For mathematical proofs or derivations of results.

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7 views

How do I prove that using OLS on de-meaned data gives the same estimates as using a dummy variable regression?

I obtained the FOCs for the dummy variable regression and know that I have to manipulate them to get the FOCs for the regression on the de-meaned data but am not sure how to go about it, as in how to ...
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1answer
41 views

Bayesian inference for the rate parameter $\lambda$ of an exponential with Accept Reject

Let a prior distribution be $$ \pi(\lambda)=\begin{cases} \frac{2\lambda}{3} & 0 < \lambda \le 1 \\ \frac{2}{3\lambda^2} & \lambda > 1 \end{cases} $$ This ...
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0answers
25 views

Modeling a time series - help in understanding the approach in a paper

The question is based on a paper titled : Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks On page 2 right above Eq(1), the authors say ...
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0answers
41 views

Proof of almost deterministic random variables [closed]

Let $X$ and $Y$ be independent random variables and suppose that $P(X + Y = c) = 1$, where $c \in \mathbb{R}$ is a constant. Prove that $X$ and $Y$ are both almost deterministic random variables ...
3
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2answers
36 views

ARCH(∞) = GARCH(p,q) proof

I am aware that the similiar question was asked already here. However, I read Bollerslev (1986) and I struggle hard with the rearrangements and substitutions he makes. Hence, there are subquestions to ...
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0answers
17 views

Are linear regression errors independent? Mean independent? Uncorrelated?

All I know is that we assume zero conditional mean (and hence zero mean) and conditional homoscedasticity (and hence homoscedasticity). When trying to prove that $E[(\hat{\beta_1} - \beta_1)\bar{u}] ...
4
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0answers
33 views

Decomposition of average squared bias (in Elements of Statistical Learning)

I can't figure out how formula 7.14 on page 224 of The Elements of Statistical Learning is derived. Can anyone help me figure it out? $$\textrm{Average squared bias} = ...
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0answers
25 views

Why is linear regression a convex optimization problem? [duplicate]

One can read everywhere that linear regression is a convex optimization problem and thus gradient descent will find the global optimum. But can someone explain how to proof that it is a convex ...
3
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1answer
39 views

Trouble understanding result in Simes (1986) $pr\{ P_{(j)} > \frac{j\alpha}{n} ; j = 1,…,n \} = 1-\alpha$

I'm referencing this paper http://www-stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/MultipleComparision/Simes86pdf.pdf On page 752, a theorem is presented which states that If ...
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1answer
63 views

Proving that cosine distance function defined by cosine similarity between two unit vectors does not satisfy triangle inequality

How to prove that the cosine distance function defined by cosine similarity between two unit vectors does not satisfy the triangle inequality?
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1answer
26 views

The projection matrix and proof of an unbiased estimator for sigma-squared

Hi, given this information we are meant to prove that the above estimator is unbiased. I understand the proof for the most part (below). What I do not understand is the intuitive reason why the ...
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0answers
27 views

How to derive conditional expection for a linear combination of independent random variables

Suppose we have the simple set up $Y = \alpha X + \epsilon$ Where $X$ and $\epsilon$ are both mean zero, variance one and independent. $\alpha$ is a constant. Trivially, $E[Y|X] = \alpha X$. By ...
3
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0answers
59 views

Prove $\sigma^2(aX+b)=a^2\sigma^2(X)$ for discrete random variable

I'm a total newbie to statistics/math in general, so please bare with me. I'm currently reading the book The Cartoon Guide to Statistics, and stumbled upon the following statement (p. 69): $$ ...
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0answers
26 views

Prove that sigmoid kernel is not a kernel function

Prove that sigmoid kernel $k(x, y) = tan h (\alpha x^T y + c)$ is not a kernel function. Why is it still used as a kernel when it does not satisfy the kernel function property? I know that I should ...
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0answers
19 views

How high dimensional t-test in Feng's article has been obtained?

I'm reading this article and I am wondering how the authors obtained this test statistic! I need to find out how they have done this! Edit: in this article Mr. Feng just says that We propose this ...
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2answers
80 views

How to find the variance of $C\hat\beta$?

