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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16 views

Independence from conditional independence

If A is independent of B given C and C is independent of B given A how can I show that A is independent of (B, C)? Also if A is independent of (B, C) is A, B, C mutually independent?
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15 views

Efficient influence function in cox proportional hazards model

I was hoping someone could help me with this problem in the cox proportional hazards model. I am given the following setup. T is a non-negative random variable with continous distribution and hazard ...
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12 views

Effect size adjustment

Suppose researchers of a paper conducted a survey where they collected water volume of several rivers before and after rainfall. Similarly, many papers also reported pre- and post rainfall water ...
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1answer
48 views

Is Poisson-Lindley an exponential family? If not, why?

$$\begin{aligned}f_Y(y_i)&=\frac{{\theta_i}^2\left(y_i+\theta_i-2\right)}{\left(1+\theta_i\right)^{y_i+3}}\\ &=\exp\ \log\left[\frac{{\theta_i}^2\left(y_i+\theta_i-2\right)}{\left(1+\theta_i\...
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14 views

minimal sufficient statistics of 1-parameter Gamma distribution

If $x_i \sim Gamma(\alpha, \alpha)$, are the minimal sufficient statistics still $\Pi_i x_i$ and $\sum_i x_i$ (same as when $x_i \sim Gamma(\alpha, \theta)$ where $\alpha \neq \theta$)? My reasoning ...
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30 views

Homework Problem Consistancy of Estimator

We have to show: Let $\theta_0$ be a k-dim vector. Show, that the following statements are equivalent: (1) $\hat{\theta}_n$ is consistent for $\theta_0$. (2) For each component $i= 1,...,k$: $\hat{\...
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13 views

How to calculate Martingale Residuals by hand?

I know we require the breslow estimator for this. But is there a concrete formula to find this by hand? Supposing we're given a data set of 6 individuals (nothing too big) and we know when each ...
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19 views

show X~$N(\mu,\sigma^2)$ is not risk-unbiased under standardized square loss function

Let X follow the $N(\mu, \sigma^2)$ distribution with parameter $\theta=(\mu,\sigma^2)$. First, I try to show X is not risk-unbiased under the standardized square loss function $𝐿(\theta,\delta)= \...
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1answer
33 views

How do you express the variogram $\gamma(u)$ in terms of correlation for a stationary process?

The Analysis of Longitudinal Data textbook by Diggle et al. (2002) mentioned twice (p48 f. and then on p82) that given the following definition of the variogram, \begin{equation} \gamma(u) = \frac{1}...
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1answer
31 views

Convergence Distribution and Probability [closed]

Suppose that $|X_n - Y_n|$ converges in probability to 0, and that $X_n$ converges in distribution to X. Show that $Y_n$ converges in distribution to X. Thanks in advance.
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1answer
39 views

Find the MRE for b of E[0,b]

$X_1,\dots , X_n$ i.i.d. from the E(0,b) distribution. Find the MRE (Minimum Risk Equivalent estimator) for b under the scale transformation group with the standardized square loss $𝐿(𝑏,\delta)=(\...
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20 views

Why are the $n^k$ ordered samples equally likely while the ${n+k-1}\choose{k}$ unordered samples are not?

Source: Blitzstein and Hwang’s Introduction to Probability. The following is said: “Consider a survey where a sample of size k is collected by choosing people from a population of size n, one at a ...
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26 views

First Principal Component Direction

I am trying to derive the first principal component direction from the definition and need help in finding which step is going wrong. Here's my attempt: $\mathbf{X} \in \mathbb{R}^{N \times p}$ is the ...
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27 views

Complete sufficient statistic of non-identical distribution: $X_i \sim EXP(i\theta)$

Problem Suppose that $X_1, \dots, X_n$ are independent $\mathrm{EXP}(i\theta)$ random variables. Find a complete sufficient statistic for $\theta$. My Attempt Since pdf of $x_i$ is \begin{equation} ...
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17 views

Assume a boosted classifier consists of weak hypotheses (aka. weak classifiers) that are each of them implemented by a threshold neuron. In that case, [closed]

Which of the following is True: Assume a boosted classifier consists of weak hypotheses (aka. weak classifiers) that are each of them implemented by a threshold neuron. In that case, the boosted ...
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12 views

Get Sample size from population mean and std dev [closed]

Your personal digital music collection has about 10000 songs, whose mean duration is 4.5 minutes and the standard deviation is 100 seconds. You recently obtained legal software that randomly packages ...
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25 views

prove the difference between mean and median is less than the variance [duplicate]

Suppose $X$ is a random variable with finite variance. Let $m$ denote the median of $X$ and $\mu$ the mean of $X$, i.e. $\mu=\mathbb{E}(X)$. Show $$(m-\mu)^2\leq\text{var}(X)$$ Intuitively this is ...
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1answer
105 views
+50

