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.

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Looking for a predictive prior for what I thought to be Bayesian Analysis

I need a prior with 2 parameters location (μ) and scale (σ) (a transformation of mean and standard deviation). My distribution is over discrete values, equally spaced with distance $c$ between ...
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3 votes
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27 views

Exact/Approximate Confidence Interval of Parameter Ratio from two Samples of iid Exponential

Suppose 2 independent samples $X_1,...,X_n \sim Exp(\lambda_1)$ and $Y_1,...,Y_m \sim Exp(\lambda_2)$, and are iid within samples. I am thinking about how to make an exact confidence interval for $\...
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Procedures to show that a process is not ergodic

I'm trying to show that a certain process is not ergodic, but as I don't have much experience, I would first like to learn how to show simple cases. We know that if a discrete stochastic process is i....
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2 answers
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Cross-validation by hand in R

Trying to teach myself cross-validation on a super simple example, linear regression. My understanding is that when I build a model via CV, it should have a lower RMSE. But I find almost identical ...
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Covariance of the empirical probability mass function

Suppose a discrete random variable $Y$ takes $k$ levels of different values $y_1,y_2,...,y_k$. Let $P(Y=y_k):=p_k$. Suppose we have $n$ i.i.d. samples of $Y$, my question is: How can we compute the ...
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3 votes
1 answer
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Boolean order statistic

Inspired from this LeetCode question. I authored this question myself however. Please review my answer for accuracy. Let's sort an array ($n \ge 1$) of $1$s and $0$s, so that all $0$s come before $1$s....
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1 vote
1 answer
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Binomial Bayesian Regression with Metropolis Hastings in R

I'm trying to implement the Metropolis Hastings algorithm in this problem but I'm having problems with the convergence. $$Y_i|\beta_0,\beta_1 \sim \text{Binomial}(m_i,\theta_i)$$ where $logit(\theta) =...
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Validating Randomization

Was the randomization successful in the table below? Why or why not? and How does one get to know this?
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Statistical inference question evaluating a joint probability mesure w.r.t. a distribution

Note: I have edited this question and delted my old question after following the guidlines to be more clear. I hope this new question is clear and well posed. I attempted this question but I got stuck ...
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1 vote
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Prove sample minimum and unbiased sample variance are independent

I have a simple random sampling (n observations) of the population $X$, whose pdf is $$f_\theta (x) = e^{-(x-\theta)} I_{(\theta,\infty)}(x) $$ and $\theta \in \mathbb{R}$. $S^2_n$ is the unbiased ...
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What does capital D above an equal sign mean?

I was beginning to self study time series and came across this notation: $\stackrel{\mathcal{D}}{=}$. I wonder what it means? (I have limited access to the textbooks for the moment, so I have to post ...
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Hypothesis testing for $\sigma^2$ in linear regression

I'm trying to figure out a way to test hypotheses about variance in the linear regression model. The hypotheses I want to test are: \begin{align*} &H_0:\sigma^2=1\\ &H_1:\sigma^2<1 \end{...
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1 vote
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Expectation of the product of multiple correlated 1-D normal variables [duplicate]

First question: is it possible to have a set of $k$ random variables $\left\{X_i\right\}$ s.t. each $X_i \sim N(0,1)$ individually, and $\text{Corr}(X_i,X_j)=\rho$, $∀i\neq j$? If those conditions are ...
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1 vote
1 answer
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What is a name of regression?

Let's say there is a set of independent variables $x_1, x_2, ..., x_n$ and a target variable $y$. A transformation is applied to the initial set of variables: $z_1=f_1(x_1,x_2,..., x_k)$, $z_2=f_2(x_{...
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UMVUE of $ \theta^2 (1- \theta ) $ X is random sample from bernoulli distribution

Let $ X_1, X_2 ..... X_n $ be a random sample from bernoulli distribution with parameter $ \theta $ , Obtain UMVUE of $ \theta^2 (1- \theta ) $ MY APPROACH I calculated that T = $ \sum X_i $ is ...
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Estimating Successful Combinations of Basketball Players

Suppose we have to analyze sports data (from our school 's informal basketball data). We have data for all games from the 2019 season, for each game we know which players were playing and which ...
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How to explain effect of outliers in logistic model through influence function

