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.
8,138
questions
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1
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18
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Unsure about assumptions of linear model with time series variables, spurious regression and periodic patterns
Background
I'm learning about time series in context of linear regression. The goal of this question is to understand how seasonality of either X or Y can affect the model.
Linear model assumptions
$...
- 251
1
vote
0
answers
18
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What is the formula to compute the height of significance line on autocorrelation plot?
Autocorrelation of a time series can be plotted in R with use of acf function. For example:
acf(ldeaths) # built-in series
I ...
- 251
0
votes
1
answer
21
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Gaps (misinterpreting perhaps) between chi square sig and Cramer’s v
I did chi square test for independence and Cramer’s V test.
I’m having problems with interpret the results.
I checked if there is a connection between gender and dropping out of school (in SPSS).
My ...
- 3
2
votes
1
answer
32
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MLE of $\beta$ and $\sigma^2$ in linear regression with Laplace errors
Say we have a linear regression model
$$
y_i = x_i^T\beta + \sigma\varepsilon_i
$$ for $i = 1, \dots, n$, and $\varepsilon_i$ is distributed according to a Laplace distribution $\mathcal{L}(0, 1)$ ...
- 193
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33
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Complete second-order model with 2 quantitative and 1 qualitative variables
I want to check if what I have for the equation for the second-order model relating y to 2 quantitative and 1 qualitative variables (3 levels) is correct:
$\mathbb{E}(y)= \beta_0 + \beta_1x_1 + \...
- 43
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0
answers
52
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Usage of Sufficient statistic for a Gamma distribution
I need some help to understand how to utilize sufficient statistic from a data.
Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I ...
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49
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Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$
Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain.
We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
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3
votes
2
answers
176
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Long-run average cost for uniform distribution
The lifetime of a device is a continuous random variable having the continuous uniform distribution $\mathrm{Unif}(0,15)$. Suppose that under an age replacement strategy a planned replacement at age $...
- 41
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0
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19
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Lasso Regression Problem [duplicate]
$\operatorname*{argmin}_\beta\{\|y-X\beta\|^2 + \lambda\|\beta\|_1$, where $X$ is orthonormal. $\beta \in \mathbb R^d$. $X = [x_1,\ldots,x_n]^T$ and $y=(y_1,\ldots,y_n)^T \in \mathbb R^n$. $X^TX=I_{d\...
- 315
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67
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Posterior distribution of shape & rate parameter in Poisson-Gamma Mixture
Currently I'm struggling to handle the following question.
Suppose $x_i,(i=1,2,\dots,n)$ follows Poisson distribution:
$$p(x_i|\theta) = \frac{\theta^{x_i}e^{-\theta}}{x_i!}, \quad x_i\in\mathbb N,\...
- 311
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1
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38
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question about outliers in a 5 term data set
the question states that no data set with only 5 terms has an outlier, and I'm stumped
you're given a set of 5 numbers in ascending order
$$
x_1, x_2, x_3, x_4, x_5
$$
I started by finding the IQR, ...
4
votes
2
answers
254
views
Expected value of decreasing function of random variable versus expected value of random variable
Given two random variables $X_1$ and $X_2$ (same sample space $\mathcal{X}$) that
$$\mathbb{E}[X_1]=\int_{\mathcal{X}}xf_1(x)dx > \mathbb{E}[X_2]=\int_{\mathcal{X}}x f_2(x)dx$$
Can we say that $\...
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2
votes
2
answers
81
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How to find probability from $E[X^n]$?
It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$
I need to find $P(|X-\frac{1}{2}| > 1)$.
What my approach is :
I have opened the modulus ...
- 195
1
vote
2
answers
36
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Interpretation of incidence rate ratio
The incidence rate ratio (and 95% confidence interval) of rotavirus gastroenteritis for the vaccine group compared to the placebo group is $0.67 (0.55, 0.82)$.
I want to know approximately how many ...
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0
answers
67
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Posterior Distribution using Jeffreys prior
I'm trying to show that if $X_1, \cdots, X_n \stackrel{iid}{\sim} N(\mu, \sigma^2)$ with unknown $\mu$, $\sigma$ and the prior $\pi(\mu, \sigma^2) \propto 1/\sigma^2$ then the posterior distribution ...
- 183
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1
answer
56
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Hastie "statistical learning" 2.28. Least squares and covariance
In Hasties book "statistical learning", just above equation 2.28, it says that $\mathbf{X}^T\mathbf{X} \rightarrow NCov(X)$ (when $N$ is large and $E(X)=0$).
