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Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

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AIC equivalent to Mallows' Cp and Mallows' Cp unbiased for test MSE

Part 1: The goal is to show that with Gaussian errors and a linear model, Mallows' $C_p$ and $AIC$ are equivalent. Using our definition of Mallows' $C_p$: $$C_p=1/n(RSS+2d\hat\sigma^2)$$ and $AIC$: $$...
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Finding MLE and MSE of $\theta$ where $f_X(x\mid\theta)=\theta x^{−2} I_{x\geq\theta}(x)$

Consider i.i.d random variables $X_1$, $X_2$, . . . , $X_n$ having pdf $$f_X(x\mid\theta) = \begin{cases} \theta x^{−2} & x\geq\theta \\ 0 & x\lt\theta \end{cases}$$ where $\theta \...
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I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
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Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
5
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119 views

Question 10.9 from Bayesian Data Analysis, what does accuracy mean here?

I'm doing an independent study in Bayesian Statistics following some chapters from BDA3. When solving the first question from Ch 10 I got stuck. It says: [If] a scalar variable $\theta$ is ...
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161 views

Simple $\chi^2$ test question

I have the following question which seems extremely easy, but the way the data are set up is causing me some uncertainty: I plan to solve this problem through finding the maximum likelihood estimate ...
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148 views

Exponential family where set of natural parameters has empty interior

In my math-stat class we have a theorem that goes: Let $\{P_\theta : \theta \in \Theta\}$ be a $k$ parameter exponential family (i.e. the density of a member of this family can be written as $f(\...
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173 views

Fisher information matrix in logistic regression

I am self-studying the basics of logistic regression. I came across this sentence: In logistic regression expected and observed information matrixes are equal I am aware that the information ...
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34 views

Can someone clearly paraphrase the following argument?

Metaculus is a site where users make and justify predictions on various questions. My question is about an estimation of the probability that a human will live to 120 years by the year 2024. I am ...
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Quiz: Determine first principal component from data-plots

We see four data plots. The goal: How does the first principal component look for each plot a-d. For plot d, it is true that both clusters have same number of datapoints. First principal component ...
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108 views

Does Cross-Validation Really Work?

This question was taken from "The Elements of Statistical Learning" by Friedman, Hastie and Tibshirani (question 7.10) Consider a scenario with N = 20 samples in two equal-sized classes, and p = 500 ...
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Statistical model with $\Gamma(\alpha_i,1)$ sample

We are given statistical sample of $X=(X_1,X_2,X_3)$, where $X_i\sim\Gamma(\alpha_i,1)$ and independent. Let $Z=X_1+X_2+X_3$ and $T$ three dimensional statistic $T:=(\frac{X_1}{Z},\frac{X_2}{Z},\frac{...
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Distribution of X-U(0,1) conditioned on sigma algebra of Y/X, where is Y is U(0,1)?

The question I have is: Define X,Y to be two independent uniform(0,1) random variables and $Z:=\frac{Y}{X}$ Compute $P(X<x|\sigma(Z))$ The answer given apparently by "straightforward elementary ...
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F Test for seeing whether different groups have same model

I'm studying for a final in econometrics, and I'm not sure I follow the reasoning in one of our practice problems. This data set is based on household expenditure surveys. $ CLOTPCT: \text{% of ...
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201 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
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141 views

Think Bayes - Chapter 7 Exercice 7.4

I'm reading this book by Allen B. Downey and trying to do the exercises http://greenteapress.com/wp/think-bayes/ I am a bit stuck at this one, 7.4. I tried looking for blogs and stuff like that where ...
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102 views

Showing independence between two functions of a set of random variables

I've been working on the following problem and I'm confused about how to get started: Let $X_1, X_2,\dots, X_n$ denote i.i.d. real valued random variables, each absolutely continuous with an ...
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474 views

Exercise on Borel Cantelli Lemma ($\lim \sup X_n/ \ln(n) =1$ a.s.) help required to rigorously write the statement

I hope this question is within the scope of this site. Please note that I have solved this Exercise, I do have doubts about my presentation though and about how to rigorously empathize on the ...
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38 views

Calculating error of MCMC algorithms?

If for example the Transitional MCMC algorithm is used (or does it matter which one?), what are the common approaches for calculating an error (some sort of distance from the actual PDF), or ...
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155 views

Sign and size of OLS bias for Tobit models

I have a question related to the sign and size of the OLS bias in the case of a Tobit model. Consider the following model (1) Sample of observations $\{X_i,Y_i\}_{i=1}^n$, i.i.d., $X_i$ is a vector ...
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362 views

Equality vs. Equality in Distribution ($t$-distribution for example)

A technical question that came up to mind as I was reading up on linear models today. Consider the $t$-distribution with $\nu$ degrees of freedom ($t_\nu$) for example. Let's say $T \sim t_{\nu}$; ...
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Independent RVs theorem: rigorous?

I am reproducing here theorem (#3.30) from "All of Statistics" by Larry Wasserman: Let X and Y have joint pdf $f_{X,Y}$ . Then $X\perp Y$ if and only if $f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y)$ for all ...
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391 views

Bayesian Analysis of Box-Cox Transformation

This problem is problem 5 in Chapter 7 of Bayesian Data Analysis, 3rd edition. Consider the Box-Cox transformation: $y_i^{(\lambda)} \sim \mathcal{N}(\mu, \sigma^2)$ where $y_i^{(\lambda)} = (y_i^{\...
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289 views

emails arriving in a Poisson process

Emails arrive according to a Poisson process with rate $λ=2/hour$. You check your inbox (instantly reading all new emails) at time $t=5$ hours and also at some uniformly distributed random time ...
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184 views

Combination of letters: some repeated letters

So I was looking for an answer on this assignment we have to program but I cannot find it anywhere. I'm a computer science student, not a statistics students. (And it isn't even for a statistics ...
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72 views

Calculating the first time a particle hits a state

Let $(X_{n})$ be a Markov chain with state space $D=(a,b,c)$ and transition matrix $$P= \pmatrix{ 0.4 & 0.6 & 0 \\ 0.5 & 0 & 0.5 \\1 & 0 & 0 \\}$$ A) Find the lim$_{n-> \...
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163 views

How do I solve this stochastic differential equation?

