Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

learn more… | top users | synonyms

11
votes
0answers
175 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
9
votes
0answers
64 views

Statistics Primer for the Unwary Mathematician

I have a new position in a biology department (after being housed in a maths department) working on cognitive and population modeling. People in my lab are asking for help with applying statistical ...
9
votes
0answers
187 views

Does $\pi$ satisfy the law of the iterated logarithm?

It is widely conjectured that $\pi$ is normal in base $2$. But what about the law of the iterated logarithm? Namely, if $x_n$ is the $n$th binary digit of $\pi$, does it seem likely (from ...
7
votes
0answers
253 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
5
votes
0answers
65 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
5
votes
0answers
66 views

Random point distribtion

How to generate numerically a set of random points $(x_1,y_1), (x_2,y_2),\cdots, (x_N,y_N)$ such that the pair-wise distances $d = \sqrt { (x_i-x_j)^2 + (y_i-y_j)^2}$, for all $ 0<i\le N, ...
5
votes
0answers
36 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray ...
5
votes
0answers
104 views

How to determine if binomial events are independent?

I have a sequence of binary experiment results, something like 1100010000100... My first hypothesis is that these events are independent, but I'd like to know if there is some way to test this. I ...
5
votes
0answers
70 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
5
votes
0answers
431 views

Idempotence and the Rao–Blackwell theorem

Original question: In the Wikipedia article on the Rao–Blackwell theorem, we read: In case the sufficient statistic is also a complete statistic, i.e., one which "admits no unbiased ...
4
votes
0answers
44 views

Decrease of entropy when iterating a random discrete function

Let $m$ be a positive integer. Let $S$ be the set of non-negative integers $x$ less than $m$, with $|S|=m$. Let $X_0$ be the discrete uniform distribution over $S$, with $P(x)=\begin{cases} 1/m & ...
4
votes
0answers
79 views

Kalman filtering correlated measurements

I would like to run a Kalman filter over a set of measurements which may (will) be correlated. Essentially, each new measurement contains (say) 90% of the same information from the previous ...
4
votes
0answers
26 views

Statistics of Lists

To start, let me say i am a programmer and not a math wiz, so this question might be very simple. I have a data set of prices that sort of looks like this ...
4
votes
0answers
72 views

Angle between $(X,Y)$ and $(E(X), E(Y)) $ where X and Y are independenyt random variables.

Suppose that X and Y are two independent random variables with known (different/same) probability distribution functions. Now consider the vector $(X,Y)$, I want to find the angle between $(X,Y)$ and ...
4
votes
0answers
105 views

Clarification in a paper

This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari. In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
4
votes
0answers
143 views

Using Bernoulli distribution approximate the $q$-th moment

Let $x$ be vector in $R^n$. Let $\pi(⋅)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Using Bernoulli (or maybe some other) distribution ...
4
votes
0answers
117 views

Existence of a UMP test for two binomial random variables

Let $X$ and $Y$ be independent random variables, distributed as Binomial($p, n$) and Binomial($p^2, m$), respectively. Does a UMP test (for fixed level $\alpha$) exist for: $H_0: p \leq p_0 \text{ ...
4
votes
0answers
414 views

Conditional expectation of the product of two independent random variables

Suppose that $a$ and $b$ are independently distributed random variables, with means; $\mu_a$, $\mu_b$ and variances; $\sigma_{a}^2$, $\sigma_b^2$, respectively. Further, let $c=ab + e$, where $e$ is ...
4
votes
0answers
129 views

Varieties and Statistics

Consider a random variable $X$ that can take on the values $0,1$ and $2$. So we have $$p_i = P(X=i), \ i = 0,1,2$$ $$\sum_{i=0}^{2} p_i = 1$$ and $$0 \leq p_i \leq 1$$ So identifying a random variable ...
4
votes
0answers
125 views

NFC SuperBowl coin toss hot streak --> hypothesis testing and power calculation

There are many Q&A's on SE related to coin tossing - the simplest stochastic process. My Q is about relating mathematics and statistics to what in biomedicine and healthcare is termed "evidence" ...
4
votes
0answers
429 views

How did Target figure out a teen girl was pregnant before her father did?

First of all I do not have a mathematics degree only a B.S. in finance so please take that into account when writing an answer. Generally what type of mathematics is involved here? And specifically ...
4
votes
0answers
412 views

Chi Squared Distribution for Maximum Likelihood

I'm beginning to work in bioinformatics and have come across some papers that utilize chi-squared distributions to make a maximum likelihood selection. Particularly in the area using 'amplicons'. I've ...
4
votes
0answers
156 views

Critical exponents and point-wise convergence

A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. ...
3
votes
0answers
48 views

Intuition behind (statistical) completeness

I was wondering if any of the members of the MSE community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from ...
3
votes
0answers
45 views

Expected minimum of a finite random walk.

So I couldn't find any resource for how to calculate the expected minimum of a random walk. Since it is such the minimum of the random variables are actually not independent as they are cumulative ...
3
votes
0answers
49 views

Conceptual questions dealing with chi-square distributions?

In my textbook they have this inequality: $$ \chi_{1-\frac{\alpha}{2}}^2 < \frac{(n-1)s^2}{\sigma^2} < \chi_{\frac{\alpha}{2}}^2$$ which later becomes this statement: ...
3
votes
0answers
80 views

How to prove that max{$X_1$,…,$X_n$} is a sufficient statistic for the Uniform distribution on [a,b]

I am having a bit of difficulty with the following: I have the Uniform distribution on [a,b] where a is known and b is unknown and $b>a$, I'd like to show that the $T=max$($X_1$,...,$X_n$) is a ...
3
votes
0answers
19 views

Confidence intervals for the variance. What if data is not noramlly distributed?

