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.
8,064
questions
0
votes
1
answer
53
views
How to tell if an estimator is unbiased? How to find expected value of an estimator?
You come up with a great idea of an estimator for $\beta_1$ in the SLR model which satisfies SLR.1 to SLR.4:$$y_i=\beta_0+\beta_1x_i+u_i$$ Given a sample $\left\{(x_i,y_i),i=1,2,3,\dots,n\right\}$, ...
1
vote
0
answers
29
views
Conditional Distribution of Multivariate Gaussian given Linear Inequalities
Consider a multivariate Gaussian $Y\sim\mathcal{N}(\mu,\Sigma)$ of dimension $n$. For fixed $c\in\mathbb{R}^n, A\in\mathbb{R}^{m\times n}$ and $c\in\mathbb{R^m}$, what is the conditional distribution ...
- 11
3
votes
0
answers
52
views
Expected value for a fair die game
You roll a fair die until the cumulative sum rolled so far is a multiple of 3. You get $1 for each roll. What is the expected amount you will get?
I started by finding the probability of the game ...
- 161
1
vote
0
answers
28
views
Convergence of estimated Survival Functions
Q1 part A&B I have so far
$$\underset{n\rightarrow\infty} {\lim} \frac{1}{n}\sum_{i=1}^nI(T_i>x)$$
since we are summing an indicator variable we can say it has a Bernoulli distribution with ...
- 11
1
vote
2
answers
292
views
Integral pdf is not 1. I don’t know what’s wrong [closed]
I differentiated cdf to find pdf and then integrated it, but the result is not 1. What’s wrong with my calculation?
CDF
$$F(x)=\begin{cases}
0&x<1\\
\dfrac{x^2-2x+2}{2}&1\le x<2\\
1&...
- 29
0
votes
0
answers
13
views
Equidispersion Property - Why and Where
I've often looked at count data models over the years. And like a lot of statistical models, they can be used/abused in cookbook style.
Identify equidispersion -> do X model (Poisson).
Identify ...
- 244
1
vote
0
answers
35
views
Determining the Identifiability of Models
I am completing exercises in the book Mathematical Statistics: Basic Ideas and Selected Topics regarding proving or disproving that a model is identifiable.
The problem I am struggling with considers $...
1
vote
0
answers
32
views
How is this a permutation test?
I am trying to read a research paper (link) that involves a permutation test. My (limited) understanding of a permutation test is that you get two samples of data, compute a statistic based on both ...
- 11
0
votes
0
answers
37
views
Posterior Distribution using a Normal Likelihood and Laplace Prior
I have the working out below but is this correct. I just want the posterior distribution of when mu=0 given x.
What I have tried is setting mu=0 after rewriting the first pdf with the summation ...
0
votes
2
answers
44
views
How can we maintain asymptotic normality with slight change?
If $(X_n-\mu_n)/\sigma_n\rightarrow_{d} N(0,1)$ (i.e., $X_n$ is $AN(\mu_n,\sigma_n^2)$), I want to show the following two statements:
(1) $X_n$ is $AN(\bar{\mu}_n, \bar{\sigma}_n^2)$ if and only if $\...
- 1
1
vote
1
answer
42
views
Newton's method and the Hessian for Softmax Logistic Regression
I'm having some trouble with optimising softmax regression via Newton's method. I'm not sure if the problem is arising with my equations for the Hessian and Gradient or with the code I've written.
For ...
- 165
1
vote
0
answers
91
views
Expectation of the realized volatility
I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix}
y_t = \exp \big( \frac{h_t}{2} \big) \...
- 41
2
votes
2
answers
156
views
How should I interpret the assumption of the regression?
I read an econometrics book which states one of the basic assumptions of regression is that
$$E(u|x) = 0$$
In another book however I see it written that
$$E(u_i|x_i) = 0$$
Are these two saying the ...
0
votes
0
answers
41
views
Why is this program not necessarily an MCMC?
