We give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT. © 2014 Elsevier B.V.

The concentration of aerosol particles, largely caused by traffic volume and often enhanced during temperature inversion episodes in the cold season, can be a concern for human health in the urban environment. This particulate matter is typically recorded as PM10, the total mass of particles below 10 μm in diameter. It is suspected that, within the PM10 class, ultrafine particles (<100nm) may be responsible for causing respiratory and cardiovascular diseases. Because of their low mass, ultrafine particles are hard to detect, and researchers try to utilize PM10 in combination with nitrogen oxides NOx and other trace gases to monitor their dynamic evolution. To meet pollution standards set by national government and European Union regulation, the city of Klagenfurt, Austria, began using calcium magnesium acetate as a deicer on 11 January 2012, hoping to literally glue pollutants to the ground and thereby reducing pollution concentrations. With the statistical methodology developed in this article, the dynamic evolution of PM10 and NOx is traced for the time period starting 4 January and ending 25 January 2012, and a change in dynamics is found. The findings are based on on-line monitoring procedures that sequentially detect structural breaks in the mean and the parameter values of an autoregressive moving average process. These are defined in terms of model residuals and one-step ahead predictors. Theoretical properties are studied, and a simulation study shows that the proposed procedures work well in finite samples. © 2014 Wiley Publishing Ltd.

In this article, novel joint semiparametric spline-based modeling of conditional mean and volatility of financial time series is proposed and evaluated on daily stock return data. The modeling includes functions of lagged response variables and time as predictors. The latter can be viewed as a proxy for omitted economic variables contributing to the underlying dynamics. The conditional mean model is additive. The conditional volatility model is multiplicative and linearized with a logarithmic transformation. In addition, a cube-root power transformation is employed to symmetrize the lagged response variables. Using cubic splines, the model can be written as a multiple linear regression, thereby allowing predictions to be obtained in a simple manner. As outliers are often present in financial data, reliable estimation of the model parameters is achieved by trimmed least-square (TLS) estimation for which a reasonable amount of trimming is suggested. To obtain a parsimonious specification of the model, a new model selection criterion corresponding to TLS is derived. Moreover, the (three-parameter) generalized gamma distribution is identified as suitable for the absolute multiplicative errors and shown to work well for predictions and also for the calculation of quantiles, which is important to determine the value at risk. All model choices are motivated by a detailed analysis of IBM, HP, and SAP daily returns. The prediction performance is compared to the classical generalized autoregressive conditional heteroskedasticity (GARCH) and asymmetric power GARCH (APGARCH) models as well as to a nonstationary time-trend volatility model. The results suggest that the proposed model may possess a high predictive power for future conditional volatility. Supplementary materials for this article are available online.

This paper is concerned with extensions of the classical Mařenko-Pastur law to time series. Specifically, p-dimensional linear processes are considered which are built from innovation vectors with independent, identically distributed (real-or complex-valued) entries possessing zero mean, unit variance and finite fourth moments. The coefficient matrices of the linear process are assumed to be simultaneously diagonalizable. In this setting, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high-dimensional setting p/n→c ∈ (0,∞) for which dimension p and sample size n diverge to infinity at the same rate. The results extend existing contributions available in the literature for the covariance case and are one of the first of their kind for the autocovariance case.

This article addresses the prediction of stationary functional time series. Existing contributions to this problem have largely focused on the special case of first-order functional autoregressive processes because of their technical tractability and the current lack of advanced functional time series methodology. It is shown here how standard multivariate prediction techniques can be used in this context. The connection between functional and multivariate predictions is made precise for the important case of vector and functional autoregressions. The proposed method is easy to implement, making use of existing statistical software packages, and may, therefore, be attractive to a broader, possibly nonacademic, audience. Its practical applicability is enhanced through the introduction of a novel functional final prediction error model selection criterion that allows for an automatic determination of the lag structure and the dimensionality of the model. The usefulness of the proposed methodology is demonstrated in a simulation study and an application to environmental data, namely the prediction of daily pollution curves describing the concentration of particulate matter in ambient air. It is found that the proposed prediction method often significantly outperforms existing methods.

This article is concerned with the spectral behavior of $p$-dimensional linear processes in the moderately high-dimensional case when both dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of $p$-dimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the $p\times p$ linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.

Statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics, and they play a central role in multivariate testing. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, such methods are relatively unexplored in the high-dimensional setting. The aim of this article is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high dimensions, a method we refer to as the spectral bootstrap. In essence, the proposed method originates from the parametric bootstrap and is motivated by the fact that in high dimensions it is difficult to obtain a nonparametric approximation to the full data-generating distribution. From a practical standpoint, the method is easy to use and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics. In addition to proving the consistency of the proposed method, we present encouraging empirical results in a variety of settings. Lastly, and perhaps most interestingly, we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics, such as the largest sample eigenvalue and others.

Image segmentation is a long-studied and important problem in image processing. Different solutions have been proposed, many of which follow the information theoretic paradigm. While these information theoretic segmentation methods often produce excellent empirical results, their theoretical properties are still largely unknown. The main goal of this paper is to conduct a rigorous theoretical study into the statistical consistency properties of such methods. To be more specific, this paper investigates if these methods can accurately recover the true number of segments together with their true boundaries in the image as the number of pixels tends to infinity. Our theoretical results show that both the Bayesian information criterion (BIC) and the minimum description length (MDL) principle can be applied to derive statistically consistent segmentation methods, while the same is not true for the Akaike information criterion (AIC). Numerical experiments were conducted to illustrate and support our theoretical findings.

This article is concerned with the spectral behavior of p-dimensional linear processes in the moderately high-dimensional case when both dimensionality p and sample size n tend to infinity so that p/n→0. It is shown that, under an appropriate set of assumptions, the empirical spectral distributions of the renormalized and symmetrized sample autocovariance matrices converge almost surely to a nonrandom limit distribution supported on the real line. The key assumption is that the linear process is driven by a sequence of pdimensional real or complex random vectors with i.i.d. entries possessing zero mean, unit variance and finite fourth moments, and that the p × p linear process coefficient matrices are Hermitian and simultaneously diagonalizable. Several relaxations of these assumptions are discussed. The results put forth in this paper can help facilitate inference on model parameters, model diagnostics and prediction of future values of the linear process.