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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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1answer
37 views

Moment inequality: $E\mid X_1 X_2 X_3\mid \leq (E(\mid X_1\mid^3)+E(\mid X_2\mid^3)+E(\mid X_3\mid^3))/3 $ for zero-mean r.v.'s?

Let $X_1, X_2, X_3$ be zero mean random variables and assume $E(\mid X_i \mid ^{4+\delta})\leq C, i=1,2,3$ where $C$ is a constant and $\delta>0$ some positive small constant. How can I show that ...
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0answers
31 views

Understanding ARMA Model calculations With an Example

Considering an ARMA (2,2) Model: $$x_t - \phi_1x_{t-1}-\phi_2x_{t-2}=a_t + \theta_1a_{t-1} +\theta_2a_{t-2}$$ where $a_t \sim NID(0, 1)$. Suppose $x_t$ is causal i.e. $$x_t= a_t + \sum^\infty_{j=1}\...
1
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1answer
33 views

Prove the MLE is an efficient estimator for $\theta$ in the context of Normal distribution

Let $X_{1},X_{2},\ldots,X_{n}$ represent a random sample whose distribution is given by $X\sim\mathcal{N}(0,\theta)$. Find the MLE of $\theta$. Then obtain the score function, find the Fischer ...
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0answers
18 views

How to improve on the preferred regression model?

After running the chow test, the F stat shows structural change is present in the model, so the unrestricted models are preferred. I am not sure how I can choose from the two regression models in the ...
0
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1answer
49 views

Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$

Let $X\sim\text{Rayleigh}(\theta^{2})$. Prove that $T_{n}$ is consistent, given that $$T_{n}(\textbf{X}) = \frac{1}{2n}\sum_{i=1}^{n}x^{2}_{i}$$ MY ATTEMPT To begin with, let us notice that \begin{...
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0answers
56 views

Calculate power for my test [closed]

I need some help with the following problem: Let's consider a density function $f_\theta(x):=\theta x^{\theta -1}$, with $x \in [0, 1]$, the null hypothesis $H_0: \theta=2$ and the alternative ...
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0answers
11 views

PAC learnability

I don't understand the solution. Everything required to compute $m$ is already given. Then why can't we apply PAC learning theory? The training error is not zero in this case so is that the reason why ...
1
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1answer
26 views

Prove that $X_{(n)} - X_{(1)}$ is an ancillary statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be an independent and equally distributed random sample whose distribution is uniform on the interval $(\theta,\theta+1)$, $-\infty<\theta<+\infty$. Then consider ...
0
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1answer
26 views

Finding distribution and checking independence of transformed normal variables

$X,Y,Z$ are three independent random variables following standard normal distribution. Consider a real function $f$ such that \begin{align}f(x)&=1 , x\geq 0 \\ &= -1, x<0\\ \end{align} Let $...
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0answers
53 views

Finding the mean and variance for pmf $P(X_i = x)=-\theta^x/x\log(1-\theta)$?

I'd like to verify that my working/thinking is correct. This is a problem from Keener's book, but the answer isn't provided. Let $X$ have distribution $P(X = x)=-\theta^x/x\log(1-\theta)$ for $x=1,2,\...
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0answers
8 views

Practicing Understanding Stochastic Differential Equations using R

Is there a book or set of notes that I can use to practice differential equations using R-Studio or Python. I don't want a solver, I want a way of visualizing them and understanding their properties. ...
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0answers
19 views

Likelihood ratio tests: a quick help on its application

To begin with, let us contextualize the problem to be proposed. Lets suppose that $f$ belongs to the exponential family. For a generic scalar model parameter $\beta$, we focus on tests of $H_{0}: \...
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0answers
13 views

Predictive Distribution in Gaussian Process Derivation

In Gaussian Process for Machine Learning (Rasmussen and Williams), on p.11, we are given the following predictive distribution: $$p\left(f_{*} | \mathbf{x}_{*}, X, \mathbf{y}\right)=\int p\left(f_{*} ...
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0answers
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 ...
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0answers
22 views

ROC curve with symmetrical kernel

I am trying to use kernels with ROC curve, and 'm succeed to plot them but now my query is about theoretical grounds, i.e. its bias, var, etc. I want to evaluate the theorems (1 & 2) in Pulit (...
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0answers
23 views

How do we obtain the canonical parameter from the variance function in the context of exponential family distributions?

