Questions tagged [mathematical-statistics]

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

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7
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2answers
3k views

Hot topics in mathematical statistics [closed]

What are some hot topics that mathematical statistics researchers are studying now?
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5answers
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What is the purpose of characteristic functions?

I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
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What is the relation between statistics theory and decision theory?

I was wondering how statistics and decision theory are related? It looks to me all the statistics problems/tasks can be formulated in decision theory. Also problems in decision theory can be ...
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Big picture on survival analysis and life data analysis

I have heard of survival analysis and life data analysis, but don't quite get the big picture. I was wondering what topics they are covering? Is it pure statistics, or just application of ...
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1answer
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Dual problem for L2 support vector machine

Here is the dual problem for L2 support vector machine: $$\max_{\alpha\in\mathbb{R}^{n}} 2\alpha^{T}y-\alpha^{T}\left(K+n\lambda Id_{\mathbb{R}^{n}}\right)\alpha$$ $$\forall i\in\left\{ 1,\ldots,n\...
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3answers
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Finding the Moment Generating Function of chi-squared distribution

I'm tasked with deriving the MGF of a $\chi^2$ random variable. I think the way to do is is by using the fact that $\Sigma_{j=1}^{m} Z^2_j$ is a $\chi^2$ R.V. and that MGF of a sum is the product ...
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7answers
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Making sense out of statistics theory and applications

I have recently graduated with my masters degree on medical and biological modeling, accompanied with engineering mathematics as a background. Even though my education program included a significant ...
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2answers
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How do I know which method of parameter estimation to choose?

There are quite a few methods for parameter estimation out there. MLE, UMVUE, MoM, decision-theoretic, and others all seem like they have a fairly logical case for why they are useful for parameter ...
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13answers
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What are the breakthroughs in Statistics of the past 15 years?

I still remember the Annals of Statistics paper on Boosting by Friedman-Hastie-Tibshirani, and the comments on that same issues by other authors (including Freund and Schapire). At that time, clearly ...
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8answers
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Is it possible to prove a null hypothesis?

As the question states - Is it possible to prove the null hypothesis? From my (limited) understanding of hypothesis, the answer is no but I can't come up with a rigorous explanation for it. Does the ...
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5answers
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Introduction to measure theory

I'm interested in learning more about nonparametric Bayesian (and related) techniques. My background is in computer science and though I have never taken a course on measure theory or probability ...
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3answers
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Proving that the squares of normal rv's is Chi-square distributed

I start with three independent random variables, $X_1, X_2, X_3$. They are each normally distributed with: $$X_i \sim N(\mu_i, \sigma^2), i = 1, 2, 3.$$ I then have three transformations, $$\...
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3answers
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p-vector and K-vector

I'm reading "The Elements of Statistical Learning" and early on there are references to p-vectors (page 10) and K-vectors (page 12). What exactly is meant by a p-vector and K-vector?
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2answers
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Is there a limit function in R?

Is there a 'limit' function in R? help(lim) or help(limit) weren't useful. And in general, is R suitable for mathematical ...
5
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3answers
466 views

References for use of symplectic geometry in statistics?

I have heard that many problems in Mathematical Statistics can be stated and solved in terms of Symplectic Geometry. Unfortunately this was a pretty vague statement and I am interested in something ...
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3answers
1k views

Which test to find out best concentration (the one having maximum effect)?

Hi my apologies first, I'm a biologist and not so good in statistics. In my study I'm studying the effect of concentration of feed on growth of a certain specimen. I have with me the different ...
5
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1answer
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Choosing the right threshold for a biometric trait authentication system

I have a biometric authentication system that is using a person's gait to authenticate them. I extract features from gait, run it through a comparison versus a template and produce a similarity score (...
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2answers
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Calculate ROC curve for data

So, I have 16 trials in which I am trying to authenticate a person from a biometric trait using Hamming Distance. My threshold is set to 3.5. My data is below and only trial 1 is a True Positive: <...
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3answers
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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 ...
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2answers
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Is there a relationship between the median of a function of random variables and the function of the median of random variables?

Background notation: RV= random variable, $\mu=$ mean $m=$ median Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV. If $f(...
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Who uses R with multicore, SNOW or CUDA package for resource intense computing?

Who of you in this forum uses ">R with the multicore, snow packages, or CUDA, so for advanced calculations that need more power than a workstation CPU? On which hardware do you compute these scripts? ...
3
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2answers
326 views

Sum of squares two ways, how are they connected?

The standard sum of squares as I know it is: $$ \sum(X-m)^2 $$ where $m$ is the mean. I ran into a different one which can be written two ways: $$ \sum(X^2) - \frac{(\sum X)^2}{n} = \sum(X^2) - m\sum ...
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What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
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4answers
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Do third order asymptotics exist?

