10 votes
Accepted

Probability of complex eigenvalues

The probability that a $n\times n$ real matrix (with elements that are independent random variables with standard normal distributions) has only real eigenvalues is given by $$ 2^{-n(n-1)/4}$$ ...
Marcel's user avatar
  • 2,510
9 votes

How to estimate the integral involving the distance function

For small $t$ the dominant contribution comes from the boundary; starting from a point $y_0$ on the boundary and integrating inwards in a perpendicular direction we have $\int_{0}^\infty e^{-r^2/t}dr=\...
Carlo Beenakker's user avatar
8 votes
Accepted

How to estimate the integral involving the distance function

The estimate you seek is reminiscent of H. Weyl's tube formula. I will give you some pointers referring for more details to section 9.3.5. of these lectures. Denote by $r$ the distance to $\newcommand{...
Liviu Nicolaescu's user avatar
6 votes
Accepted

Can samples be compressed?

No, the inequality is not guaranteed. One example with $k=2$ is for $g(x_1,x_2) = OR(x_1,x_2)$. Then $f_g(1;\theta) = 1-(1-\theta)^2$, and $\mathcal{I}_g(\theta) = 4/(2\theta-\theta^2)$. We have $\...
Yoav Kallus's user avatar
  • 5,926
5 votes
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Generalization of van der Corput's estimate on oscillatory integrals

For convenience of notation write $\Phi(x) = \lambda_1 x^\alpha + \lambda_2 x^\beta$. The general argument is based on the integration by parts argument $$ \int_a^b e\circ\Phi = \int_a^b \frac{1}{2\pi ...
Willie Wong's user avatar
  • 37.4k
4 votes

An alternative proof of Bayesian Cramer-Rao

In line with Deane's comment, this is an "answer" that also uses the Cauchy-Schwarz inequality but does so in a way that you might find more natural. I'll use different notation than yours (sorry; ...
Tom Leinster's user avatar
  • 27.2k
4 votes

Using Fisher Information to bound KL divergence

I'm not sure if this is still of interest to you, but I think it is possible to get some reasonable bounds if you are okay with dropping the factor of $\frac{1}{2}$. Here's my work, which can be ...
Gabe K's user avatar
  • 5,364
4 votes
Accepted

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

Let $n:=M$. Since you say "the denominator follows a Chi-square distribution with $2M$ degrees of freedom", it appears that you assume the $x_i$'s to be iid standard complex Gaussian; anyway, if you ...
Iosif Pinelis's user avatar
4 votes
Accepted

Proving bounds on analytic functions using only the Taylor expansion

There is no such general method. You cannot see directly from the Taylor series that $\sin x$ is bounded on the real line, or that $\exp z$ is bounded on the negative ray. Of course what I stated is ...
Alexandre Eremenko's user avatar
4 votes
Accepted

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

IEEE transactions on Information Theory comes to mind. Their specifications of topics is broad. Fisher information regularly appears in papers there. The IEEE Transactions on Information Theory is a ...
kodlu's user avatar
  • 10.1k
4 votes
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Form of minimax estimator

$\newcommand\P{\mathcal P}\newcommand\N{\mathbb N}\newcommand\de{\delta}$You wrote: Hence I additionally assume that $\mathcal{P}$ is permutation-invariant, in which case I conjecture that all of the ...
Iosif Pinelis's user avatar
3 votes

How to estimate the integral involving the distance function

\begin{aligned} & \int_{0}^{+\infty} e^{-\frac{s^{2}}{t}} \mathcal{L}_{n-1}(\{x \in \Omega \mid d(x, \partial \Omega)=s\}) d s \\ =& \int_{0}^{+\infty} e^{-\frac{s^{2}}{t}} \mathcal{L}_{n-1}(\...
katago's user avatar
  • 543
3 votes
Accepted

Lower bound of q pochhammer symbol

$$\prod (1-x_i)=1-x_1-x_2(1-x_1)-x_3(1-x_1)(1-x_2)-\ldots \geqslant 1-\sum x_i, \forall x_i\in [0,1],$$ use this for $x_i=1/n^i$.
Fedor Petrov's user avatar
3 votes
Accepted

Estimating the average of two gaussians' mean

The maximum likelihood estimator (MLE) for $(\mu_1,\mu_2)$ is $(X,Y)$. So, by the functional invariance of the MLE (that is, simply by definition), the MLE of $g(\mu_1,\mu_2):=(\mu_1+\mu_2)/2$ is $g(X,...
Iosif Pinelis's user avatar
3 votes
Accepted

Estimating the size of the remainder in a random partition

Here is a heuristic that I am sure can be made rigorous. The $x_i$'s can be written recursively as $x_{i+1}=U_{i+1}x_i$, where the $U_i$ are independent Unif[0,1] random variables. In particular, $x_n=...
Anthony Quas's user avatar
  • 22.5k
3 votes

Is the min function ever an unbiased estimator for the mean?

