1
vote
1answer
28 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
2
votes
0answers
23 views

Alternatives to Fisher information

The Fisher information matrix is defined as the following: $$\mathcal{I}(\theta)=E[(\frac{\partial \log f(x;\theta)}{\partial \theta})^2]=-E[\frac{\partial^2 \log f(x;\theta)}{\partial \theta ...
0
votes
0answers
24 views

Measure Theory vs. Decision Theory - problem classification

I am having trouble classifying my problem, and I am seeking some guidance on book advice. I don't know if I have measure-theory problem and/or a decision-theory problem (or other field). I want to ...
0
votes
2answers
42 views

What is the meaning of Common Support here

I am reading a notes in statistical inference, and I am constantly being confused about the term 'common support', i hardly find any definition of this,here is an example, 'Suppose S is a space of ...
6
votes
2answers
143 views

What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
0
votes
1answer
63 views

Integrating with the indicator function of some random variables

I have the following problem. Suppose $X_1, X_2, \cdots, X_{n} $ are independent random variables which are all distributed in $U [ a, b]$. Define a new random variable $$N_{x, y} = \sum_{k=1}^y ...
1
vote
3answers
57 views

Gaussian random variable in $\mathbb{R}^n$ question

Let $X=(X_1,...,X_n)$ is a Gaussian random variable in $\mathbb{R}^n$ with mean $\mu$ and covariance matrix $V$. I want to show that we can write $X_2$ in the form $X_2 = aX_1 + Z$, where $Z$ is ...
1
vote
1answer
102 views

Show $\psi$ and $\Delta$ are identifiable

Let $X_1$,...,$X_m$ be i.i.d. F, $Y_1$,...,$Y_n$ be i.i.d. G, where model {(F,G)} is described by $\hspace{20mm}$ $\psi$($X_1$) = $Z_1$, $\psi$($Y_1$)=$Z'_1$ + $\Delta$, where $\psi$ is an unknown ...
0
votes
1answer
19 views

About the variance and a connected integral

Is given a positive measure $\mu$ such that $\mu(\mathbb{R}^+)<+\infty$. Is it generally true that: $$\int_0^\infty x^2 d \mu < + \infty \space\space ^{?}\iff^{?} \int_0^\infty \left(x ...
3
votes
2answers
38 views

$L^1$ norm of product of independent random variables

I am trying to show that $\|XY\|_1 = \|X\|_1\|Y\|_1$ for $X,Y$ independent random variables, where $\|X\|_1 = \int{|X| d\mathbb{P}}$. I have a feeling that this result is intuitive, but could anyone ...
1
vote
1answer
54 views

A simpler proof of evaluating the limsup of a standard normal sequence

Let $\{X_n\}$ be a sequence of i.i.d. random variables following the standard normal distribution. We need to prove that $$\limsup_n\left(\frac{X_n}{\sqrt{2\log n}}\right)=1$$ This is quite easy to ...
1
vote
1answer
85 views

Prove random vector with covariance matrix $\Sigma$ has non-degenerate distribution iff $\Sigma$ is positive definite

Let $X$ denote a $d$-dimensional random vector with covariance matrix $\Sigma$ satisfying $|\Sigma| < \infty$. Prove $X$ has non-degenerate distribution iff $\Sigma$ is positive definite. ...
4
votes
1answer
129 views

Question from “An introduction to measure theory” by Terence Tao [duplicate]

If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} < \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is ...
0
votes
0answers
39 views

Exchangeability of R.V.'s

Let $X_0$,$X_1$,....,$X_n$ denote i.i.d. real valued r.v.'s. For every definition of $Y_1$,...,$Y_n$ below, say whether or not $Y_1$,...,$Y_n$ are exchangeable and justify your answer. $Y_j$ = ...
0
votes
1answer
72 views

If a sequence of random matrices converge in probability, do their elements also converge?

Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ ...
2
votes
1answer
83 views

Prove that $ \mathsf{E}[g(X)] = \int_{- \infty}^{\infty} G(t) \varphi(t) \, d{t} $.