In Chapter 1, Handbook of Markov Chain Monte Carlo, Geyer writes
Suppose you have a computer program
...
- 135
0
votes
0
answers
16
views
Equation for all frequent itemsets
Consider that the database has only three transactions {<a1,a2,...,a100>,
<a1,a2,...,a50>, <a1,2,...,a25>}. Suppose min_sup = 2. Give the equation for all
frequent itemsets. Also ...
1
vote
1
answer
53
views
Calculation of Posterior distribution numerically
For calculating posterior probabilities numerically, I did not understand that why is in the following codes they have divided by 0.001 in the denominator to calculate ...
- 47
1
vote
0
answers
30
views
Show: scaling transformation $Y_i=D^{-1}(X_i-\bar{X})$ can be written as $Y = HX_{n\times p}D^{-1}$
Show that the scaling transformation $\textbf{y}_i=D^{-1}(\textbf{x}_i-\bar{\textbf{x}})$ can be written as $Y = HX_{n\times p}D^{-1}$ where $H$ is the centering matrix $(I_n-\frac{1}{n}\textbf{11}')$....
- 187
1
vote
1
answer
101
views
Poisson Regression and Linear Regression give the same error
I'll use the data azdrg112 from COUNT package. The los will be the response variable while ...
- 75
1
vote
0
answers
52
views
Appropriate statistical test with pre\post experiment
This might be very simple so excuse me in advance. I'm trying to test if the difference between five test scores is statistically significant in a pre\post experiment with three different times, as ...
- 11
0
votes
0
answers
23
views
Two contradicting derivations of the Covariance Matrix for Linear Regression
I am looking to compute the variance covariance matrix for the standard linear regression coefficients $\hat{\beta}$ when:
$$Y = X \beta + \epsilon $$
and $\epsilon \sim N(0,\sigma^2)$. I have derived ...
- 610
0
votes
0
answers
23
views
Show that values produced by a random variable form the sample space of a new random variable
The set of values X($\omega_{k}$) of a random variable on $\Omega$ form the points of a new sample space $\Omega_{X}$. Show that $\omega$ is a random variable on $\Omega_{X}$ (i.e. no information has ...
- 1
4
votes
3
answers
192
views
Prove that $\mathbb{E}\vert X \vert^{a}<\infty$ iff $\sum\limits_{n=1}^{\infty}n^{a-1}\mathbb{P}(\vert X \vert \geqslant n)<\infty$
Suppose $X$ is a random variable, $a>0$ is a constant. Prove that $\mathbb{E}\vert X \vert^{a}<\infty$ iff $\sum\limits_{n=1}^{\infty}n^{a-1}\mathbb{P}(\vert X \vert \geqslant n)<\infty$.
...
- 81
1
vote
0
answers
36
views
Orthonormal Basis assumption in PCA derivation [migrated]
I'm doing the Mathematics for Machine Learning course on Coursera (Course 3, Week 4). I am trying to understand the derivation of PCA.
Specifically from:
$J =\frac{1}{N} \sum_{n=1}^{N}\Vert \sum_{j=M+...
- 349
0
votes
0
answers
37
views
Find the conditional PDF of a multivariate normal distribution given a constraint [duplicate]
Problem to solve
We have a vector of random variables $\textbf{X}=(X_1,X_2)$ issued from a bivariate normal distribution. In particular, $\mu = \begin{bmatrix} 2 \\ 3 \end{bmatrix}$, $\Sigma = \begin{...
2
votes
1
answer
56
views
Cook's distance for GLM
In Applied Logistic Regression by Hosmer, Lemeshow and Sturdivant (2013), the formula for Cook's distance in a logistic regression is given as,
$$\Delta\beta_j = \frac{r_{sj}^2h_j}{1-h_j}$$
where $r_{...
