For mathematical proofs or derivations of results.

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18 views

How is Central Limit Theorem affected when hypothesis space is restricted?

Some people told me that Central Limit Theorem and other convergence theorems break when we restrict space of distributions to those that are effectively computable. How true is this statement, and ...
4
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1answer
42 views

Partition data into two sets such that the difference of their variance is minimal

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ ...
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1answer
30 views

Conceptual proof that conditional of a multivariate Gaussian is multivariate Gaussian

I understand the arithmetic derivation of the PDF of a conditional distribution of a multivariate Gaussian, as explained here, for example. Does anyone know of a more conceptual (perhaps, co-ordinate ...
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4answers
40 views

Variance of linear combinations of correlated random variables

I understand the proof that $$(aX+bY) = a^2Var(X) +b^2Var(Y) + 2abCov(X,Y), $$ but I don't understand how to prove the generalization to arbitrary linear combinations. Let $a_i$ be scalars for ...
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1answer
38 views

How to prove the properties of penalized likelihood estimator in Fan and Li (2001) paper

I'm reading through Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties". In Page 2 near bottom right corner, they proposed three properties that a ...
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62 views

Mathematics / Statistics [closed]

Today I was thinking how I'm bad at making statements, I know who does statistical cares more about the practical part, but knowing how to do demonstrations in the theoretical part is important? I ...
4
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1answer
41 views

Generalized inverse solution to system of linear equations proof

I'm going through a set of course notes for an introduction to the theory of linear models class at my university. Unfortunately, the professor who wrote this note set is no longer at this school, and ...
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2answers
88 views

Is $\text{Cov}(|a|,|b|)\geq \text{Cov}(a,b)$?

The above seems intuitively true, (where $|a|$ refers to the absolute value of $a$), but I'm struggling to prove it - would be very grateful for either a proof or a reference.
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1answer
39 views

Proof for the sampling variance of the Neyman Estimator

I'm going through Imbens and Rubin's new book and I just for the life of me can't figure out 1 minor detail in their proof for the sampling variance of the Neyman estimator $\bar{Y}^{obs}_{t} - ...
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1answer
105 views

Transformation of a random vector

I am very familiar with the formula $$f_{Y}(y) = f_{X}(x(y))\left|\dfrac{\text{d}}{\text{d}y}[x(y)]\right|$$ where $X$ and $Y = g(X)$ is a one-to-one transformation of $X$. (I use $x(y)$ to mean "$x$ ...
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2answers
131 views

A strange step on a proof about the distribution of quadratic forms

The following theorem comes from the 7th edition of "Introduction to Mathematical Statistics" by Hogg, Craig and Mckean and it concerns the necessary and sufficient condition for the independence of ...
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0answers
42 views

Proving a “well-known” result regarding the distribution of a normally distributed random variable

In an important project work, I would like to include a "proof" of the following, but have unfortunately been unable to readily compute it myself. I am aware that this is a flaw on my part, but ...
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1answer
44 views

Is a covariance matrix composed of matrixes derived from separate samples guaranteed to be positive definitive?

I have two samples that partially overlap on the variables they describe. The samples are taken from more or less the same population, and show similar values on the overlapping variables. Based on ...
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1answer
58 views

Don't understand identity in proof for unbiased sample variance

Wikipedia gives the following proof why to use Bessel's correction for the unbiased sample variance: \begin{align} E[\sigma_y^2] & = E\left[ \frac 1n \sum_{i=1}^n \left(y_i - \frac 1n ...
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1answer
22 views

Mini-batch Stochastic Gradient Descent: Question about proof in paper

I'm going through the JMLR paper on distributed mini-batch Stochastic Gradient Descent and I have a question about a part of the proof. They use the notation: $$ F(w) = \mathop{\mathbb{E}}_{z}[f(w, ...
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1answer
31 views

Random variables, inequalities involving them and medians thereof (II)

I have a continuous function $A$ for which it holds that: $$x\in S\implies A(x)<0$$ $$x\in T\implies A(x)>0$$ Of course, $S$ and $T$ don't intersect. Now consider two continuous ...
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1answer
57 views

Random variables, inequalities involving them and medians thereof

I have three facts which I know to be true and I am wondering if together they logically imply a fourth one (or not!). Consider $F_x$ and $G_y$ two continuous, uni-modal distributions on ...
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0answers
19 views