So I have a second order stationary process $Y(t), \infty < t < \infty$ which has a continuous sample function, mean $\mu_Y = 1$ and covariance function $r_Y(t) = e^{-|t|}, -\infty < t < \...
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Deriving the maximum likelihood for a generative classification model for K classes

In Christopher Bishop's book "Pattern Recognition and Machine learning", there is the following question: Consider a generative classification model for $K$ classes defined by the prior class ...
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Finding $Var(S^2), E(S^4),$ and unbiased estimator for $\sigma^4$ from random, normal samp

Let $X_1,...,X_n$ be a random sample of size $n$ from the normal distribution $N(\mu,\sigma^2)$ and let $S^2$ be the sample variance. (a) Find $V(S^2)$ and derive $E(S^4)$. (b) find an unbiased ...
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Neyman-Pearson lemma: critical region and hypothesis testing

Let $X_1,X_2,...,X_n$ be i.i.d r.v's with common p.d.f. $$ \mbox f(x)=\frac{x^5e^{-x/\theta}}{5!\theta^6} $$ where $\theta$ > 0. Show that the Neyman-Pearson lemma produces a test of $H_0: \...
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Splitting a variable with nominal and numeric values

I have a variable that has both numeric and nominal components. The source has a documentation which helps in identifying which is which and for splitting into their proper components. I will do ...
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121 views

Finding the limiting distribution

I was given an exercise to do that sounded something like this: The Arizona football team scored $45$ goals in $19$ games in the 2007/08 season. If $y_i$ denotes the number of goals scored in the $i$-...
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289 views

Deriving priors for MCMC implementation

I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution. The posterior distribution is generated from a ...
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61 views

Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...
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Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
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59 views

Check Computation of MME and MLE

Let $X_1$, . . . , $X_n$ be i.i.d random variables having pdf $$f(x\mid\theta) = (\theta+ 1)x^{\theta}I_{(0,1)}(x)$$ where $\theta \gt−1$ (a) Give a MME of $\theta$ based on the first ...
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29 views

Show that there is no efficient estimator for the variance of a normal distribution using properties of the exponential family

I want to prove the statement in the title using the following statement from Wikipedia: it was proved that efficient estimation is possible only in an exponential family, and only for the natural ...
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Can an asymptotically efficient estimator be biased?

In "Theory of point estimation" by Lehmann and Casella (1998) there is the following definition: It is also said that So terms of the asymptotically normal sequence of estimators can be ...
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39 views

Find unbiased estimators for $\lambda$ and $\lambda^2$.

For the spatial homogeneous Poisson process, find unbiased estimators for $\lambda$ and $\lambda^2$. Attempt: Since the homogeneous Poisson process is over an area, how i would i go about ...
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45 views

How can marginalizing over intermediate variable give arbitrarily complex distribution

In this web page has the following statement regarding an inference model $$ q(z_2,z_1|x) = q(z_2|z_1) q(z_1|x) $$ "Although we are still sticking to Gaussians for all of the factorized ...
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Deriving Autocorrelation Structure for Binary Markov Chain

I'm trying to derive the autocorrelation structure of a Binary Markov Chain with \begin{align} Pr(s_t=1 | s_{t-1}=1) &= q \\ Pr(s_t=0 | s_{t-1}=1) &= 1-q \\ Pr(s_t=0 | s_{t-1}=0) &= p \\ ...
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572 views

Generate Beta distribution from Uniform random variables

I need to generate random numbers from Beta distribution using random variables from Uniform distribution. If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$, and If $Y_1+Y_2&...
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Implicit hypothesis testing: mean greater than variance and Delta Method

I am struggling with a hypothesis test between the mean and variance of a sample of i.i.d Gaussian random variables. This (self-study) question arises in the context of the Delta Method (first or ...
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274 views

Concentration of maximum of subexponential random variables

I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection: \...
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186 views

Voting between classifiers : How to prove it works?

Assume m independent binary classifiers with probability $p$ to be correct $p>0.5$. Show that the probability of a voting, e.g. decision is made by the majority of classifiers is correct with ...
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29 views

Comparison between variance of $|x|$ and $x$ for the symmetric distribution

For a symmetric distribution, how the following inequality holds which is given by my teacher: $V(|X|)>V(X)$ What I think is that it should be opposite since for a symmetric distribution the mean ...
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Finding PDF from CDF

I just got really unsure, can someone confirm/rectify? I have the CDF defined as $F(x)= \begin{cases}0, &\text{if}~x < 0,\\ 4x^2 &\text{if}~ 0 \leq x < \frac{1}{4} \\ 1-\frac{4}{3}(1-x)^...
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Trying to pick the coin with the highest bias toward heads after uneven numbers of trials of 2 coins

I'm a stats noob and this isn't a homework question, but I'm just looking for a way to approach problems like this anyway - not an actual answer. Question: Two random coins are selected from a bag ...
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72 views

Tricky Conditional Expectation

A group of 21 women and 14 men have a certain disease with probability p, independently. If we know that exactly 5 of the people have the disease, what is the expected number of women who have the ...