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1
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0answers
24 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
0answers
8 views

Least squares / residual sum of squares in closed form [migrated]

In finding the Residual Sum of Squares (RSS) We have: \begin{equation} \hat{Y} = X^T\hat{\beta} \end{equation} where the parameter $\hat{\beta}$ will be used in estimating the output value of input ...
2
votes
1answer
68 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 ...
4
votes
1answer
64 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
19 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
83 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
49 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
42 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
27 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
512 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
53 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 ...
4
votes
1answer
70 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
42 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
83 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
122 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
38 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
113 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
20 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
366 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
88 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
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1answer
88 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
65 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
39 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 ...
1
vote
3answers
260 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
62 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
310 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
61 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
185 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) ...
7
votes
2answers
259 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
87 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
146 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
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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 ...
1
vote
1answer
57 views

How to show that $\mathrm{mgf}$ $M(s)$ and $\mathrm{pgf}$ $P(s)$ are related?

Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $s∈(-s_0,s_0),s_0>0$. ...
1
vote
0answers
87 views

Questions about the order statistics of uniform distributions

I refer to the Simes (1986) paper found here. In this setting, $P_{(1)}$ through $P_{(n)}$ are the order statistics of $n$ independent Uniform$[0,1]$ random variables and, for $0\le \alpha \le n$, ...
1
vote
1answer
106 views

Distance between the product of marginal distributions and the joint distribution

Given a joint distribution $P(A,B,C)$, we can compute various marginal distributions. Now suppose: \begin{align} P1(A,B,C) &= P(A) P(B) P(C) \\ P2(A,B,C) &= P(A,B) P(C) \\ P3(A,B,C) &= ...
0
votes
0answers
70 views

Econometrics with proofs [duplicate]

Which econometrics book, or any other reference, do you believe provides the most rigorous view of the subject, preferably with proofs and full-blown discussion of mathematical topics? I mean a book ...
4
votes
1answer
100 views

Basic proof of mixture models

I need to proof that $X$ follows a distribution $F$ with probability $1-p$ and a distribution $G$ with probability $p$ if, and only if, its distribution function is: $(1-p)F + pG$ Can anyone give me ...
1
vote
0answers
74 views

How to calculate the probability $P(A' \cup B')$ and related properties?

Let $A$ and $B$ denote two events, and suppose that $\text{P}(A) = 0.2$, $\text{P}(B) = 0.3$, and $\text{P}(A\cap B) = 0.1$. Are the following computations correct? $\rm{P}(A \cup B) = \rm{P}(A) ...
0
votes
0answers
36 views

There are two students born exactly in one day of year. How to prove? [duplicate]

I remember once in our Statistics class, our professor mentioned an interesting problem, that is: In our class, consisting of 30 students, I bet that there are two students born on an exact day of ...
5
votes
3answers
215 views

Question about a normal equation proof

How can you prove that the normal equations: $(X^TX)\beta = X^TY$ have one or more solutions without the assumption that X is invertible? My only guess is that it has something to do with generalized ...
1
vote
1answer
159 views

Prove (or disprove) that this function is a kernel

I devised a distance function similar to this form $d(x,y) = \sum_{i = 1}^{n-1} b(x_i, y_i,x_{i+1}, y_{i+1}) $ with $b(x_i, y_i,x_{i+1}, y_{i+1}) = 0 \mbox{ if } x_i \leq 0 \vee y_i \leq 0 \vee ...
0
votes
0answers
118 views

ACF proof for ARMA(1,1) model

I need to derive a proof for the autocorrelation function (ACF) of a stationary, invertible ARMA(1,1) model, that shows that the ACF on the first lag depends on both the $\theta_1$ and $\phi_1$ ...
2
votes
1answer
562 views

Diagonal elements of the projection matrix

I am having some problem trying to prove that the diagonal elements of the hat matrix $h_{ii}$ are between $1/n$ and $1$. Suppose that $Range(X_{n,k})=K $ the number of columns of our matrix of data ...
4
votes
1answer
195 views

Limit of c.d.f. of Poisson($n$) when $n$ goes to infinity

I'm trying to prove that $\lim_{n\rightarrow\infty}F_{X}(n)=1/2$ when $X\sim \text{Poisson}(n)$ without success. Could someone help me ?
4
votes
0answers
93 views

Is it possible to have an estimator that is unbiased and bounded?

I have a parameter $\theta$ which lies between $[0,1]$. Let us say that I can run an experiment and obtain $\hat{\theta} = \theta + w$, where $w$ is a standard Gaussian. What I need is an estimate of ...
3
votes
1answer
965 views

KL divergence between two multivariate Gaussians

I'm having trouble deriving the KL divergence formula assuming two multivariate normal distributions. I've done the univariate case fairly easily. However, it's been quite a while since I took math ...