Inference in Dirichlet process mixtures via collapsed Gibbs sampling

I need to cluster some data $\{x1, \ldots, x_n\}$ through a Dirichlet process mixture model. Consider the following Dirichlet process mixture model, in which the base measure is a $NIW(\mu_0, \...
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1answer
93 views

Showing $X\sim \operatorname{Poi}(\lambda)$ is minimax

Assume that $X$ has $\operatorname{Poisson} (\lambda)$ distribution and the loss function is $\ell(\lambda,a)=\frac{(\lambda-a)^2}{\lambda}$. Now, I want to show that $X$ is minimax. A hint that is ...
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2answers
85 views

Show that classification tables do not always correlate with goodness of fit for logistic regression

Background I am reading the textbook Applied Logistic Regression by David Hosmer, specifically chapter 4, which discusses logistic regression model assesment of fit. Hosmer gives an interesting ...
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5answers
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Is a neural network essential for deep learning?

I received preliminary materials on deep learning in my class. It was written as follows. This raised me the question of the basic meaning of the word deep learning. Deep learning is a machine ...
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12 views

Does PPO update across all training data?

I'm a bit confused by the update terminology for PPO. Generally when I see epoch it means a step that uses all training data. In the PPO paper it says Optimize surrogate L wrt θ, with K epochs and ...
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1answer
36 views

Applying a Hessian matrix to a logistic function in R

I'm using the following code to implement the logistic regression function so I may get the result for that of a Hessian matrix. I start with the function defined as $\frac{1}{(1+e^{-x})}$ called &...
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2answers
84 views

Beginner Bayesian question - which statement is false?

I'm working my way through Statistical Rethinking as a beginner Bayesian and am struggling with one of the concept-check questions. Book is here but is paywalled, so I'll also link to the solutions ...
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1answer
53 views

convergence rate of sample covariance matrix

I have a question about deriving the rate of convergence of sample covariance matrix. For the sake of simplicity, we can assume that our sample $\{ X_i\}_{i=1}^{n}$ is i.i.d. (I known we can relax ...
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0answers
37 views

Proving that two statistics are equivalent in sufficient statistics sense

I'm trying to understand an example from Mathematical Statistics by Bickel and Doksum, 2nd edition. In Example 1.5.4 (continued) part, I'm having hard time figuring out why expressing $t_2$ by $L_X(0,...
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1answer
32 views

What is a “constant best fit”?

I'm working on this homework problem: Text of problem In Question 7, you fitted a simple linear regression model $y_i=\beta_0+\beta_1x_i+\varepsilon_i$, where $\varepsilon_i\sim N(0,\sigma^2)$ are i.i....
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1answer
51 views

MCMC Gamma Distribution

I am applyig a MCMC simulation with a Gamma distribution. I am trying to simulate the rainfall in a city using data collected during 1000 days. First step is to simulate the "data colleceted ...
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2answers
147 views

Show that the maximum of $x_1,…,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. (From definition)

Problem Show that the maximum of $x_1,...,x_n \sim \mathrm{Uniform}(0,\theta)$ is a sufficient statistic for $\theta$. Background This question has been asked before, but most answers tackle the ...
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18 views

What methods can I use to validate my source/input data?

For example, a ticker's price coming from a stock exchange or earnings from a company's financial statements. How can I validate that it's trustworthy as an input into a model? In my career, I've just ...
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14 views

Separating hyperplane and soft-margin classifier

I have some questions that I'm supposed to answer, but I have no idea where to begin. Any hints or solutions would be helpful. I feel like I don't have enough information about the dataset to answer ...
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58 views

What's the joint conditional distribution of 2 random variables?

What's the joint conditional distribution of 2 random variables? Please provide a reference for the definition. (I forgot if any 2 continuous random variables necessarily have a well-defined joint pdf....
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23 views

Compute probability of an event by integrating over continuous random variable

Let $B$ be a continuous random variable. Let $K$ be an event. Am I right to think it does not necessarily make sense to say '$P(K)=\int_{b \in \mathbb R}P(K|B=b)f_B(b)$'? My guess: Well based on ...
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29 views

Show that if $Y \sim X + \delta(X)$, then $G(\cdot)= F(\cdot-\Delta)$ implies $\delta(x) \equiv \Delta$

Problem is from Bickel - Mathematical Statistics, which I am working through on my own. Suppose that $Y \sim X + \delta(X)$ where $X\sim N(\mu,\sigma^2)$ and $\delta$ is continuous. Suppose also that $...
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20 views

Get Continuous Distribution from Discrete Variable: Problem 6.77 of Wackerly, Mendenhall, Schaeffer, 5th Ed