Problem Let $ X $ be a real-valued random variable with characteristic function $ \varphi $. Suppose that $ g: \mathbb{R} \to \mathbb{R} $ satisfies $$ \forall x \in \mathbb{R}: \quad g(x) = ...
0
votes
1answer
133 views

Characteristic function

Question: Let $X_1$ and $X_2$ denote independent real-valued random variables with distribution functions $F_1$, $F_2$, and characteristic functions $\varphi_1$, $\varphi_2$, respectively. Let Y ...
2
votes
2answers
116 views

Characteristic function of $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$

Let X denote a real-valued random variable with an absolutely continuous distribution with density function $p(x) = \frac{1}{2} e^{-|x|}$, $-\infty < x < \infty$. Find the characteristic ...
2
votes
1answer
41 views

Finding $E(X^r\mid Y)$ of an exponential function

Let $(X,Y)$ denote a two-dimensional random vector with an absolutely continuous distribution with density function $$p(x,y) = \frac{1}{y}\exp(-y), \qquad 0 < x < y < \infty.$$ Find ...
4
votes
1answer
98 views

Stable Convergence in Distribution - Martingale CLT problem (Lemma 3.1 in Hall and Heyde)

I'm studying Hall and Heyde's (1980) book on martingale limit theory. In their Lemma 3.1, they seem to use the identity \begin{equation} \mathrm{E}\left({\exp{(itZ)}\mathbb{1}_A}\right) = ...
0
votes
1answer
33 views

$p(X = c)=1$ then $E(X) = c$

Let $X$ be an aleatoric number. If $X \equiv c$ then $E(X) = c$. But if $p(X = c)=1$ how can I show, starting from the axioms of expectation or easy properties (e.g. Chebyshev inequality), that $E(X) ...
0
votes
0answers
48 views

How to prove that if $X_n \to X$ in distribution then $X_n = O_p(1) $

The title says it all: How to prove that if $X_n \to X$ in distribution then $X_n = O_p(1) $ My idea was to approach this with some measure theory: If not $X_n = O_p(1) $, then there exists an ...
1
vote
1answer
65 views

Convergence almost everywhere?

Consider the following two statements about a random sequence $X_n$: (1) $X_n \stackrel{a.e.}{\rightarrow} X$. (2) $\mathrm{P}\{|X_n-X|>\epsilon, \ i.o.\} = 0, \ \forall \epsilon>0$. (a.e. ...
0
votes
2answers
282 views

densities being absolutely continuous wrt Lebesgue measure

I'm reading an article with an assumption similar to: "The density $f(.)$ exists and is absolutely continuous with respect to Lebesgue measure". I don't understand this assumption because $f$ is not ...
1
vote
1answer
72 views

How to understand $\frac{dP}{dQ}$, where $P, Q$ denote two distributions?

I am currently reading a paper named Estimating Individualized Treatment Rules Using Outcome Weighted Learning by Zhao et al., where they wrote an equation $$\frac{dP^D}{dP}=\frac{I(a=D(x))}{P(A=a)}$$ ...
1
vote
2answers
143 views

One question regarding r.v independence

I just encounter independence in a Statistics course, I get stuck in this question for a long time..any help will be extremely appreciated. First one is, if $X_1, X_2, X_3...X_k$ (finitely many) are ...
1
vote
1answer
61 views

Necessary condition for pairwise sufficient statistic [duplicate]

I'm struggling to prove the following. If $T:\left(X,\mathbf{A}\right)\rightarrow\left(Y,\mathbf{B}\right)$ is a pairwise sufficient statistic for a set $\left\{\mu_0,\mu_1,\mu_2\right\}$ of three ...
0
votes
2answers
66 views

Is the limit of a sequence of B-measurable functions itself B-measurable?

Let $\left(\Omega,\mathcal{A}\right)$ be a measurable space and let $\mathcal{B}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Let $g,f_1,f_2, f_3,\dots$ be real-valued functions with domain $\Omega$ ...
2
votes
1answer
335 views

Book on Measure Theoretic Statistics

I'm looking for a book, preferably a good one, on statistics from a rigorous, measure theoretic point of view. Ideally, this book should be introductory in nature and cover no more nor less than a ...
0
votes
0answers
39 views

Time Series: existence of moments $\Rightarrow$ existence of distribution?

