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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Correlation between spatial variables

I am trying to understand what type of statistical test I can use to check if two or more variables that vary spatially are correlated. Suppose I have data acquired inside a company building, e.g., ...
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41 views

Studying Signal Processing [on hold]

I'd Like to ask two questions : What is the difference between studying Signal processing (both Deterministic and statistical) in Department of Electrical Engineering versus Department of Mathematics ...
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24 views

Success Ranking Methodology [on hold]

I'm trying to calculate the success of a couple of students but they aren't all in sync with their attended exam count. These are the data I have right now: ...
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0answers
20 views

Mathematics of simple performance testing [migrated]

I have a set of sorted tables T that have known but different dimensions. There are two types of functions in this system: f(T) g(T, n), where n is an integer parameter. ... and two types of costs ...
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1answer
43 views

How to extend Dirichlet distribution to Dirichlet process

For a Dirichlet process, there are two parameter $\alpha$ and $H$, and the Dirichlet process $X$ is defined as $$(X(B_1),\cdots,X(B_n))\sim Dir(\alpha H(B_1),\cdots,\alpha H(B_n))$$ ...
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18 views

Merging regions of function with similar mean and deviation using statistical test

I have got a question related to statistical tests that I would like to use in a new algorithm I am developing. Given an action space $x$, the algorithm would identify the regions in the function ...
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27 views

Distribution of Wishart Sample Eigenvalues for Multiple Roots

I am interested in finding an asymptotic approximation to the latent roots $l_1>\dots>l_p$ of a white noise Wishart matrix $nS\sim W_p(n,I)$ as $n\rightarrow\infty$ (where $p$ is fixed). In ...
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1answer
52 views

Unbiased sample from a product

Let $X = (x_1,\ldots,x_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod_{i=1}^n x_i$ Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that on average considers only a ...
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2answers
270 views

Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for. I'm interested to know ...
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0answers
18 views

based on statistics- related to median,mean and mode of a grouped data [closed]

In a frequency distribution , the mode and mean are 26.6 and 28 respectively ,find out the median.
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2answers
596 views

Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
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21 views

Optimize the distribution if it is left unsmoothed

I have a question about distribution. Let see my problem The paper said that the distributions p and q are left unsmoothed, so we can ignore Kernel density. But I don't understand what is left ...
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1answer
57 views

Distance between two distribution of image

I am looking for a common distance method to compare two distribution (ex: histogram of image). Please suggest to me some common method to do it. I found some method ex: Bhattacharyya distance , K-L ...
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2answers
228 views

Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line? $x_0 = n$ $x_t$ is a uniformly random integer between 1 and ...
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88 views

derivative of log determinant of the product of matrices

I'm dealing with a multivariate gaussian ($n$-dimensional) with a particular parametrization. In detail, mean is equal to $AB\Theta$, where $A$ is an $n\times r$ matrix, $B$ is an $r\times r$ matrix, ...
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1answer
24 views

Multivariate Rayleigh [closed]

What is the closed form formula (pdf) for a multivariate Rayleigh distribution. Is it - $x^T \Sigma^{-1} x \times \exp(\frac{-x^T \Sigma^{-1} x}{2})$ How do you prove it is from the exponential ...
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70 views

Convergence of Maximum Likelihood Estimator

I apologize for the basic question. If $\{p_\theta(x): \theta\in K\subseteq\mathbb{R}\}$ is a smooth family of distributions, then the MLE $\hat{\theta}_n,$ under suitable regularity conditions ...
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1answer
109 views

Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
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52 views

Weighted Kaplan-Meier estimator

Let two samples $(T_1, \ldots ,T_n)\sim F$ and $(C_1, \ldots ,C_n)\sim G$ are given, but not observed. Instead we observe $\tilde T_i = \min (T_i, C_i)$ and $\Delta _i = \mathbf{1}(T_i \leq C_i)$, ...
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1answer
83 views

Markov chain Monte Carlo: why is non-reversible MC MC not as popular?

I am new to methods for simulating Markov chains in order to sample from the target, unknown distribution. After a couple days of reading, I found out that even though people have realized that ...
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58 views

Maximizing the “uniformity” of a probability measure, with constraints, via path length minimization

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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1answer
157 views

How to check if a symmetric random variables is the difference of two iid symmetric random variables

I have the continuous symmetric random variable $X$ in $\mathbb{R}$. If I know its distribution function $F(x)$ what are the conditions on $F(x)$ so that $X=Y_1 - Y_2$ where $Y_i$ are also iid ...
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2answers
211 views

Gaussian expectation of an exponentiated outer product

Given a normal random column vector $\mathbf{x} \sim N(\mu, \Sigma)$, I need the expectation, $$ E\left[ \exp(\mathbf{xx}^\top)\right]$$ where $\exp(\cdot)$ is element-wise exponential function (not ...
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1answer
39 views

How are two tailed p values (especially) and one tailed p values useful given the following? [closed]

So I'm a self-learner which is always dangerous because I don't have anything to test if I am understanding things correctly, so I wanted to ask what is wrong/right with my assumptions. When reading ...
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0answers
38 views

Can anything be said of the correlation of X and Y / X? [closed]

I apologize in advance if I overstep my (relatively minimal) statistical knowledge. I am looking at two random variables X and Y, and am unhappy with the correlation between the two. On a whim, I ...
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50 views

Quantile as solution to minimization problem

I posted this on Math Stack Exchange, but since I got no response, I'm trying my luck here. I'm studying basics of quantile regression now and I have trouble proving that $\tau-$th quantile of ...
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1answer
390 views

Is there a mistake in Vapnik's “Basic Lemma”?

