# Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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### Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
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### Do random variables follow the same algebraic rules as ordinary numbers?

In the comments on my answer to a recent question about the sum of random variables, I came across a link to the Wikipedia article on the ratio distribution, and noticed the following peculiar claim ...
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### How to show that a sufficient statistic is NOT minimal sufficient?

My homework problem is to give a counterexample where a certain statistic is not in general minimal sufficient. Irrespective of the details of finding a particular counterexample for this particular ...
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### MLE, regularity conditions, finite and infinite parameter spaces

The problem I have is in figuring out why the MLE is no longer consistent in countable parameter spaces under conditions specified below. The set up is as follows: we are consider a parameters space ...
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### Finding the distribution of sample range for a Beta population

Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables having density $$f(x)=2(1-x)\mathbf1_{0<x<1}$$ I am trying to derive the distribution of the sample range $R=X_{(n)}-X_{(1)}$. The ...
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### Expected squared distance from origin of training points vs. test points

This is from Exercise 2.4 (Page 39) of Elements of Statistical Learning: The edge effect problem discussed on page 23 is not peculiar to uniform sampling from bounded domains. Consider inputs drawn ...
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### score function of bivariate/multivariate normal distribution

I tried to obtain the score vector (1st derivative of density function w.r.t. parameters) of multivariate normal distribution. Density function of multivariate normal distribution: \begin{align*} f(x)...
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### Time evolution of a Bayesian posterior

I have a question regarding the time evolution of a quantity related to a Bayesian posterior. Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$, The data generating ...
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### Reference Request: Information Geometry for Ridge Regression

I am reading the book "regression estimators" by Gruber 2010 where he uses this technique to compare Ridge Regressors, however he concentrates on deriving the mathematical results without giving any ...
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### Taylor Series and Multivariate Delta Method

I asked this question on https://math.stackexchange.com/ but did not get any answer. Sorry for cross posting. I'm trying to understand delta method for matrices and vectors to find the variance-...
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### What does “def” above an equals sign mean?

I am reading this: https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf and on equation (17), there is a def on top of the equal sign. What does this mean?
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### French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
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### Are there unbiased, non-linear estimators with lower variance than the OLS estimator?

Consider an ordinary least squares model, $$y = \beta X + \epsilon \qquad \epsilon\sim N(0, \sigma)$$ The Gauss-Markov theorem tells us that the ordinary least-squares (OLS) estimator is the minimum-...
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### How to prove the properties of penalized likelihood estimator in Fan and Li (2001) paper

I'm reading through Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties". In Page 2 near bottom right corner, they proposed three properties that a ...
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544 views