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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2
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1answer
87 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 ...
0
votes
1answer
65 views

Gaussian expectation of a 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 ...
-2
votes
0answers
51 views

Convergence of empirical random variable

Let $X$ be a RV on the real line, of probability measure $P_X$, and let $X_n$ for $n=1,...,N$ be an iid sample from $P_X$. The Glivenko-Cantelli theorem says that the empirical measure, $P_N$, ...
-2
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0answers
21 views

Can a very bad Coefficient of determination (R squared value) not be indicative of model performance? [migrated]

Thanks in advance for the advice. I am trying to build a generalized linear model that has many predictors. The R squared value of the model is quite low (.21), but when I use the model to predict ...
1
vote
1answer
36 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 ...
0
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0answers
24 views

sample from a probability distribution

I'm sampling from a function that is proportional to the probability distribution using the Metropolis Hasting algorithm. What is the state of art algorithm to do this task? Thanks
1
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0answers
34 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
47 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 ...
0
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0answers
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 ...
5
votes
1answer
386 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 ...
3
votes
1answer
158 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 ...
0
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0answers
44 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 ...
0
votes
1answer
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
1answer
203 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
1answer
54 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 $\theta$ ...
0
votes
1answer
33 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 ...
1
vote
0answers
87 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
0answers
56 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
1answer
103 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 ...
1
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0answers
193 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
1answer
143 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
1answer
96 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 ...
0
votes
1answer
67 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 ...
2
votes
1answer
95 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
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0answers
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 ...
2
votes
1answer
208 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 ...
0
votes
1answer
69 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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0answers
33 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 ...
5
votes
3answers
170 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 ...
2
votes
2answers
86 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 ...
2
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0answers
100 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
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0answers
125 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 ...
5
votes
0answers
128 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 ...
3
votes
1answer
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 ...
0
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0answers
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
1answer
36 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 ...
2
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0answers
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
1answer
61 views

Averaging function of sum of variables using central limit theorem

I'm trying to evaluate an integral of the following form $$\int \prod_i \left[ dx_i \,P(x_i) \right] \; f \Big( \frac{1}{N} \! \sum_{i=1}^N x_i \Big)$$ and I know that the distribution of $x$ is ...
5
votes
1answer
240 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 ...
1
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0answers
40 views

Efficient evaluation of multidimensional kernel density estimate

Edit I have copied this discussion to the stats community site here, since I feel it is more relevant. Please feel free to close this in due course. I've seen a reasonable amount of literature about ...
2
votes
2answers
118 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
1
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0answers
52 views

Whether r.v. with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$ is overdispersion?

When discrete r.v. $X$ is not Poisson distributed and ${\rm{Var}}X,EX < \infty $, I want to know whether r.v. $X$ with p.g.f. $\exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)],({q_i} \in ...
4
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0answers
137 views

Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all $n\geq3$, the function: ...
0
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0answers
26 views

How to generalize uncertainty coefficient to set-valued classes?

This question is the reason I asked How to estimate the entropy of a distribution on a power set? Proficiency (AKA uncertainty coefficient) is an information-theoretic measure of predictor quality, ...
-1
votes
1answer
54 views

Finiteness of “novel variance” from a kernel on a compact space [closed]

Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
1
vote
3answers
115 views

How to estimate the entropy of a distribution on a power set?

Given a probability distribution $(X,p)$, its entropy is defined as $H=-\sum_{x\in X} p(x)\log p(x)$. Given a sample of observations $x_n,n=1..N$, one can estimate $p(x)=\frac{\#\{i:x_i=x\}}{N}$ and ...
1
vote
1answer
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 ...
-4
votes
1answer
79 views

Is it possible to determine if these random numbers are not really random? [closed]

I've been given a big ordered list of integer numbers. Looks like this : 10 -11 -3 -6 -10 -1 ..... ..... ..... Allegedly, these values are random from -12 to +12 However, there has been ...
1
vote
2answers
155 views

Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...
5
votes
1answer
180 views

Measures which exhibit the “uncorrelated implies independent” property

Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...