**2**

votes

**1**answer

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 ...

**2**

votes

**0**answers

171 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

532 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

70 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

684 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

683 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 ...

**-1**

votes

**2**answers

77 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

359 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

422 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

106 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

390 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

405 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

419 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

56 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 ...

**7**

votes

**1**answer

794 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

416 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

104 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 ...

**15**

votes

**2**answers

590 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

254 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

181 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

284 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

238 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

251 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

154 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} ...

**2**

votes

**1**answer

608 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

98 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

680 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

129 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 ...

**0**

votes

**1**answer

144 views

### Estimation of mean of the unimodal symmetric distrubution - is “sample mean” the best estimate ?

Consider probability distribution which is unimodal, symmetric and mean value exists. (Of course due to symmetry mean will coincide with mode (position of the maximum)).
Question Is sample mean ...

**11**

votes

**2**answers

477 views

### Covariance of INID order statistics [closed]

In the IID case, it is known that all order statistics are positively correlated.* Thus, we know that $$\text{Cov}(X_{(i)},X_{(j)}) \geq 0.$$ Is this known in the INID (independent, non-identically ...

**1**

vote

**1**answer

257 views

### Product of probability densities of the form x^{-t} exp (-ax)

I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, ...

**1**

vote

**1**answer

194 views

### Kalman Filter…Denoising measurement data to track objects

Hi Everyone,
I am about to implement a Kalman Filter in a software.
I found this very helpful article here:
http://bilgin.esme.org/BitsBytes/KalmanFilterforDummies.aspx
The example helps a lot, ...

**0**

votes

**2**answers

363 views

### Interpolating a “manifold” between two points

Edit: I have reworded the question.
This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional ...

**2**

votes

**0**answers

163 views

### Expectation of a multivariate Gaussian over a plane

For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :
$E[X|X^Tb = c]$
...

**1**

vote

**1**answer

205 views

### Extending Wald's equation to two classes of i.d. random variables?

I try to adopt Wald's equation to a slightly more complex problem. In fact, after a full day, I found some solution now, but it has a confusing argument in the middle. Perhaps somebody can help me at ...

**1**

vote

**2**answers

226 views

### Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...

**2**

votes

**2**answers

306 views

### Total variation distance between a Poisson and a distribution with known mean/variance

Suppose that $\mu$ is the law of a Poisson distribution of mean 1, and that $\nu$ is the law of an unknown distribution on the non-negative integers, though I do know that its mean and variance are ...

**0**

votes

**1**answer

84 views

### Can one combine (join) probabilities from 2 aspects of a related process?

Consider 2 related aspects of a process for prices in a financial market:
time &
return.
Time
Say I've identified a distribution that reasonably models the occurrence of the lengths of price ...

**0**

votes

**1**answer

315 views

### central limit theorem for binomial random variable

I'm confused about applying central limit theorem to Bernoulli random variables. Let
$X_i=\frac{n}{\sqrt{n-1}}(Z_i - \frac{1}{n})$ where $Z_i$ is iid Bernoulli($\frac{1}{n}$). Then $E[X_i]=0$ and ...

**3**

votes

**3**answers

369 views

### What is the name for a non-normalized distribution?

For some analysis work with probability distributions, I remember a common trick being to drop the "integrate to 1" requirement, so the set becomes closed under addition and is more convenient to work ...

**0**

votes

**0**answers

87 views

### Credibility theory. Negative between group variance estimate

In the Buhlman-Straub credibility model I get a relatively small (in relation to the within group variance estimate) and negative between group variance estimate.
The reason for the estimate being ...

**1**

vote

**0**answers

148 views

### Universal Correlation measure — ranking correlations

I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.
Correlation of the series changes over time and across ...

**0**

votes

**1**answer

154 views

### marginal parameter estimation in copula with copula (dependence) parameter known

I've posted this already in stats.stackexchange. I'm not sure what the rules are for cross-posting but mathoverflow seems to be more active.
Suppose we have data $x_i, i=1,2,3,...n$ that are ...

**2**

votes

**0**answers

105 views

### Quantifying the amount of structure in a data set via random matrix theory

Given a data matrix, $M \in \mathbb{R}^{n \times p}$, I am interested in methods quantifying the amount of structure in present in $M$.
I've found a few approaches, but I would like to learn more ...

**1**

vote

**3**answers

491 views

### Estimating parameters of a mixture of normal distributions.

I want to estimate the parameters $\mu_i$ and $\sigma^2_i $ of a countable mixture of Gaussians with assumed equal weights, variance and identically spaced means. I intially thought that the Fourier ...

**0**

votes

**0**answers

99 views

### What are Effective Regression Techniques for Linguistic Analysis of Linked Data?

I am in the early stages of a problem that involves parsing a large number ($\approx 5 \times 10^9$) of documents (web pages) and estimating values from them. In particular I need to identify pages ...

**1**

vote

**0**answers

104 views

### time derivative of the median of a stochastic process

Suppose you have a cumulative distribution that is changing with time, namely $ P_t(x) $. Assume $ P_t $ is monotone increasing and smooth enough so that we can define $ x_t(P) = P_t^{-1} $. We want ...

**7**

votes

**4**answers

1k views

### Recent impressive combinatorial developments in probability theory

In the preface to the second edition of Daniel Stroock's book "Probability Theory: An Analytic View", there is this striking claim (on p. xv)
... I suspect that, for at least a decade, the most ...

**2**

votes

**2**answers

5k views

### Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...