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
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8
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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 ...
8
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1answer
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 ...
7
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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 ...
7
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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 ...
7
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1answer
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:...
6
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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 ...
6
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1answer
91 views

Maximum Likelihood Estimator of $P(Y_1=1)$ where $Y_i=1$ if $X_i>0$ and $0$ otherwise, given $X_1,\dots,X_n\sim N(\theta,1)$

This is part(a) of exercise 6 of Chapter 9 from Wasserman's All of Statistics. Let $X_1,\dots,X_n\sim N(\theta,1)$. Define $Y_i=\begin{cases} 1 &\text{ if }X_i>0 \\ 0 &\text{ if }X_i\le 0....
6
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123 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 \...
6
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1answer
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 ...
6
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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 ...
6
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0answers
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 ...
6
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1answer
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,...
5
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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 ...
5
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1answer
73 views

Functions of continuous random variables

Let Y be an exponential random variable with parameter $\tau > 0$. Compute the cdf and pdf of $F_W$ where $W = Y^3$ The solution states the cdf as $1 - e^{\frac{-y^\frac{1}{3}}{t}}$ because $F_Y =...
5
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1answer
59 views

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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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 ...
5
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1answer
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 ...
5
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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 ...
5
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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 ...
5
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159 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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36 views

How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
4
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0answers
125 views

Test for Lipschitz continuity (is there some?)

Let $x_1, \dots, x_n$ be a random sample from a distribution $D$. Say, I want to test whether $F(z)$, the cdf of $D$, is Lipschitz continuous, i.e. there exists $L$ such that $F(z + \delta) - F(z) \...
4
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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 ...
4
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0answers
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(...
4
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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 ...
4
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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 ...
4
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1answer
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 $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
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1k 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&...
4
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1answer
256 views

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 ...
4
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1answer
135 views

What is the likelihood function of this random variable (beta distribution parameterizing a Bernoulli distribution)?

This is related to an earlier self-study question of mine. The setup is that there are $N$ individuals, indexed by $i$, and two time periods. Individuals choose whether to "invent" something in the ...
4
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447 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: \...
4
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2answers
3k views

Difference between contextual anomaly and collective anomaly

Contextual and collective anomalies are defined as follows (source): Contextual Anomalies. If a data instance is anomalous in a specific context (but not otherwise), then it is termed as a contextual ...
4
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0answers
35 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 ...
4
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0answers
120 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 ...
4
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1answer
454 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
4
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0answers
224 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-...
4
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1answer
91 views

Bayes net probability question

I've made this Bayes net based on a problem and I'm trying to find the probability of W but I'm stuck. I know I probably have to use Bayes theorem backwards through to find $P(W)$, but I'm not sure ...
4
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0answers
324 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. ...
4
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0answers
113 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$...
4
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2answers
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 $k_2$ ...
4
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1answer
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 ...
4
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0answers
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 ...
4
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0answers
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 ...
4
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0answers
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 ...
4
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2answers
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 ...
4
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0answers
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 $T \sim t_{\nu}$; ...
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2k views
4
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0answers
88 views

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}{...
4
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0answers
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 ...
4
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0answers
483 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^{\...