Dempster (1958, 1960) proposed a non-exact test for the two-sample significance test when the dimension of data is larger than the degrees of freedom. He raised the question of what statisticians should do if traditional multivariate statistical theory does not apply when the dimension of data is too large. Later, Bai and Saranadasa (1996) found that even when traditional approaches can be applied, they are much less powerful than the non-exact test when the dimension of data is large. This raised another question of how classical multivariate statistical procedures could be adapted and improved when the data dimension is large. These problems have attracted considerable attention since the middle of the first decade of this century. Efforts towards solving these problems have been made along two directions: the first is to propose special statistical procedures to solve ad hoc large-dimensional statistical problems where traditional multivariate statistical procedures are inapplicable or perform poorly, for some specific large-dimensional hypotheses. The family of various non-exact tests follows this approach. The second direction, following the work of Bai et al. (2009a), is to make systematic corrections to the classical multivariate statistical procedures so that the effect of large dimension is overcome. This goal is achieved by employing new and powerful asymptotic tools borrowed from the theory of random matrices, such as the central limit theorems in Bai and Silverstein (2004) and Zheng (2012).

Recently, research along these two directions has become very active in response to an increasingly important need for analysis of massive and large-dimensional data. Indeed, such “big data” are nowadays routinely collected owing to rapid advances in computer-based or web-based commerce and data-collection technology.

To accommodate such need, this monograph collects existing results along the aforementioned second direction of large-dimensional data analysis. In Chapters 2 and 3, the core of fundamental results from random matrix theory about sample covariance matrices and random Fisher matrices is presented in detail. Chapters 4–12 collect large-dimensional statistical problems in which the classical large sample methods fail and the new asymptotic methods, based on the fundamental results of the preceding chapters, provide a valuable remedy.

In Chapter 2 a number of estimation problems were introduced where monotonicity of functions can be taken into account in the estimation process. In this chapter asymptotic properties of monotone estimators will be derived. In Section 3.1 various methods for proving consistency of monotone estimators will be described. In Section 3.2, the pointwise limit behavior of the Grenander estimator will be derived heuristically. In particular, the typical rate of convergence of the estimator, n−1/3, will emerge from heuristic calculations. In order to make the heuristics rigorous in concrete problems, properties of (the derivative of) convex minorants of functions are needed. Important properties, especially the switch relation, will be reviewed in Section 3.3. The empirical process theory needed to make the convergence of certain processes precise is introduced in Section 3.4. In Section 3.5, empirical process theory is applied to derive the asymptotic distribution of the isotonic inverse estimator in a deconvolution problem defined in Section 2.4. Using the switch relation of Section 3.3 and empirical process results from Section 3.4, the pointwise asymptotic distribution of the Grenander estimator is rigorously derived in Section 3.6. An alternative approach to settle the asymptotic distribution theory proceeds via the theory of martingales. Section 3.7 states some important results from that theory. Using these results, in Section 3.8 local asymptotic properties of the maximum likelihood estimator in the current status model are derived. Various limit distributions are encountered in this chapter, related to convex minorants of processes related to Brownian motion. One of these, the Chernoff distribution, is discussed in Section 3.9. In Section 3.10 results on the concave majorant of Brownian motion and Brownian bridge are stated and discussed.

Consistency

In this section, some general methods are discussed that can be used to prove consistency of nonparametric estimators in monotone estimation problems. The first is rather direct. It is based on the explicit construction of estimators as derivative of a convex minorant of a random set of points.

Research on nonparametric estimation under shape constraints started in the 1950s. Papers such as Ayer et al., 1955, and Van Eeden, 1956, appeared on estimation of functions under the restriction of monotonicity or unimodality, more generally called isotonic estimation. An isotonic estimator is an estimator that is computed under an order restriction, where the order can be a partial order. The order restriction can also be imposed on the derivative of the estimator, so an estimator of a convex function (in dimension one or higher), which is itself also convex, is also called an isotonic estimator.

A summary of the early work was given in the well-known book by Barlow et al., 1972, on isotonic regression. Originally, the focus was on defining and constructing estimators satisfying these order constraints. As an example, in Grenander, 1956, it is shown that the (nonparametric) maximum likelihood estimator (MLE) of a monotone decreasing density can be constructed as the left-continuous slope of the least concave majorant of the empirical distribution function. Developing asymptotic distribution theory for these isotonic estimators turned out to be rather difficult. Nonnormal limit distributions appear and rates of convergence are slower than the square root of the sample size. This behavior is now commonly classified as belonging to the area of nonstandard asymptotics. In the case of the mentioned Grenander MLE, the rate of convergence of this estimator (evaluated at a fixed point, under some local assumptions) is the cube root of the sample size. Moreover, the nonnormal asymptotic distribution of the estimator is (after rescaling) the so-called Chernoff distribution, which is (up to a factor 2) the distribution of the derivative of the greatest convex minorant of two-sided Brownian motion with parabolic drift, evaluated at zero.

Research on isotonic regression received new impetus in the 1990s when it became clear that it was the right setting for studying (nonparametric) MLEs of the distribution function in inverse problems.