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1
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1answer
63 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
106 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
108 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
25 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
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2answers
102 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
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1answer
32 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
169 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
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1answer
67 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
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2answers
70 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
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0answers
33 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
110 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
80 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
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0answers
23 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
265 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
63 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
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0answers
27 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
72 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
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0answers
65 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
90 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
32 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
138 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
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0answers
80 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
98 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
38 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( ...
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3answers
536 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
90 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
83 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
54 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
121 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
237 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
41 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
199 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
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0answers
34 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
556 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
116 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
131 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
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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
49 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 ...
3
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3answers
357 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
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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
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2answers
367 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
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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
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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
329 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
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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 ...