Comparing regressions: usual regressor vs regressed-out regressor

I'm comparing the regression coefficients between 2 models: Model 1: $$ Y = \beta_1X_1 + \beta_2X_2 + u $$ Model 2: $$ Y = \beta_1'X_1' + \beta_2'X_2 + v $$ where $X_1' = ...
3
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0answers
89 views

Proof of Kolmogorov-Smirnov test

Could someone provide me a reference, preferably a book, where I can find detailed proofs and explanations of the Kolmogorov-Smirnov test (including the two-sample variant) and the derivation of the ...
0
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1answer
29 views

Proof that points change clusters less often as iterations proceed in k means

Is there a way that to prove the following: In k-means clustering, as the iterations proceed, the data points tend to stay in their existing clusters, overall, because the replacement of the centroid ...
5
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1answer
54 views

“Only if” and “if” direction in Kolmogorov's Existence Theorem

The title of this question sounds somewhat grand, but all I'm asking is whether there's a typo in the book I'm reading. The book is "A First Look at Rigorous Probability Theory," by Jeffrey Rosenthal ...
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1answer
72 views

Time-series and autocorrelation inequality

I am having problems proving for a weakly stationary process $\{X_t : t\in T\}$: $\rho_X(2)\geq 2 (\rho_X(1))^2-1$ where $\rho_X(j)=corr(X_t, X_{t+j})$. So far I have shown that $-1\leq ...
3
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1answer
107 views

Proving a distribution is a member of the simple exponential family

Does anyone have any tips/ideas/method for proving that a distribution is a member of the simple exponential family (SEF)? Or is the process unique to each distribution? For example, I am trying to ...
2
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0answers
65 views

Issue with the proof of PCA

I found a very nice PCA proof over here PCA_proof and I'm trying to understand it (I don't know what Langrange multipliers are so I'm trying my best). From the second page of the previous link, ...
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3answers
90 views

Proof that the probability of one RV being larger than $n-1$ others is $\frac{1}{n}$

This is a follow-on from my previous question about samples from a distribution. Suppose $X_1 \ldots X_{n-1}, X_n$ are random variables all following some fixed distribution $D$. How do I prove that ...
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0answers
129 views

What's the Mode of a Bivariate Poisson Distribution?

I have been looking at the bivariate Poisson distribution and I am wondering if there is close form expression for the mode of this distribution. I know the mode of the univariate Poisson distribution ...
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22 views

Are these three different ways of expressing the optimal value function $V^*$ the same? (reinforcement learning)

My question didn't really fit on the title but its the following are the three following equations actually the same: $$V^*(s) = \underset{\pi}{max}V^{\pi}(s)$$ and $$V^*(s)=R(s)+ \underset{a \in ...
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1answer
391 views

Expectation, Variance and Correlation of a bivariate Lognormal distribution

If $Y \sim N(\mu,\sigma^2)$ is normally distributed, then $X=\mathrm{e}^Y$ is lognormally distributed. To get the log-$\mu$ and log-$\sigma$ of this lognormal distribution you calculate $$\sigma^2 = ...
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1answer
21 views

prove that SVM chooses the bisecting line of nearest support vectors?

I have trouble solving the problem 3.18 from "pattern recognition" by "Sergios Theodoridis, ‎Konstantinos Koutroumbas" the problem is : Show that for the case of two linearly separable classes the ...
5
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1answer
167 views

Explanation of formula for median closest point to origin of N samples from unit ball

In Elements of Statistical Learning, a problem is introduced to highlight issues with k-nn in high dimensional spaces. There are $N$ data points that are uniformly distributed in a $p$-dimensional ...
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106 views

proving regression with dummy variables gives same estimates as separate models

Let ($x_{i1}$, $x_{i2}$, ..., $x_{id}$, $y_i$), $i = 1,..., n$ be an i.i.d. multivariate sample and furthermore assume each observation belongs to one of possible $K$ categories. Assume for each ...
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1answer
131 views

Show that if $X \sim Bin(n, p)$, then $E|X - np| \le \sqrt{npq}.$

Currently stuck on this, I know I should probably use the mean deviation of the binomial distribution but I can't figure it out.
2
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1answer
79 views

How to prove Bernoulli distribution belongs to the exponential family

According to a book, a distribution belongs to the exponential family if it can be written in the form of I wrote the Bernoulli distribution as $\exp\Big(y \log\,[{\mu}/{(1-\mu)}] + ...
13
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2answers
415 views

Uniform random variable as sum of two random variables

Taken from Grimmet and Stirzaker: Show that it cannot be the case that $U=X+Y$ where $U$ is uniformly distributed on [0,1] and $X$ and $Y$ are independent and identically distributed. You should not ...
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2answers
277 views

Is there an elegant/insightful way to understand this linear regression identity for multiple $R^2$?

