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8 votes
2 answers
330 views

q-Means and the mode of a distribution

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that \begin{equation} \int_{\mathbb{R}} |x| f(x)\, dx < \infty, \end{equation} and ...
Maurizio Barbato's user avatar
4 votes
1 answer
213 views

Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution. When I measure the convergence rate ...
Anton's user avatar
  • 101
4 votes
0 answers
75 views

Marginalization of Wishart distribution

Consider the following Wishart distribution $$ f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1} $...
RenatoRenatoRenato's user avatar
3 votes
2 answers
168 views

A definite integral related to sample variances of bivariate Gaussians

This integral is needed to obtain the joint distribution of the sample variances of a random sample from a bivariate Gaussian distribution. For details on the joint distribution of the sample means, ...
Chee's user avatar
  • 984
3 votes
1 answer
942 views

Cumulative integral of the Marchenko-Pastur density for Wishart eigenvalues

I can't find any work on the cumulative density function of the well-known Marchenko-Pastur density for the eigenvalues of a standard Wishart Matrix as its dimension goes to $\infty$, i.e. $$F(\beta)=...
blacklist's user avatar
3 votes
0 answers
286 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
Econ's user avatar
  • 31
2 votes
1 answer
636 views

Sufficient condition for function of conditional probability density to be increasing

Let $Y$ and $W$ be two jointly distributed random variables; $Y$ takes values on $(y_1,y_2)$ and $W$ takes values on $(w_1,w_2)$. The conditional probability density of $W$ given $Y$ is given by $f_{W|...
Ararat's user avatar
  • 143
2 votes
1 answer
122 views

Analytical solution for a double integral involving logistic functions and Gaussian distributions

I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...
Charles's user avatar
  • 31
2 votes
0 answers
341 views

Marginalizing multivariate normal over defined interval

Hello everyone, I am trying to obtain an analytic expression for the following Gaussian integral $$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
amanoel's user avatar
  • 21
1 vote
1 answer
613 views

Integral of the product of a gaussian pdf and cdf

I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. ...
Kurt Z.'s user avatar
  • 11
1 vote
1 answer
147 views

Proving that an integral related to order statistics is increasing in a certain parameter

Let $f$ and $F$ denote, respectively, the pdf and cdf of a probability distribution on $\mathbb R$. Take any natural $n\ge3$ and any real $a$ and $c$ such that $a\le c$. Does it always follow that $$...
carlogambino's user avatar
1 vote
0 answers
251 views

Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
Anton's user avatar
  • 101
1 vote
0 answers
100 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 ...
Anton's user avatar
  • 101