Tagged Questions

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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Hierarchical (Recursive) Random Walk Model

Consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/...
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Drawing graph with some different metric values [closed]

I have some metrics but all of them have different values. Some are decimals, some are integers and some are large numbers. Assume the metrics are (average values): - metric1 - 1500 - metric2 - 0....
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Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
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Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...
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Discretization of a continuous distribution

For a research project I work with continuous distributions, like the normal distribution. In my use case however the random variable Z generally follows a normal distribution, though it can only take ...
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Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...
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Is there any Monte carlo or statistical approach to variational integral problems?

I am just shooting in the dark: From brain data imaging we have integrals of the form $L(D):=\int_{\Omega}(\left \| A_{tensor}(D)-\widehat{A}\right \|+\sqrt{|\gamma(D)|})d\Omega$, where we minimize ...
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Probability of collision of some family of hash functions

Given $x$ and $y$ in $\mathbb{R}$, and let $\mathcal{H} = \{ h \mid \mathbb{R} \to \mathbb{N} \}$ be a family of hash functions where $h(x) = \left\lfloor x + \sum^C_{i=1} U_i \right\rfloor$ for some ...
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L1 distance between gaussian measures

L1 distance between gaussian measures: Definition Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
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information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
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What is the mathematics behind the random experiment which produces the data with this strange property?

I have a following scenario. there is a huge collection of data resulting from a random experiment $E$ (I do not say random variable yet, for reasons that you will need to explain in your answer). Let ...
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Nonparametric estimation in diffusion

Fan and Wang In the above paper, the Authors provide estimators for the squared spot volatility process $\left(\sigma^{2}_{t}\right)_{t\geq 0}$. My question is how to find estimators for the process ...
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Generalizing an expected increase in autocorrelation near a bifurcation point to a system of ODE

Near a bifurcation point, a stochastically forced dynamical system should show an increase in autocorrelation and variance. This is due to critical slowing (a loss in resilience to perturbations). ...
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Why is the classical secretary problem about ranks?

This relates here: http://math.stackexchange.com/questions/1820997/why-is-the-classical-secretary-problem-about-ranks You want to stop optimal in a sequence of items presented sequentially, that is ...
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The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk. I want to figure out the necessary ...
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System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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Estimating $E[X ; A]$ where $A$ is, e.g., an inter-quantile range

Estimating $E[X]$ from i.i.d. copies $X_1,X_2,\dotsc$ of a random variable $X$ with unknown distribution $P$ is well studied, obviously. When $X$ has extremely large variance, the Monte Carlo ...
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connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field $$c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.$$. What can be said ...
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In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
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Moments of function of Poisson process

(I'm new to Poisson processes, so please edit if my terminology is incorrect.) Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...
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Convergence of an rcll process along a random subsequence

I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...
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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 ...
Let $F_n$ be the empirical distribution obtained from an i.i.d. sample of the distribution $F:R ^d \to [0, 1]$. Kiefer (1961) shows that the convergence of the empirical distribution is like P\left(... 2answers 5k views Why 1.5IQR whiskers in boxplot? [closed] Hi math people. I'm in the process of analyzing some data that I collected through an experiment. The data are (somewhat) normally distributed and I represent the different data-sets using boxplot, ... 0answers 104 views Limit theorem : reproduce a proof with an adaption from discrete to continuous time Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes". The setup is as follows: We have a Levy-Process X_t fullfilling the CLT \begin{align} \sqrt{t}(X_t/t-E(... 0answers 64 views Basic Monte Carlo Integral Approximation On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let $$I = \int_D g(\textbf{x})d\textbf{x},$$ where D \subset \... 0answers 47 views Literature on transformed Gaussian matrices I am considering real n-by-m matrices of the following type: M=SM^\prime,\\ M^\prime_{ij}\sim^{iid}N(0,1).  Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...
Is there a name for the following process? Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...