Questions tagged [self-study]
A routine exercise designed to test one's knowledge; often 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.
8,168 questions
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Deriving the normal equations by differentiating the matrix form
So, if we have $RSS(\beta) = (\mathbf{y}-\mathbf{X}\beta)^T(\mathbf{y}-\mathbf{X}\beta)$ and differentiatiate with respect to $\beta$, and set it to zero in order to minimize it, how do we get $\...
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11
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Sequential probability ratio test for variance
An exercise problem from Hogg and McKean's book "Introduction to Mathematical Statistics" is the following
Exercise 8.4.1. Let $X$ be $N(0,\theta)$ and, in the notation of this section, let $...
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Likelihood ratio test for samples from two different populations
This is basically exactly same as the post here. The answer given over there was not accurate as the null distribution specifically states $\mu_0=\mu_1=0.$
I have attempted it but am not sure if the ...
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1
answer
39
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Can my solution to a counting problem be corrected?
There was an interesting question in my stats class:
There are 5 scientists, 2 of which are Statisticians ($N_{a}$) and 3 are Biologists ($N_{b}$). How many ways can we choose 3 scientists, such that ...
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Condition for quadratic form to have a chi-squared distribution
According to Seber and Lee's textbook theorem 2.8: If $\mathbf{Y} \sim N_n(\mathbf{0},\Sigma)$ and $A$ is symmetric then $\mathbf{Y}^TA\mathbf{Y} \sim \chi^2_r$ if and only if $r$ of the eigenvalues ...
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1
answer
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Maximum value of Friedman's randomized block statistic
I'm reading about Friedman's statistic from the 3rd edition of Nonparametric Statistical Methods (Hollander, Wolfe and Chicken). On page 302 they have an exercise problem:
Show directly, or ...
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118
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Which significance test to use for a difference of means? How to be sure?
[Updated] I have two data sets containing all official recorded games played in the USA for this one sport. One data set is for the women's teams and the other for the men's teams. I'm trying to ...
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Finance: investment analysis
What is the correlation coefficient if beta of stock is 1.714, standard deviation stock is 12% and standard deviation market is 20%?
I was tackling this problem and i used the correlation coefficient ...
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2
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1
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60
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What exactly is the definition of a UMP unbiased test?
I am solving exercises from section 8.3 of Hogg and McKean's "Introduction to Mathematical Statistics." I cannot proceed because the authors have not formally defined UMPU test even though ...
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0
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29
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X-Pareto Distribution
I have an assignment in which I have to show that Weibull Pareto belongs to exponential family and then mean and variance of term $a(x)$.
$g(x)=\frac{\beta c}{x}\{\beta \log (\frac{x}{\theta})\}^{c-1}\...
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Comparing two samples with same number of observations and having same mean but different variances
Given two different set data samples have same mean and same number of observations; if their variances are same, what can be concluded? Also if both variances are different what can be concluded?
...
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1
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Different parametrizations of the exponential family
I found in this article https://arxiv.org/pdf/1607.06450 , formula 10, a parametrization of the exponential family that I think can be written like this:
$$P(t|\eta,s)=e^{\eta t/s}e^{-g(\eta)/s}e^{c(t,...
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1
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Covariance matrix residuals equation
I'm to show that the covariance matrix of the residuals are
$$Cov(\mathbf{Y}-\mathbf{\hat{Y}}) = \sigma^2(\mathbf{I}-\mathbf{H})$$
I know that $Cov(\mathbf{Y}) = \sigma^2\mathbf{I}$ and that $Cov(\...
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1
answer
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Proof in "Linear regression analysis" by Seber and Lee
In appendix result A.2.2, the authors want to show that:
If $A$ is any matrix and $P$ and $Q$ are any conformable nonsingular matrices, then $rank(PAQ) = rank(A)$
They start the proof by giving the ...
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2
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Concept of complete sufficient statistic for the cdf $F(x)$
I am studying Hogg and McKean's "Introduction to Mathematical Statistics." At the end of section $7.7$ where they talk about completeness, sufficiency etc for multi-parameter case, theny ...
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1
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Deriving the score of a diffusion model under DDIM
I am trying to understand how the linear relationship between the diffusion noise prediction model $\epsilon_\theta(x_t)$ which predicts noise added to a sample and the score function is derived $$\...
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Motivation behind the technique to find MVUE of $3\theta_2^2$
This question if from Hogg and McKean's "Introduction to Mathematical Statistics."
Exercise 7.7.11.
Let $X_1,X_2,\cdots,X_n$ be a random sample from a $N(\theta_1,\theta_2)$ distribution.
(a)...
