For mathematical proofs or derivations of results.

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1
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0answers
8 views

True or false? “sum of an m-strongly convex and a convex function is m-strongly convex” [migrated]

I would like to know if the following conjecture is true or false? If $f(x) = g(x) + h(x)$ where $g$ is m-strongly convex and $h$ is convex, then $f$ is m-strongly convex. NOTE: For a ...
3
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0answers
57 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): $$ ...
0
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0answers
12 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 ...
0
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0answers
17 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 ...
0
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2answers
71 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
70 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 ...
1
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2answers
31 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 ...
1
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0answers
18 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
30 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 ...
7
votes
2answers
348 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 ...
3
votes
1answer
76 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
36 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?
1
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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 ...
1
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0answers
18 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
votes
0answers
41 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 ...
0
votes
0answers
28 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
votes
0answers
31 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
votes
1answer
53 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
votes
0answers
30 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} ...
0
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0answers
17 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
votes
2answers
61 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
votes
0answers
28 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
votes
1answer
34 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
63 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 ...
0
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0answers
7 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?
0
votes
1answer
121 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
votes
1answer
30 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
157 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
votes
1answer
176 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 ...
0
votes
3answers
86 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 ...
1
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0answers
51 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
votes
2answers
260 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, ...
1
vote
1answer
59 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
30 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
votes
1answer
40 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
votes
1answer
33 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 ...
1
vote
1answer
62 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.
-1
votes
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
votes
4answers
1k 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 ...
4
votes
1answer
48 views

Partition data into two sets such that the difference of their variance is minimal

Suppose there are $n$ data values $x_1<x_2<\ldots<x_{n-1}<x_n$,and I've found a partition number $k$, such that $$ ...
2
votes
1answer
78 views

Conceptual proof that conditional of a multivariate Gaussian is multivariate Gaussian

I understand the arithmetic derivation of the PDF of a conditional distribution of a multivariate Gaussian, as explained here, for example. Does anyone know of a more conceptual (perhaps, co-ordinate ...
1
vote
4answers
562 views

Variance of linear combinations of correlated random variables

I understand the proof that $$(aX+bY) = a^2Var(X) +b^2Var(Y) + 2abCov(X,Y), $$ but I don't understand how to prove the generalization to arbitrary linear combinations. Let $a_i$ be scalars for ...
3
votes
1answer
51 views

How to prove the properties of penalized likelihood estimator in Fan and Li (2001) paper

I'm reading through Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties". In Page 2 near bottom right corner, they proposed three properties that a ...
1
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0answers
73 views

Mathematics / Statistics [closed]

Today I was thinking how I'm bad at making statements, I know who does statistical cares more about the practical part, but knowing how to do demonstrations in the theoretical part is important? I ...
4
votes
1answer
123 views

Generalized inverse solution to system of linear equations proof

I'm going through a set of course notes for an introduction to the theory of linear models class at my university. Unfortunately, the professor who wrote this note set is no longer at this school, and ...
4
votes
2answers
101 views

Is $\text{Cov}(|a|,|b|)\geq \text{Cov}(a,b)$?

The above seems intuitively true, (where $|a|$ refers to the absolute value of $a$), but I'm struggling to prove it - would be very grateful for either a proof or a reference.
1
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1answer
67 views

Proof for the sampling variance of the Neyman Estimator

I'm going through Imbens and Rubin's new book and I just for the life of me can't figure out 1 minor detail in their proof for the sampling variance of the Neyman estimator $\bar{Y}^{obs}_{t} - ...
3
votes
1answer
112 views

Transformation of a random vector

I am very familiar with the formula $$f_{Y}(y) = f_{X}(x(y))\left|\dfrac{\text{d}}{\text{d}y}[x(y)]\right|$$ where $X$ and $Y = g(X)$ is a one-to-one transformation of $X$. (I use $x(y)$ to mean "$x$ ...
6
votes
2answers
137 views

A strange step on a proof about the distribution of quadratic forms

The following theorem comes from the 7th edition of "Introduction to Mathematical Statistics" by Hogg, Craig and Mckean and it concerns the necessary and sufficient condition for the independence of ...
2
votes
0answers
56 views

Proving a “well-known” result regarding the distribution of a normally distributed random variable

In an important project work, I would like to include a "proof" of the following, but have unfortunately been unable to readily compute it myself. I am aware that this is a flaw on my part, but ...