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2
votes
0answers
52 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 ...
0
votes
0answers
34 views

Proof of probability density function? [closed]

Is there any website or any book which you can recommend to me for learning the proof of all PDF (distribution function)?
1
vote
1answer
31 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
votes
3answers
314 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 ...
9
votes
2answers
181 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
votes
1answer
20 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 $f$, $$\begin{align} \nabla_{\theta}\mathbb{E}[f(x)] &= \mathbb{E}[f(x) \nabla_\theta p(x)] \end{align}$$
1
vote
1answer
32 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
votes
1answer
60 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
votes
2answers
49 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 ...
5
votes
0answers
49 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 ...
6
votes
2answers
197 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 ...
1
vote
1answer
42 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 ...
1
vote
0answers
43 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 ...
0
votes
2answers
54 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
votes
1answer
39 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 ...
0
votes
2answers
39 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 ...
27
votes
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
votes
0answers
201 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
vote
1answer
75 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
votes
1answer
341 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?
1
vote
2answers
131 views

Multivariate Bayesian formula

I got there example graphs bishop's PRML (8.2.1) 1. a <- c -> b $$ p(a,b,c) = p(a|c)p(b|c)p(c) --(1)\\ p(a,b) = \sum_c p(a|c)p(b|c)p(c) --(2) $$ Q1: Can I use a new graph to represent the ...
1
vote
1answer
43 views

[Revised]Proving the expected \bold{density} of being the Nth order statistics is decreasing in sample size

(Sorry that I've previously formulated the question in a wrong way, which confused everyone including myself. This is a better version of the question. Thanks!) Here's another order statistics ...
3
votes
2answers
126 views

Proving some properties of expected first order statistics with respect to sample size

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as $E(\mathcal{O}^n_1)= ...
1
vote
1answer
44 views

Differentiating Shannon's entropy [closed]

Can somebody please show the steps of how differentiation of Shannon's entropy yields the following result? $H = -\sum_{l=0}^{L-1} p(l)\log_2[p(l)]$ The result of differentiating is $H_m = ...
6
votes
2answers
339 views

Deriving the bivariate Poisson distribution

I've recently encountered the bivariate Poisson distribution, but I'm a little confused as to how it can be derived. The distribution is given by: $P(X = x, Y = y) = ...
0
votes
1answer
74 views

Derive the mean of a discrete probability distribution

I am reading inference statistics by casella and berger. They are deriving the general formula for the probability distribution like that: ...
5
votes
2answers
88 views

How would one formally prove that the OOB error in random forest is unbiased?

I have read this statement many times but have never come across a proof. I would like to try to produce one myself but I'm not even sure on what notation to use. Can anyone help me with this?
0
votes
0answers
59 views

PCA proof needed for proportion of variance explained by L PCs = mean R-square from regression on PC scores

I observed the following relation and would like to know where I can find a general proof for this: Assume a data matrix $A = [a_{ij}]_{t x k}$. 1) Perform principal component analysis (PCA) using ...
2
votes
1answer
123 views

If a random variable V is independent of two independent random variables X and Y, how to prove that V is independent of X + Y?

This is question 3.8.4 of An Introduction to Mathematical Statistics and Its Applications, 5th Edition, by Larsen and Marx. This is not homework for a class I am taking now, but might someday be for ...
2
votes
1answer
92 views

Justification for the Bootstrap Percentile Interval

The following is a proof of the validity of the (bootstrap) percentile interval taken from Larry Wasserman's "All of Statistics". $\theta^*_{\alpha/2}$ and $\theta^*_{1-\alpha/2}$ denote the ...
2
votes
0answers
26 views

Independent event proof

I'm doing a stats past paper for my first year exam, and I'm having some trouble answering the following question. Could someone please help me? Thanks 'Show that if 3 events $\{A, B, C\}$ are ...
20
votes
3answers
2k views

How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $\sigma^2 = \frac{\sum_{i=1}^N (x_i - \mu)^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about ...
4
votes
2answers
305 views

Independence of a linear and a quadratic form

How can I prove the following lemma? Let $\mathbf{X}^ \prime$ = $ \left[ X_1 , X_2 , \ldots, X_n \right]$ where $ X_1, X_2, \ldots X_n $ are observations of a random sample from a distribution which ...
3
votes
1answer
67 views

Proof that the weighted sum of $n$ PDFs is a valid PDF

Let $f_i(y)$ for $i = 1, \ldots, n$ be valid PDF’s, and let $a_i ∈ (0, 1)$ be constants, such that $\sum_{i=1}^n a_i= 1$. Show that the function $f(y) = \sum_{i=1}^n a_i\, f_i(y)$ is a valid PDF. If ...
0
votes
0answers
32 views

Conditional independence proof seems too obvious, am I looking at it incorrectly?