I am assuming the following definition of the exponential family: \begin{align*} f(y,\theta,\phi) = \exp\left\{\phi[y\theta - b(\theta)] + c(y,\phi)\right\} \end{align*} In such context, the variance ...
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0answers
10 views

If I wanted to see if their was a difference between two groups and paired each person with someone of similar age what test would I do?

I was thinking of doing a paired samples t-test, since we are pairing specific individuals from each group with each other. Is this correct?
2
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1answer
28 views

Showing $\delta'(X)$ is a Bayes estimator of $\theta^k$ for specified prior

Problem: Define $\delta(X) = \frac{\sqrt{n}}{1+\sqrt{n}}\bar{X}_n + \frac{1}{2(1+\sqrt{n})}$ We assume $\bar{X}_n|\theta \sim Bin(\theta,n)$. It is known that $\delta(X)$ is the Bayes estimator of $\...
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4answers
117 views

Probability of absolute value of a sum of random variables

Consider two random variables $X$ and $Y$, and let $b$ be a real number. Show that $$P(\mid X+Y\mid>b)\leq P(\mid X+Y\mid>b,\mid X\mid>b/2)+P(\mid X+Y\mid>b,\mid Y\mid>b/2).$$ I'm ...
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1answer
29 views

How to assign a prior distribution to a loading matrix that has restrictions?

I came across the paper Fast Variational Bayesian Linear State-Space Model. They work with the following model: $$\begin{align} {\bf{x}}_n &= {\bf A} {\bf{x}}_{n-1} + {\text{noise}} \\ {\bf{y}}_n &...
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1answer
25 views

How is an ANOVA table calculated when using continuous predictors only?

An ANOVA can be described as a regression with dummy variables. You could for example calculate the sums of squares treatment in an ANOVA table using the coefficients from a linear model ...
1
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0answers
15 views

How can I do t.test when I don't have a null value?

I have a dataset containing 20,000 records of home sales. I'm asked Choose a numerical variable from your dataset. 1. Calculate a 98% confidence interval for the mean of this variable. I chose the ...
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1answer
23 views

Sufficient Statistic and Unbiased Estimate in Exponential Family

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he deals with sufficient statistics in exponential distributions ...
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0answers
48 views

Volatility forecast with GARCH(1,1)

I am having trouble with this question: $Y_t = \sigma_t \epsilon_t$ $\sigma^2_t = 0.003+0.41Y^2_{t-1}+0.53 \sigma^2_{t-1}$ and I am given that $\sigma^2_T = 0.01$ and $Y_T = 0.2$. I am asked to ...
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0answers
20 views

Understanding PCA from a linear transformation perspective

I've came across several great questions and answers here regarding PCA, but I would like to have a look at it from a linear transformation perspective. Let's say I have a (demeaned) data matrix $X$ ...
3
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1answer
46 views

Proof of Rao Blackwellization

I am reading this classic paper (Information and the Accuracy Attainable in the Estimation of Statistical Parameters) by CR Rao where he introduces the idea of minimizing the variance of an unbiased ...
2
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1answer
46 views

Long-run variance of ARMA(p,q)

Assume you have $A(L)y_t = B(L)e_t$ and $e_t$ is a zero mean white noise with variance $\sigma^2$. Why is the long-run variance of $y_t$ equal to $\sigma^2\left(\frac{B(1)}{A(1)}\right)^2$? I know ...
1
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1answer
21 views

What does the PCA().transform() method do?

I've been taught to think of the PCA as change of basis technique with a cleverly chosen basis. Let's say my initial data is a $m\times n$ matrix $X$ where $m$ is a number of features and $n$ is a ...
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2answers
54 views

Is the zero truncated Poisson Distribution part of the Exponential Family? [duplicate]

This is the density of a truncated Poisson: $$P(X = x \mid X > 0) = \frac{\lambda ^ x e^{- \lambda} }{x ! \left ( 1 - e^{- \lambda} \right )}$$ To show that it's member of the Exponential ...
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0answers
41 views

find pdf (using bivariate transformation) of X/Y where X,Y ~ uniform(0,1) independent

I know how to find the distribution of X/Y when they are independent uniform(0,1) by drawing the integration area. Correct answer is: $$ P(X/Y \leqslant t) = \\ \frac{1}{2} t, t\leqslant1\\1-\frac{...
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0answers
50 views

OLS basic doubt

In a multivariate OLS model, $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$$ Is my estimator for $\beta_1$ given by: $$\hat \beta_1 = [X_1'X_1]^{-1} X_1'Y$$ or: $$\hat \beta_1 = [X_1' ...
0
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0answers
23 views