This might come to you as a bit silly, because normally we are used to the vice-versa question. But here is what I have: a nonlinear time-series model, for which I can derive by infinite backwards ...
0
votes
1answer
60 views

Coefficient of variation

Let $Q=\{q_1,\ldots,q_n\}$ ($n\in\mathbb N$, $n>1$) a collection of elements and $d:\,Q\times Q \longrightarrow \mathbb R^+$ a distance between a pair of elements (as a measure of similarity). ...
1
vote
1answer
126 views

Conditional Expectation and identically independent distributed random variables.

I have a problem that can be resolved if i show that $$E(\varepsilon_k\mid\sigma(\varepsilon_1,\ldots,\varepsilon_{k-1}))=E(\varepsilon_k)$$ where $\varepsilon_1,\ldots,\varepsilon_k$ $\sim ...
2
votes
0answers
178 views

conditional expectation and order statistic

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space.Let $\ X=(X_1,..,X_n)$ a random vector, with$\ n$ independents random variables whose law is $\mu$ on $\mathbb{R}$. We define ...
3
votes
1answer
184 views

Relationship between two random variables?

What is the relationship between a random variable obeying the subexponential distribution defined here and a random variable $X$ satisfying $P\left(\left|X\right|>t\right)\le\alpha e^{-\beta t}$ ...
0
votes
1answer
137 views

Conditional expectation independence

I'm working on some statistics project and am not getting further because of some stupid prediction that doesn't want to be 0. That's why I was wondering if maybe the following holds: Suppose we have ...
0
votes
1answer
143 views

Measure-theoretic view of expectation of a Bernoulli sequence

Problem: I have a good understanding of basic Bernoulli and Binomial RVs, but this was foundational work in statistics. I am attempting to try and apply my (minimal but increasing) knowledge of ...
1
vote
1answer
1k views

Weighted average vs. weighted mean

Is there a formal difference between weighted average and weighted mean? I get corrected to the latter if I type in the former in wikipedia, and then there is a lot of stuff about the name "average" ...
3
votes
2answers
309 views

Show that $P$ is Countably additive problem.

Let $F$ be the field consisting of the finite and the cofinite ($A$ is cofinite if $A^c$ is finite) sets in an infinite and countable $\Omega$, and define $P$ on $F$ by taking $P(A)$ to be $0$ if $A$ ...
3
votes
1answer
148 views

expectation and group of permutations

Let $r_i, i=1,\ldots,m$ be random variables with $P(r_i=1)=P(r_i=-1)=1/2$. let $b_i, i=1,\ldots,m$ be real numbers. I should calculate $E\left(|\sum_{i=1}^m b_ir_i|^4| \sum_{i=1}^m r_i=0\right)$ using ...
1
vote
1answer
193 views

Definition of rate of convergence for a sequence of measurable mappings

If a sequence of measurable mappings defined from a measure spaces to another measure space converges in different modes (see Wikipedia and John Cook's site), I wonder if there are some concepts ...
2
votes
2answers
687 views

Monotonic transformation of continuous random variable are continuous

This appeared as a throwaway statement in a proof - that a strictly monotonic (increasing) transformation of a continuously distributed random variable (I am assuming that this means that the ...
3
votes
1answer
165 views

Variational Distance vs. maximum norm

Suppose I have vector $x^t \in \mathbb{R}^n, x_i > 0$ that is a random variable in $t$. I define a measure $D(x) := \max_{i,j} |x_i - x_j|$, which essentially is the maximum discrepancy of any two ...
0
votes
1answer
329 views

Does this induce a proper probability distribution on the space of covariance matrices?

Say I define a probability distribution on $P$ dimensional symmetric matrices such that the diagonals are strictly positive but the off diagonals are unrestricted (except for the symmetry constraint). ...