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.

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11
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105 views

Number of moves necessary to solve rubic's cube by pure chance

Suppose, random moves are made to solve rubic's cube. A move consists of a $90$-degree-rotation of some side. The starting position is also random. What is $E(X)$, where $X$ is the number of moves ...
11
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242 views

Kähler Geodesics

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

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
9
votes
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816 views

Maximum likelihood and sufficient statistics

$$f_T(t;B,C) = \frac{\exp(-t/C)-\exp(-t/B)}{C-B}$$ where our mean is $C+B$ and $t>0$. so far i have found my log likelihood functions and differentiated them as follows: $$dl/dB = ...
7
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80 views

Hottest Days of The Year

Recently, there has been much talk in the media of it being the hottest day of the year so far. It has always seemed to me that there are likely many more of these in the northern hemisphere than the ...
7
votes
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586 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 ...
7
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311 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, $\phi_i$ ...
6
votes
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80 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
6
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95 views

Finding an upper bound for $\frac{d}{d\theta}\beta^*(\theta)|_{\theta=\theta_0}$

Suppose that a random variable X has a distribution depending on a parameter $\theta$, $\theta \in \Theta$, and consider a test of hypothesis $H_0: \theta = \theta_0$ versus the alternative $H_1: ...
5
votes
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74 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 & ...
5
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100 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
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58 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
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135 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} ...
5
votes
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91 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: ...
5
votes
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85 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
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316 views

Why is the partition function able to describe the whole system?

No matter what the real system or subject is, if there is a partition function $Z$, then these kind of identities hold $$\langle X\rangle=\frac{\partial}{\partial Y}\left(-\log Z(Y)\right).$$ If one ...
5
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169 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. ...
4
votes
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17 views

A good, self-study statistical computing book

I'm looking for a book an introductory statistical computing that has proofs for the methods as well as examples. I'd like proofs that are about the same level as (or lower than) proofs in Statistical ...
4
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22 views

Intuitive explanation of requirement for achieving the Cramer Rao Lower Bound

this question relates to the requirement for achieving CRLB. I know that for a random sample $Y_1, \ldots, Y_n$, an estimator $U$ of $g(\theta)$ is MVUE (i.e. it is unbiased and also ...
4
votes
0answers
82 views

Can the limit of the MSE of an estimator be infinity?

Is it ever possible for the limit of the MSE of an estimator be infinity? I was doing an exercise and it turns out that the estimator is consistent but the limit of the MSE is infinity, so I am ...
4
votes
0answers
47 views

An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
4
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76 views

Using Jensen's inequality to prove the Cauchy distribution has no mean

I can see that there is no mean because $\int x / \pi(1+x^{2})$ does not converge from -inf to inf. But my prof hinted at using Jensen's inequality for the proof. $$f(E(X)) \le E(f(X))$$ How can I ...
4
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44 views

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for ...
4
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68 views

Exponential distribution unbiased estimator

Let $$X_1, \ldots, X_n \overset{iid}{\sim} Exp(\lambda), \quad \lambda > 0$$ The Maximum-Likelihood-Estimator is given by $$\widehat{\lambda} = \frac{1}{\frac{1}{n}\sum_{i=1}^{n}{X_i}} = ...
4
votes
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94 views

Probability help! Am I even doing this right?

I am really bad with probability, so I just want some explanations and help with this problem (and probably many more to come!) and I also want to know if I am on the right track. Thank you! Lyme ...
4
votes
0answers
87 views

Expectation of the absolut value of the determinant of a random matrix

Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable $$X := |\det A|$$ Can ...
4
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53 views

Bayesian linear regression cost function

I am studying classification using linear regression . Now, I want to map it in Bayesian regression. Let talk about binary classification using linear regression again. Assume that I have a set ...
4
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43 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
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0answers
150 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
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182 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
81 views

Multiplicative version of Mcdiarmid's inequality?

Suppose you have $n$ i.i.d. random variables taking values in $\{0,1\}$, and $X$ represents their sum. Then you can use a Chernoff bound to control the deviation of $X$ from its expectation. The ...
4
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578 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
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145 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
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0answers
164 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
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0answers
460 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 ...
3
votes
0answers
19 views

Regression with many discrete and continuous predictors and few rows

I want to do regression on a dataset. It has one continuous dependent variable that I want to predict. It has many categorical and some continuous predictors. It only has a few rows. A simplified ...
3
votes
0answers
28 views

How to perform nonlinear regression with regressors affected by gaussian error?

I am trying to calibrate a sensor and I have a data set consisting of several observations of a 3-dimensional vector $X_i$, with $X_i=w_i + \epsilon_i$ where $w_i$ is the value that the sensor ...
3
votes
0answers
47 views

Sufficient statistics and UMVUE for joint poisson, bernoulli

Given a pair $(X,Y)$ of r.v.s such that: $$X \sim \text{Poisson}(\lambda)\quad \text{and}\quad Y \sim B(\frac{\lambda}{1+\lambda})$$ with $X,Y$ independent, determine a one-dimensional ...
3
votes
0answers
45 views

On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
3
votes
0answers
41 views

calculating $E(X_{(i)}| \sum_{i=1}^5 X_i)$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
3
votes
0answers
43 views

Weak convergence of order statistics

I've encountered the following problem: Let $U_1,...,U_n$ be iid uniformly distributed on $[0,1]$ and let $U_{n(k_n)}$ denote the $k_n$-th order statistic where $k_n$ is chosen, s.t. ...
3
votes
0answers
40 views

What is the appropriate statistical test to see if a quantity has been distributed differently into discrete bins?

Say I have $10^6$ balls, $3$ bins $A,B,C$, and $2$ machines $X$ and $Y$ that distribute the balls into the bins according to an internal set of rules (i.e. a probability distribution). If I run both ...
3
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0answers
36 views

How big should the sample size be to disprove this article?

There is a new poker computer that is claimed to be unbeatable. http://www.theguardian.com/science/2015/jan/08/poker-program-cepheus-unbeatable I beat this computer on my first try today but my ...
3
votes
0answers
55 views

when can I say there is a relationship between events?

I am by no means a math expert, but I am analyzing very large weather data data for a computer science course .I am taking away all the weather related issues out of the analyzing and just looking ...
3
votes
0answers
42 views

What distribution would describe this?

I start with 100 eggs, 10 of them being broken. I randomly select eggs without replacement until they are all split into baskets of 10 eggs each. Here's what I know: Best case scenario all 10 bad ...
3
votes
0answers
37 views

Rao-Cramer lower bound regularity condition and dominated convergence

Let $(\mathcal{X}, \mathcal{F}, (\mathbb{P}_\vartheta)_{\vartheta \in \Theta})$ be a statistical model dominated by a sigma-finite measure $\mu$ with Likelihood-function $L(\vartheta, x)$ which is ...
3
votes
0answers
49 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
3
votes
0answers
207 views

How does 2D kriging interpolation work?

I have a grid of points Example ...
3
votes
0answers
58 views

Estimating parameters for a binomial

First of all I'd like to precise that I'm not an expert of the subject. Suppose to have two random variables $X$ and $Y$ that are binomial, respectively $X\sim B(n_1,p)$ and $Y\sim B(n_2,p),$ note ...
3
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0answers
27 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...