Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies.

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3
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2answers
283 views

Probability distribution for two-state system that depends on residence time

I am a statistical physicist, and I've come across a problem that I don't know how to solve. I believe my issue lies with how to formulate it mathematically. I'd be very grateful for any assistance, ...
0
votes
3answers
411 views

Probability that one RV will exceed many others

Assume the $1 \times N$ vector $\mathbf X = [X_1, X_2, \ldots , X_N]$ contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume another random variable ...
1
vote
0answers
115 views

Placing Bounds on Correlation/Covariance Through Correlation with an Intermediate Variable

I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is ...
1
vote
1answer
120 views

Transition time in finite voter model

I believe the following problem is related to something called the "voter model" in statistics. This is not my area of expertise so please forgive me if the answers turn out to be well known. ...
1
vote
1answer
120 views

ROC curve with repeated measures

Hi, I have some repeated measures data, one measurement a day for three days in a row, and the measured variable looks normally distributed. I have two groups, the "really ill" and the "not ill after ...
3
votes
1answer
183 views

Exact sampling from 2D Ising model where coupling is constant?

What progress has been made towards sampling from the 2D lattice Ising model with the following Hamiltonian: $H=-J\sum_{\langle i,j \rangle}S_iS_j - \sum_i b_iS_i$ Where the first sum runs over all ...
2
votes
1answer
366 views

Expectation of the trace of an inverse of a random matrix

Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. ...
2
votes
1answer
228 views

The first eigenvalue of a branching process matrix

Let $M$ be the real square matrix of a typed branching process, such that $M_{ij}$ is the expected value of offspring of type $j$ emanating from type $i$. We know that if the first eigenvalue if $M$ ...
0
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1answer
52 views

On Variance Break detection

Say that the variance of a constant mean scalar stochastic process can take finite number of values. The problem is to detect the the point of break in variance as observation data comes in. I tried ...
2
votes
1answer
215 views

Estimator for sum of independent and identically distributed (iid) variables

This interesting question was asked at http://math.stackexchange.com/questions/231455/estimator-for-sum-of-independent-and-identically-distributed-iid-variables a while ago but got no answers. The ...
3
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0answers
92 views

A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
0
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0answers
165 views

Spectral densities and their corresponding covariance functions.

Hey guys, I'm currently doing a course in stochastic processes and have come across something that has been wrecking my mind for a while. So, let's say that I have some even, symmetric function ...
3
votes
1answer
417 views

The average number of people that can sit on a bench of a given length.

Let me explain what I mean: The width of the average person varies, perhaps with a normal distribution. Given a specific variance, how many people (on average) can sit side-by-side on a bench of a ...
0
votes
1answer
303 views

Expected value with a kronecker product and Gaussian distributional assumption

What is the expected value, $ \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right]$ where $Z \sim N(0, \sigma^2I) $? The kronecker product is where the confusion ...
0
votes
1answer
158 views

How many ways we know to join two line segments with a smooth transitional function?

This topic was created to discuss how many ways we know to create piecewise linear functions with smooth transitions between the phases. An alternative is presents by Bacon & Watts (1971): the ...
0
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1answer
378 views

Has the controversy about *fiducial distribution* been settled? [closed]

Has the controversy about the correct meaning of Fisher's notion fiducial distribution meanwhile been settled? And are there newer applications than quoted in the following literature? G.P. Klimov: ...
9
votes
3answers
437 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) \, ...
15
votes
1answer
429 views

The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
0
votes
1answer
129 views

Expression for the square of the correlation of two Gaussian variables as an expectation value

Dear all, I might just be blind, so forgive me if it is a trivial question. Given two normally distributed variables $x_1$ and $x_2$ (with zero mean), their correlation $c$ can be estimated from the ...
2
votes
1answer
104 views

How to calculate average lifespan of a new population?

What's the best estimate one can make about the average lifespan of a new population? For instance, let's say an alien kind of life came to earth and we're able to breed then. We then get 1000 aliens ...
3
votes
1answer
171 views

Is the Binomial Expectation of a Multivariate Convex Function Convex in the Vector p?

Let $\mathbf{p}=(p_1,\dots,p_m)$ be a vector in $[0,1]^m$ and let $\mathbf{X}=(X_1,\dots,X_m)$ be a vector of independently-distributed binomial random variables such that $X_i\sim ...
0
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0answers
89 views

Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following: $$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...
5
votes
3answers
468 views

Concentration of sum of pairwise squared Euclidean distances of random vectors

Let $X_1, \ldots, X_n $ be independent random vectors in $B(0, D) \subset R^d$ ($\ell_2$ ball of radius $D$ centered at the origin). I am trying to find the concentration of the following quantity ...
2
votes
0answers
180 views

Is connected correlation/cumulant expansion additive?