Assuming a Gauss-Markov model such that $H_o$: $C\beta$ = $d$, how do I prove that the variance of $C\hat\beta$~ $N(C\beta, \sigma^2C(x'X)^-C')$ ? My Work...Which I Know is Not Correct, when ...
2
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1answer
74 views

Binary treatment with covariates

I am stuck on problem, asking me to show that in the model: $$ Y = \beta_0 + \beta_1T + \Gamma X + u $$ Where Y is the outcome, T is a treatment indicator and X are a set of controls (pre ...
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2answers
34 views

Proving that weighted probabilities lie in the range 0 to 1

How to prove that: $$P_{avg} = \dfrac{w_{1}P_{A} + w_{2}P_{B} + w_{3}P_{C}}{w_{1} + w_{2} + w_{3}}$$ lies between 0 and 1, where $P_{i}$ corresponds to probability scores and $w_{j}$ are real ...
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0answers
24 views

Proving Orthogonal Properties of Projection Matrix

I am working through a multi-part proof of how orthogonal projection matrices give specific results from their properties. I read through the Gauss-Markov model theory to get a start. This is only a ...
3
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1answer
38 views

Prove that sum of uniform distribution (-1,1) is also uniform (-n,n)? [duplicate]

If $d_i \in U(-1,1)$ (uniform distribution between -1 and 1 - not sure what the canonical notation is for this), then it seems intuitive that $\sum_{i=1}^n d_i \in U(-n,n)$ and thus ...
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2answers
358 views

Problem with proof - why exponentially smoothed time series is biased

I'm working through the proof why the exponential smoothing is a biased estimator of a linear trend. The book is trying to describe the expected value of an exponentially smoothed time series. It's ...
4
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1answer
435 views

Proof of convergence of k-means

For an assignment I've been asked to provide a proof that k-means converges in a finite number of steps. This is what I wrote: In the following, $C$ is a collection of all the cluster centres. ...
1
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1answer
50 views

Proof that square of a standard normal r.v. has Chi-Square Distribution using MGF's

Supposes $Z \sim N(0,1)$. We know that $Z^{\top}\!Z\sim\text{Chi-Square}(1)$. Does the proof for this concept require the use of moment generating functions/method of moments per say?
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1answer
85 views

Find the distribution of $|v_0^*w_0|^2$ in this specific case

Suppose I have a set of complex Gaussian (with zero mean and unit variance) i.i.d. vectors $w_0,w_1,\ldots,w_k$, each of which have dimension $n \times 1$. We define matrix $W=[w_1,\ldots,w_k]$. For ...
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0answers
22 views

How to determine the variance of an autocorrelation estimator?

In reference to the hint: the calculated expected value from problem 2 was found to be: Where the variables changed slightly due to where the image was found, however l = h in the question, and r ...
2
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0answers
48 views

Understanding Autocovariance under Gaussian Random Process

I'm recently been trying to understand time series better,and would really appreciate if someone can show me this: I found this online under a lecture slide by J. McJames of Portland Univ., and I ...
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0answers
46 views

How to prove the unbiased and biased estimator of autocovariance function?

These two estimators are commonly referenced as sample autocovariance functions. I'm curious how you're to show the first is an unbiased estimator, while the second is a biased one. And how would ...
2
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0answers
36 views

Hoes does using Tukey's test correct for multiple comparison problem?

I am curious about the intuition behind the Tukey's HSD. I know that it is designed for post-hoc test(WHEN and HOW part), but I want to know underlying theory that justifies its usage(WHY part). To ...
3
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1answer
102 views

How to prove or disprove this function is valid kernel?

I have the following function $$ K(x, y) = \begin {cases} 1, & if ||x - y||_2 \le 1 \\ 0, & otherwise \end{cases} $$ I'd like to prove (or disprove) that it's a valid kernel function. In ...
3
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0answers
34 views

Kendall's tau derivation from first principles?

I know that Kendall's tau is given by: $$\tau = P[(x_1-x_2)(y_1-y_2)>0]-P[(x_1-x_2)(y_1-y_2)<0]$$ However I cannot see how this gives: $$\tau = {2 \over n^2-n} \sum_{1\leq i<j\leq n} ...
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0answers
20 views

Pandey and Dubey estimator.

I am studying sampling theory Pandey and Dubey(1988) proposed the following product estimator. $\bar y_{PD} = \bar y \left( \frac{\bar x + C_x}{\bar X +C_x}\right)$ And its Mean square error is ...
0
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2answers
66 views

Gaussian Process Proofs and Results

I am building a model based on Gaussian processes and want to assume something like as my sample size $n$ gets large my prediction error goes to 0. In other words,a re there any proofs or theorems ...
0
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0answers
33 views

Is a Monte Carlo approximation to a consistent estimate itself a consistent estimate?