Expectation of a product of random variables problem

Let $U$ be a random variable with uniform distribution over $[0,1]$ and a bivariate random vector $(Z,T)$ defined by $$(Z,T)=(0,0)1_{\{U\geq 2\alpha\}}+(Q_X(U),Q_Y(U))1_{\{U< 2\alpha\}}$$ where $...
1
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1answer
22 views

How much matrix differentiation background for Seber and Lee's Linear Regression Analysis

How self-contained is Seber & Lee's textbook 'Linear Regression Analysis'? It seems to assume knowledge of matrix differentiation.
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1answer
68 views

Stochastic Sequence; Compute $\lim_{n\to\infty}$

Let $(X_n)n≥1$ be a stochastic sequence of i.i.d. random variables, each $X_n$ with values in the set {1, 4, 8, 16} and probability distribution: $P[X_n = 1] = 1/6, P[X_n = 4] = 1/4, P[X_n = 8] = 1/3,...
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1answer
64 views

Multiple linear regression: am I interpreting the methodology right?

This is a follow-up question to 1 and 2. So we have the normal linear model \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*} where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\...
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0answers
18 views

law of iterated expectations doubt

Is it correct to do so: $E(X^{-1} Y) = E_x(E(X^{-1} Y \vert X) = E_x(X^{-1} E(Y \vert X)) $ Are we allowed to use LIE in case of non linear functions like $X^{-1} $
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0answers
13 views

i.i.d. random vectors [duplicate]

If $(X_1,Y_1), (X_2,Y_2)$ are independent random vectors having the same joint distribution function $F$, then is it correct to say: $E(X_1)=E(X_2)$ and $E(Y_1)=E(Y_2)$ (the same for variance); Both, ...
1
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1answer
43 views

regression basic doubt

If have a simple bivariate regression model: $ Y_i= x_i \beta + \epsilon_i $ where $i$ are the number of observations. How do I test for the hypothesis that the OLS coefficient $\beta$ does not ...
3
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1answer
57 views

Why do we need to determine the distribution of $\textbf{Y}$ in the multiple linear regression problem?

Once again, here I am. Given the multiple linear regression model \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*} where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\textbf{I})$ ...
1
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1answer
36 views

Testing whether two categorical variables have identical coefficients

I am currently doing an exercise question asking me to construct a model to test whether the coefficients of two categorical variables ($X_2$ and $X_3$) are same in R. Specifically, these two ...
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1answer
31 views

Question about the Multiple Linear Regression: why and how does it work?

I know this question is quite simple and maybe quite naive as well, but I would like to get some help. The general linear model can be expressed as \begin{align*} \textbf{Y} = \textbf{X}\beta + \...
0
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1answer
65 views

How to choose the best method to generate random values [closed]

In my specific case, I have a pdf that has no closed form, and I want to generate random values ​​of this distribution. It depends on a summation that goes to infinity (coming from a poisson process) ...
1
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1answer
37 views

Gradient clipping just before averaging

A typical way of implementing mini batch learning is by calculating the gradients of every element within the mini batch and then average all of these element's gradients to come up with the final ...
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1answer
42 views

No population? How to answer such question?

Say the givens are: probability of a battery being defective is = 0.40 being not defective = 0.60 STD = 1.5 hours mean = 7.5 hours The question states: if 5 batteries are randomly selected what is ...
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0answers
35 views

Is assigning an inverse-Wishart distribution to a diagonal matrix problematic?

I'm reading the paper Bayesian Vector Autoregressions by Thomas Wozniak. He considers the model $$y_t = \mu + A_1 y_{t-1} + \cdots A_k y_{t-k} + u_t$$ where each $y_i$ is a $N$-vector, each $A_j$ is a ...
1
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1answer
39 views

Stationary Process Ergodicity

Can you give me an example of a stationary nonergodic stochastic process that is time continuous?
2
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1answer
68 views

pdf from a set of conditional pdfs

I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
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0answers
15 views

Condensing values of categorical data

Beginner ML question here. I have a dataframe with a categorical column, a lot of the values are slightly different but essentially mean the same thing. Here's an example of such values: ...
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0answers
42 views

How can I fit a multiple linear regression model in R if the value for beta coefficient of each predictor is given? [closed]

I've got an exercise question asking me to fit a multiple linear regression model in R when the values of coefficients are given. I don't know how to do it. specifically, I have 5 predictors in my ...
0
votes
1answer
59 views

Computing Variances by Conditioning

I have trouble with the first part of this problem (Please take a look at the image below). This is an example problem from my old textbook years ago and I have had trouble understanding: How Y is ...