Say X is a free field or a Gaussian random variable. Then I want to analyse the connected correlation, $<(X + a (X^2 - \langle X^2 \rangle))^n>_c$ I think that for $n \geq 4$ there are no ...
2
votes
1answer
132 views

Optimization problem

I'm trying to solve a very practical optimization problem and I think I hit a dead-end. There are $N$ products ($N \sim 50$). Each product can have a price $p_i$ in range between 1 and 40 dollars. ...
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1answer
147 views

Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
4
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2answers
313 views

Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation: $$\mathbb{E}[(c+e^X)^{-n}]$$ where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...
2
votes
1answer
302 views

Probability Density Optimization

I am working on an optimization problem which I am stuck on towards the end. Essentially, I have two probability density functions in $\mathbb{R}^2$, call them $q(x,y)$ and $p(x,y)$, now I define ...
1
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1answer
78 views

Estimate which random variable has highest expectation

Problem: You are given a sample of size $m$ from $n$ independent normally distributed random variables. Expectations and standard deviations of the random variables are unknown. Estimate, which ...
3
votes
1answer
124 views

Statistical properties of principal components and their convergence rates.

Hello everyone, I'm interested in doing statistical tests on properties of principal components, but none of the literature I've found so far seems quite right for my purposes. Many articles present ...
5
votes
2answers
421 views

Is the Binomial Expectation of Convex Function Convex in p?

Suppose $X$ has a binomial distribution with success probability $p$ and $n$ trials and let $h(\cdot)$ be a positive convex real-valued function. Is the function $g(p)=\mathbb{E}[h(X)\ |\ p]$ convex ...
0
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0answers
248 views

Markov transition probabilities and negative binomial distribution.

A realization of a Markov process generates a sequence of interval lengths between transition from one state to another. A natural way of modeling the distribution of the lengths is as a negative ...
8
votes
1answer
611 views

What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
1
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0answers
133 views

Random walk conditioned on sum and last step

Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$. ...
8
votes
0answers
254 views

Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
6
votes
2answers
363 views

Most inconsistent ranking

A matrix of $k$ rows and $n$ columns is filled with the numbers $1,2,\ldots,k$ such that the following conditions are satisfied: Every column contain all the numbers form 1 to $k$ without ...
0
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0answers
182 views

Expectation of sample variance

Hi all, Just a quick question - I want to make sure I'm not missing anything obvious here! I'm trying to evaluate $E(S^2 \mid \bar{X} = \bar{x})$, where $X_1,\ldots,X_n$ are i.i.d. Normal($\mu, \sigma ...
2
votes
1answer
266 views

Derivative of a random process

Consider $w(t)$ as Guassian random process, with $w(t)$ being $\mathcal{N}(\mu,\sigma)$ and i.i.d for all t. I consider applying a (stochastic)derivative operation to the random process. What is the ...
2
votes
0answers
186 views

What machine learning algorithm is appropriate for predicting one time-series from another?

I have eye-tracking data on two subjects -- a teacher, and a student. It's in the form (x, y, time), so there is a series of these for each subject. What the teacher looks at influences what the ...
2
votes
0answers
42 views

probability particle paths .two sets of parameters

Consider a paticle is going from a to b At the first part of its journey is one set of mean and standard deviation for a normal distribution. Then it keeps going contingent upon a second normal ...
1
vote
0answers
73 views

calculating how much to oversell given an acceptable risk (statistics)

I have a shared resource with a finite capacity (let's say 100), and I have usage data (2 hours average of samples taken each 20 seconds). I accept a risk of 10% per year to reach the capacity. ...
5
votes
1answer
131 views

Do there exist (almost surely) $C^{\infty}$-smooth Gaussian random fields?

Let $d \ge 1$. Do there exist Gaussian random fields on $\mathbb R^d$ which are (almost surely) $C^{\infty}$-smooth, but which are not analytic? If so, what are necessary and sufficient conditions ...
1
vote
0answers
131 views

CIMDO Methodology

Hi guys, I'm doing a quantitative finance research project (which includes quite a fair bit of probability and statistics) that is based on the following papers: "Systemic Risk and Sovereign Debt in ...
1
vote
2answers
266 views

Uniform law of large numbers for martingale difference

Let $\xi_{tn}(\theta),t=1,\dots,n$ be a real-valued martingale difference array indexed by a parameter $\theta \in \Theta \subset R$, where the set $\Theta$ is compact. Now, for all fixed $\theta \in ...
0
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2answers
191 views

Are all variables in a set of random variables independent if all pairs are independent?

If I have a sequence of random variables $X_1, X_2, \ldots, X_n$ (possibly infinite) such that all pairwise cdf's are factorized: $$F(X_i, X_j) = F_i(X_i) F_j(X_j)$$ for all pairs $(X_i, X_j)$, does ...
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0answers
80 views

Markov renewal process with failure?

I hope this question is not too elementary for this site, and that it contains a sufficient degree of detail. I have a problem where I want to model sequences of variable length $\boldsymbol{e}_i = ...
2
votes
0answers
196 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
5
votes
0answers
152 views

Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
0
votes
1answer
74 views

Taking the partial derivative of the t-CDF with respect to the degrees of freedom

I am trying to find the maximum likelihood estimate of the parameters for the t-copula. Ideally I'd want to use a gradient-based method for optimization. However, I am having some difficulty in ...
2
votes
1answer
144 views

Moments of random matrices - when are they finite

I need to evaluate the moment $$\mathbb{E} (AX)^n,$$ where A is an NxN Hermitian square matrix, and X is $$X=ZZ^{\ast},$$ where $Z=\mu+Y$, where $\mu$ is mean of $Z$ and $Y$ is a zero-mean complex ...