2
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1
answer
62
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Proving a matrix multivariable result from Freedman's "Statistical models"
It is given that $X$ is $n\times p$ with rank $p < n$. $Y$ is $n \times 1$. $\hat\beta=(X'X)^{-1}X'Y$ and $e=Y-X\hat\beta$. He asks us to show that if $\gamma$ is $p\times 1$, then, $$||Y-X\gamma||^...
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3
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How to make two perfectly negatively correlated growing Geometric Brownian Motion (GBM) series? (Impossibility)
Intro
I am self studying in Youtube the course MIT 18.S096 Topics in Mathematics w Applications in Finances and in the following lecture min 34:50 by Dr. Jake Xia is studied the efficient frontier of ...
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3
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Probability/Standard Normal Distribution Homework Help
I am so stuck on this problem in my homework, and was wondering if someone could offer any tips on how to approach it?
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2
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Multiple Linear Regression -- Proving B1=B2=B3=B4
Given y = Bo +B1X1 +B2X2 +B3X3+ B4X4 + e
Question asks: Use general linear hypothesis to show how to test:
a) Ho: B1=B2=B3=B4
b) Ho: B1=B2, B3=B4
Given the solution in the photo, I'm seeking an ...
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1
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Maximum Likelihood Estimation for Pairs of Observations
I have $n$ pairs of observations $(x_i,y_i)$, where each $y_i$ is distributed according to $\text{Pois}(\theta x_i)$, and I wish to do a maximum likelihood estimation for $\theta$ only based on this ...
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How to interpret cross tabulation with the hypothesis so that it is not confusing?
(This is not a homework looking for answers but rather like a discussion on the methodology)
The hypothesis is: older people are less likely to speak a foreign language.
chi square test result is ...
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Prediction Intervals With Hierarchical Regression Model
I'm reading this data analysis book by Gelman and Hill and am trying to understand predictions with hierarchical models. On page 273 they are demonstrating making new predictions for an already ...
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1
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49
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Help needed in understanding the convolution of two distributions
I'm reading Introduction to Probability Models by Sheldon Ross, 12th edition. On page 57, it says:
It is often important to be able to calculate the distribution of $X + Y$ from the distributions of $...
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39
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Minimal and overcomplete exponential families
I am studying Kevin Murphy's Probabilistic Machine Learning: Advanced Topics, version of June 26, 2024. Right now, I am having trouble with some concepts about exponential families.
Definition
...
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29
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Did I understand the concept of Hypergeometric Random Variable correctly?
I am having some issues understanding the concept of a hypergeometric random variable. I was studying this concept from the book, "Introduction to Probability Models" by S Ross. The thing ...
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Sufficient statistic of a function of the parameter
In Hogg and McKean's "Introduction to Mathematical statistics", there is a section ($7.6$) on functions of a parameter. Purpose is to find the MVUE of function of a parameter. He uses ...
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Pseudo-regression as a way of checking MANOVA-assumptions
I stumbled across an online tutorial (a YT-series for supposedly a graduate course in statistics) for implementing a MANOVA-test in R. Before carrying out the test, important assumptions are being ...
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Complete and sufficient statistic when only the maximum of the data is observed
I found the joint PDF, but I was unable to apply factorization afterwards.
How can i solve this problem?
The joint PDF is given by:
$
F(z, \Delta) = \left(f_X(z) F_Y(z)\right)^\Delta \left(f_Y(z) F_X(...
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Suppose a die is rolled repeatedly. Let X be the no. of rolls needed to get all the faces appear for the 1st time. Find the probability mass function
Let a die be such that $P(\{i\})=P_i, i=1,...,6.$ Suppose the above die is rolled repeatedly and independently. Let $X$ be the no. of rolls needed to get all the faces appear for the 1st time. So, $\...
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Motivation behind this exercise problem on complete sufficient statistic
This is from Hogg and McKean's "Introduction to Mathematical Statistics" Chapter 7 (Sufficiency), section 7.4 (Completeness and Uniqueness).
Exercise 7.4.10.
Let $Y_1 < Y_2 < \cdots &...
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How to evaluate estimation of treatment effect under wrong linear model ignoring replicated measurements?
A Simulation study with the following task is given:
Do 1000 simulation runs based on simulated data from the model below to explore the following issue:
How well is the coefficient of $x_1$ (...
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1
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Algorithm to enumerate distribution of $n$ balls in $m$ bins?
I am reasoning through a kind of sample space. My inquiry will be assisted by an algorithm which can enumerate—ideally with some kind of ordering—the possible ways to distribute $n$ balls among $m$ ...
