All Questions
Tagged with pr.probability st.statistics
1,135 questions
1
vote
1
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502
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nonnegative series expansion of nonnegative functions
The title says it all! When using orthogonal series expansions like the Gram-Charlier expansion to approximate probability density function, a big problem (making this approach less usefull and less ...
- 2,600
2
votes
2
answers
956
views
An Easy Sanov-Type Theorem for Markov Chains?
First, the (simple!) setup:
I have a Markov chain X t on some finite state space Ω with stationary distribution π, and a function f from Ω to R. I'd like to estimate the integral of ...
- 263
1
vote
1
answer
434
views
A point process for modeling location of trees in an infinite forest?
I am looking for an example of a stationary, infinite point process on $\mathbb R^n$ with a few simple properties. I would not be surprised to discover that there is a well-studied, canonical process ...
- 2,155
1
vote
2
answers
744
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Order statistics: probability random variable is k-th out of n when ordered.
Given a random variable $X_1$ drawn from a distribution with cdf $F$, and random variables $X_2, \cdots,X_n$ drawn from another distribution with cdf $G$, what is the formula for the probability that $...
- 11
2
votes
3
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571
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How does the Dirichlet process work?
Hi, i'm looking to get into nonparametric bayesian techniques but I'm having problem understanding what's going on in the definition of the Dirichlet process or how it works. So what does P ~ DP(&...
- 23
-1
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1
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75
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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, ...
- 8,512
2
votes
0
answers
422
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Generalizations of Gram-Charlier and Edgeworth series?
I am looking for references pertaining to, and/or help in deriving, generalizations of the Gram-Charlier and Edgeworth series for non-Gaussian reference probability distributions.
I would like to ...
- 1,890
1
vote
1
answer
356
views
Statistical inequality
Let $X$ be a finite discrete variable and $X\ge0$. Is it true that
$$16\operatorname{Var}(X) \le \left[8{\mathbb E}(X) + \operatorname{Range}(X)\right]\operatorname{Range}(X)$$
where $\operatorname{...
- 95
1
vote
1
answer
454
views
An infinite Gaussian mixture with mixing parameters being also Gaussian
A finite Gaussian mixture with $k$ components has a probability distribution function $p(y|\mu_1,...,\mu_k, \sigma_1, ..., \sigma_k, \pi_1, ..., \pi_k)=\sum_{j=1}^{k} \pi_j\mathcal{N}(\mu_j, \sigma_j^...
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4
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386
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Recovering a function from a set of approximations
We assume that we have a finite set of agents with approximate knowledge about a certain function, and from this collection of approximations we want to recover the actual value of the function.
More ...
- 1,228
0
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1
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801
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Information criteria for ridge regression
Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$...
- 177
1
vote
2
answers
791
views
Likelihood function for sequential random variables
Context
Consider the following sequential data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence.):
$Y_1 = X_1^' \beta + \epsilon_1$
...
- 346
9
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0
answers
2k
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Has the Lie group preserving a probability distribution been used in Bayesian statistics?
For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define
$$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$
Here $\operatorname{STO}(n)$ denotes ...
- 15.4k
3
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0
answers
213
views
Find a minimum entropy code for a simple gibbs random field.
Just to make precise what I am talking about, I will include the definition of a minimum entropy code. I will then define the precise markov random field I am asking about.
In the rest of this ...
- 131
3
votes
1
answer
203
views
Bounds on tails with moments
A sort of continuation of Comparing distributions with moments
Suppose I have some estimates of the moments of a non-negative random variable $X$: $$\log \mathbb{E}(X^n) = n \log n + (\beta-1)n + O(\...
- 275
2
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0
answers
1k
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Problem with Pearson correlation coefficient. [closed]
I have two random variables X and Y. X follows a power law distribution. I know its generating function G(x). I also know the Pearson correlation coefficient of X and Y. How do I find the generating ...
- 31
1
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0
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101
views
What is the range of a positive random variable after whitening?
Let ${\bf x}\in\mathbb R^N$ be a positive multivariate random variable, i.e.
$$x_i\in [0,\infty).$$
What is the range after whitening, i.e. the range of ${\bf y} = \sqrt{C}^{-1}{\bf x}$ with the ...
- 51
0
votes
2
answers
257
views
Efficient computation of $E\left[\frac{1}{1+\sum_iX_i}\right]$ where $X_i$ is RV with Bernoulli distribution with different probabilities
Suppose we have the random variables $X_1, \ldots, X_n$ that have Bernoulli distributions with the (possibly different) probabilities $p_1, \ldots, p_n$. For example, $X_1$ = 1 with probability $p_1$ ...
- 21
1
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0
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555
views
How to obtain tail bounds for a linear combination of dependent and bounded random variables?
Hi everyone,
Note: This question is a general case and edited version of my previous question ``How to obtain tail bounds for a sum of dependent and bounded random variables?''.
I am looking for ...
- 197
4
votes
0
answers
153
views
A simplified MCMC / MH algorithm. Are there known convergence results?
Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
- 241
0
votes
2
answers
327
views
Copulas and time series
Please, can anybody give a reference(s) to some good recent review papers about copulas and time series?
- 2,600
4
votes
3
answers
286
views
Medium-Sized Calculations and Organization
This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
- 63
2
votes
0
answers
271
views
Convergence of sample mean
I have a two-index succession of real-valued random variables $x_{t,n}$ such that $\lim_{n\to\infty} x_{t,n} = x_t$, for all $t$ and suitable limit r.v. $x_t$.
I would like to prove that $$\lim_{n\to\...
