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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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 ...
12
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2answers
465 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 ...
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0answers
52 views

How to get a regression curve for an x-y value pair where a single x can have multiple y values in the set?

I am trying to generate a regression curve for a set of x-y values where a single x value can have multiple y values. Although in the end, only one y value should be the correct one and rest all ...
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1answer
230 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 $x>0$, ...
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1answer
180 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, ...
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2answers
341 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 ...
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0answers
157 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 : $E[X|X^Tb = c]$ ...
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1answer
195 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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215 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 ...
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2answers
265 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 ...
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83 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 ...
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292 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 ...
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3answers
365 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 ...
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84 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 ...
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126 views

Better estimate than Sample mean?

Today I came a cross an old math thing I did some years ago. I remembered that I didn't find a similar result from anywhere at that time but forgot the thing altogether. So now I would like to ask if ...
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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
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1answer
148 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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103 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 ...
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3answers
435 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 ...
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0answers
95 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 ...
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99 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 ...
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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 ...
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2answers
4k 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 ...
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204 views

prewhitening (whitening transform) in terms of expected-value-wr-sigma-algebra

I'm trying to understand the mathematics of prewhitening a little better. (See http://en.wikipedia.org/wiki/Whitening_transformation, e.g.) Taking the conditional expectation of an RV with respect to ...
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1answer
341 views

MCMC with progressive demollification of delta distributions

Edit: I simplified the example to a canonical case for clarity. Given an integral $\int_{\Omega}{g(\mathbf{x})}$ with a well-posed integrand $g(\mathbf{x})$ defined on some multidimensional space ...
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86 views

Limiting distribution of the cardinal of a Markovian set

Let $S_1=\lbrace u_1 \rbrace$ where $u_1$ is a random uniform drawing on $[0,1]$. To build $S_{n+1}$ draw $u_{n+1}$ uniformly on $[0,1]$ (independently from previous draws) and draw $v_{n+1}$ ...
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196 views

How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are $m$ independent $d\times d$ random matrices and let $\overline{X} = \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices ...
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1answer
359 views

Null hypothesis test for independent but not identically distributed samples

I'm trying to figure out the best statistical test to use for an edge case I've run into: trying to figure out the likelihood of the null hypothesis for a set of samples that each (potentially) come ...
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0answers
352 views

Find closed form for comparison of two binomial random variable: solve inequality

Hi, Dear All, I come up with this problem, which I think for a long time without a good answer. Suppose two independent random variables $X \sim \mathrm{Binomial}(n, p)$ and $Y \sim ...
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0answers
187 views

Optimal Winsorizing Cutoffs for a stratified proportional to size population estimator.

I'm not familiar with the subject, but here's my question. Is there any reference in the literature of Winsorized Type II for total population estimator in a estratified with clustering proportional ...
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210 views

First Table of Random Numbers

What was the first table of random numbers of any sort? The best I can do is Tippett and Pearson's Random Sampling Numbers of 1927. Can anybody identify an earlier table? Thanks for any ...
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1answer
224 views

Stieltjes convolution with white noise

I'm looking for a reference that would discuss a Stieltjes convolution between a wiener process and a function of bounded variation. Additionally, I had a question about this sort of convolution. Is ...
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1answer
534 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
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4answers
277 views

Avoiding overfitting by averaging polynomials fit to part of the data?

I was thinking about the problem of overfitting data. Suppose you have a hundred data points sampled from an unknown function (call this the training set). You could try fitting a ...
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3answers
1k views

Bayesian statistics for pure mathematicians

Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. ...
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114 views

Positive estimator

Suppose that one knows how to generate (independent) random samples $X_1, X_2, \ldots$ distributed as the random varable $X$ with $\mathbb{E}[X]=\mu \in \mathbb{R}$. It is then easy to construct an ...
0
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1answer
127 views

multivariate linear regression with dependent noise terms?

What is it //called// when you are doing linear regression on the problem: $ Y = AX+BZ $ where you are given observations Y and X and are assuming Z is independent Gausssians? If you do max-Likelihood ...
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111 views

Types and Typical sequences

Dear colleagues, Joint types can often be given in terms of the type of x and a stochastic matrix \begin{equation} V:X\rightarrow Y \end{equation}such that $ P_{x,y}(a,b)=P_{x}(a)V(b|a)$ for every ...
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0answers
93 views

Maximum of the norm of k-averages of n iid random d-dimensional vectors

Suppose $X_1, ... X_n$ are i.i.d. random vectors in $d$-dimensional space (i.e., $R^d$) with continuous centrally symmetric density function $f(\cdot)$ (i.e., symmetric with respect to the origin). ...
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0answers
277 views

Bounding the probability that a random variable is maximal

Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$. I am looking for ...
4
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1answer
185 views

Hyperplane arrangements and covering numbers

Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$. For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$. For a point $x ...
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127 views

Conditioning over Conditional probability? (also: $\phi$-mixing sequences)

For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows: $$ \phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - ...
5
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1answer
313 views

Computing the correlation between two vectors without divulging them

Alice and Bob respectively know a vector of $N$ real numbers $u$ and $v$. They would both like to know $\rho = \langle u,v \rangle/N$ but Alice does not want Bob to gain anymore information about $u$ ...
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2answers
261 views

Convolutive noise removal

I have the time domain signal $$ u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t) $$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...
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0answers
55 views

Statistic for goodness of fit to a multidimensional distribution with geometric tails?

Cross post from Stats.SE: http://stats.stackexchange.com/questions/29601/statistic-for-goodness-of-fit-to-a-multidimensional-distribution-with-geometric Essentially a reference request, since someone ...
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1answer
76 views

multimodal circular model

Hi, can someone provide me with a list of probability models that is akin to Von Mises but consists multiple (potentially infinite) modes that takes into account attractors in the entire 2-D spatial ...
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2answers
432 views

Vapnik-Chervonenkis dimension of lines in the plane

I'm having some problems with this problem concerning VC dimensions ( http://en.wikipedia.org/wiki/VC_dimension ), I hope for some helping input. Given a set $L$ of $n$ lines in the plane, define a ...
2
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2answers
305 views

My overdetermined linear system gives both bad and good estimates. Why ?

Hello to everyone. What the question means is that different ways of expressing the same relation between the data and unknown variables produce really weird fit results: The problem: I have the ...
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2answers
275 views

Measuring the independence between the components of a stochastic process

In a context of blind source separation (e.g. you want to extract the voice of a singer from a song), many approaches consist in maximizing the independence between the components of a certain ...
3
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1answer
104 views

exact simulation of a large sample histogram

Say I want to create a histogram of $N$ samples from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins, ...