Why is this true?
$Cov(X)$ is ...
- 1
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0
answers
25
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PDF of difference of uniform distributions [duplicate]
Main questions are in bold but feel free to correct me if I'm wrong somewhere else. As far as possible, I need both intuition and formal explanation.
Let $X \sim Uniform(a,b)$ and $Y \sim Uniform(c,d)$...
1
vote
1
answer
34
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FInding a complete and sufficient statistic
I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class:
Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
- 315
1
vote
0
answers
35
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Understanding Schoenfeld Residuals
I'm reading the original paper by Schoenfeld from 1982. It is very laconic.
Few parts that I don't understand:
Why can we substitute $\hat \beta - \beta$ with the Score, and why is the score equal to ...
- 3,403
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30
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GLM link function, variance Function, and dispersion for linear regression
we were recently introduced in class to GLM's and I am still trying to wrap my head around link functions, variance functions and dispersion paramters. In particular we were asked re-write the linear ...
- 315
1
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0
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38
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What to do in Box-Jenkins framework when time series has deterministic trend and seasonality?
I'm self-studying time series and I'm puzzled by apparent lack of consistency between :
the "classical" decomposition of time-series
and the Box-Jenkins methodology.
Concerning the ...
0
votes
0
answers
37
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BIvariate Normal and Conditional Expectation
I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X ...
- 315
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votes
1
answer
37
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F-distibution only depends on degrees of freedom
I am reading ISLP Chapter 3.2 on multiple linear regressions. They define the F-statistic as $S_1$, the variance of sample 1 from population 1, divided by $S_2$, the variance of sample 2 from ...
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0
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52
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Two-proportion hypothesis test with variances unknown
"A group of test subjects are split into two groups. 1000 people are given a vaccine, and 15 of them get the disease. 800 people are given a placebo, and 60 of them get the disease. Test the ...
- 207
3
votes
1
answer
66
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How to show in R, using a simulation, that when sampling from a normal distribution, the sample mean and sample variance are independent
What is the best way to show that when sampling from a normal distribution, the sample mean and sample variance are independent?
I know the theory behind this result, I would like to show it using a ...
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49
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F-Statistic and F-distribution [duplicate]
I am reading ISLP chapter 3.2 about multiple linear regressions.
F-statistic seems like to be a statistic calculated per sample. But also we know that it is just and estimation for the stat on whole ...
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1
vote
0
answers
41
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Finding probability involving dependent random variables [closed]
Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between ...
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2
votes
1
answer
66
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Exercise involving Bayes' Theorem
Consider the problem
During exam time, in a certain school, only 25% of the teachers warn their students in writing that they are not allowed to get up to ask questions during exam time.
in writing to ...
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3
votes
2
answers
91
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Find $P(Y>\frac{5}{4})$ when joint pdf of $X$ and $Y$ is given
The joint pdf of $X$ and $Y$ is given as
$f(x,y)$ = $\frac{1}{2x^2y}$ , $1<x<∞$ and $\frac{1}{x}<y<x$
Then find $P(Y>\frac{5}{4})$
Since the limits of $Y$ have $X$ in them, I am not ...
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votes
3
answers
482
views
How do I compute a probability from the MGF?
I have a random variable $X$ with moment generating function:
$$m_X(t) = \frac{2}{9} + \frac{e^{-t}}{9} + \frac{e^{-2t}}{9} + \frac{2e^{t}}{9} + \frac{e^{2t}}{3}.$$
I want to find the probability $\...
- 195
4
votes
1
answer
140
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Comparison of confidence intervals: bootstrap & exact resampling
Consider data $X_1,...X_n$ generated from a probability distribution $F$ with density $f$.
I'm interested in constructing confidence intervals for a parameter say, $\theta(F)$. Via Monte Carlo ...
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3
votes
1
answer
136
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If X ~ Bin(2, 1/2) and Y ~ Bin(3,1/3), then find P(2X+3Y=13)
The 2 random variables have different parameters p, therefore cannot be added. I understand that
2X ~ Bin(4,1/2) and 3Y ~ Bin(9,1/3) [by additive property of binomial distribution with same parameter ...
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0
votes
1
answer
28
views
Principal Component Analysis and Relation to the SVD of a matrix [duplicate]
We are learning about Principal Component analysis in our class, and I having trouble understanding how to compute the principal component given a matrix. For example, here is the matrix we were given....
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7
votes
1
answer
94
views
Power of two-sample z-test
In a pilot study with two groups, the control distribution has the Mean1 = 90, SD1 = 5 and the treatment distribution has Mean2 = 85, SD2 = 5. The null hypothesis of the test is that the sample ...
