Questions tagged [estimation-theory]

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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/...
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15 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{...
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14 views

sample complexity of hypothesis testing with non-uniform prior

Given two hypothesis $$\mathcal{H}_0:\; x_i\underset{iid}{\sim} p_0(x), i=1,\cdots,n\\ \mathcal{H}_1:\; x_i\underset{iid}{\sim} p_1(x), i=1,\cdots,n$$ with priors $p$ and $1-p$ ($p<1/2$) ...
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33 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 ...
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  • 101
1 vote
0 answers
27 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 ...
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0 answers
123 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)=...
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2 votes
1 answer
113 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) ...
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1 vote
1 answer
76 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 $X,Y$....
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  • 608
1 vote
1 answer
85 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 ...
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  • 608
2 votes
0 answers
178 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;...
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  • 101
4 votes
1 answer
305 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}^\...
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19 views

Explicit bound computation for Kalman filter covariance $\|P_{k|k}\|$ as a function of $\|P_0\|$

My Goal: Given and initial covariance $P_0$, a Kalman filter updates covariance $P_{k|k}$ according to a nonlinear update equation. I am looking for a bound $B=B(\|P_0\|)$ such that $\|P_{k|k}\|\leq ...
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1 vote
0 answers
69 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 ...
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1 vote
0 answers
35 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 ...
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2 votes
0 answers
55 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 $\mu$, ...
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  • 173
0 votes
1 answer
102 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 ...
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2 votes
0 answers
131 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);...
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2 votes
1 answer
80 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 ...
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1 vote
0 answers
54 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 ...
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1 vote
0 answers
49 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 ...
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2 votes
1 answer
133 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 ...
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2 votes
0 answers
55 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 ...
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3 votes
1 answer
114 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,\...
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6 votes
3 answers
502 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 ...
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-1 votes
1 answer
76 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 ...
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1 vote
0 answers
52 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 $y=Bx+ η$, ...
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6 votes
1 answer
346 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 ...
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3 votes
0 answers
105 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,...
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3 votes
0 answers
113 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^{...
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  • 1,025
1 vote
0 answers
35 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,\; ...
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-1 votes
1 answer
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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 ...
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2 votes
0 answers
263 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. ...
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1 vote
0 answers
57 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 ...
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  • 91
-2 votes
2 answers
242 views

Lower bound of q pochhammer symbol [closed]

How one could prove, that q pochhammer symbol $(1,1/n) = \prod_{k = 1}^{\infty}(1-\frac{1}{n^k}) \geq 1 - \frac{1}{n-1}$
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0 votes
1 answer
111 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(\...
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5 votes
1 answer
85 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 ...
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-2 votes
1 answer
84 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 $y \sim \mathcal{N}(0,1)$.
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0 votes
1 answer
48 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 ...
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3 votes
1 answer
77 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", ...
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4 votes
0 answers
472 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, ...
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2 votes
1 answer
148 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,
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2 votes
1 answer
450 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 ...
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3 votes
1 answer
157 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, ...
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3 votes
1 answer
105 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) ...
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  • 482
2 votes
2 answers
472 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 ...
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  • 599
1 vote
0 answers
88 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 = ...
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2 votes
0 answers
51 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(\...
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  • 121
1 vote
1 answer
140 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: $$...
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  • 237
1 vote
0 answers
90 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 ...
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  • 11
1 vote
1 answer
120 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 ...
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