The influence function of the MLE for the logistic model is $IF(y,x,\beta)=M^{-1}(y-F(\beta^Tx))x$, where $y\in \lbrace 0,1\rbrace, x\in \mathbb{R}^p$, $F(z)=e^z/(1+e^z)$ with $F'$ denoting its ...
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Derivation of the SVM regression problem

EDIT: The terms at the end will not get cancelled as they involve different variables: $\alpha$ and $\alpha^*$, etc. Now the post is corrected. I will leave this question here in case someone is as ...
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root n consistency of parameter in mixture distribution

We have iid observations $\{X_i\}_{i=1}^n$ from CDF $\theta G + (1- \theta)H$ where $\theta \in (0,1)$ is unknown. Find a $\sqrt{n}$ consistent estimator for $\theta$ using the observations. Note that ...
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2 votes
1 answer
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Generalized difference-in-differences without unit specific effects

Can I get some help in answering the following task? I figured it is generalized difference-in-differences (DiD) (the task is called "RCT and DiD"), but I don't know how to circumvent the ...
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0 answers
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In a shipment of 20 engines, history shows that the probability of any one engine proving unsatisfactory is 0.1. can you please provide the answer?

What is the probability that the second engine is defective given the first engine is not defective? From the result, draw the conclusion if the first and second engines are dependent or independent. ...
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1 vote
1 answer
43 views

T-distribution problem

I am working through the exercises of Martin Sternstein's Statistics (Barron's College Review Series) ... struggling with Exercise 4 in Chapter 14. Given are the weight lifting strengths of 9 people ...
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1 answer
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Hazard ratio with time-varying covariates

I am confused about time-varying covariates in Cox models. I believe this refers to the fact that in the normal situation, you can calculate a hazard ratio for $X$, assuming $X$ for an individual is ...
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$X=Y_{1}+Y_{2}$, given $R^2$ of regression of $Y_{1}$ on $X$, find $R^2$ of regression of $Y_{2}$ on $X$

Using National time series data for $i=1,...,n$, I run Least Squares Linear relationship of $Y_{1}$=Consumption on $X$, where $X$=disposable income, obtaining $\hat Y_{1}=a_{1}+b_{1}X$. I also run the ...
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1 vote
2 answers
81 views

Sufficient condition for $ \sigma_{X}^{2} \leq \sigma_{Y}^{2}$

Suppose $X$ and $Y$ are random variables whose expected values are $\mu_X$ and $\mu_Y$, and variances are $\sigma_{X}^{2}$ and $\sigma_{Y}^{2}$, respectively. Also, we suppose $F_x$ and $F_Y$ are the ...
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Question regarding the Bayes estimator of a parameter $\theta$ which has a continuous distribution

Let $c > 0$ and \begin{equation} L(\theta,a)=\left\{ \begin{array}{@{}ll@{}} c|\theta-a|, & \text{if}\ \theta < a \\ |\theta-a|, & \text{if}\ \theta \ge a \quad. \end{array}\...
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1 vote
0 answers
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How can any sample be drawn from a continuous distribution? [duplicate]

Since the probability of a continuous variable $X$ assume any particular value is always zero, how can any sample be obtained from such distribution? How can we obtain, e.g., the set of observations $\...
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Assessing the effectiveness of a teaching tool by paired t-test

You have been asked to assess the effectiveness of a new teaching tool. A sample of 25 students from your classes have been tested before and after the provision of the tool, so each student has two ...
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2 votes
1 answer
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power and sample size estimation

For a one-sided test: $$H_0: \mu_1 \leq \mu_2 \ H_1: \mu_1 >\mu2$$ $$\alpha=0.025,\beta=0.1$$ How to calculate the sample size needed? Is this information enough to calculate the sample size?
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1 vote
0 answers
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No previous information in a clinical trial [closed]

I'm conducting a clinical trial with a treatment group and a placebo group and calculated the sample size based on a previous study. However, I would like to add two more new placebo groups with ...
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9 votes
1 answer
183 views

Finding the MVUE of the center of a circle of unknown location

Is there a known analytic solution for finding the minimum variance unbiased estimator of a disk of an unknown location given that a sample of $n$ points was drawn uniformly and randomly from the disk ...
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1 vote
1 answer
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Finding the non-zero region of a marginal

Basic question but: what happens to the region on which a pdf is non-zero when a bivariate is integrated to get a marginal? The example I'm working on (course problem booklet for a mathematics BSc ...
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3 votes
1 answer
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Calculate sample size for clinical trial without raw data

Usually, we calculate the sample size needed for a clinical trial based on a former clinical trial which shows there is a significant effect. What if there is no such a trial? What if there is no ...
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6 votes
1 answer
104 views

What is Galton's paradox?

