Playing Several Patterns Against One Another

We revisit the game in which each of several players chooses a pattern and then a coin is flipped repeatedly until one of these patterns is generated. In particular, we demonstrate how to compute the probability of any one player winning this game, and find the distribution of the game's duration. Our presentation is an extension (and perhaps a simplification) of the results of Blom and Thornburn

Yet Another Proof of Sylvester's Determinant Identity

In 1857 Sylvester stated a result on determinants without proof that was recognized as important over the subsequent century. Thus it was a surprise to Akritas, Akritas and Malaschonok when they found only one English proof - given by Bareiss 111 years later! To rectify the gap in the literature these authors collected and translated six additional proofs: four from German and two from Russian. These proofs range from long and "readily understood by high school students" to elegant but high level. We add our own proof to this collection which exploits the product rule and the fact that taking a derivative of a determinant with respect to one of its elements yields its cofactor. A differential operator can then be used to replace one row with another

Finding an ARMA(p,q) model given its spectral density or its correlogram

An ARMA model can be fully determined based on either its spectral density, or its correlogram, i.e. a formula for computing the corresponding k th serial correlation for any integer k. In this article we describe how to find, given one of these three ways of specifying the model, the other two

An Illustrated Introduction to the Truncated Fourier Transform

The Truncated Fourier Transform (TFT) is a variation of the Discrete Fourier Transform (DFT/FFT) that allows for input vectors that do NOT have length $2^n$ for $n$ a positive integer. We present the univariate version of the TFT, originally due to Joris van der Hoeven, heavily illustrating the presentation in order to make these methods accessible to a broader audience

Formula to evaluate a limit related to AR(k) model of Statistics

Computing moments of various parameter estimators related to an autoregressive model of Statistics, one needs to evaluate several expressions of the type mentioned in the title of this article. We proceed to derive the corresponding formulas

Accurate distribution of X^{T}X with singular, idempotent variance-covariance matrix

Assume that X is a set of sample statistics which follow a special case Central Limit Theorem, namely: as the sample size n increases the corresponding distribution becomes multivariate Normal with the mean (of each X) equal to zero and with an idempotent variance-covariance matrix V. It is well known that X^{T}X has (in the same limit), a chi-squared distribution with degrees of freedom equal to the trace of V. In this article we extend the above result to include the corresponding (1/n)-proportional corrections, making the new approximation substantially more accurate and extending its range of applicability to small-size samples

Three competing patterns

Assuming repeated independent sampling from a Bernoulli distribution with two possible outcomes S and F, there are formulas for computing the probability of one specific pattern of consecutive outcomes (such as SSFFSS) winning (i.e. being generated first) over another such pattern (e.g. SFSSFS). In this article we will extend the theory to three competing patterns

Asymptotic Distribution of Centralized rr When Sampling from Cauchy

Assume that $X$ and $Y$ are independent random variables, each having a Cauchy distribution with a known median. Taking a random independent sample of size $n$ of each $X$ and $Y$, one can then compute their centralized empirical correlation coefficient $r$. Analytically investigating the sampling distribution of this $r$ appears possible only in the large $n$ limit; this is what we have done in this article, deriving several new and interesting results

Parsimonious Skew Mixture Models for Model-Based Clustering and Classification

In recent work, robust mixture modelling approaches using skewed distributions have been explored to accommodate asymmetric data. We introduce parsimony by developing skew-t and skew-normal analogues of the popular GPCM family that employ an eigenvalue decomposition of a positive-semidefinite matrix. The methods developed in this paper are compared to existing models in both an unsupervised and semi-supervised classification framework. Parameter estimation is carried out using the expectation-maximization algorithm and models are selected using the Bayesian information criterion. The efficacy of these extensions is illustrated on simulated and benchmark clustering data sets

Improving Accuracy of Goodness-of-fit Test

It is well known that the approximate distribution of the usual test statistic of a goodness-of-fit test is chi-square, with degrees of freedom equal to the number of categories minus 1 (assuming that no parameters are to be estimated -- something we do throughout this article). Here we show how to improve this approximation by including two correction terms, each of them inversely proportional to the total number of observations