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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207 views

Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...
0
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1answer
16 views

Is it possible to find an asymptotic distribution for the LRT without the ML estimators being consistent?

I'm reading a comment(last page) to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the test may be ...
5
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1answer
200 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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0answers
13 views

Dirichlet distribution: Normalization of alpha values [migrated]

I'm a programmer and currently trying to apply the Latent Dirichlet Allocation algorithm by Blei et al. on a text mining problem. I am using a library called gensim for this, which takes, among ...
1
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5answers
864 views

Estimating the number of clusters

For a collection of points in $\mathbb{R}^n$, is there a statistic that I can compute which will estimate the number of clusters with some level of confidence?
6
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1answer
141 views

Closure of random rotations

Are matrix Fisher random variables closed under multiplication? For those unfamiliar with the jargon, let me unpack the terms above and repose my question. This is a question about probability ...
5
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3answers
259 views

A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal: (1) Извѣстія Физико-математического общества при Казанском университете I am surprised by the ...
17
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5answers
2k views

Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?

After having read Gunnar Carlsson's http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf I feel enthusiastic to use some topological data analysis (TDA) methods ...
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0answers
43 views

Statistics, probability [closed]

A statistician-gone-mad has concocted the following multi-part experiment. For the first part of the experiment, a fair, seven sided die is rolled and the upper-most facing number is noted. If the ...
3
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2answers
268 views

Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem. Let ...
0
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0answers
30 views

Variance along the regression line [closed]

I'm doing computational geometry on 3D surfaces, a $z = f(x,y)$ type of situation, where I have a set of gradients associated with a vertex for which I want to be able to measure if the gradients make ...
16
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3answers
1k 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 ...
2
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1answer
59 views

Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...
2
votes
1answer
164 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 ...
0
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1answer
71 views

Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...
1
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0answers
43 views

Is there an efficient algorithm for sampling from the negative hypergeometric distribution? [closed]

I'm writing a small statistics library currently. One of the algorithms I'm implementing has two variants: one that samples the hypergeometric distribution and one that samples the negative ...
3
votes
2answers
72 views

Is a function of complete statistics again complete?

suppose $T$ is a complete stats for a parameter $\theta$. Is any function $f(T)$ again complete? It sounds weird but the definition seems to confirm that $f(T)$ is indeed complete..
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0answers
15 views

Significance of Eigenvectors of a Covariance matrix [migrated]

In PCA and in many other problems of machine learning we use Eigenvectors of covariance matrix of the data. How do we visualize Eigenvectors of Covariance matrix? The Principal Eigenvector ...
0
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1answer
236 views

Size of KL-divergence neighbourhoods

I am new here. I was reading another post here and this got me wondering what can be said about the size of the following kl divergence neighborhoods. Consider these two kl-divergence neighbourhood ...
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0answers
28 views

Bounds on Product of CDF or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
2
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0answers
27 views

Mean and variance of a general multivariate skew normal distribution

I have a problem about a general multivariate skew normal distribution. There is a $p\times 1$ vector, $\mathbf{y}=(\mathbf{y}_1',\mathbf{y}_2',\ldots,\mathbf{y}_n')',p>n$, which has the density as ...
0
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2answers
93 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 ...
2
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1answer
43 views

Linear least squares with unordered response variable

In the classical linear regression model one considers the equation $$ y = X \beta + \epsilon.$$ I was wondering whether there are also results when the ordering of the response variable $y$ is not ...
0
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0answers
54 views

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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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
50 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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0answers
20 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 ...
0
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0answers
31 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 ...
2
votes
2answers
600 views

Proof of no unbiased estimation of standard deviation.

It is well known that for iid random variables $X_1, \ldots, X_n$ with variance $\sigma^2$ that $$\frac{1}{n-1} \sum_{i=1}^n (X_i - \overline X)^2$$ gives an unbiased estimator for $\sigma^2$, but ...
2
votes
1answer
56 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 ...
5
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2answers
280 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 ...
8
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2answers
609 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 ...
2
votes
2answers
235 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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0answers
22 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 ...
2
votes
1answer
60 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 ...
5
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0answers
73 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
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 ...
3
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1answer
112 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 ...
2
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0answers
56 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 $\Delta _i = \mathbf{1}(T_i \leq C_i)$, ...
2
votes
2answers
220 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 ...
0
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0answers
59 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
93 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 ...
5
votes
2answers
3k views

Something like mathoverflow in other sciences [closed]

Are the sites similar to mathoverflow in other sciences related to mathematics? statistics, computer science, physics, economics, etc? Let me explain what I mean by "similar": those are sites devoted ...
0
votes
1answer
332 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 ...
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1answer
41 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 ...
0
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0answers
56 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 ...
2
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1answer
2k views

The difference between Principal Components Analysis (PCA) and Factor Analysis (FA)

I am trying to understand the difference between PCA and FA. Through google research, I have come to understand that PCA accounts for all variance, while FA accounts for only common variance and ...
3
votes
1answer
163 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 ...
1
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1answer
116 views

forward algorithm Hidden Markov Model

I am studying the the forward-backward algorithm used in Hidden Markov Models. I understand that that you are trying to propagate through a sequence (and the available states) to find the most ...