Thomas J. Fisher

Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix

Many applications require an estimate for the covariance matrix that is non-singular and well-conditioned. As the dimensionality increases, the sample covariance matrix becomes ill-conditioned or even singular. A common approach to estimating the covariance matrix when the dimensionality is large is that of Stein-type shrinkage estimation. A convex combination of the sample covariance matrix and a well-conditioned target matrix is used to estimate the covariance matrix. Recent work in the literature has shown that an optimal combination exists under mean-squared loss, however it must be estimated from the data. In this paper, we introduce a new set of estimators for the optimal convex combination for three commonly used target matrices. A simulation study shows an improvement over those in the literature in cases of extreme high-dimensionality of the data. A data analysis shows the estimators are effective in a discriminant and classification analysis.

New Weighted Portmanteau Statistics for Time Series Goodness of Fit Testing

We exploit ideas from high-dimensional data analysis to derive new portmanteau tests that are based on the trace of the square of the mth order autocorrelation matrix. The resulting statistics are weighted sums of the squares of the sample autocorrelation coefficients that, unlike many other tests appearing in the literature, are numerically stable even when the number of lags considered is relatively close to the sample size. The statistics behave asymptotically as a linear combination of chi-squared random variables and their asymptotic distribution can be approximated by a gamma distribution. The proposed tests are modified to check for nonlinearity and to check the adequacy of a fitted nonlinear model. Simulation evidence indicates that the proposed goodness of fit tests tend to have higher power than other tests appearing in the literature, particularly in detecting long-memory nonlinear models. The efficacy of the proposed methods is demonstrated by investigating nonlinear effects in Apple, Inc., and Nikkei-300 daily returns during the 2006–2007 calendar years. The supplementary materials for this article are available online.

A new test for sphericity of the covariance matrix for high dimensional data

In this paper we propose a new test procedure for sphericity of the covariance matrix when the dimensionality, p, exceeds that of the sample size, N=n+1. Under the assumptions that (A) 0~ for i=1,...,16 and (B) p/n->c ~. Our simulation results show that the new test is comparable to, and in some cases more powerful than, the tests for sphericity in the current literature.

A Cauchy estimator test for autocorrelation

This article presents a new test for serial correlation in an observed stationary time series. Rather than using the traditional portmanteau tests based on the sample autocorrelation function, we propose a test based on the Cauchy estimator of correlation. A goodness-of-fit statistic for fitted autoregressive moving average models is also derived and the asymptotic distribution of this statistic is quantified. The test can be employed using either this asymptotic distribution or by using Monte-Carlo quantiles. The small sample behaviour is studied via simulation and the Monte-Carlo-based test seems to be more precise. The method is demonstrated on monthly asset returns for Facebook, Incorporated.

GAL1-SceI directed site-specific genomic (gsSSG) mutagenesis: a method for precisely targeting point mutations in S. cerevisiae

BackgroundPrecise targeted mutations are defined as targeted mutations that do not require the retention of other genetic changes, such as marker genes, near the mutation site. In the yeast, S. cerevisiae, there are several methods for introducing precise targeted mutations, all of which depend on inserting both a counter-selectable marker and DNA bearing the mutation. For example, the marker can first be inserted, and then replaced with either a long oligonucleotide carrying the mutation (delitto perfetto) or a PCR fragment synthesized with one primer containing the mutation (SSG mutagenesis).ResultsA hybrid method for targeting precise mutation into the genomes uses PCR fragments as in SSG mutagenesis together with a CORE cassette devised for delitto perfetto that contains the homing endonuclease SceI. This method, termed gsSSG mutagenesis, is much more efficient than standard SSG mutagenesis, allowing replacements to be identified without extensive screening of isolates. In gsSSG, recombination between the PCR fragment and the genome occurs equally efficiently regardless of the size of the fragment or the distance between the fragment end and the site of marker insertion. In contrast, the efficiency of incorporating targeted mutations by this method increases as the distance between the mutation and the marker insertion site decreases.ConclusiongsSSG is an efficient way of introducing precise mutations into the genome of S. cerevisiae. The frequency of incorporating the targeted mutation remains efficient at least as far as 460 bp from the insertion site meaning that a single insertion can be used to create many different mutants. The overall efficiency of gsSSG can be estimated based on the distance between the mutation and the marker insertion, and this efficiency can be maximized by limiting the number of untargeted mutations. Thus, a single insertion of marker genes plus homing endonuclease cassette can be used to efficiently introduce precise point mutations through a region of > 900 bp.

Cross-Correlation Matrices for Tests of Independence and Causality Between Two Multivariate Time Series

An often-studied problem in time series analysis is that of testing for the independence of two (potentially multivariate) time series. Toeplitz matrices have demonstrated utility for the related setting of time series goodness-of-fit testing—ergo, herein, we extend those concepts by defining a nontrivial block Toeplitz matrix for use in the setting of independence testing. We propose test statistics based on the trace of the square of the matrix and determinant of the matrix; these statistics are connected to one another as well as known statistics previously proposed in the literature. Furthermore, the log of the determinant is argued to relate to a likelihood ratio test and is proven to be more powerful than other tests that are asymptotically equivalent under the null hypothesis. Additionally, matrix-based tests are presented for the purpose of inferring the location or direction of the causality existing between the two series. A simulation study is provided to explore the efficacy of the proposed methodology—the methods are shown to offer improvement over existing techniques, which include the famous Granger causality test. Finally, data examples involving U.S. inflation, trade volume, and exchange rates are given. Supplementary materials for this article are available online.