From One Thousand Exercises in Probability I found the exercise: Galton's paradox. You flip three fair coins. At least two are alike, and it is an even chance that the third is a head or a tail. ...
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$E(SN)$ for aggregate claim amount $S$, $S=X_{1}+...+X_{n}, X_{i}$ are iid [duplicate]

Consider the following model for aggregate claim amounts $S$: $S=X_{1}+X_{2}+...+X_{N}$ where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
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1 vote
1 answer
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When does a difference in means not capture the true treatment effects vs a regression with pre-treatment controls?

A question from Gelman - Regression & Other Stories... In answer to my own question...my understanding is that a difference in means should not capture the treatment effect when there are pre-...
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0 votes
1 answer
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Bayesian model comparison with systematic error

Two parameters $(x,y)$ were measured for 3 different objects, wielding the following results: $$\{ (x,y) \}= \{ (1,3), (3,5), (5,9) \}$$ Knowing that the error in the estimation of the values $y$ is ...
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question of a Poisson random effect model

Assume a Poisson random effect model (with a random intercept) $$\log E(Y_{it}|b_i,x_i)=\beta_0+\beta_1x_i+b_i$$ $$b_i \sim N(0,\tau^2)$$ How to derive the population-averaged effect of one unit ...
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0 votes
1 answer
52 views

football probability

I have the following problem: Suppose that the probability of a football team (T) to win the match with 2-1 is p1 and, given that T scores a goal, the probability that the best player of the team ...
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0 votes
0 answers
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find a suitable statistical method to analyze Yes/ No type qualitative data

I have done a survey on finding the applicability of Co- management system on a reservoir. This survey contains 8 main points. Under one main point, I included some Yes/No questions (No of questions ...
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1 vote
0 answers
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Why does my ARIMA predictions on monthly data form a straight line?

For short detail, the goal was to forecast using 51 monthly observations of KPI of project implementations which I aggregated by sum from 463 observations from about 4 years of data (May 2017 to July ...
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Confidence interval for the parameter of a random variable

I am studying statistics, and I came across a problem that I am not sure if what I am doing is correct or if there is some other way to do this. Any feedback is appreciated. I divided in 2 problems, ...
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0 votes
1 answer
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How to add change (in unemployment rate) to dependent variable

I'm an Econometrics beginner and I'm sorry if the title is confusing. I basically have a variable “unemp” which gives me the unemployment, the dependent variable of this model should be the change in ...
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2 votes
0 answers
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Tail-equivalence implying same domain of attraction

Suppose two distributions F and G that have the same extreme point ($x^F = x^G$) and $$\lim_{x \to x^F}\frac{\bar{F}(x)}{\bar{G}(x)} = c \in (0, \infty)$$ Show that F and G belongs to the same domain ...
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How to test for significance with t-test and interaction variables? [duplicate]

I need to answer this question for my Econometrics class: "Does the gender gap differ significantly in urban versus non-urban areas? Test at the 5% level". I think I need to use the t-test ...
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0 answers
24 views

Standardization in Machine Learning Models

Please answer this question in two contexts: Context 1 - Performance: Which models are sensitive AND which models are insensitive to standardization? Why? Are there any edge-cases in which ...
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0 votes
0 answers
26 views

What's wrong with this conditional probability working?

A past exam paper for my course (BSc Mathematics, second-year module in statistical inference and modelling, unpublished) has a question, Let $(X,Y)^T$ be a bivariate random variable with joint ...
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How to determine uncertainty of data from a bayesian posterior distribution

I am a bit confused as to how we determine the uncertainty of a set of data from Bayesian Analysis. In my specific case, I am asked the following: Assume $f(x,x_0)$ as the correct model for the ...
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0 votes
1 answer
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Bayesian update with the shifted and scaled data

Suppose I have data $y$ (N observations) which follows a normal distribution: $y \sim N(\alpha+\beta*\mu,\sigma^2)$ while $\alpha$ and $\beta$ are known parameters. I want to update $\mu$ and the ...
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0 votes
2 answers
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determining the sample size for quality test

It has been a long time since I did statistics in uni and I have a "simple problem" that I need solving. I need to determin for a sample size for a limited population of documents ( lets say ...
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