answers

61
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

346
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

159
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\mid\mathbf{x}] = \mu(\...

5
votes

1
answer

98
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

89
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

50
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

82
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", ...

4
votes

0
answers

603
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

152
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,