I'm given the following question about conditional independence: "Suppose we have four random variables a, b, c, d. Prove that, if a is conditionally independent of b and c given d, then a is ...
0
votes
1answer
162 views

Why does eigenvalue decomposition of a correlation matrix maximizes possible variance?

I was reading up on principal component analysis and I was wondering how does eigenvalue decomposition of the correlation matrix maxmimizes the possible variance that is captured? Can someone refer to ...
2
votes
0answers
118 views

Prove that this exponential kernel is positive definite

Let $x,y\in R^d$ and $d:R^d\times R^d \rightarrow R$ a metric on $R^d$ be given. The exponential kernel is defined by: $k(x,x')=e^{−αd(x,x')}$ where $α>0$. The kernel matrix is defined as the ...
2
votes
1answer
82 views

How do principal components change upon the addition of new data?

How do the components from PCA change on addition of new data (i.e., $\frac{d(PC_1(x))}{d({\rm var}(x))}$? I am looking for any mathematical formulas and proofs (since it would be easy to ...
6
votes
1answer
101 views

Coverage Proof of Confidence Intervals

The confidence interval for the mean of a random variable $Y$ has coverage $1-\alpha$ which I am trying to show. Starting from $$\widehat{E(Y)} - ...
0
votes
0answers
65 views

Finding parameter bias under omitted variable, with variance covariance notation

Dear CrossValidated community, Can anyone help me to prove the bias in a given parameter of a regression when there is omitted variable? I know to do it using matrices and matrix algebra. For ...
7
votes
4answers
405 views

Prove the equivalence of the following two formulas for Spearman correlation

From wikipedia, Spearman's rank correlation is calculated by converting variables $X_i$ and $Y_i$ into ranked variables $x_i$ and $y_i$, and then calculating Pearson's correlation between the ranked ...
1
vote
0answers
167 views

Proof that omitted variable bias may lead to endogeneity

I am looking for a proof that omitted variable bias (OVB) in OLS regression may lead to endogeneity. I have found many examples here and out there on how to prove that a given parameter $b_{j}$ (where ...
6
votes
1answer
126 views

Quantile regression estimator formula

I have seen two different representations of the quantile regression estimator which are $$Q(\beta_{q}) = \sum^{n}_{i:y_{i}\geq x'_{i}\beta} q\mid y_i - x'_i \beta_q \mid + \sum^{n}_{i:y_{i}< ...
5
votes
1answer
104 views

E[g(Y)] proof question

This is one of the theorems in my stats text, and I need some help understanding the proof. How can the summand($g_{i}$) be out of its summation sign when multiplying? I thought you can never ...
1
vote
0answers
48 views

Proof for mean and variance of a multidimensional half normal distribution

Given a $p$-dimensional random variable $\xi \sim \mathcal{N}\left( 0, I_{p} \right)$ and a $p\times p$ dimensional diagonal matrix $\Lambda$ with diagonal entries $\lambda = \left( ...
11
votes
3answers
575 views

The concept of 'proven statistically'

When the news talk about things been 'proven statistically' are they using a well-defined concept of statistics correctly, using it wrong, or just using an oxymoron? I imagine that a 'statistical ...
0
votes
0answers
106 views

Having trouble understanding netflix RBM

According to this paper (pdf), the energy function of the restricted Boltzmann machine (RBM) is defined as: and the paper shows that the conditional probability of softmax unit is: I'm having ...
3
votes
1answer
37 views

Least squares with equal predictors

I am trying to figure out how to start this problem: We have a multivariate regression of Y on X. Show that if we have tied values for X, we can replace those by a single set and use the mean for ...
5
votes
1answer
97 views

Sufficiency of order statistics

I am told the following proof is incorrect, but I cannot understand why. Consider $X_{(1)}, \ldots, X_{(n)}$ are the order statistics of a random sample of size $n$. I want to show that the order ...
0
votes
1answer
40 views

Sum of squares proof where $N=n_{UC}+n_{TX}$

$TX$ is variable that indicates treatment status ($TX=1$ if the patient gets the new treatment, and $0$ otherwise, and $UC = 1 - TX$ indicates they got the standard treatment). Of $N$ patients, ...