This chapter discusses optimal uniform rate of convergence for the nonparametric estimators of a density function or its derivatives. It describes the optimal uniform rate of convergence of an arbitrary estimator of T(f) for various choices of F. It is called the optimal rate of convergence, if it is both a lower and an achievable rate of convergence.

This chapter discusses multivariate display and discusses a few methods that are either exemplary of a class of others or are sufficiently useful and unique to warrant inclusion. Polygon plots as well as other sorts of iconic displays based upon nonmetaphorical icons are useful for the qualitative conveyance of information and are made more useful if the icons are displayed in a position that is meaningful. Trees are the only display methods that do a good job at depicting the covariance structure among the variables. A powerful use of a graphic display is to present information in a nonlinear way. Thus, the exploded diagrams of automobile transmissions show clearly as to which pieces go where and indicate clearly as to which orders of assembly are possible and which are not. The complex charts of population, often found in statistical atlases, provide many stories of immigration trends.

This chapter discusses fixed sample and the sequential procedures of testing and estimation and describes classical and Bayesian approaches to the change-point problem. It presents Bayesian and maximum likelihood estimation of the location of the shift points. The Bayesian approach is based on modeling the prior distribution of the unknown parameters, adopting a loss function and deriving the estimator, which minimizes the posterior risk. The chapter discusses this approach with an example of a shift in the mean of a normal sequence. The estimators obtained are generally nonlinear complicated functions of the random variables. From the Bayesian point of view, these estimators are optimal. The maximum likelihood estimation of the location parameter of the change point is an attractive alternative to the Bayes estimators. Regression relationship can change at unknown epochs, resulting in different regression regimes that should be detected and identified.

This chapter discusses Cramér–Rao inequality. In its simplest form, the Cramér–Rao inequality asserts that under regularity conditions, the variance of an unbiased estimator of a parametric function is at least the square of the derivative of that function divided by the Fisher information in the sample for all values of the unknown parameter. The Cramér–Rao inequality holds with the information in the sample equal to the information in a single observation times the expected sample size. The Cramér–Rao inequality holds for almost every value of the parameter under the sole condition that information be definable. An a.e. version of the Cramér–Rao inequality can be established for sequential estimators under the condition that the information in a single observation be positive.

This chapter discusses sequential rank tests. It reviews some properties of the sample distribution function. The non-linear renewal theorem is used to determine the asymptotic distribution of the excess over the boundary. The chapter describes a sequential probability ratio test.

This chapter discusses ranks and order statistics. It describes the normal approximation and the computation of asymptotic efficiencies for the two-sample rank statistics. To approximate the rank statistic by a sum of independent random variables, no fewer than six remainder terms to tend to zero, each for its own particular reason.

This chapter discusses M and N plots and describes a method for scatterplotting four-dimensional data. Conventional scatterplots use dots on a rectangular coordinate system to graphically represent two-dimensional vectors. An M and N plot represents (M+N)-dimensional vectors by plotting M coordinates in one coordinate system and N coordinates in a second coordinate system. The two dots representing a point are connected by a straight line segment. Thus, scatterplots are 2 and 0 plots. A 1 and 0 plot—the dot plot—is sometimes used to plot one-dimensional data, plotting each point as a dot. There are two ways to draw M and N plots of two-dimensional data: the conventional scatterplot (2 and 0 plot) and the 1and 1 plot. A 1 and 1 plot represents a two-dimensional point (x,y) by 2 dots and a line segment on a pair of parallel coordinate axis. The basic ingredients of a 2 and 2 plot are a pair of coordinate axes and a segment. With a PRIM 9-1ike graphical device, it is possible to draw 3 and 3 plots of six-dimensional data.

This chapter presents a nonhomogeneous Markov model of a chain-letter scheme. The role of the model in the legal proceedings is to demonstrate that the promises made by the promoters are inherently deceptive and constitute misrepresentation because it is virtually impossible for the majority of participants to make anywhere near the large sums indicated by the promotional material. Many participants can lose all or a part of their entrance fee. As a participant's earnings depend on the time of entrance, defendant promoters can produce a few winners to testify that they did well. The probabilistic approach shows that the proportion of winners is small and most participants can lose money. The chapter discusses the basic properties of the process and its limiting diffusion approximation. The transition properties of this conditional chain present two difficulties when the state variable is appropriately normalized: the increments neither have bounded conditional expectations nor their conditional variances are bounded away from zero. These difficulties are overcome by modifying the chain beyond constant boundaries and then showing that the probability of the chain moving outside these boundaries is sufficiently small so as to make the effect of this modification negligible. A suitable probability inequality for the constant boundary crossing event is obtained for this purpose using a rebounding effect in the transition probabilities of the chain.

This chapter discusses some recursive residual rank tests for change-points. A general class of recursive residuals is incorporated in the formulation of suitable rank tests for change-points pertaining to some simple linear models. The asymptotic theory of the proposed tests rests on some invariance principles for recursively aligned signed rank statistics. The chapter presents allied efficiency results along with the asymptotic properties of the proposed tests and describes testing procedures for a possible change in location or regression relationship occurring at an unknown time-point between consecutively taken observations.

This chapter discusses the general theory of optimal stopping for Brownian motion. It explains why many sequential decision problems in discrete time lead to stopping problem when they are formulated in continuous time. The solution of the limiting form of the problem can be constructed, in principle, by taking the union of certain open sets and this approach is closely related to the comparison technique. The chapter describes several Markov processes, each of which is capable of producing a sequence of rewards, and the problem of choosing just one of them at each stage so as to maximize the total discounted expectation. Each separate process has an index, depending on its state, and the optimal allocation policy is then defined by choosing the process with highest index at every stage. There is a strong connection between multi-armed bandits and stopping problems because the index for a single Markov process must be determined by solving a family of stopping problems.