**2**

votes

**1**answer

155 views

### Maximization of specific Likelihood function

N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum ...

**8**

votes

**1**answer

111 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**-2**

votes

**0**answers

18 views

### Individual contribution to a global change in a ratio [on hold]

I have about four thousand records, each consisting of an entity ID and four variables that represent two data points from two different years. x2010 and x2013 are counts of all cases for each entity. ...

**0**

votes

**1**answer

32 views

### Supremum of centered jointly generalized chi-square random variables

Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...

**3**

votes

**1**answer

73 views

### assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...

**-1**

votes

**0**answers

27 views

### Bayesian inference on gamma distribution

The likelihood of an observation $x$ under a gamma distribution is
$$L(x | \alpha, \beta) \propto \beta^\alpha x^{\alpha-1} \frac {\exp(-x\beta)} {\Gamma(\alpha)}$$
Suppose I have some observations ...

**21**

votes

**1**answer

4k views

### 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 ...

**2**

votes

**1**answer

1k views

### a problem with “Lower bound of Wilson score confidence interval”

Hi,
I am trying to calculate the 'Most helpful' review. I found a solution few days ago on Calculating the "Most Helpful" review
I have a new problem now..
Let's say I have two items. The ...

**5**

votes

**5**answers

142 views

### Generate Bernoulli vector with given covariance matrix

I am from different background, so please forgive me if the answer is so well known.
Let $C=(c_{ij})$ be a given $n\times n$ matrix. Do we have a way to generate samples of random Bernoulli vectors ...

**12**

votes

**4**answers

1k views

### How long for a simple random walk to exceed $\sqrt{T}$?

Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$.
What is an expression for the probability distribution of ...

**2**

votes

**7**answers

2k views

### Estimating the mean of a truncated gaussian curve

Say I have a black box generating data samples, and I want to estimate the parameters of the black box from the samples.
The black box works like this: it has a parameter m (a real number), and to ...

**5**

votes

**2**answers

69 views

### Gaps between descending order statistics

Let $\{X_{1},X_{2},\cdots,X_{n}\}$ be a random sample of size $n$. Denote $(X_{(1)},X_{(2)},\cdots,X_{(n)})$ to be its descending order statistics. Define gap $g_{i}(n)$ to be ...

**3**

votes

**1**answer

366 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 ...

**1**

vote

**0**answers

48 views

### Finding an error estimation for the De Moivre–Laplace theorem with Stirling's formula

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De ...

**3**

votes

**1**answer

79 views

### $\int_0^t f(s)\,dB_s$ normally distributed, mean and variance

Suppose that $f(t)$ is a (non-random) continuous function on $[0, \infty)$. Let$$Z_t = \int_0^t f(s)\,dB_s.$$
How do I see that $Z_t$ is normally distributed?
What is the mean and variance?
I need ...

**9**

votes

**1**answer

762 views

### Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...

**1**

vote

**0**answers

32 views

### Adding weights to the Brier score

Fix $n > 0$, and consider the space $\cal P$ of probability functions defined over the Boolean closure of a fixed $\cal S = \{ s_1, \ldots, s_n \}$. The Brier score of $P \in \cal P$ at $s_i \in ...

**1**

vote

**1**answer

77 views

### KL divergence Inequality

I am trying to find a proof for the following inequality, but I did not get anywhere following the references from the paper I was reading.
Consider two probability measures $P$ and $Q$ both ...

**6**

votes

**1**answer

184 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
...

**0**

votes

**1**answer

484 views

### probability mass function fitting [closed]

I have a probability mass function of some experimental data who's log looks like the following: (please ignore the fact that it is not normalized)
![alt text][1]
[image shack image removed]
...

**5**

votes

**2**answers

122 views

### Reference to iterated logarithm law and Smirnov law of empirical CDF

I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws.
Let ...

**3**

votes

**1**answer

116 views

### Practical bounds for the Wasserstein distance in 2 dimensions

Let $X_1,\dots,X_n$ be a set of independent samples of a distribution $\mu$ on the unit square, let $\hat\mu_n$ be the empirical distribution on the points $X_1,\dots,X_n$, and let ...

**2**

votes

**0**answers

95 views

### Convergence rate of Pearson correlation matrix

I am interested in (rather sharp if not the finest) tail/concentration bounds for the Pearson correlation matrix: let $X_1,\ldots,X_N \sim \mathcal{N}(0,1)$ be correlated random variables; let ...

