**1**

vote

**1**answer

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

**1**answer

117 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

**2**answers

384 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**

votes

**0**answers

217 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

**1**answer

552 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**

vote

**0**answers

128 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

**0**answers

250 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

**2**answers

359 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**

votes

**0**answers

179 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 ...

**1**

vote

**1**answer

238 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 ...

**1**

vote

**0**answers

184 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 ...

**1**

vote

**0**answers

41 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

**0**answers

72 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

**1**answer

128 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

**0**answers

125 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

**2**answers

241 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**

votes

**2**answers

190 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 ...

**0**

votes

**0**answers

78 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

**0**answers

180 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

**0**answers

149 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

**1**answer

69 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

**1**answer

143 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 ...

**2**

votes

**0**answers

163 views

### Does Multiplicative Version of Azuma's Inequality Hold?

It is known that there are multiplicative version concentration inequalities for
sums of independent random variables. For example, the following
multiplicative version Chernoff bound.
Chernoff ...

**3**

votes

**1**answer

489 views

### Expectation of random matrix inverse

Given a $K\times M$ matrix $X$, where $M\gg K$, comprising independent complex Gaussian random variables, each one with mean
$$E[X_{k,m}]=B_{k,m}$$
and variance
$$Var[X_{k,m}]=\Sigma_{k,m}$$
define ...

**2**

votes

**0**answers

68 views

### “Soft” Voronoi cells or statistical criterias

It is probably some basic statistics question, but...
Informally 1: How to choose "criteria", such that it will guarantee that error decision probability is less than "epsilon", and maximize ...

**4**

votes

**2**answers

657 views

### Interesting thesis topic on statistical inference that is sufficiently mathematical

Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on ...

**2**

votes

**1**answer

622 views

### Which clustering algorithm could I use to group 2D points that are arranged over a time series? [closed]

I am a software developer struggling to understand which clustering method/algorithm would be most appropriate to spatially group 2-dimensional point data (x,y) that is arranged over a time series (an ...

**0**

votes

**2**answers

76 views

### What is the likehood function in the noise free observation case

In the nonlinear Bayesian Tracking problem, if we consider the noise exists only in the state equation : x[k] = f(x[k-1],v[k-1]) where vk-1 here is an iid process noise sequence
And we suppose that ...

**8**

votes

**2**answers

337 views

### Rescaling positive definite matrices to force a unit eigenvector

Hello,
Let $X'X$ be a positive definite matrix and let $\mathbf{1}$ denote the vector of ones.
I'm hoping to construct a positive, diagonal matrix $W$ such that
$$(W X'X W) \mathbf{1} = ...

**4**

votes

**1**answer

413 views

### Prove an inequality related to moments

I am reading a paper and stuck with an inequality used in that paper.
$\varepsilon^n=(\varepsilon_1^n, \varepsilon_2^n,\ldots,\varepsilon_n^n)^T$ is a vector of i.i.d. random variables with mean 0 ...

**2**

votes

**1**answer

101 views

### Role of statistical estimation in formal proof

Consider the following scenario: There is some mathematical constant $c$ that you want to compute. You don't have a formal proof for any particular value of $c$, but you have some sound statistical ...

**11**

votes

**1**answer

349 views

### Applications of the Giry monad in probability and statistics

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...

**6**

votes

**2**answers

398 views

### When is a space of measures a measurable space?

Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...

**0**

votes

**0**answers

405 views

### Gradient Descent for Primal Kernel SVM with Soft-Margin(Hinge) Loss

Given the primal objective
$$F({\bf a})=L\sum_{i,j}a_{i}a_{j}k(x_i,x_j) + \sum_{i}max(0, 1-y_i \sum_{j}a_jk(x_i,x_j)$$
for the soft margin SVM, where ${\bf a}=(a_1,...,a_N)$, N being the number of ...

**1**

vote

**0**answers

54 views

### Standard Errors for Two-Step MLE Procedure for Computationally Intensive Likelihood Functions

Suppose we have a likelihood function, $L(\theta_{1},\theta_{2},\theta_{3};X_{1},X_{2})$
where $\theta_{1}...\theta_{3}$ are sets of parameters and $X_{1}$
and $X_{2}$ are data. The model is fully ...

**6**

votes

**1**answer

731 views

### What's the maximum entropy probability distribution given bounds [a,b] and mean?

What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?
For example:
if a=0, b=1, ...

**6**

votes

**2**answers

394 views

### Strange pattern in rounding errors?

This will look at first like a posting about trigonometry, then maybe about statistics, then finally about peculiarities of either
a certain random process; or
the pseudorandom number generator that ...

**0**

votes

**0**answers

103 views

### Sampling without replacement: probability for total successes from successes in sample?

Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...

**14**

votes

**2**answers

541 views

### Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...

**0**

votes

**2**answers

240 views

### Creating composite rank [closed]

Problem: Suppose that $K$ different students are ranked based on $N$ different parameters (such as Physics marks, English marks, Biology marks, IQ etc). The rank under each parameter can be repetitive ...

**4**

votes

**1**answer

177 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 ...

**0**

votes

**0**answers

40 views

### Using symmetries of a r.v.'s distribution to boost samples and possibly do variance reduction

Suppose, for example, you are simulating samples from a (multivariate) Gaussian with mean zero and covariance $\Gamma=BB^T$. If you had generated a sample $x$, you could generate more (dependent) ...

**2**

votes

**4**answers

276 views

### Estimate number of distinct items

This question is very similar to this unanswered one http://math.stackexchange.com/questions/242607/estimate-number-of-distinct-items .
Suppose I have a large array of $n$ integers and I want to ...

**3**

votes

**1**answer

222 views

### convex combination of two covariance estimates

I am interested in covaraince matrix estimation. In brief: I have two estimates of the covariance matrix, and now I want to form a bona fide convex combination of the two.
Background: I have studied ...

**3**

votes

**1**answer

249 views

### Deciding whether or not an exponentially distributed random variable exists in a set via the use of a “black box” function

I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ...

**0**

votes

**0**answers

145 views

### Singular Value Decomposition Question

Hi: I'm new to this list so apologies if I do anything wrong with my question.
Suppose I have a matrix $Y$ whose SVD can be decomposed as
$Y = U_{0}\Sigma_{0}V_{0}^{*} + U_{1}\Sigma_{1} ...

**1**

vote

**2**answers

512 views

### Choice of Lipschitz constant for proximal gradient optimization

I'm trying to use proximal gradient methods (Forward-Backwards Splitting and FISTA) to minimize a function
$f(B) = \frac{1}{2}|| XB - Y||_F^2 + \frac{\gamma}{2}||B C^T||_F^2$, where $X \in ...

**0**

votes

**0**answers

97 views

### power law distribution of time events

Suppose you have the logs of a web server. In these logs you have tuples of this kind:
user1, timestamp1
user1, timestamp2
user1, timestamp3
user2, timestamp4
user1, timestamp5
...
These ...

**2**

votes

**2**answers

615 views

### Singular Value Decomposition of Noisy Matrices

I am an engineer who makes measurements of a variable over a grid
of, say, $m\times n$. Since these are actual measurements, the true
values are always corrupted by noise, and what I measure is a ...

**3**

votes

**0**answers

126 views

### Find a minimum entropy code for a simple gibbs random field.

Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about.
In the rest of this ...