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0
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24
views
Does this state space model make sense?
I'm working on a problem,
Consider that a times series $\{y_t\}$ is generated from an $\text{ARIMA}(1,1,1)$ model, so that $$y_t-y_{t-1}=\alpha(y_{t-1}-y_{t-2})+\epsilon_t+\gamma\epsilon_{t-1},$$ ...
- 589
1
vote
1
answer
47
views
What modeling approach should i use AR, MA, ARMA?
those are my ACF and PACF plots for my time series after two differentiating. I watched a couple of tutorials but I cannot figure out what method I am supposed to use. Also, an interpretation of the ...
- 143
1
vote
0
answers
119
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Distribution of $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ [closed]
Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution ...
- 365
3
votes
2
answers
186
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Calculating $E[(\sum X_i)^4]$
Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
- 365
1
vote
3
answers
97
views
Asymptotic normality implies consistency
I'm trying without success to solve the following exercise in my econometric textbook:
Show that $\sqrt{N}\left(\widehat{\beta_1} - \beta_1 \right) \xrightarrow{d} \mathcal{N}(0,a^2)$, where $a^2$ is ...
1
vote
1
answer
27
views
How to manually evaluate a parameter estimate using the first few values of a time series?
I'm working on a problem (I paraphrase),
Consider the $\text{AR}(2)$ model $$y_t=\alpha y_{t-1}-(1-\alpha)y_{t-2}+\epsilon_t, \hspace{1em} \epsilon_t\sim N(0,\sigma^2).$$ The conditional least ...
- 589
1
vote
0
answers
43
views
Deriving Sample Size for two samples t test
I am trying to figure out what sample size recommendations to make in this case when the samples have different sizes. Since their sizes are different the variance of each will be different.
Suppose ...
- 315
4
votes
2
answers
79
views
In R, why are the results of summary(lm(y~-1))$sigma^2 and mean(y^2) same?
Consider the following data and the model:
y <- c(9,3,-2,7,6,12)
lm(y~-1)
summary(lm(y~-1))$sigma^2
[1] 53.83333
I expected ...
- 309
4
votes
1
answer
205
views
Finding the MLE for a piecewise function
I am attempting to find the MLE for the two parameters (i.e. $\alpha, \beta$) in the following piecewise function, but I am slightly confused as to how to proceed through this question:
$$
P(X_i \leq ...
- 315
0
votes
0
answers
44
views
How to derive OLS estimates of beta by minimizing the error sum of squares?
For the model: $Y_i=\beta_0+\beta_1X_i+\epsilon_i, \epsilon_i \sim^{iid} N(0,\sigma^2)$ for $i=1,...,n.$ Let $X_i >0$ and $\sigma^2_i=\sigma^2X_i$. $\tilde{Q}(\beta)=\sum^{n}_{i=1}\sigma_{i}^{-2}\{...
- 315
3
votes
1
answer
89
views
Simple vs. Composite Hypothesis Question
I am looking at one of the questions in the qualifying exam, where a multiple logistic regression model is being fitted, where the response is about dying.
The question says that there are multiple ...
- 43
1
vote
0
answers
66
views
How to deal with a indicator function in a likelihood? [duplicate]
Question: A panel study followed 25 married couples over five years. One item of interest is the relationship between divorce rates and the various characteristics of the couples. For example, the ...
- 11
3
votes
1
answer
72
views
How to write the AR polynomial for a model containing a constant?
I'm working on an exercise,
Consider the autoregressive time series model $$y_t=c+\frac{1}{3}y_{t-1}-\frac{1}{2}y_{t-2}+\epsilon_t,\tag{$\ast$}$$ where $\epsilon_t$ is white noise with variance $1$. ...
- 589
0
votes
0
answers
35
views
How to draw samples from two correlated Negative Binomial variables?
Data and problem description: my data is the number of corner kicks of home team, away team, total corner kicks and corner kicks difference. Below is code for data and the plots (assuming the number ...
- 57
0
votes
0
answers
21
views
Single or Multiple Bivariate analysis?
For the data structure, we are fitting the multilevel model to the data. Before fitting the model, we are eager to do bivariate analysis so that we can keep those independent variables in the ...
- 309
1
vote
0
answers
74
views
Correct interpretation of Bayesian credible intervals
I'm starting to learn about Bayesian statistics for my own personal education (this is not a homework question). Something puzzles me a bit about the interpretation of credible intervals.
From the few ...
- 57