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1 vote

Assessing the effectiveness of a teaching tool by paired t-test

Clearly, this is a trick question. It would be inappropriate to assess performance before and after using the teaching tool to conclude something about the effectiveness of the teaching tool. You ...
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3 votes

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

Here is a homework problem from Mark Schervish's Theory of Statistics that addresses a similar question: Let $(X_1, Y_1),\dots,(X_n, Y_n)$ be conditionally IID with uniform distribution on the risk ...
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0 votes

power and sample size estimation

Suppose that this is a one-sided, one-sample t test. You want to detect whether $\mu_2 - \mu_1 = \Delta = 2$ or more and you estimate that the population standard deviation is $\sigma = 10.$ You are ...
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1 vote
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Finding the non-zero region of a marginal

Roughly stated, the support of the marginal is made of all the $x$'s for which the joint in $(x,y)$ is positive for some $y$'s (plus some caution about positive measure of the set of such $y$'s). More ...
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2 votes
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Calculate sample size for clinical trial without raw data

It would be unusual (if not irresponsible) for a clinical trial to be conducted on a new drug in a complete vacuum of information. You need to try to find a suitable sample size for such a trial, ...
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determining the sample size for quality test

We have $N$ documents, $E$ of those documents have errors. We don't know how many errors that there are, but if we assume that these errors may have been introduced randomly with probability $p$ for ...
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4 votes

What is Galton's paradox?

The "paradox" here arises from sneaking in an implicit insinuation of independence that does not actually hold, which allows the argument to lead you to a wrong answer with a series of ...
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1 vote
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Bayesian model comparison with systematic error

I believe that your intuition is correct. From the problem description, I interpreted the systematic error $y_{sys}$ as being an unknown parameter that would be contributing additively to the ...
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0 votes
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When does a difference in means not capture the true treatment effects vs a regression with pre-treatment controls?

There is no confounding in your example because the treatment was generated completely independently of the predictor of the outcome. You basically implemented a randomized trial. In such a scenario, ...
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PACF for MA(1) process

From Time Series Analysis, 1990 by William W. S. Wei book \begin{eqnarray*} \phi_{kk} = \frac{ \det \begin{pmatrix} 1 & \rho_1 & \rho_2 & \cdots & \rho_{{k}-2} & \...
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1 vote
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question of a Poisson random effect model

Start with your first expression: $$\log E(Y_{i}|b_i,x_i)=\beta_0+\beta_1x_i+b_i$$ The effect of a one-unit change in $x_i$ is easy to write: $$\log E(Y|x_i+1,b_i) - \log E(Y|x_i,b_i) = \beta_0+\...
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0 votes

football probability

How can I find the probability of the best player score the first goal? This is given in the question "given that T scores a goal, the probability that the best player of the team scored it is ...
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0 votes

How to add change (in unemployment rate) to dependent variable

First, you need to look into values, I am not sure what is the type unemployment variable here. Let's assume it contains two values 0 and 1 where 0 mean employed and 1 is unemployed. And economic ...
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0 votes

How to determine which variables are statistically significant in multiple regression?

Yes, you should look at the last column which contains the p-value parameter. Usually, we consider that if p-value < 0.05 for a certain variable then it is significant and has some relationship ...
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Exercises concerning a linear regression model with parameter $\beta$

I assume the $x_i$'s are fixed constants. Consider uncorrelated but heteroscedastic errors , i.e. $$\operatorname{Cov}(\varepsilon_i,\varepsilon_j)= \begin{cases}\sigma_i^2 &,\text{ if }i=j \\ 0 &...
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Bayesian update with the shifted and scaled data

You can do a standard update for distribution parametrized by $\mu’ = \alpha + \beta\mu$. Next, given that it's a location-scale transformation you can just transform it. But your likelihood is ...
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11 votes

Antithetic method for monte carlo when bounds of the integral are infinite

The point to the method of antithetic variates is to improve on such direct sampling methods. What can make it work well is to re-express the integral as an expectation of something with respect to a ...
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determining the sample size for quality test

I originally thought this could be done with a binomial approximation, but I've since changed my mind. Because you have a finite sample (N=200 documents) then we can do Bayesian inference to determine ...
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9 votes
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Antithetic method for monte carlo when bounds of the integral are infinite

Your code does not correspond to your description of the problem, so it is not surprising that you are not getting the results you expected. The integral you want to approximate is $$ \int_0^\infty e^{...
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3 votes
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Determining distribution based on first three even moments

Take a linear combination of $X^2,X^4$ and $X^6$: $$g(X)=aX^6+bX^4+cX^2$$ Choose scalars $a,b,c$ such that $E\left[g(X)\right]=0$ and $g(X)\ge 0$ almost surely. This would imply $g(X)=0$ almost surely ...
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Why is SATE different to Treatment effect (difference in means)?

You made an error in your simulation. The difference in means estimator is unbiased for the ATE. The error you made is in simulating t (i.e., $\tau$). The question ...
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1 vote

Efficient influence function in proportional hazards model

Write $\mathbb{P}_n$ as the empirical expectation. The estimator $\hat\psi = \mathbb{P}_n I(T>t_0)$ satisfies that $$\sqrt{n}(\hat\psi - S(t_0)) = \sqrt{n}\mathbb{P}_n [I(T>t_0)-S(t_0)] + 0,$$ ...
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1 vote
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Expected Prediction Error for 0-1 Loss Function

Figured this out by writing the sum explicitly: The expected conditional loss given by selecting a class $g$ is given as $\sum P(G_i \neq g|X=x)$, which is effectively equivalent to $1-P(g|X=x)$.
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3 votes
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Why is it called a ratio scale?

In an interval scale, the intervals are equal. The increase in temperature from 10 degrees to 20 degrees is the same increase from 20 degrees to 30 degrees. The intervals are equal. That's not true ...
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