Miklós Csörgő

Quantile processes with statistical applications

A Preliminary Study of Quantile Processes A Weak Convergence of the Normed Sample Quantile Process Strong Approximations of the Normed Quantile Process Two Approaches to Constructing Simultaneous Confidence Bounds for Quantiles Weak Convergence of Quantile Processes in Weighted Sup-Norm Metrics and Further Strong Approximations On Bahadurs Representation of Sample Quantiles and on Kiefers Theory of Deviations Between the Sample Quantile and Empirical Processes Quadratic Forms of the Quantile Process. Weighted Spacings and Testing for Composite Goodness-of-Fit Strong Approximations of the Quantile Process of the Product-Limit Estimator An Invariance Principle for Nearest-Neighbor Empirical Density Functions A Nearest- Neighbor Estimator for the Score Function.

20 Nonparametric methods for changepoint problems

Publisher Summary Changepoint problems have originally arisen in the context of quality control, where one typically observes the output of a production line and would wish to signal deviation from an acceptable average output level while observing the data. When one observes a random process sequentially and stops observing at a random time of detecting change, then one speaks of a sequential procedure. Otherwise, it is observed that a large finite sequence for the sake of determining possible change during the data collection. Such procedures are described in terms of asymptotic results, and can be called as nonsequential procedures. Sequential and nonsequential procedures are usually based on parametric or nonparametric models for changepoint problems, allowing at most one change (AMOC) or, possibly, more than one change. This chapter focuses on the nonparametric AMOC setting and discusses non-sequential nonparametric AMOC procedures. A large number of nonparametric and parametric modelling of AMOC problems result in the same test statistic, general rank statistics with quantile and Wilcoxon type scores whose asymptotics are described in terms of a two-time parameter stochastic process, U-statistics type processes which are considered for the nonparametric AMOC problem, and detect change in the intensity parameter of a renewal process.

A new method to prove strassen type laws of invariance principle. II

SummaryA new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that

Central limit theorems for Lp-Norms of density estimators

{{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} \mathord{\left/ {\vphantom {{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }}} \right. \kern-\nulldelimiterspace} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }} = 0

Strong approximation of additive functionals

with probability 1, for any g3>0, where Sn=X1 + ... +Xn, Xi is a sequence of i.i.d.r.v.s with P(Xi<t)=F(t), and F(t) is a distribution function obeying (i), (ii) and W(n) is a suitable Wiener-process. Strassen in [1], proved (under weaker conditions):

Asymptotic confidence bands for the Lorenz and Bonferroni curves based on the empirical Lorenz curve

S_n - W\left( n \right) = O\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)