**17**

votes

**3**answers

967 views

### Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...

**4**

votes

**3**answers

390 views

### Expectation of $(c+e^{N(0,\sigma^2)})^{-n},\, n>0$

I would like to know if there's a way to compute or approximate the following expectation:
$$\mathbb{E}[(c+e^X)^{-n}]$$
where $X=N(0,\sigma^2)$ and $n,c>0$ (you can also assume that $n$ is a ...

**5**

votes

**3**answers

3k views

### measure of quality of curve fit

I am interested in a measure for the quality of fit to a curve which would distinguish the two cases shown in the following image (without addressing the fact that incidentally the right one has more ...

**1**

vote

**2**answers

67 views

### Is it possible to find an asymptotic distribution for the LRT without the ML estimators being consistent?

I'm reading a comment(last page) to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the test may be ...

**2**

votes

**1**answer

83 views

### Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...

**0**

votes

**0**answers

25 views

### Consistency of M-estimators when the constraint set also has to be estimated

Let $K \subset \mathbb R^n$ compact and convex. Also let $H$, $G_i, \; i \in \{1,\dotsc,m\} $: $K \to \mathbb R$ be convex functions.
Assume we have the following convex optimization problem:
$$
...

**0**

votes

**0**answers

35 views

### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...

**0**

votes

**0**answers

21 views

### A book on discriminant analysis

Can anyone suggest a good book on discriminant analysis - comprehensible and detailed? (Kendall and Stuart write about the subject too concisely.)
Thanks in advance.

**0**

votes

**0**answers

49 views

### Best measure for curve similarity

I would like to measure similarity between two curves represented by two arrays of points.
The similarity measure should not depend on the size of these shapes. Two similar shapes but have different ...

**1**

vote

**2**answers

104 views

### Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in ...

**0**

votes

**0**answers

85 views

### Probability of substring given string production probabilities

I originally posted this question on the Math StackExchange, but have not received answers there and thought it might be more appropriate to post it here.
Let $\Sigma$ be an alphabet and let $y = x_1 ...

**3**

votes

**4**answers

3k views

### Linear Regression Coefficients W/ X, Y swapped

Let's say I have a linear regression model of the form $ y = B_x x + I_x + \epsilon $, where $B_x$ is the beta coefficient of the $x$ term, $I_x$ is the intercept term and $\epsilon$ is additive, ...

**2**

votes

**1**answer

190 views

### Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a ...

**4**

votes

**0**answers

153 views

### Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...

**0**

votes

**0**answers

39 views

### convergence of empirical distribution of random vectors

Given
(a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal
entries $A_{ij}\sim \mathcal N(0, 1/n)$; and
(b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...

**2**

votes

**4**answers

17k views

### How do I convert a uniform value in [0,1) to a standard normal (Gaussian) distribution value?

I have uniform value in [0,1). I'd like to transform it into a standard normal distribution value, in a deterministic fashion.
What I'm confused about with the Box-Muller transform is that it takes ...

**2**

votes

**1**answer

108 views

### Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities

Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...

**2**

votes

**1**answer

112 views

### Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...

**0**

votes

**1**answer

172 views

### How are epidemic models simulated in case of mobility?

I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...

**0**

votes

**1**answer

212 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to ...

**9**

votes

**2**answers

4k views

### Coin Pusher Game

While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...

**3**

votes

**2**answers

473 views

### An Upper Bound for the Average of Top Order Statistics

The following problem arises when we try to bound the expected offline optimal value of a simple online assignment problem with random values and unit weights, by its deterministic approximation.
The ...

**0**

votes

**0**answers

23 views

### Mixture model: optimization vs regression

Consider a sample $\mathcal D = \{T_n\}_{n=1}^N$ of independent random variables, s.t.:
$$
p(T_n) = p_n(T) = \sum _{m=1}^Mp_n(\mathcal C_m)p_n(T\mid \mathcal C_m) = \sum _{m=1}^Mw_{nm}q_m(T)
$$
I will ...

**0**

votes

**0**answers

46 views

### Stationarity and Regression

apologies this might turn out to be a bit on the simple side, but I've been thinking this through and haven't quite found the right approach.
Suppose I have a bunch of time series (say ...

**24**

votes

**12**answers

10k views

### why is it so cool to square numbers? (in terms of finding the standard deviation)

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...

**1**

vote

**1**answer

56 views

### Rate-Distortion theory: What is the distribution of distortion on an optimal encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error ...

**6**

votes

**4**answers

101 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq ...