Questions tagged [statistical-inference]
The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.
3,986 questions
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Double negatives in hypothesis test conclusions
I am grading hypothesis tests for an introductory statistics class and students occasionally give the following conclusion after rejecting the null hypothesis:
Since $H_0$ is rejected, there is not ...
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Mixture Model Expectation
We know that if an i.i.d. sample is drawn from $p_{\theta}=\text{Ber}(\theta)$, $\theta\in (0,1)$ then
$$\mathbb{E}_{p_{\theta}}[\bar{X}] = \theta,$$
where $\bar{X}$ denotes the sample mean.
Now, ...
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An MLE Asymptotic Normality problem with i.n.i.d. data
Suppose $n \in \mathbb{N}$. Suppose $s_0 > 1$ and $\xi_j \sim N (0, j^{- s_0}
+ n^{- 1})$, $j = 1, 2, 3, \ldots, n$. Let $\hat{s}_n$ be the maximum
likelihood estimator of $s_0$. Is $\hat{s}_n$ ...
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Log-linear model for 2 way contingency table
I have some confusion on part c of the problem.
Our null hypothesis is $$H_0:\pi_{1j}=\pi_{2j}=\pi_{3j}=\pi_{4j}\\\forall j$$
Should our log-linear model be $$logu=u+uT+uR$$ or $$logu=u+uR$$
where uR ...
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Is the variance of a posterior distribution always less than the variance of the prior?
A prior distribution is given by distribution $f_\theta(\theta)$, with variance $\sigma^2$.
A posterior distribution is $g_\theta(\theta)=h(x,\theta)\cdot f_\theta(\theta)$, where $x$ is our sample.
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Predicting 50% range
It's been years since I last used what I remember from statistics class (and decades since the class).
Small data set:
...
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Confidence Interval for Reliability Weighted Samples
I'm trying to do statistical inference on a home poker game. I have calculated the winnings per hour, and I want to create a confidence interval for the variable winnings per hour, in say dollars.
The ...
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Expectation under true distribution with mixture samples
Let $X_1, X_2, \dots , X_n$ be an i.i.d. sample from the mixture distribution
\begin{equation} \label{eqn:mixture distribution}
p_{\epsilon,\theta} = (1 - \epsilon)p_{\theta} + \epsilon \delta,
\end{...
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definition of estimator in random variable estimation
I am studying random signals and noise (a course for EE students, but mathematical and formal), and have a question about the definition of an estimator (in the context of estimating a random variable ...
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Reference request: Formal treatment of statistics
I have started reading about Mathematical Statistics with the goal to better understand part of the foundations of Data Science. At this stage, I am particularly interested in statistical inference. ...
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Calculating Expected Coin Tosses
Let's say I had 2 coins, one with p = 2/3 and other p = 1/3 for head, how many trials are need to correctly identify what coin I was tossing with, with 99% accuracy?
I need help with the solution, Is ...
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Sampling from two distributions to determine which is which
Peter Winkler wrote the following in his book "Mathematical Puzzles (revised edition)":
As it turns out, it’s a theorem that in trying to determine which is which of two known probability ...
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Show that powers of an MVUE is an MVUE
Question is in the title. Given that $\delta:=\delta(\mathbf X_n)$ is MVUE (minimum variance unbiased estimator) of a scalar parameter $\theta$, we are asked to show that for all natural numbers $k$, $...
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Minimization of expectation of random variable
Suppose $X$ is a random variable with $\mathbb{E}\left[X\right] = \mu $ and $\mathrm{Var}\left( X\right) = \sigma^{2} $. For which value of $a>0$ is the value of $$\mathbb{E}\left[\left(aX - \dfrac{...
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How many samples do you need to invert a modular arithmetic table?
A toy problem of interest in machine learning is modular arithmetic, where for a positive integer n we have input integers $(a, b)$ where each a, b maps to a unique integer in the range $\varepsilon $ ...
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