Let $A(x)$ be a consistent estimate of some population quantity $A_0$, where $x$ is the data and there are $N$ observations. However, $A(x)$ is difficult to calculate directly, but can be ...
2
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1answer
36 views

A supposedly straight forward proof

The authors of this paper claim that equation (1) is equivalent to equation (2). I really don't see how this is possible. There even go further to say, "it is easy to check". Can anyone help?.
1
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1answer
64 views

Understanding a proof

The following is a proof of the existence of M-estimates from the book of Maronna et al. "Robust Statistics". While I understand their strategy, I am having trouble comprehending a step on their ...
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0answers
9 views

Prove that when a Birth-death chain has a stationary distribution it satisfies the detailed balance equations

I had an attempt at proving this using mainly materials from youtube lectures however my final answer is still a bit off: Have I missed something with my attempt here?
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1answer
227 views

Spectral density function of AR(1) process

I'm studying the derivation of the spectral density function of an AR(1) process. Starting from its autocovariance function, we have that: $$\gamma_0 = \frac{\sigma^2}{1-\alpha_1 ^2}$$ and $\gamma_k ...
0
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1answer
36 views

What is my missing assumption is sum of variances?

In this answer, it says that in general the sum of the variances is not equal to the variance of the sum. I tried to work it out by myself, and I think I got a different result, namely that the ...
1
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0answers
301 views

Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
8
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1answer
321 views

Clustering — Intuition behind Kleinberg's Impossibility Theorem

I've been thinking about writing a blog post on this interesting analysis by Kleinberg (2002) that explores the difficulty of clustering. Kleinberg outlines three seemingly intuitive desiderata for a ...
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3answers
106 views

Proof of Variance Formula for Central Chi-Squared Distribution

I wanted to know what the proof for the variance term in a central chi-squared distribution (degree n) is. I know that the answer is 2n, but I was wondering how to derive it. Here's my attempt so ...
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0answers
53 views

Prove that the event '$\lim Y_n$ is finite' is in the tail $\sigma$-algebra of $Y_1, Y_2, …$

Given random variables $Y_1, Y_2, Y_3, ...$, let $\tau = \bigcap_{m\geq1} \sigma(Y_{m+1}, Y_{m+2}, ...)$ be their tail sigma-algebra. For convenience, $\tau_m \doteq \sigma(Y_{m+1}, Y_{m+2}, ...)$. ...
4
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2answers
269 views

Are products of independent random variables independent?

Let $Z_0, Z_1, Z_2,...$ be independent and identically distributed such that $P(Z_n = 1) = P(Z_n = -1) = 1/2$ for $n = 0, 1, 2, ...$ Let $X_0 = Z_0$, $X_1 = X_0 Z_1$, $X_2 = X_1 Z_2$, ... Are $X_0, ...
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1answer
74 views

Is Minimax Linkage a Lance-Williams hierarchical clustering?

I found the following article on "Hierarchical Clustering With Prototypes via Minimax Linkage". It is stated in Property 6 that Minimax linkage cannot be written using Lance–Williams updates. ...
0
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0answers
31 views

Can any distribution be fully described by the [infinite] set of all its moments? $\{E[x^n]\}_{n\in N}$ [duplicate]

Is it possible describe any distribution uniquely by the infinite set of all it's moments $\{E[x^n]\}_{n\in N}$ If yes, does that include discrete, truncated, etc. distributions? If not, is this ...
0
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1answer
44 views

Regression Tree Impurity

It seems pretty obvious that if I currently have a tree with a certain total impurity, then by splitting it again optimally, I can never end up with a greater total impurity. It seems similar to the ...
0
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1answer
44 views

Variance of Wilcoxon Rank-Sum Test (no ties)

$\newcommand{\E}[1]{\mathbb{E}\left[#1\right]} \newcommand{\r}[1]{r\left(#1\right)} \newcommand{\P}[1]{\mathbb{P}\left(#1\right)} \newcommand{\Var}[1]{\text{Var}\left[#1\right]}$ Given two sets of ...
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1answer
83 views

Where can I find a clear derivation of backpropagation through a Convolutional Neural Network?

Any links to books, articles or papers would be appreciated, or even a written explanation.
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1answer
55 views

Expectation of a random variable and the indicator random variable proof

I need to show that $E[T 1_A] = E[T|A]P(A)$. What I've got so far is $E[T1_A] = \int_{\Omega}T.1_A dP = \int_AT dP$. Now I know that $\int_AdP = P(A)$ however I'm lost as to how do I get the ...
6
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4answers
2k views

How to Prove that an Event Occurs Infinitely Often (Almost Surely)?

Exercise: There is a fair 6-sided die and a biased coin that has probability p > 0 of coming up heads on each toss. The die gets rolled infinitely often, and whenever you roll a 6, you then ...