I have a concern about the "Basic Lemma" which Valdimir Vapnik states and proves in his 1998 book Statistical Learning Theory (ch. 14.3, pp. 574–76): It seems like a certain coefficient should have ...
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1answer
162 views

An efficient method to find the MLE of the combination of two point processes

I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity $$\lambda(t) = \mu$$ For ...
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64 views

Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition. What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...
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1answer
68 views

higher-level independence of three or more correlated RVs

I'm hoping for some help in nailing down a vague idea about independence. It starts with finding the expectation of a product of three RVs (or more, but I'll stick to three for now). These are not ...
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1answer
203 views

two correlated processes

I apologize if this question is not placed in the right place. But I am having a hard time to figure it out. It would be greatly appreciated if some one could help me out. Assume that there are two ...
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1answer
59 views

What is the problem with this model parameter estimation algorithm?

In a statistical model with parameters $\theta$ and unobserved laten variables $Z$, the model likelihood is $$L(\theta;X)=Pr(X|\theta)=\sum_ZPr(X,Z|\theta)$$ The standard way to estimate $\theta$ ...
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1answer
33 views

finite mixture of order statistics

Let $F(u)$ be a n-degree polynomial continuous distribution function in $[0,1]$, with $F(0)=0$, $F(1)=1$, that is $F(u)=\sum_{i=1}^{i=n} a_i u^i$. My question is: is that kind of distributions ...
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96 views

random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...
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58 views

Distribution of the Gram Matrices

Let $\mathbf{X}$ be an $m\times m$ random matrix full rank matrix, having the density function $f_{\mathbf{X}}(X)$. Also, let $\mathbf{W}$ be a deterministic $k\times m$ matrix of rank $k$ and ...
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1answer
106 views

Gibbs sampler with linear constraints

My problem concerns the estimation of truncated multivariate normal distributions under constraints. Let $X_1$ and $X_2$ two random variables following normal distributions ...
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215 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 ...
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1answer
161 views

Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$. Is it true that: If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...
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1answer
97 views

Expected number of samples above certain value of a normally distributed variable with a given sample mean

Suppose $n$ values, $X_1,...,X_n,$ are generated by a random number generator with normal distribution $N(0,1).$ Suppose that the (sample) mean of $X_1,...,X_n$ is $\mu.$ What is known about the order ...
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1answer
88 views

Compound Poisson process and central limit theorem [closed]

If I have a compound Poisson process $$Y(t) = \sum_{i=1}{N(t)}D_{i}$$ where $ \{\,N(t) : t \geq 0\,\}$ is a Poisson process with rate $\lambda$, and $ \{\,D_i : i \geq 1\,\}$ are i.i.d random ...
2
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1answer
96 views

Distribution of the Gram matrix

Let $\mathbf{X}$ be an $m\times k$ random matrix ($m>k$) of rank $k$, having the density function $f_\mathbf{X}(X)$. What is the distribution of $\mathbf{Y}=\mathbf{XX}^T$? Basically my question is ...
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19 views

Minimal rectangular confidence regions

For a given multivariate pdf $f$ (mainly the gaussian one) I'm looking to compute a minimal rectangular confidence region for a given level $\alpha$. For example, I would like to solve problems of the ...
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1answer
209 views

Probability distribution of uAv…

Consider the complex domain ℂ. If U and V are 2 unitary random matrices and A is a deterministic matrix. What is the distribution of $u^HAv$ ( or $||u^HAv||^2$) where : u is a column vector of U. v ...
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1answer
77 views

Third order central moment of a positive linear combination of log-normal random variables

What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables? It seems to be a common notion that the skewness of random variables ...
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33 views

Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction. Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
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174 views

Constructing a Bernoulli random variable for ratio of Bernoulli weights

$X$ and $Y$ are Bernoulli random variables with weights $0 < \alpha < 1$ and $0 < \beta < 1$. Is it possible to construct a sampler for the Bernoulli random variable with weight ...
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2answers
89 views

Sampling from maximally skewed stable distribution

I am reading a paper which refers to a maximally skewed stable distribution $F(x;1,-1,\pi/2,0)$ . Is there an efficient way to sample from this distribution? If $X$ has distribution ...
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0answers
101 views

Hilbert Schmidt Operators and the Conditional Expectation Operator

Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
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127 views

Doubts about Bayes' Theorem [closed]

I meet one problem on the probability and statistic theory. "Assume given a measure space $(X,S)$ with three probability measure $\mu_1,\mu_2,\lambda$ on the space. And there exsit functions ...
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134 views

Inverse moment of the number of inversions of a permutation

Let $\pi$ be a permutation of $\{1,2,...,n\}$. A pair of elements ($\pi_i$,$\pi_j$) is called an inversion if $i$ $>$ $j$ and $\pi_i$ $<$ $\pi_j$. The total number of inversions in $\pi$ is ...