Statistics

Motivated by the interest of observing the growth of cancer cells among normal living cells and exploring how galaxies and stars are truly formed, the objective of this paper is to introduce a rigorous and effective method for counting point-masses, determining their spatial locations, and computing their attributes. Based on computation of Hermite moments that are Fourier-invariant, our approach facilitates the processing of both spatial and Fourier data in any dimension.

Working with complex data is one of the important updates to the 2014 ASA Curriculum Guidelines for Undergraduate Programs in Statistical Science. Infusing 'authentic data experiences' within courses allow students opportunities to learn and practice data skills as they prepare a dataset for analysis. While more modest in scope than a senior-level culminating experience, authentic data experiences provide an opportunity to demonstrate connections between data skills and statistical skills. The result is more practice of data skills for undergraduate statisticians.

In this paper, we provide a Graph Fourier Transform based approach to downsample signals on graphs. For bandlimited signals on a graph, a test is provided to identify whether signal reconstruction is possible from the given downsampled signal. Moreover, if the signal is not bandlimited, we provide a quality measure for comparing different downsampling schemes. Using this quality measure, we propose a greedy downsampling algorithm. Most of the prevailing approaches consider undirected graphs, and exploit the topological properties of the graph in order to downsample the grid, while the proposed method exploits spectral properties of graph signals, and is applicable to directed graphs, undirected graphs, and graphs with negative edge-weights. We provide several experiments demonstrating our downsampling scheme, and compare our quality measure with measures like normalized cuts.

Demand for data science education is surging and traditional courses offered by statistics departments are not meeting the needs of those seeking training. This has led to a number of opinion pieces advocating for an update to the Statistics curriculum. The unifying recommendation is computing should play a more prominent role. We strongly agree with this recommendation, but advocate the main priority is to bring applications to the forefront as proposed by Nolan and Speed (1999). We also argue that the individuals tasked with developing data science courses should not only have statistical training, but also have experience analyzing data with the main objective of solving real-world problems. Here, we share a set of general principles and offer a detailed guide derived from our successful experience developing and teaching a graduate-level, introductory data science course centered entirely on case studies. We argue for the importance of statistical thinking, as defined by Wild and Pfannkuck (1999) and describe how our approach teaches students three key skills needed to succeed in data science, which we refer to as creating, connecting, and computing. This guide can also be used for statisticians wanting to gain more practical knowledge about data science before embarking on teaching an introductory course.

In statistics education, the concept of population is widely felt hard to grasp, as a result of vague explanations in textbooks. Some textbook authors therefore chose not to mention it. This paper offers a new explanation by proposing a new theoretical framework of population and sampling, which aims to achieve high mathematical sensibleness. In the explanation, the term population is given clear definition, and the relationship between simple random sampling and iid random variables are examined mathematically.

The traditional measurement theory interprets the variance as the dispersion of a measured value, which is actually contrary to a general mathematical concept that the variance of a constant is 0. This paper will fully demonstrate that the variance in measurement theory is actually the evaluation of probability interval of an error instead of the dispersion of a measured value, point out the key point of mistake in the traditional interpretation, and fully interpret a series of changes in conceptual logic and processing method brought about by this new concept.

We introduce a stochastic version of Taylor's expansion and Mean Value Theorem, originally proved by Aliprantis and Border (1999), and extend them to a multivariate case. For a univariate case, the theorem asserts that "suppose a real-valued function f has a continuous derivative f ??on a closed interval I and X is a random variable on a probability space (Ω,F,P) . Fix a?�I , there exists a \textit{random variable} ξ such that ξ(?)?�I for every ??��?and f(X(?))=f(a)+ f ??(ξ(?))(X(?)?�a) ." The proof is not trivial. By applying these results in statistics, one may simplify some details in the proofs of the Delta method or the asymptotic properties for a maximum likelihood estimator. In particular, when mentioning "there exists θ ??between θ ^ (a maximum likelihood estimator) and θ 0 (the true value)", a stochastic version of Mean Value Theorem guarantees θ ??is a random variable (or a random vector).

In an increasingly data-driven world, facility with statistics is more important than ever for our students. At institutions without a statistician, it often falls to the mathematics faculty to teach statistics courses. This paper presents a model that a mathematician asked to teach statistics can follow. This model entails connecting with faculty from numerous departments on campus to develop a list of topics, building a repository of real-world datasets from these faculty, and creating projects where students interface with these datasets to write lab reports aimed at consumers of statistics in other disciplines. The end result is students who are well prepared for interdisciplinary research, who are accustomed to coping with the idiosyncrasies of real data, and who have sharpened their technical writing and speaking skills.

The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form log(1+1/d) , where d=1,2,...,9 . Such a law keeps elusive for over one hundred years because it was obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of the nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. We reveal that the first digit law is originated from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

A long memory process has self-similarity or scale-invariant properties in low frequencies. We prove that the log of the scale-dependent wavelet variance for a long memory process is asymptotically proportional to scales by using the Taylor expansion of wavelet variances.