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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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MLE, regularity conditions, finite and infinite parameter spaces

The problem I have is in figuring out why the MLE is no longer consistent in countable parameter spaces under conditions specified below. The set up is as follows: we are consider a parameters space ...
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132 views

Rao-Blackwellization in variational inference

The Black box VI paper introduces Rao-Blackwellization as a method to reduce the variance of the gradient estimator using score function, in section 3.1. However I don't quite get the basic idea ...
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103 views

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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1k views

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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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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218 views

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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125 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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163 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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152 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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69 views

Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
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29 views

Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
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33 views

Uniform distribution on the simplex. - Thomas cover

I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...
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248 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 ...
4
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839 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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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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201 views

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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116 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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134 views

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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275 views

Estimating the intercept and its variance in a moving average model

Let $y_1,y_2,\dots,y_{10}$ be a time series generated by $$y_t=\delta+\epsilon_t+\theta\epsilon_{t-2}$$ where $\epsilon_t$ is white noise with $E[\epsilon_t]=0$, $Var(\epsilon_t)=\sigma^2$ and $\...
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30 views

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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209 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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280 views

Interpretation of Kaplan-Meier Curve that doesn't go to 0

I'm making a survival analysis and I founded a little strange the survival curve founded by the Kaplan-Meier estimator. The curves are below Here each colour represents a group and time is in days. ...
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108 views

Comparison of Difference of Expectations of Conditional Variances

I want to show (if possible) that $$\mathrm{E}[\mathrm{Var(Y|X_1, X_2)}] - \mathrm{E}[\mathrm{Var(Y|X_1)}] \geq \mathrm{E}[\mathrm{Var(Y|X_1, X_2, X_3)}] - \mathrm{E}[\mathrm{Var(Y|X_1, X_3)}] \tag 1$...
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157 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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107 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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508 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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0answers
170 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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387 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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Show that MLE of $\lambda = \frac{n-T_n}{S_n+cT_n}$

$X_i$ are i.i.d exponential, mean $\lambda^{-1}$ for $1 \leq i \leq n$ and, the values are measured such that $X_i = c$ if $X_i \geq c$ and $X_i$ otherwise. Show that MLE of $\lambda = \frac{n-T_n}{...
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55 views

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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449 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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305 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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190 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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129 views

Diferrencing vs Moving Average

Moving Average and differencing a series can both be used to remove seasonality. Does the difference of these two lie in the model they are used? Moving Average used in classical decomposition and ...
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73 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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169 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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173 views

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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123 views

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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4k views

Metropolis-Hastings within Gibbs sampling

Suppose we have the following classical normal linear regression model: $$y_i = \beta_1 x_{1i} + \beta_2x_{2i} + \beta_3x_{3i} + e_i$$ where $e_{i} \sim iid.N(0, \sigma^2)$ for all $i = 1, 2, \cdots,...
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124 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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293 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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33 views

UMVUE for $g(p) = \mathbb{E}_p[X^2]$, where X follows a geometric distribution

I have a random variable X with pmf $$p_\lambda(x) = (1-p)^{x-1}p, \ \ x = 1,2,3,\ldots, \ \ p \in (0,1)$$ and I am trying to find a UMVUE for $$g(p) = \mathbb{E}_p[X^2]$$. Here is my attempt so ...
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38 views

Algebraic Manipulations in Mann-Whitney-Wilcoxon Test Statistics

Given Let $\Delta > 0 $ be positive real number. Consider the Wilcoxon-Mann-Whitney upper tail test $H_0: \Delta \leq 0 \,\,\, \text{vs} \,\,\, H_a: \Delta > 0$ aimed at testing the difference ...
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36 views

Require understanding regarding the concept of restricted estimators

I was reading "The Elements of Statistical Learning Book by Jerome H. Friedman, Robert Tibshirani, and Trevor Hastie" where I encountered the following: The part tells us that the RSS criterion will ...
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81 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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0answers
63 views

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=(\...