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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2
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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 ...
2
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0answers
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
1answer
529 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
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0answers
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
2answers
681 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
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1answer
682 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
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2answers
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
2answers
358 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
1answer
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
1answer
105 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
1answer
387 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
2answers
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 ...
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0answers
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
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0answers
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
1answer
792 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
2answers
415 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
0answers
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
2answers
589 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
2answers
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
1answer
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
0answers
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
4answers
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
1answer
237 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
1answer
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
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0answers
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
1answer
603 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
0answers
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
2answers
676 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
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0answers
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
1answer
143 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
2answers
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
1answer
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
1answer
193 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
2answers
361 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
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0answers
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
1answer
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 ...
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2answers
225 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
2answers
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
1answer
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
1answer
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
3answers
368 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
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0answers
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
0answers
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
1answer
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 ...
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0answers
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
3answers
488 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
0answers
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
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0answers
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
4answers
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
2answers
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 ...