Most asymptotic results in statistics prove that as $n \rightarrow \infty$ an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood ...
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1answer
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Regression Proof that the point of averages (x,y) lies on the estimated regression line

How do you show that the point of averages (x,y) lies on the estimated regression line?
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1answer
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Statistics Proof that $E[(X-Y)^2] = 0$

If X and Y are standardized variables and are perfectly positively correlated with respect to each other, how can i prove that $E[(X-Y)^2] = 0$?
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4answers
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Can you say that statistics and probability is like induction and deduction?

I've read through this thread, and it looks to me like it can be said that: statistics = induction ? probability = deduction ? But I am wondering if there might be some more details on the ...
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1answer
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Large deviations proof question

Below is part of the proof of large deviations result. K is cumulant generating function. Can anyone explain how the last step follows? This is page 157 of McCullagh's "Tensor Methods in Statistics"
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3answers
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Empirical relationship between mean, median and mode

For a unimodal distribution that is moderately skewed, we have the following empirical relationship between the mean, median and mode: $$ \text{(Mean - Mode)}\sim 3\,\text{(Mean - Median)} $$ How ...
4
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2answers
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What are good resources/criteria for judging human bias in data collection?

I've just been given a stack of polling data to analyse. Some of the questions are obviously leading or present subtle incentives (for the poller or polled) for specific answers. Of other questions I'...
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5answers
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Standard reference for classical mathematical statistics?

Can anyone recommend some books that are considered to be standard references for classical (frequentist) statistics? IE, fairly comprehensive, and also, been around for a while so that typos and ...
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2answers
337 views

How to prove the exponential expansion recursive form?

Consider the following exponential expansion form: $$ \exp\left[\sum_{k=1}^\infty \gamma_k x^k\right] = \sum_{j=0}^\infty\delta_j x^j $$ where $\gamma_k$'s are known, and $\delta_0=1$, $$\delta_{j+1}...
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Why does the supremum of the Brownian bridge have the Kolmogorov–Smirnov distribution?

The Kolmogorov–Smirnov distribution is known from the Kolmogorov–Smirnov test. However, it is also the distribution of the supremum of the Brownian bridge. Since this is far from obvious (to me), I ...
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1answer
891 views

Intuition for higher moments in circular statistics

In circular statistics, the expectation value of a random variable $Z$ with values on the circle $S$ is defined as $$ m_1(Z)=\int_S z P^Z(\theta)\textrm{d}\theta $$ (see wikipedia). This is a very ...
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3answers
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Non-trivial bound for $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$

How to find a non-trivial upper bound on $E[\exp(Z^2)]$ when $Z \sim {\rm Bin}(n, n^{-\beta})$ with $\beta \in (0,1)$? A trivial bound is obtained for substituting $Z$ with $n$. A background on this ...
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13answers
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Simple algorithm for online outlier detection of a generic time series

I am working with a large amount of time series. These time series are basically network measurements coming every 10 minutes, and some of them are periodic (i.e. the bandwidth), while some other aren'...
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2answers
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What types of data analysis do not count as statistics?

When does data analysis cease to be statistics ? Are the following examples all applications of statistics ?: computer vision, face recognition, compressed sensing, lossy data compression, signal ...
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1answer
411 views

Lies, Damn Lies and Statistics [closed]

Is there something about statistics that lends itself to this sort of saying, or is it just that people will say anything to support their case, and this includes citing irrelevant or incomplete ...
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8answers
15k views

What is Bayes' theorem all about?

What are the main ideas, that is, concepts related to Bayes' theorem? I am not asking for any derivations of complex mathematical notation.
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21answers
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What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
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7answers
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How do you convey the beauty of the Central Limit Theorem to a non-statistician?

My father is a math enthusiast, but not interested in statistics much. It would be neat to try to illustrate some of the wonderful bits of statistics, and the CLT is a prime candidate. How would you ...
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19answers
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Mathematical Statistics Videos

A question previously sought recommendations for textbooks on mathematical statistics Does anyone know of any good online video lectures on mathematical statistics? The closest that I've found are: ...
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6answers
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Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
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0answers
479 views

Testing (and proving) the randomness of numbers [duplicate]

Possible Duplicate: Testing random variate generation algorithms What's a good way to test a series of numbers to see if they're random (or at least psuedo-random)? Is there a good statistical ...
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1answer
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What are some good frameworks for method selection?

I have been looking into theoretical frameworks for method selection (note: not model selection) and have found very little systematic, mathematically-motivated work. By 'method selection', I mean a ...
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9answers
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What is meant by a "random variable"?

What do they mean when they say "random variable"?

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