I am writing an essay about confidence intervals for the variance and there is a lot of information available under assumption that our data is normally distributed, but there is not much said about ...
3
votes
0answers
40 views

The sum of Gaussian functions

Suppose there is a normal distribution and the Gaussian function is $F(x)=\exp(-c\|x-b\|^2)$ where $c$ is a constant and $x,b\in \mathbb{R}^N$, b means the mean value. ...
3
votes
0answers
184 views

Relation between standard deviation and mean in random processes

In a Poisson distribution the square of the standard deviation $\sigma$ is equal to mean $\mu$ ($\sigma^2=\mu$) and in a binomial distribution $\sigma ^2=\mu\,(1-p)$ (with $p$ the probability of ...
3
votes
0answers
44 views

How many items to acquire to get a full collection? (random chance involved)

The problem: My daughter wants to have a collection of certain Kinder Surprise toys. There're 10 different toys in this series. Assuming the chances of getting each toy are equal, how many Kinder ...
3
votes
0answers
58 views

Calculating that confidence that pairs of lightbulbs are independently illuminated.

So, you're sitting in a dark room, and on the far wall you see $n$ lightbulbs mounted above plaques numbered $1$ through $n$. There is a lightswitch on the arm of your chair. Every time you flip the ...
3
votes
0answers
86 views

Mathematics Courses for an Economist

I am an Economist and I am interested in further developing my mathematical knowledge and skills. I would like to get your opinions on the topics that I should cover and which are also important for ...
3
votes
0answers
84 views

SLLN of Markov chains .

Let $X_1$, $X_2$,... be a finite state, irreducible and aperiodic Markov chain with initial state $X_0=i$. It is known that \begin{equation} \mathrm{P}\Big\{\lim_{n\rightarrow\infty} ...
3
votes
0answers
107 views

Integral of a max function over a hypersphere; expected max z score

Is there a closed-form expression for the integral of max($x_1,..., x_n$) over the ($n-2$)-dimensional hypersphere, {$x \in \mathbf{R}^n$: $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i^2 = 1$}? I come ...
3
votes
0answers
90 views

Inequality of covariances between a bivariate normal vector and its indicator functions

Why holds for a standardized bivariate normal vector $Z:=(Z_1,Z_2)$ that \begin{equation} |\operatorname{cov}(Z_1,Z_2)|\geqslant |\operatorname{cov}(1\{Z_1\leq u\},1\{Z_2\leq u\})|? \end{equation} ...
3
votes
0answers
56 views

Convergence of a Subordinator.

Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace expoent given by $$ \Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda x}\right) \nu\left( ...
3
votes
0answers
85 views

Expected gap between two consecutive order statistics

Consider a random variable $X\sim_X(x)$ with $\operatorname{supp}{p_X}=[a,b]\subset{\mathbb R}$, and let $X_1, X_2, \ldots, X_n$ be $n$ $i.i.d.$ samples from $p_X(x)$, then we have their associated ...
3
votes
0answers
22 views

How to simulate a sequence of partial sums $(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),$ given some properties.

I want to generate/simulate a sequence of partial sums. $$(X)_n(w) = \sum\limits_{i=1}^n (Y_i(w)-Y_{i-1}(w)),\text{ for }1 \leq n \leq 100$$ Let $W$ be a random variable such that: $W \thicksim ...
3
votes
0answers
146 views

History of odds making in sports betting

Can anyone provide a reference to the history of odds making in sports betting? In many cases, certain odds are set and then adjusted as people make bets. However, I am having difficulty tracing the ...
3
votes
0answers
66 views

Simple question about histogram

Wikipedia article about histograms says following: A histogram is a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area ...
3
votes
0answers
139 views

Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
3
votes
0answers
221 views

Intuition for Fisher information metric

In statistical maniolds $S=\{p_\theta\}$,$\theta=(\theta_1,\dots,\theta_n)$, the Riemaanian metric usually defined is the Fisher information metric $$g_{ij}(\partial_i,\partial_j)=\int \partial_i(\log ...
3
votes
0answers
84 views

Who established the word “ Degree of freedom ” in statistics?

I wonder who is the first one that established and applied the word : "degree of freedom" in statistics? Why he/she need degree of freedom in the calculation of many statistical values?
3
votes
0answers
183 views

Multivariate Gaussian equivalent for a Gaussian integration identity.

For a one-dimensional x, $$\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}dx=\frac{1}{2}\int_{-\infty}^{\infty}e^{-x^{2}}dx$$ This can be shown through integration by parts. There is a good derivation of ...
3
votes
0answers
67 views

Does this count as a Monte Carlo simulation?

Let's say I have a group of robots that walk on a 11x11 grid of tiles in four directions, N, S, E, W, and each robot has different probability distribution functions that assign different ...
3
votes
0answers
273 views

On the empirical mean and variance of a Poisson i.i.d. sample

Let $X_1, X_2, \ldots, X_n$ be a random sample from a Poisson($\lambda$) distribution. Let ($\bar{X}$) be their sample mean and $s^2$ their sample variance. Show that ...
3
votes
0answers
80 views

Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...
3
votes
0answers
44 views

Help with Statistical Analysis

I have been asked to conduct a small survey and then analyse the results of the said survey for a class project. I decided to conduct an anonymous survey of voters for the 2012 presidential elections. ...
3
votes
0answers
139 views

Expected number of cumulative distinct values when sampling with replacement from a changing population over time

I'm trying to estimate the cumulative number of distinct values when sampling with replacement from a changing population of integers over time. Concretely (and forgive my awful notation here), I'm ...