2
votes
3answers
114 views

Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
5
votes
1answer
259 views

Is this inverted integral transform valid?

I have the following transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ with the following conditions: $f(x)$ and $F(y)$ must ...
1
vote
0answers
83 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
7
votes
0answers
441 views

American put option pricing by “binomial trees”

Dear MO World, I'm teaching a financial mathematics course and have found a fascinating (to me) numerical phenomenon and wonder if anyone has studied it, or knows anything similar. I'll try and give ...
9
votes
3answers
442 views

Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form $$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...
2
votes
2answers
243 views

Computing hypergeometric function of matrix argument

In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...
6
votes
1answer
150 views

Algorithm for numerically approximating the Prokhorov metric?

Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures? Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
3
votes
1answer
107 views

exact simulation of a large sample histogram

Say I want to create a histogram of $N$ samples from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins, ...
1
vote
0answers
296 views

Monte Carlo sampling high dimensions with the halton sequence?

Referring to the Halton Sequence, Swiler et al 2006 state that In cases where a large number of input variables are sampled, Robinson and Atcitty recommend using a leaped sequence, where the ...
5
votes
3answers
420 views

Traceless GUE : Four Centered Fermions

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel \[ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} ...
4
votes
3answers
606 views

Quanitative de Moivre–Laplace theorem (reference request)

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution: $${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi ...
5
votes
1answer
361 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
0
votes
0answers
276 views

Estimating a multinomial sum

I have the following sum \begin{equation} \sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda} ...
2
votes
1answer
292 views

Numeric problem when evaluating log of a pdf

In maximum likelihood estimation, one typically needs to compute the log (natural log) of probability values. When a probability, say $p(x)$, becomes so close to zero, $log(p(x))$ returns -Inf. What ...
3
votes
5answers
537 views

Numerical Solution to Inverse Integral (Pseudo Random Number Generation)

If I have an arbitrary positive monotonically decreasing function $f(x), x \in [0,\infty]$, is there an 'efficient' method for finding $y$ in: $r = \int\limits_0^y f(x) dx $ for a known $r \in [0, ...
6
votes
1answer
686 views

Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs

Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
-2
votes
1answer
698 views

Determine noise distribution [closed]

I'm trying to solve the following least squares problem: $\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$ where $Ax = b$ and $\tilde{b} = b + w$ Question: How do I determine which probability ...
1
vote
1answer
302 views

Distribution on permutations derived from probability of pairwise orderings

A followup question to Probability estimates for pairwise majority votes - I think it doesn't actually give an answer in any terribly precise sense, but it would give something I'd be happy to use in ...
10
votes
0answers
651 views

Constructive aspects of Caratheodory's theorem in convex analysis

Let me paraphrase Caratheodory's theorem in a probabilistic setup: Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
7
votes
1answer
498 views

Sampling from Sine Kernel and Airy Kernel

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples ...