**0**

votes

**0**answers

33 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**

votes

**1**answer

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 ...

**0**

votes

**1**answer

24 views

### Bivariate skew-normal distribution [on hold]

I'm using the Gaussian distribution as a weight function for solving pde's.
I'm interested in skewing the function. For one-dimensional problems, it was easy to
derive the resulting skewed Gaussian ...

**0**

votes

**1**answer

66 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 ...

**-3**

votes

**0**answers

63 views

### What are the odds in the card game war that the game will start and finish on the first battle? [closed]

What are the odds? My 6 year old daughter and I played war. On the first card we both played sevens and the battle bagan. We played three face down and one up and they matched again. Again, we ...

**3**

votes

**1**answer

150 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**

vote

**1**answer

100 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 ...

**5**

votes

**1**answer

375 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 ...

**0**

votes

**0**answers

9 views

### How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [migrated]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution:
$$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$
Is there any ...

**-1**

votes

**0**answers

9 views

### MPE and MAPE averages [migrated]

I am a computer science graduate with a minor in mathematics, however it has been a long while since taking any statistical courses.
I am looking at calculating the Mean Percent Error and Mean ...

**0**

votes

**1**answer

304 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 ...

**1**

vote

**1**answer

126 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**

votes

**0**answers

32 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 ...

**1**

vote

**0**answers

28 views

### What are the assumptions of linear regression (both simple and multiple) [migrated]

I know this is a basic question, but I get slightly different answers everywhere I look, a quick example is that I learnt at Uni that the errors of the dependent variable have to be normally ...

**2**

votes

**0**answers

196 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**1**

vote

**1**answer

82 views

### Can I test many p-values with KS or AD

(Sorry if this is a noob question. I'm a mathematician learning statistics.)
I would like to know if it's sound (or advisable) to test many p-values against the continuous uniform distribution using ...

**3**

votes

**1**answer

104 views

### Estimating total variation distance from a given distribution

Given a known distribution supported on a finite set of $n$ elements with probabilities $p_1, \dots, p_n$ and an access to an unknown distribution $q$ is it known what is the number of samples from ...

**1**

vote

**1**answer

196 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 ...

**0**

votes

**1**answer

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 ...

**0**

votes

**1**answer

202 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 ...

**2**

votes

**1**answer

52 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 $\theta$ ...

**1**

vote

**0**answers

25 views

### Natural exponential family distribution with special moment structure

I encountered a problem related to a research project I was doing. Basically, it asks to find out natural exponential family distributions for which the variance function is the expectation of a ...

**6**

votes

**4**answers

3k views

### Constructing Bernoulli random variables with prescribed correlation

For which $n \times n$ correlation matrix $C$ can one construct Bernoulli random variables $(B_1, \ldots, B_n)$ with correlation $C$ ?
Following the approach described in this MO thread, one can ...

**0**

votes

**1**answer

32 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 ...

**2**

votes

**2**answers

84 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 ...

**5**

votes

**3**answers

470 views

### Concentration of sum of pairwise squared Euclidean distances of random vectors

Let $X_1, \ldots, X_n $ be independent random vectors in $B(0, D) \subset R^d$ ($\ell_2$ ball of radius $D$ centered at the origin). I am trying to find the concentration of the following quantity ...

**0**

votes

**1**answer

64 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 ...

**1**

vote

**0**answers

187 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 ...

**1**

vote

**0**answers

84 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 ...

**1**

vote

**0**answers

55 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 ...

**1**

vote

**1**answer

100 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 ...

**0**

votes

**1**answer

95 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 ...

**5**

votes

**3**answers

2k views

### uneven spaced time series

Let $(t_k), k \in \mathbb{N}$, be an increasing sequence of real numbers ($t_{k-1} < t_k$) and $(X_{t_k}$) be a sequence of real numbers indexed by $(t_k)$. Such a sequence is sometimes called a ...

**2**

votes

**1**answer

94 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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**2**answers

128 views

### References for Poisson and Lexis trials

I have been trying to find more information on Poisson and Lexis trials (generalizations of Bernoulli trials), but I have failed to find anything outside of MathWorld (I went through a number of ...

**2**

votes

**1**answer

207 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 ...

**2**

votes

**0**answers

32 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 ...

**1**

vote

**0**answers

82 views

### Two questions on conditional independence [closed]

I have two simple questions about conditional independence.
Suppose two random variables $X$ and $Y$ are independent. Then I want to show that they are independent given any arbitrary random ...

**2**

votes

**0**answers

21 views

### How to get the Expectation of the normalization of some log-normal-distributions?

Problem Definition:
Suppose that a random variable of multivariate Gaussian distribution $X \sim N(\Sigma,\mu)$, $\Sigma$ is the covariance matrix, and $\mu$ is the mean. For each $x_i$ from $X$, $x_i ...

**1**

vote

**1**answer

962 views

### Unbiased estimate of the variance of an *unnormalised* weighted mean

I have a follow-up question to this one:
unbiased estimate of the variance of a weighted mean
Specifically, how do I generalise the result given here (and on Wikipedia) for the unbiased sample ...

**5**

votes

**3**answers

169 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 ...

**1**

vote

**2**answers

1k views

### Inversion of Moment-generating functions (aka Laplace transform of prob dist)

I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by
\begin{equation}
M_X(t) = \text{E} \exp(tX)
\end{equation}
Since I have ...

**2**

votes

**0**answers

95 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 $f_X$, $f_W$ ...

**1**

vote

**0**answers

121 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 ...

**16**

votes

**4**answers

1k 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 ...

**5**

votes

**0**answers

125 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 ...

**0**

votes

**0**answers

48 views

### Linear Bounds on estimation error

Consider a markov chain on discrete state space $\mathbb{S} = \left\{1,2,..,S \right\}$, with transition probability matrix defined as $A = [a_{ij}]_{S \times S}$ where $a_{ij} = ...

**-1**

votes

**1**answer

35 views

### Express $cov(X^2,Z)$ in terms of means, variances, and covariance of $X$ and $Z$? [closed]

Suppose $X$ and $Z$ are random variables. Can the covariance of $X^2$ with $Z$ be expressed in terms of the means, population variances, and covariance of $X$ and $Z$ alone?
My attempts at solving ...

**5**

votes

**1**answer

233 views

### Central limit theorem for independent random variables, with a Gumbel limit

Consider independent random variables $Y_i$, $i>0$, such that $\mathbb{E}(Y_i)\approx \frac{1}{i}$ and $\text{Var}(Y_i)\approx \frac{1}{i^2}$, where $\approx$ means asymptotically equivalent up to ...