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Question About Approximating the Variance of the Sample Mean for an AR(1) Process
$
\newcommand{\on}[1]{\operatorname{#1}}
\newcommand{\ol}[1]{\bar{#1}}
\newcommand{\Cov}{\on{Cov}}
\newcommand{\Var}{\on{Var}}
$
Problem Statement: Suppose that the time series data $\{x_i:i=1,\dots,N\...
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1
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Expected Value for Complex-Valued Random Variable
This question is part of Exercise 3.14 in the book The Analysis of Time Series: An Introduction with R, 7th Ed., by Chatfield and Xing.
Problem Statement: Suppose $\theta$ is uniformly distributed ...
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0
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Expectation of uniform distribution conditioned on an interval [duplicate]
I am trying to understand the concept of conditioning on an event better. To do so, I've cooked up the following toy problem then tried to generalize it in the context of uniform distribution.
Suppose ...
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How can I rebase a series to a different base year?
I have three monthly data set for import prices.
First one --> Base year = 2003 (2003=100), data period: 2003 M1 - 2012 M12.
Second one --> Base year = 2010 (2010=100), data period: 2010 M1 - ...
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Likelihood ratio test -- Multiparameter multinomial distribution
This is a problem from Hogg and McKean's "Introduction to Mathematical Statistics" (Exercise $6.5.11.$)
Problem Statement
Let $n$ independent trials of an experiment be such that $x_1,x_2,\...
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1
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166
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Uniform prior and poisson likelihood, what posterior distribution will be produced?
If i have a uniform distribution over a fixed specified and a finite range, and a Poisson likelihood distribution, what posterior will be produced?
The likelihood has this form $$P(\pmb{X}|
\pmb{\...
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How to solve the ARCH effect problem in estimating linear bivariate regression model?
I estimated a linear bivariate regression model by OLS method.
I did the ARCH effect test. And there is the presence of ARCH effect in residuals.
How can I deal with the presence of ARCH effect while ...
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Have I constructed the Neyman-orthogonal score correctly?
I am trying to construct a Neyman-orthogonal score for the Poisson m-estimator, using section 2.2 of Chernozhukov et al. (2018). Have I done this correctly? If so, can somebody please help me show/...
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1
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Understanding the proof of Rao-Blackwell Theorem
I'm having problems on understanding the Rao-Blackwell theorem. In particular I don't understand why the resulting estimator is the one with minimum variance between ALL unbiased estimators of the ...
2
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0
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57
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No Existence of Efficient estimator
I need to prove that given $(X_1,...,X_n)$ from the density $$\frac{1}{\theta}x^{\frac{1}{\theta}-1}1_{(0,1)}$$ no efficient estimator exists for $g(\theta)$=$\frac{1}{{\theta}+1}$.
I have shown that ...
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Equivalence of $\sum_{i=1}^3 \sum_{j=1}^3 k_j Cov[y_i, x_j]$ and $\sum_{i=1}^3 k_j\sum_{j=1}^3 Cov[y_i, x_j]$ [duplicate]
We define two correlated random variables $Y_i$ and $X_j$ and say we have this sample, $Y_i:\{y_1,y_2,y_3\}$ and $X_j:\{x_1,x_2,x_3\}$ for the convenience of illustration.
I want to calculate $Cov[\...
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Question on ARCH effect Test
I try to conduct ARCH effect test.
I have a time series (Global price of Brent crude oil) which follows AR(3).
First of all, I estimate AR(3) model for the Oil price.
$$Y_t = \alpha_0 + \alpha_1 Y_{t-...
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1
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With $X$ and $Y$ being two independent $\text{Bernoulli(1/3)}$ rvs, show whether $U = |Y-X|,~V = X+Y$ are independent or not
Let $X$ and $Y$ be two independent $\text{Bernoulli(1/3)}$ random variables. Define random variables $U$ and $V$ as $$U = |Y-X|, \hspace{5mm} V = X+Y$$ Are $U$ and $V$ independent?
I am new to the ...
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2
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Methods for Detecting outliers in a time series
I have a question on detecting the outliers in a time series like PPI, CPI, inflation,...etc.)
Which method should I use? How can I precisely detect these outliers in a test or a method?
Please ...
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0
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19
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derivative of Logistic Regression with sigmoid func [duplicate]
I am having difficulty figuring out, why I get different answer from the professor. we are tasked with finding the deriative of the logistic regression cost function with the sigmoid function:
$$L(w│D)...
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2
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0
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54
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derivative of Logistic Regression (sigmoid) [closed]
I am having difficulty figuring out, why I get different answer from the professor. we are tasked with finding the deriative of the logistic regression cost function with the sigmoid function:
$$ L(w│...
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