Browning-related oxygen depletion in an oligotrophic lake

ABSTRACT In recent decades, terrestrial dissolved organic matter (DOM) has increased in many northeastern North American and European lakes and is contributing to long-term browning. We used a long-term dataset (1988–2014) to study the consequences of browning-related decreased water transparency on dissolved oxygen dynamics in 2 small temperate lakes in Pennsylvania, USA, that differ in their dissolved organic carbon concentrations. The oligotrophic (“clearer”) lake has low productivity and historically oxygenated deep waters. The mesotrophic–slightly dystrophic (“browner”) lake also has relatively low productivity but historically anoxic deep waters. We examined whether browning coincided with changes in summer dissolved oxygen dynamics, with a focus on deep-water oxygen depletion. In the clearer lake, we found that minimum oxygen concentrations decreased by ∼4.4 mg L−1 over the 27-year period, and these changes were strongly associated with both decreased water transparency and increased water column stability. We also found a shallowing of the maximum dissolved oxygen depth by ∼4.5  m and anoxic conditions established in more recent years. In the browner lake, the metrics we used did not detect any significant changes in dissolved oxygen, supporting the prediction that vertical temperature and oxygen patterns in clearer lakes may be more sensitive to increasing DOM than darker lakes. Anoxia is traditionally considered to be a consequence of anthropogenic nutrient loading and, more recently, a warming climate. We show that browning is another type of environmental change that may similarly result in anoxia in oligotrophic lakes.

A Cheap Trick to Improve the Power of a Conservative Hypothesis Test

ABSTRACT Critical values and p-values of statistical hypothesis tests are often derived using asymptotic approximations of sampling distributions. However, this sometimes results in tests that are conservative (i.e., understate the frequency of an incorrectly rejected null hypothesis by employing too stringent of a threshold for rejection). Although computationally rigorous options (e.g., the bootstrap) are available for such situations, we illustrate that simple transformations can be used to improve both the size and power of such tests. Using a logarithmic transformation, we show that the transformed statistic is asymptotically equivalent to its untransformed analogue under the null hypothesis and is divergent from the untransformed version under the alternative (yielding a potentially substantial increase in power). The transformation is applied to several easily-accessible statistical hypothesis tests, a few of which are taught in introductory statistics courses. With theoretical arguments and simulations, we illustrate that the log transformation is preferable to other forms of correction (such as statistics that use a multiplier). Finally, we illustrate application of the method to a well-known dataset. Supplementary materials for this article are available online.

A Review of: “Practical Multivariate Analysis, Fifth Edition, by A. Afifi, S. May, and V. A. Clark”

Many statistical textbooks present methods in a logical order from the viewpoint of learning the techniques and basic theory but ignore the fundamental aspect of performing the analysis. These texts tend to fall into two categories as either mathematical statistics books or simplified imitations that leave out some of the more difficult mathematics. When real-world data are analyzed, they tend to be ideal or contrived examples to demonstrate a specific point and do not represent the difficulties generally experienced in practice. Practical Multivariate Analysis attempts to bridge these gaps by presenting a functional account of typical univariate and multivariate procedures and data analysis. Seven real-world datasets are described in the opening chapter and analyzed throughout the book, each bringing its own set of nuisances and difficulties that are typical in practice. Two datasets are prominent: a longitudinal study of depression in Los Angeles County and one of lung function based on certain demographics and environmental influences. The text is comprised of 18 chapters separated into three parts. The first section describes the basics of data preparation, entry, and analysis. The second part recaps the methods from applied regression analysis, as it is one of the most widely used statistical tools in data analysis. The third section provides an overview of practical multivariate and related techniques for use in the life and health sciences. The text assumes the reader has a basic understanding of typical statistical procedures and some rudimentary competency in a statistical software package. The text includes some details for the packages R, S-PLUS, SAS, SPSS, Stata, and STATISTICA. The first five chapters provide the basis for the procedures described in the other two sections. In particular, Chapter 1 provides examples of multivariate data and an overview of the methodology described in the text. Chapter 2 describes the standard Stevens classifications of variables to characterize data. Chapter 3 provides an overview of the different computer software packages and includes an exceptional description of data entry and management that is lacking in most statistical texts I’ve read. Chapter 4 briefly describes transformations of data and graphical techniques typically included as part of a residual analysis. Traditional descriptive statistics are highlighted in Chapter 5. Throughout, the authors utilize easy-to-read graphs and tables summarizing the methods and their implementation in the various software packages. Chapter 6 begins statistical analysis by outlining simple linear regression, for both fixed and random predictor variables. Depending on your experience level, the