Tagged Questions

0
votes
2answers
36 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
43 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
79 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
47 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
46 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 $\Omega$ ...
1
vote
1answer
68 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
30 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
42 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
76 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
97 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
127 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
0answers
113 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
122 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 ...
0
votes
1answer
651 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
222 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 $A$ ...
0
votes
0answers
97 views

Normalization/comparison-when the observed cardinality is a random variable, over finite sets

Given, $p_{ij}=\frac{|A_i|+|A_j|}{|A_i\cup A_j|}$ for sets $A_i$,$A_j$ $\forall i\not=j \in [1,n] $ and given the fact that $|A_i \cap A_j|>0$ for all $A_i,A_j$: it is clear that $p_{ij}$ > 1. ...
3
votes
1answer
121 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
163 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
476 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
123 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
191 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). ...