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
$$ 2^{-n(n-1)/4}$$
...
- 2,552
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=\...
- 188k
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{...
- 34.7k
5
votes
Accepted
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 ...
- 39k
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; ...
- 27.7k
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 ...
- 6,001
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 ...
- 128k
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 ...
- 91.8k
4
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,...
- 128k
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 ...
- 10.4k
4
votes
Accepted
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 ...
- 128k
4
votes
Accepted
Almost sure convergence of double averages of IID random variables
Let us give a name to the partial sums
$$
S_{P,Q}(f)=\frac 1{PQ}\sum_{i=1}^P\sum_{j=1}^Q f(X_i,Y_j).
$$
and define the functions
$$
f_1\colon x\mapsto \mathbb E\left[f(x,Y_1)\right]-\mu, \quad, f_2\...
- 3,958
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}(\...
- 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 $x_i=1/n^i$.
- 109k
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=...
- 23.2k
3
votes
Accepted
How to study the convergence of the sample mode for arbitrary probability spaces
Concerning strategy 2: The set $2^S$ of all subsets of the finite set $S$ of the $K$ letters is finite. So, the only (say) Hausdorff topology over $2^S$ is the discrete one. Moreover, we have $\hat\...
- 128k
3
votes
Accepted
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
$\newcommand\si\sigma$The inequality in question fails to hold if e.g. $\mathcal N=[n]:=\{1,\dots,n\}$, the $h_j$'s are i.i.d. exponential random variables, and $n=4$. This follows because then
$$E\...
- 128k
2
votes
Moments of Matrix Gamma distribution
A good source on this topic is chapter 7 of Roob Muirhead's book, "Aspects of Multivariate Statistical Theory", or chapter 3 of Peter Forrester's "Log-Gases and Random Matrices". In particular, there ...
- 2,552
2
votes
Moments of Matrix Gamma distribution
The matrix Gamma distribution (as defined in Wikipedia) $MG_p(\alpha,\beta,\Sigma)$ is nothing but the Wishart distribution $W_p\left(2\alpha, \frac{\beta}{2}\Sigma\right)$. The moments and the ...
- 2,560
2
votes
Reconstructing the number of distinct elements from a random projection
Note first, that we may assume $n=D$ and all elements of the sequence distinct.
To obtain a lower bound, we can start by considering the expectation.
Note that we can write $Z = \sum_{i=1}^{k} Z_i$, ...
- 736
2
votes
maximum likelihood estimation of X is better than that of f(X)?
I interpret the question as follows:
Given are two pdf's, $p_0$ and $p_1$, on $\mathbb{R}^d$ and a function $f\colon \mathbb{R}^d \to \mathbb{R}^k$. For any $x\in\mathbb{R}^d$, let
\begin{...
- 128k
2
votes
Accepted
Showing that $\sum_{n=0}^\infty (4n+1)q^{\left (\frac{4n+1}{2}\right)^2} - \sum_{n=1}^\infty (4n-1)q^{\left (\frac{4n-1}{2}\right)^2} \geq 0.1$
Let
$$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{...
- 128k
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 ...
- 54.2k
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 ...
- 14.2k
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 ...
- 128k
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 $(...
- 128k
2
votes
Accepted
Calderón–Zygmund/$L^p$ estimates for the linear heat equation
This is not a complete answer.
I've been searching for this kind of reference few months ago (see this post of mine).
Here is what I have collected.
In these notes you have a multiplier approach (...
- 3,425
2
votes
Upper Bound for $\mathbb{E}\left[\max_{j \in \mathcal{N}} h_{j}\right]$
For $\{H_j\}_{j∈N}$ independent Gamma random variables as $H_j > 0$ then $max_{j\in N} H_j \leq \sum_{j\in N} H_j$. As $H$ are independent then $$\mathbb{E}[max_{j\in N} H_j] \leq \sum_{j\in N} \...
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