# 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.

1,722 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
254 views

### Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=(\rho^{|i-j|})$

Suppose $X\sim N_3(0,\Sigma)$, where $\Sigma=\begin{pmatrix}1&\rho&\rho^2\\\rho&1&\rho\\\rho^2&\rho&1\end{pmatrix}$. On the basis of one observation $x=(x_1,x_2,x_3)'$, I ...
964 views

### How to show that a sufficient statistic is NOT minimal sufficient?

My homework problem is to give a counterexample where a certain statistic is not in general minimal sufficient. Irrespective of the details of finding a particular counterexample for this particular ...
136 views

### 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 ...
164 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 ...
705 views

### An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the solution:...
110 views

### Understanding equation used by Hastie et al

I am trying to recreate FIGURE 3.6 from Elements of Statistical Learning. The only information about the figure is included in the caption. I am not clear on what the equation on the Y-axis means ...
91 views

128 views

### Help understanding a paragraph in Kadane's book Principles of uncertainty

Consider two infinite sequences of indicators of events, $s_1$ and $s_2$, with respective relative frequencies $l_1$ and $l_2$, where $l_1\neq l_2$. Let $A$ be the indicator of an event not an ...
576 views

### Exercise on Borel Cantelli Lemma ($\limsup X_n/ \ln(n) =1$ 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 ...
1k views

### 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 ...
244 views

### Sufficient statistics for $\mu_1 - \mu_2$

If $X_1, ..., X_n$ is a random sample from $X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $\sigma^2$ is known,...
83 views

### French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
73 views

219 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 ...
1k views

### Basu's Theorem Proof

I am having trouble with the proof of Basu's theorem... specifically, I'm not sure about the $\theta$s in the expectations below: Let $T$ be a complete sufficient statistic. Let $V$ be an ancillary ...
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 ...
164 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 ...
159 views

79 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 ...
44 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(...
34 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 ...
380 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 ...
80 views

### Convergence in Distribution for i.i.d. data

Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
1k views

60 views

### What statistical test do I need?

Say I have $N$ light bulbs. Whevener one breaks down, I immediately fix it. $k_0$ of these $N$ light bulbs do not break down during this year, $k_1$ break down (and get fixed) once, and $k_2$ ...
505 views

### Likelihood of LDA compared to logistic regression

I've come across an interesting exercise. We are given four classification models for binary response and a $d$-dimensional independent variable: A Linear Discriminant Analysis model where the ...
172 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 ...
111 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 ...
192 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 ...
54 views

### Time series and images : difference and terminology

A time series is an ordered collection of random variables. Considering a one-dimensional time series $A_i = {a_{i1},a_{i2},\ldots,a_{it}}$ where $t$ denotes the time index. So, the time series is a ...
412 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 $T \sim t_{\nu}$; ...
2k views

### How is the confidence interval for variance component in “lmer” function computed by the command “confint”?

Here is the R code : ...
88 views