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Questions tagged [mathematical-statistics]

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

Mathematical statistics is the study of statistics from a mathematical point of view, concerned with formal definitions and general results.

References

The following threads contain extensive references for studying mathematical statistics:

The following journals are dedicated to research in mathematical statistics:

7765 questions
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What is the curse of dimensionality?

Specifically, I'm looking for references (papers, books) which will rigorously show and explain the curse of dimensionality. This question arose after I began reading this white paper by Lafferty and Wasserman. In the third paragraph they mention a…
khoda
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Error terms vs Innovations

I noticed that we sometimes call the error terms "innovations". I do not understand if this is in special situations or if these terms can be used one for another. Then, another question is "why do we call error terms "innovations"? thanks
user49186
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How to use CDF and PDF statistics for analysis

This may be too much of a general question but I hope I can find help here. I am starting a RA job in my university and my topic will be related to Internet Traffic Analysis. I am fairly new to the world of analysis but I guess in the world of…
sfactor
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How can I compute $\int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$ in closed form?

How can one evaluate the expectation of the squared normal CDF in closed-form? $$\mathbb{E}\left[\Phi\left(aZ+b\right)^{2}\right] = \int_{-\infty}^{\infty}\Phi\left(az+b\right)^{2}\phi(z)\,dz$$ Here, $a$, $b$ are real numbers,…
Andrei
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The sum of two symmetric random variables is symmetric

If $X$ and $Y$ are two random variables with probability density functions which are symmetric around their respective means, their sum, $X+Y$, has a probability density function which is symmetric around its mean as well. Could someone offer a…
Radu
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Does learning thorough statistical theory require learning analysis?

Does learning thorough statistical theory requires learning analysis before that? I looked at the textbook for statistical theory. So far I don't know if analysis is required, but I think I have heard analysis is a prerequisite. Should I learn…
070701
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Hot topics in mathematical statistics

What are some hot topics that mathematical statistics researchers are studying now?
Craig Feinstein
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The only spherical and independent density is normal!

A well-known result is that the only density that is both spherical and independent error is normal: more precisely Let $e_i$ be errors, If the joint probability density satisfies $$f_n(e_1,e_2, ..., e_n) = f_1(e_1)f_1(e_2)...f_1(e_n)$$…
KH Kim
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Will quantum computing allow new statistical techniques?

I just read that you can now buy a quantum computer (albeit that there has only been one sold so far!). Will quantum computing have any applications in statistics? {edit - for the purposes of the question let's assume that eventually quantum…
Andrew
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Limiting moments and asymptotic moments of a statistic

From Casella's Statistical Inference: Definition 10.1.7 For an estimator $T_n$, if $\lim_{n\to \infty} k_n Var T_n = \tau^2 < \infty$, where $\{k_n\}$ is a sequence of constants, then $\tau^2$ is called the limiting variance or limit of the…
Tim
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Creating a recommender system for shoe sizing

(Sorry if I have mis-categorized or titled this question.) Suppose you have shoe size data for a bunch of people across a variety of shoe brands. So, one person might have Nike = 11, Reebok = 10.5, Adidas = 11, Converse = 11.5, and so on. You might…
Eric Angle
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Why the statistical model of sampling with replacement is incomplete?

Suppose $(\mathscr{X,B,P})$ is a statistical model and $(\mathscr{X,B,P})^n$ is the corresponding model for sampling with replacement. My text book states that its incompleteness allows multiple unbiased estimators, where problems such as which…
Ziyuan
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Misunderstanding the chi squared distribution

I believe there is a misconcept in my mind about the $\chi^2$ and/or standard normal distribution. Hence, I would like you to help me to understand what does it means that the $\chi_k^2$ distribution is a sum of $k$ independent, squared, standard,…
Stefano Fedele
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Unbiased estimator of $X^k$ given independent unbiased estimators of $X$

Suppose we are interested estimating $X^k$, and we have access to independent unbiased estimators $Y_i$ for $i = \{1,2,\dots, k\}$, i.e., $\mathbb{E}[Y_i] = X$. A straightforward way to construct an unbiased estimator of $X^k$ is to define $Y^{(k)}…
fool
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What are U-type statistics?

In an article, I recently came across the mention of first and second order U-type statistics without further detail. Does anyone know what U-type statistics are? References will be highly appreciated.
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