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3
votes
2answers
84 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
28 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
153 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
50 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: I do not get the step before the result. I would ...
5
votes
2answers
63 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
31 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 ...
0
votes
0answers
14 views

Linear algebra - easy proof [migrated]

This one comes from Gilbert Strang's Linear Algebra. Pick any numbers x+y+z = 0. Find an angle between v=(x,y,z) and w=(z,x,y). Explain why v*w/||v||*||w|| is always -0.5.
2
votes
1answer
109 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
78 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
22 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 ...
17
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
262 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
62 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
24 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
61 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
63 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
76 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
89 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
31 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 ...
5
votes
2answers
125 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
78 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
93 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
101 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
35 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
530 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
86 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
36 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
81 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, ...
1
vote
1answer
53 views

Asymptotic distribution of Kernel density estimator

For my research I am looking for proof of the asymptotic distribution of the univariate Kernel density estimator as proposed by Rosenblatt 1956 and Parzen 1962. A proof is for example given here and ...
1
vote
1answer
116 views

Linear Combination of multivariate t distribution

I am looking for a resource where i can find derivation of the linear combination of multivariate t distribution. Does anyone here know any good site or place (s)he can point me to? I am trying to see ...
0
votes
1answer
205 views

Proof of consistency of Maximum Likelihood Estimator(MLE)

I would appreciate some help comprehending a logical step in the proof below about the consistency of MLE. It comes directly from Introduction to Mathematical Statistics by Hogg and Craig and it is ...
1
vote
1answer
39 views

Convergence in probability of a product of RVs

I have found the following very useful theorem and I would appreciate some help comprehending it fully. Theorem Let $\{X_n \} $ be a sequence of random variables bounded in probability and let $ ...
0
votes
1answer
185 views

Why is GARCH(1,1) = ARCH(infinity)?

How do you prove that GARCH(1,1) = ARCH(infinity)? Need some guidance on how to start on this.
0
votes
0answers
30 views

Limiting Distribution of a Polya-Eggenberger Urn with Stopping Rule

I am trying to find literature on the limiting distribution of a Polya-Eggenberger Urn with a stopping rule contingent on the proportion of the two colors in the urn. To clarify, let's say that the ...
1
vote
1answer
525 views

Proof that regression residual error is an unbiased estimate of error variance

Consider the least squares problem $Y=X\beta +\epsilon$ while $\epsilon$ is zero mean Gaussian with $E(\epsilon) = 0$ and variance $\sigma^2$. I need to prove that $\frac{V(\hat{\beta})}{N-(n+m)}$ ...
3
votes
1answer
107 views

Variance of a sample - proof

On page 72 of Introductory Statistics, A Conceptual Approach Using R (Routledge, 2012), the authors first compute the variance of a sample of size $n$ using: ...
1
vote
1answer
115 views

Covariance with conditional expectation

Suppose $X$ and $Y$ are random variables, $E(Y^2) < \infty$ and $\varepsilon = Y - E(Y|X)$ so that $Y = E(Y|X) + \varepsilon$. Given that $E(\varepsilon | X) = E(\varepsilon) = 0$, show ...
3
votes
0answers
67 views

How to prove that a t-distribution can be written as a ratio distribution?

If $X \sim N(0,1)$ and $Y \sim \chi^2(n),$ then it's "known" that $Z = X/\sqrt{Y/n}$ is $t$ distributed. Is there anywhere a proof for this? That in the end one can see the $t$ distribution?
0
votes
0answers
48 views

Proving that the best minimum squared predictor for a stationary ARMA model is the conditional mean

I am trying to prove that the best minimum mean squared predictor of $r_{t+l}$ for a stationary ARMA model is the conditional mean $E[r_{t+l}|r_t,r_{t-1},...]$ Attempt: $$r_t = \psi(B)a_t = a_t ...
2
votes
3answers
331 views

Problem with proof of Conditional expectation as best predictor

I have an issue with the proof of $E(Y|X) \in \arg \min_{g(X)} E\Big[\big(Y - g(X)\big)^2\Big]$ which very likely reveal a deeper misunderstanding of expectations and conditional ...
1
vote
0answers
67 views

Show $\mathrm{V}(\bar{y}) = \sigma^2/n\cdot (N-n)/(N-1)$

How do we show that $$\mathrm{V}(\bar{y}) = \frac{\sigma^2}{n}\cdot \frac{(N-n)}{(N-1)}$$ I think this is the sample variance for simple random sampling without replacement. I have not seen this ...
0
votes
2answers
351 views

Proof that $\mathrm{E}(s^2) = \sigma^2 \cdot N/(N-1)$

What's the derivation for expected value for sample variance for a sample taken from simple random sampling without replacement, i.e., how do we show that $$\mathrm{E}(s^2) = \sigma^2 ...
1
vote
0answers
64 views

Equality in linear regression [closed]

I am looking for some hints to prove the following equality: $$ y^{\top}y - y^{\top}X(X^{\top}X)^{-1}X^{\top}y = \dfrac{\det(L^{\top}L)}{\det(X^{\top}X)}, $$ where $y$ is a $n\times 1$ vector, $X$ is ...
2
votes
2answers
188 views

Proof for E[X|X] = X

I saw from the lecture that $$E[X|X]=X$$ where $X$ is a random variable. But when I am trying to prove it formally, I cannot reach to this conclusion. $$E[X|X]=\int{xf_{x|x}(x) ...
8
votes
2answers
315 views

Sum of two normal products is Laplace?

It is apparently the case that if $X_i \sim N(0,1)$, then $X_1 X_2 + X_3 X_4 \sim \mathrm{Laplace(0,1)}$ I've seen papers on arbitrary quadratic forms, which always results in horrible non-central ...
2
votes
2answers
89 views

Is it true that $F_U(U(\bar\omega)) = F_X(z)$ implies $z=X(\bar\omega)$?

For a statistics class, I have to prove a result which leads me to the following question. If I can show that it is true, my proof is done. So here is the question. Suppose that $U: \Omega ...
4
votes
1answer
148 views

Looking for a proof of this statement

Let the data matrix $\pmb X$ be a collection of $n$ $p$-vectors $x_i\sim\mathcal{E}_p(\mu,\pmb \varSigma)$ where $\mathcal{E}_p(\mu,\pmb \varSigma)$ is a square integrable, continuous elliptical ...
0
votes
0answers
41 views

Show that the cross-product term $\hat y'r$ from $ y'y$ is 0 [duplicate]

Show that the cross-product term \hat $y'r$ from $y'y$ is 0. I see that $y'y = ( \hat y + r)'( \hat y + r) = \hat y' \hat y + \hat y'r + r' \hat y + r'r$ But not sure what identity can be used here ...
0
votes
0answers
18 views

If $n$ order stats are iid from Uniform(0,1), why does dividing by the highest order stat give $n-1$ order stats iid from Uniform(0,1)? [duplicate]

As the title states: If $P_{(1)}, ... ,P_{(n)}$ are order statistics of $n$ independent uniform $(0,1)$ random variables, why are $P_{(1)}/P_{(n)} ..... P_{(n-1)}/P_{(n)}$ also order statistics of ...