In linear regression I have come across a delightful result that if we fit the model $$E[Y] = \beta_1 X_1 + \beta_2 X_2 + c,$$ then, if we standardize and centre the $Y$, $X_1$ and $X_2$ data, ...
2
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1answer
22 views

Gradient of expected transformed outcome with respect to distribution parameters

If $x$ has density $p(x \mid \theta)$ why is it true that for some function $f$, $$\begin{align} \nabla_{\theta}\mathbb{E}[f(x)] &= \mathbb{E}[f(x) \nabla_\theta p(x)] \end{align}$$
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1answer
39 views

Sample covariance mean-corrected vector proof

Prove that $$(n-1)S = X^TX -{1\over{n}}(X^T\vec1)(\vec1^TX) = X^TX-n\vec{\bar x}\vec{\bar x}^T$$ My attempt so far goes like this $$S = {1\over{n-1}}X_m^TX_m$$ Edit: Where $X_m$ is the ...
2
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1answer
74 views

Rewrite instrumental variables estimator into formula with covariances?

In the book Microeconometrics of Cameron and Trivedi, they write the IV estimator as $\widehat{\beta}_{IV} = \frac{Cov[z,y]}{Cov[z,x]}$, formula (4.49) on p. 99. They say that they derived this from ...
3
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2answers
126 views

Poisson as a limiting case of negative binomial

I was reading "Maximum Likelihood Estimation for the Negative Binomial Dispersion Parameter" by Walter W. Pieogorsch, and in the intro it says the Poisson distribution is a limiting case of negative ...
8
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1answer
112 views

Is the negative binomial not expressible as in the exponential family if there are 2 unknowns?

I had a homework assignment to express the negative binomial distribution as an exponential family of distributions given that the dispersion parameter was a known constant. This was fairly easy, but ...
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2answers
301 views

Proving the LATE Theorem of Angrist and Imbens 1994

Assume we have a binary instrument $Z_i$ which can be used to estimate the effect of the endogenous variable $D_i$ on the outcome $Y_i$. Suppose the instrument has a significant first stage, it is ...
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1answer
72 views

Why constrain mean and standard deviation when proving Gaussian is maximum differential entropy pdf?

I'm reading Bishop's Pattern Recognition and Machine Learning. In chapter 1.6: Information Theory (page 53) when trying to derive the maximum differential entropy pdf from the definition of continuous ...
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0answers
93 views

Average within-cluster distance using divisive clustering

I have to prove that the average within-cluster distance for 10 data points cannot increase when going from 1 cluster to 2 clusters (divisive clustering). Intuitively, it seems obvious that this is ...
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3answers
154 views

Proving Linear Estimator (beta) is BLUE?

In the book Statistical Inference pg 570 of pdf, There's a derivation on how a linear estimator can be proven to be BLUE. I got all the way up to 11.3.18 and then the next part stuck me. After ...
3
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1answer
49 views

Equivalence of random effects via likelihood and smoothed splines

Some fake data: X = runif(1000) ff = rep(1:10,100) E = rnorm(1000) y = x+e+f f = as.factor(ff) When you fit a model like ...
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2answers
40 views

Stuck on a proposition

Suppose four numbers $\{a,b,c,d\}$ where $a$ and $c$ random variables from a continuous distribution with support on $\mathbb{R}$. Does $b\neq d$ imply $|a−b|−|c−d|+a−c\neq 0$ almost surely? I ...
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6answers
2k views

How can I analytically prove that randomly dividing an amount results in an exponential distribution (of e.g. income and wealth)?

In this current article in SCIENCE the following is being proposed: Suppose you randomly divide 500 million in income among 10,000 people. There's only one way to give everyone an equal, 50,000 ...
4
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0answers
535 views

Cantelli's inequality proof

I am trying to prove the following inequality: EDIT: Almost immediately after I posted this question, I discovered that the inequality I am being asked to prove is called Cantelli's inequality. When ...
1
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1answer
89 views

Proof of asymptotic variance

How do you prove that $X_n - E[X_n] = O_p(\sqrt{Var(X_n)})$ It's used in my textbook and I don't know where they get it from.
8
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1answer
826 views

Proof / derivation of skewness and kurtosis formulas

Can anyone explain to me where the formula of skewness or kurtosis comes from? (I mean its derivation.) What's the logic behind it? Who proved it?