Problem Statement: $\newcommand{\szdp}[1]{\!\left(#1\right)}$ Let $v$ denote the volume of a three-dimensional figure. Let $Y$ denote the number of particles observed in volume $v,$ and assume that $Y$...
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72 views

For Y∼ Uniform (−1,1) Prove (a) $Y_{n}\xrightarrow{L}Y$

For $Y \sim$ Uniform$(-1,1)$ $ Y_{n}= \begin{cases} \text{Y if } > \vert Y \vert \leq 1-\frac{1}{n}\\ \text{n if } \vert Y \vert > >1-\frac{1}{n}\\ \end{cases} $ Prove (a) $ Y_{n}\...
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31 views

Suppose X1∼U(0,1) and X 2|X1 =x1 ∼U(0,x1) are uniform random variables. Compute probability of (X1+X2≥1)

The answer to this problem is (1-ln2). I am getting 0.5 which is not even close. Any kind of hints or even suggestive reading would be helpful as I am getting a lot of doubt in problems of the same ...
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1answer
40 views

Expectation of negative moment $E[k^{-1}]$ for zero-truncated Poisson distribution

Can we simplify $$E[K^{-1}] = \frac{e^{-\lambda}}{1-e^{-\lambda}} \sum_{k=1}^\infty k^{-1} \frac{\lambda^k}{k! }$$ to compute or estimate $E[K^{-1}]$ when $K$ is a zero truncated Poisson or binomial ...
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36 views

Multiple Regression Coefficients

In section 3.2.3 of Elements of Statistical Learning (Link), there's this statement on multiple regression coefficients on Page 54 we have shown that the $j^{th}$ multiple regression coefficient is ...
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0answers
37 views

Asymptotic distribution of $S_n$

I am reviewing some exams from Hansen's webpage and came across this question that unfortunately doesn't have a suggested solution. In the regression/projection model $$y_i=x_i'\beta+e_i$$ $$E(x_ie_i)=...
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22 views

Sufficient statistic for a given distribution from exponential form

Given a particular form, i can verify whether it is sufficient statistic or not using $\frac{p_\theta(x_1,x_2...x_n)}{p_\theta(T(x_1,x_2...x_n))}$ is independendent of $\theta$ then i can say $T(\bar ...
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16 views

resampling of imbalanced dataset with only binary predictors and target

I am trying to classify an indicator of health as 0 and 1. I have an imbalanced dataset (0 : 5700, 1:1700) where all the values are binary (0 and 1 only for all features and target). I applied many ...
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1answer
27 views

Precision issue in normal distributions - Graphics Calculators and R Calculations [closed]

I am using R for make some stuff of normal distribution calculation for test some examples that use Texas Instruments TI-83 Plus Graphing Calculator (thats looks like and ad) I dont have it and i dont ...
4
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0answers
30 views

Bayes Theorem. If conditional on $\theta$ ,$ X $ and $Y$ are $Normal ( \theta ,1 ) $ then $Y |X$ is $Normal (X,1)$ [closed]

If conditional on $\theta$ ,$ X $ and $Y$ are $Normal ( \theta ,1 ) $ $\theta$ is real ( unknown) and $f ( y |\theta)$ is the pdf of $y$ and $f ( x |\theta)$ is the pdf of $x$. How can we show ...
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2answers
55 views

Using binomial approximation for calculating probability

I am trying to solve the problem but stuck with 'at least' and 'at most', question is: In a shipment of 20 engines, history shows that the probability of any one engine proving unsatisfactory is 0.1 a)...
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1answer
27 views

Uniform distribution and probability

Let $Y \sim \mathcal{U}(0,4)$. If 20 independent random samples are extracted, what is the probability that in at least 5 of them $Y > 2$? My attempt was: the required probability should be given ...
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19 views

Calculate the minimum amount of student to be in a class for it to be more than 50% likely that two of them end up with the same ID digits

Question: Suppose a professor assigns exam seating using the last 4 digits student's ID. How large could a class be before it is more likely than not that at least two students will not be able to ...
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0answers
29 views

Estimator of a censored exponential

I am trying to self-study the MIT OpenCourseware course on Statistics, here: https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-applications-fall-2016/syllabus/ On problem set 4, question 3 ...
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15 views

How to derive t-distribution using sympy?

I'm studying inference and I would like to understand better how classical distributions were derived. I manage to derivate the t-distribution using sympy but my result is a bit different from whats ...
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1answer
45 views

How to know which AR model is reprentative of stock prices?

I have an exercise like this: Consider the following three AR models that a researcher suggests might be a reasonable model of stock market prices: $$\begin{align} p(t) &= p(t−1) + u(t) \\[6pt] p(...

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