- 57
2
votes
1
answer
243
views
Recommendations for study material for mathematical statistics
I am currently preparing for a qualifying exam in mathematical statistics and am looking for good resources for self-study. I have access to old exams which I have been slowly working on; however, I ...
1
vote
1
answer
51
views
maximum likelihood estimator of regression coefficient
Consider the following simple linear regression model:
$$ y_i = (-1)^i \cdot \beta_1 + \epsilon_i \quad \text{where} \quad i = 1, \ldots, n $$
here $\epsilon_i$'s are i.i.d. $N(\beta_0, 1) \text{ ...
- 115
1
vote
0
answers
50
views
Stochastic volatility model- autocorrelation of $\Delta \ln(y_t^2)$
for the following stochastic volatility model : $y_t = exp(\frac{\alpha_t}{2}) \epsilon_t $
where $\epsilon_t \sim NID(0,1) $
$\alpha_{t+1} = \phi \alpha_t + (1-\phi)\mu +\eta_t$
where $\eta_t\sim NID(...
- 41
2
votes
1
answer
52
views
The difference of $\sum_{i=1}^{n}X_{i}$ and $\sum_{i=1}^{n}X_{(i)}$
Here is a exercise from *Mathematical Statistics. Jun Shao. Second edition. EX2.20
Let $X_1,..., X_n$ be $i.i.d.$ random variables having the exponential distribution $E(a,\theta)$, $a\in R$, and $\...
- 81
2
votes
0
answers
58
views
How can I find the unconditional variance of this process?
Let $y_t = \Delta p_t$ denote a time series of asset returns, where $p_t$ are logarithmic prices. $y_t$ is generated by a heteroskedastic MA(1) process
\begin{aligned}
y_t &= z_t+\theta z_{t-1}, \\...
- 41
1
vote
1
answer
48
views
Predictive Distribution in Gaussian Process for Machine Learning
I am reading Gaussian Process for Machine Learning equation 2.9, where it is deriving the predictive distribution
$$p(f_* | \mathbf{x}_*, X, \mathbf{y}) = \int p(f_* | \mathbf{x}_*, \mathbf{w}) p(\...
- 11
3
votes
1
answer
71
views
Let $N(t)$ is a poisson process with rate $\lambda$, $T^* \sim \operatorname{Exp}(\lambda^*)$, find the expectation of $N(\min(t, T^*))$
Currently, my approach is to split $N(\min(t, T^*))$ like the following by the law of total expectation.
\begin{align*}
&E(N(\min(t,T^*))) = E(N(t \wedge T^*)) \\
= {}&E(N(t\wedge T^*) \...
- 133
1
vote
1
answer
38
views
Probability generating function [closed]
X and Y are independent random variables having poisson distribution with parameters a and b respectively. By using probability generating function, prove that X + Y have a poisson distribution and ...
- 11
0
votes
0
answers
27
views
How to calculate lambdas from Bivariate Poisson distribution?
I have three random variables X1, X2, X3 which follow independent Poisson distributions with parameters λ1, λ2, λ3 >0 and then the random variables X = X1 + X3 and Y = X2 + X3 jointly follow a ...
- 75
0
votes
0
answers
16
views
How to determine significance for a Corrado rank test in an event study?
I apologise if this is a stupid question (it feels stupid tbh). I am currently doing an event study and my abnormal returns are not normally distributed. I am now in the process of performing a ...
3
votes
1
answer
52
views
Simulating iid mean zero Matérn processes
I'm currently learning FDA (functional data analysis), following "Introduction to Functional Data Analysis" by Kokoszka & Reimherr. Here it is exercice 1.5:
1.5 The Matérn covariance ...
0
votes
0
answers
27
views
Generalized likelihood ratio test for a left-truncated exponential distribution [duplicate]
I am doing self study in statistical inference and am rather confused about how to approach generalized likelihood ratio test (GLRT) problems. I am trying the traditional approach by definition and ...