- 20.2k
1
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1
answer
321
views
"Bridging" uniform and "mass" distributions
Foreword. The original formulation of this problem was inaccurate; chamomille and Didier Piau came up with a simple example which would not solve the problem in its accurate formulation. Sorry for my ...
- 37
1
vote
0
answers
98
views
Small ball probabilities for functions of correlated normals
Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
- 111
1
vote
1
answer
303
views
Log concavity of noncentral chi-square
I am trying to check if the noncentral chi-square distribution is log-concave in its noncentrality parameter. Specifically, given
$p(y ; \lambda, \sigma^2) = \frac{1}{2\sigma^2}\exp\left(-\frac{y+\...
- 11
1
vote
0
answers
336
views
Normalized correlation with a constant vector
I am confused how to interpret the result of preforming a normalized correlation with a constant vector. Since you have to divide by the standard devation of both vectors (reference: http://en....
- 171
2
votes
2
answers
521
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A variant of the hypergeometric distribution - in the literature?
I have been working on a problem in combinatorics that makes use of the following discrete distribution.
Let $a_{1}, a_{2},..., a_{N}$ be any binary sequence of of length $N$ with $n$ ones and $m$ ...
- 201
0
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0
answers
160
views
Two Different Representations of Multivariate Bernstein Polynomials
In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following:
$$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in \{0,\dots,n\}^m}...
- 103
1
vote
0
answers
100
views
Distribute Monte Carlo samples among dimensions
Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
- 101
0
votes
1
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285
views
Is there a monotone coupling of Dirichlet random variables?
Let $X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$ and $Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$, where all $a_i$ and $b_i$ are positive. Is there a natural coupling between $X$ ...
- 13
6
votes
2
answers
428
views
how to sample a conditioned diffusion
there are several reasons why we could be interested in sampling conditioned diffusions:
if we observed a diffusion at discrete time and want to do some kind of inference on the parameters of the ...
- 2,133
1
vote
0
answers
69
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 {\rm{...
- 113
3
votes
2
answers
255
views
Correcting bias in samples selected by a prediction
Here is the scenario:
I'm trying to find as many golden tickets as I can, so that I can sell them to kids that want to go on a tour of Wonka's chocolate factory.
Fortunately, I have a machine that ...
- 269
1
vote
3
answers
332
views
Is ERNIE output skewed by statistical tests?
ERNIE is a hardware random number generator used to generate winning Premium Bond numbers in the UK. Wikipedia says: "ERNIE's output is independently tested each month by an independent actuary ...
- 221
1
vote
1
answer
152
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 ...
- 203
5
votes
0
answers
506
views
Missing mass estimate
Let $S$ be a finite set with probability distribution $P$. Define the random variable $m_i$ to be the "missing mass" after seeing $i$ iid samples from $S$ under $P$. That is, $m_i$ is the total mass ...
- 6,541
2
votes
1
answer
258
views
How would one extend the Brier score to an infinite number of forecasts?
Is there a neat way to use something like the Brier score to score an infinite set of forecasts/outcomes?
- 367
1
vote
0
answers
153
views
Sampling without replacement: probability for total successes from successes in sample?
Consider drawing $n$ balls from an urn containing $N$ balls, of which $m$ are red. If i know $N$, $m$ and $n$ i can use the hypergeometric distribution to calculate the probability that my sample ...
1
vote
1
answer
499
views
A question about Chapter 12 (Vapnik-Chervonenkis Theory) of 'A Probabilistic Theory of Pattern Recognition'
Hi,
Can anyone familiar with the book 'A Probabilistic Theory of Pattern
Recognition' or the theory described help me out?
See quote from chapter 12, 'Vapnik-Chervonenkis Theory', of 'A
...
- 113
4
votes
2
answers
258
views
near independence of markov chain observations at high lags
I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would ...
- 549
0
votes
2
answers
595
views
univariate prior corresponding to weighted sum of L1 and L2 penalties?
Is there a univariate probability distribution $p_{\lambda,\alpha}(\beta)$ over the reals, parameterized by $\lambda > 0$ and $1 >= \alpha >= 0$, such that $p_{\lambda,\alpha} \propto \exp(-\...
- 103
1
vote
0
answers
397
views
Random Walk vs Branching process
1) Let us consider the set of all $N!$ permutations of the $N$ elements ${1, 2, . . . ,N}$. In the random state, each permutation of these elements occurs
with probability 1/N!. The probability $Pm(N)$...
3
votes
1
answer
412
views
Sparse representation of a distribution with independent and correlated variables
Here's what I'm trying to do:
Imagine a probability distribution over $\mathbf{R}^2$, $P(x,y)$. I can approximate $P(x,y)$ with set of $N$ points $\{(x,y)_i\}$ drawn from $P$. By approximate, I mean ...
- 1,902
10
votes
0
answers
391
views
Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing
Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...
- 101
1
vote
1
answer
242
views
Measuring the randomness in random numbers
I'm looking to write a program to investigate a few random number algorithms. Basically I am looking to see if the spread of numbers is indeed randomly distributed enough. What kind of statistical ...
- 111
2
votes
0
answers
530
views
About generalization of stirling numbers of the second kind
Hello,
The Stirling numbers of the second kind count how many ways can a set of $k$ elements be partitioned into $n$ non-empty classes, with $k=n,n+1,\dots$.
My question is: Is there a ...
2
votes
0
answers
548
views
What will be the distribution of harmonic mean of two correlated gamma random variables?
Suppose there are two correlated random variables $X_1$ and $X_2$ both are gamma distributed but having different shape and scale parameters with correlation coefficient $\rho$. What will be the ...
- 133
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
0
votes
1
answer
101
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 ...
- 61