# Questions tagged [estimation-theory]

The estimation-theory tag has no usage guidance.

108
questions

0
votes

1
answer

88
views

### Calderon-Zygmund/$L^p$ estimates for the linear heat equation

Let $C_r$ denote the open cylinder
$$
C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\}
$$
and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation
$$
\...

0
votes

0
answers

20
views

### What is the Fisher information matrix of the von Mises-Fisher distribution?

Assuming the von Mises-Fisher distribution as
$$f_{p}(\mathbf{x}; \boldsymbol{\mu}, \kappa) = C_{p}(\kappa) \exp \left( {\kappa \boldsymbol{\mu}^\mathsf{T} \mathbf{x} } \right),$$
where $\kappa \ge 0$,...

2
votes

1
answer

111
views

### Almost sure convergence of double averages of IID random variables

Let $ \{X_i\}_{i=1}^{P} $ and $ \{Y_j\}_{j=1}^{Q} $ be two sequences of independent and identically distributed (i.i.d.) random variables. $X_i$ and $Y_j$ are independent between all pairs of $i$ and $...

2
votes

1
answer

114
views

### Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression

Consider the multivariate regression model
$$Y = XB + E$$
where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...

1
vote

0
answers

145
views

### conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...

1
vote

1
answer

334
views

### Form of minimax estimator

Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...

1
vote

0
answers

90
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

63
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

756
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

445
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

85
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

280
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

58
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

117
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

126
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

85
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

345
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

19
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

99
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

34
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

130
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

348
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

97
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

768
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

222
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

337
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

80
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

70
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

124
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

121
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

138
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

258
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

71
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

95
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

153
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

89
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

129
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

656
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

185
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

133
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

425
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

111
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

238
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

53
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

140
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

378
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

449
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

167
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(\...