**-2**

votes

**0**answers

26 views

### Let X be a Geometric (1/4) and Y be a Geometric (1/2) be two independent random variables [on hold]

Let X be a Geometric (1/4) and Y be a Geometric (1/2) be two independent random variables.
Obtain the conditional distribution of Y , given that X - Y = 1. The Answer I got was one. However, I just ...

**0**

votes

**0**answers

14 views

### 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 ...

**1**

vote

**0**answers

20 views

### 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). ...

**3**

votes

**1**answer

307 views

### 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 ...

**4**

votes

**1**answer

71 views

### 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-...

**-2**

votes

**0**answers

45 views

### Deriving PDF from Lorenz Curve

My data is organized as bins, with the total value of the variable (wages) specified. i.e For example, for people with income between $10k-$30k, total wages add up to $15 million (say).
This data can ...

**1**

vote

**0**answers

56 views

### 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 ...

**0**

votes

**1**answer

29 views

### 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 ...

**0**

votes

**0**answers

32 views

### System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$
$$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...

**3**

votes

**0**answers

36 views

### 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 ...

**1**

vote

**0**answers

48 views

### Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...

**10**

votes

**1**answer

377 views

### The geometric median of a solid triangle

Let $\Omega\subset \mathbb R^n$ be a compact subset of dimension $n$. Define the geometric median on $\Omega$ as the point $m_{\Omega}\in \mathbb R^n$ such that the integral $\int_{\Omega}|x-m_{\Omega}...

**1**

vote

**1**answer

105 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**1**

vote

**0**answers

34 views

### Covariance of order statistics [closed]

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I have also posted on math.stackoverflow, but given that this seems to be ...

**1**

vote

**0**answers

48 views

### Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...

**0**

votes

**0**answers

26 views

### Product of lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.
Consider the corresponding log-normal random variables: $...

**5**

votes

**1**answer

122 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial \theta^...

**4**

votes

**1**answer

55 views

### Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If $|h(w_1,...

**0**

votes

**0**answers

33 views

### Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the $[0,1)\...

**1**

vote

**0**answers

74 views

### 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, ...

**2**

votes

**1**answer

79 views

### What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?

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(...

**1**

vote

**0**answers

103 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(...

**2**

votes

**0**answers

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 ...

**0**

votes

**0**answers

63 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
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset \...

**7**

votes

**1**answer

107 views

### Choosing a sample based on where the density function is highest

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 $f$. ...

**2**

votes

**0**answers

43 views

### A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals (...

**3**

votes

**1**answer

110 views

### Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?

Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?

**3**

votes

**2**answers

163 views

### 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 ...

**0**

votes

**0**answers

24 views

### Adaptive refinement of integral domain

In electromagnetics we need to calculate the radiated power which is defined as something like
$P_r=\int_0^{2\pi}\int_0^{\pi}R(\theta,\phi)\sin{\theta}d{\theta}d\phi$
We already have $R(\theta,\phi)...

**3**

votes

**1**answer

110 views

### Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition
Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are $L^...

**0**

votes

**0**answers

19 views

### Error propagation with black boxes: add uncertainty in quadrature, or use a weighted standard deviation?

I have a measurement $x$ with a known uncertainty $\sigma_m$.
I have a black box that can take an error-free measurement $x$ and produce a value $y$ with a known uncertainty $\sigma_{b}$ (which is ...

**1**

vote

**0**answers

38 views

### 2-step sampling from a conditional density

The setting is as follows:
We are given two random variables $X : \Omega \to \mathbb{R}$ and $\Theta : \Omega \to T$ for some 'parameter space' $T \subset \mathbb{R}$, and
1) we know the density of $...

**18**

votes

**4**answers

920 views

### Applications of algebraic geometry to machine learning

I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this ...

**1**

vote

**0**answers

24 views

### Robust weighted estimator of location

Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, ...

**1**

vote

**0**answers

57 views

### Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...

**1**

vote

**1**answer

188 views

### 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
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...

**3**

votes

**0**answers

78 views

### An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...

**3**

votes

**1**answer

135 views

### Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...

**3**

votes

**1**answer

106 views

### Learn a distribution from distributions on samples

There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...

**0**

votes

**0**answers

34 views

### A functional's expectation using both known and unknown pdf

Suppose we have a random variable $X$ with a known distribution $f$ over an interval $[a,b]$ and another r.v $Y$ over the same interval but with an unknown distribution $g$. We also have a functional $...

**3**

votes

**0**answers

90 views

### How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...

**2**

votes

**1**answer

96 views

### An Inequality Regarding the Squared Conditional Variance

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. ...

**5**

votes

**3**answers

273 views

### The mean of points on a unit n-sphere $S^n$

A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the great-...

**1**

vote

**0**answers

44 views

### Sum of Log AR(1) processes

I have two AR(1) processes that look like this:
$x_t=\rho x_{t-1}+\eta_t$
and
$y_t=\rho y_{t-1}+\epsilon_t$
where $0<\rho<1$ and $\eta_t \sim N(0,\sigma^2_{\eta})$ and $\epsilon_t\sim N(0,\sigma^...

**1**

vote

**1**answer

98 views

### Do there exist random variables that force transitivity of dependence? [closed]

In general, statistical dependence is not transitive. If $Y$ and $X_{1}$ are dependent, and $Y$ and $X_{2}$ are dependent, then $X_{1}$ and $X_{2}$ are NOT necessarily dependent.
However, in some ...

**6**

votes

**1**answer

149 views

### 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 $x \in \mathbb{R}^2$ ...

**0**

votes

**0**answers

46 views

### Maximal Correlation with Weak Gaussian Perturbation

Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have density with respect to Lebesgue measure) with joint distribution $P_{XY}$. The maximal correlation $\rho_m(...

**2**

votes

**1**answer

78 views

### Covariance matrix as optimization problem solution?

I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|_{l^2}^2](:= \int_{\...

**5**

votes

**2**answers

146 views

### Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian

This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a ...