Suppose $X_1,\ldots,X_n \sim \text{i.i.d.} \operatorname{Uniform}(\theta,0),$ where $\theta$ could be any negative number. Then $\min\{X_1,\ldots,X_n\}\cdot (n+1)/(2n)$ is a better unbiased estimator ...
Michael Hardy's user avatar
2 votes

Moments of Matrix Gamma distribution

A good source on this topic is chapter 7 of Roob Muirhead's book, "Aspects of Multivariate Statistical Theory", or chapter 3 of Peter Forrester's "Log-Gases and Random Matrices". In particular, there ...
Marcel's user avatar
  • 2,510
2 votes

Moments of Matrix Gamma distribution

The matrix Gamma distribution (as defined in Wikipedia) $MG_p(\alpha,\beta,\Sigma)$ is nothing but the Wishart distribution $W_p\left(2\alpha, \frac{\beta}{2}\Sigma\right)$. The moments and the ...
Stéphane Laurent's user avatar
2 votes

Reconstructing the number of distinct elements from a random projection

Note first, that we may assume $n=D$ and all elements of the sequence distinct. To obtain a lower bound, we can start by considering the expectation. Note that we can write $Z = \sum_{i=1}^{k} Z_i$, ...
assaferan's user avatar
  • 736
2 votes

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

I interpret the question as follows: Given are two pdf's, $p_0$ and $p_1$, on $\mathbb{R}^d$ and a function $f\colon \mathbb{R}^d \to \mathbb{R}^k$. For any $x\in\mathbb{R}^d$, let \begin{...
Iosif Pinelis's user avatar
2 votes
Accepted

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

Let $$s_q(N):=\sum_{n=0}^N f_q(n),\quad t_q(N):=\sum_{n=1}^N g_q(n),$$ where $$f_q(n):=(4n+1)q^{(4n+1)^2/4},\quad g_q(n):=(4n-1)q^{(4n-1)^2/4}.$$ We want to show that $$s_q(\infty)-t_q(\infty)\overset{...
Iosif Pinelis's user avatar
2 votes
Accepted

How to detect, track and map a Markov chain

For a given state $i$, row $i$ of the transition matrix gives the transition probabilities $P_{ij}$ from $i$ to $j$, $j=1..n$ (the number of states). This is a probability distribution, and the ...
Robert Israel's user avatar
2 votes
Accepted

DKW inequality for $L^1$-norm

The exponent 2 on $\epsilon$ cannot be improved by passing to the $L^1$ norm. Consider $X_i$ i.i.d. uniform in $[0,1]$. Let $C$ be a constant, and denote $\varepsilon_n= C/\sqrt{n}$. Let $A_n $ be the ...
Yuval Peres's user avatar
2 votes
Accepted

Singular Fisher information matrix and existence of unbiased estimators

$\newcommand\th\theta\newcommand\ol\overline$Responsding to your comments: "About the collinear sensors in my question: I understand that for $y\ne0$, the model is non-identifiable. But, what I ...
Iosif Pinelis's user avatar
2 votes
Accepted

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

$\newcommand\th\theta\newcommand\si\sigma\newcommand\p\partial\newcommand\ol\overline$There is no reason to get confused here. Indeed, that "the first term of the derivative does not depend on $(...
Iosif Pinelis's user avatar
1 vote
Accepted

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

$\newcommand\si\sigma$Clearly, the best estimator of $\mu_1$ is $X$, no matter what $\si_1$ and $\si_2$ are. Similarly, the best estimator of $\mu_2$ is $Y$, no matter what $\si_1$ and $\si_2$ are. So,...
Iosif Pinelis's user avatar
1 vote
Accepted

Spline Interpolation error of higher degree

The following paper by de Boor suggests that this is the case, although he develops the proof only up to degree 6 splines. de Boor, C., On the convergence of odd-degree spline interpolation, J. ...
Iddo Hanniel's user avatar
1 vote

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

Perhaps the simplest approach is to consider the problem as the calculation of a functional derivative. Let's put $$ (a_1, \ldots, a_n, b_1, \ldots, b_n)\triangleq (\mathbf{a}, \mathbf{b})\in\Bbb C^{...
Daniele Tampieri's user avatar
1 vote

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

It can be obtained that the MMSE of $Y$ is given by $\hat{Y}=\frac{\sigma_y}{\sigma_y+\sigma_v}Z=kZ$, and the covariance of $\hat{Y}$ is $k^2(\sigma_y+\sigma_v)$. Now suppose $X=t\hat{Y}$, to ensure $...
Jing Zhou's user avatar

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