- 150
0
votes
0
answers
22
views
Prove second derivative of loglikelihood is Fisher information [duplicate]
Problem
Prove below equation , fisher information come from negative of second derivative of likelihood.
$$\begin{align*}
\mathcal{I}(\theta) = \mathbb{E}_{x\sim p(y;\theta)}\left[-\nabla^2_{\theta'} \...
- 135
1
vote
0
answers
26
views
Wald Test on two exponential samples
Based on independent samples of data $X_i \sim$ Expon$(\lambda_1)$, $i = 1, \cdots, n,$ and $Y_j \sim$ Expon$(\lambda_2)$, $j = 1, \cdots, n$ (both samples of the sample size $n$), find the Wald test ...
- 11
0
votes
0
answers
32
views
How to prove EM algo convergence?
Original Problem
I'm looking at problem-set in cs229 about EM algorithim.
As my understanding $\ell_{\text{semi-sup}}(\theta)$ is
$$
\ell_{\text{semi-sup}}(\theta) = \sum^m_{i=1} \log \sum_{z^{(i)}} ...
- 135
1
vote
0
answers
75
views
Geometric intuition of kernel trick
I would like to understand better the geometry underlying the Kernel trick with the Gaussian Kernel. In particular my question is:
How the Kernel trick can be interpreted geometrically, in particular ...
- 840
2
votes
2
answers
89
views
If I draw the variable "m" from M~N(0, 1), then draw "x" from N(m, 1), what is the distribution of "X" (not X|M)
Step 1:
I draw the observation "m" from M~N(0, 1) (i.e. a Normal distribution with mean 0 and variance 1)
Step 2:
Then I draw "x" from N(m, 1) (i.e. a Normal distribution centered ...
- 185
0
votes
1
answer
24
views
One simple question related to probability equations convertion
Why
$$\begin{align*}
l(\theta) &= \sum^m_{i=1} \log p(x^{(i)};\theta) \\
&= \sum^m_{i=1} \log \sum_{z(i)} p(x^{(i)},z^{(i)};\theta)
\end{align*}$$
Find this equation from cs229-2018 problem-...
- 135
0
votes
0
answers
12
views
Time series data [duplicate]
Is the following M4_regular_matrix time series data? I am a bit confused since The rows indicate locations although the columns consist of time. We actually ...
- 327
1
vote
1
answer
31
views
Time Series plot in R [closed]
The following R codes produce one graph for time series, while the next chunks of codes produce 8 graphs. Why?
...
- 327
17
votes
3
answers
2k
views
What is the expected number of children until having the same number of girls and boys?
A couple decides to keep having children until they have the same number of boys and girls, and then stop. Assume they never have twins, that the "trials" are independent with probability 1/...
- 171
0
votes
1
answer
40
views
Assessing model performance [closed]
Suppose there are $n=100000$ locations. At each location,there are observations of dependent variable $Y$ for 30 years. That is, we have time series data and this data have $100000$ rows and $30$ ...
- 327
2
votes
1
answer
61
views
Correct notation when proving that $\hat{\beta_1}$ is linear
When reviewing lecture slides for the proof that $\hat{\beta_1}$ is linear in OLS-regression my teacher posted the following on the lecture slides:
$$\hat{\beta_1}=\frac{\sum(X_i-\bar{X})Y_i}{\sum(X_i-...
- 23
0
votes
0
answers
42
views
Are innovations $0$ for the first term in a differenced series?
I'm working on an unassessed course problem,
Consider a time series consisting of quarterly observations of temperature in a city. The seasonally differenced time series $\{x_t\}$ with $D=1$ is ...
- 535
0
votes
0
answers
14
views
How to test a MA process with extra terms for invertibility?
I'm working on an unassessed course problem (paraphrased for brevity),
Consider the time series model $$y_t=\alpha+\beta t+\epsilon_t$$ where $\epsilon_t$ is a white noise and $\alpha,\beta$ are ...
- 535