Bayesian sequential change-point detection problem is studied for scalar and vector-valued data, where each component can experience a sudden change. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. For wide classes of stochastic processes, with or without nuisance parameters, asymptotically pointwise optimal (APO) stopping rules are obtained, translating the classical concept of Bickel and Yahav to sequential change-point detection. These APO rules are attractive because of their simple analytic form, straightforward computation, and weak assumptions.

Application of new methods in environmental science, finance, epidemiology, and energy disaggregation will be shown. These models often involve nuisance parameters, time-dependence, nonstationarity, and rather complex prior distributions. Proposed APO rules can operate under these conditions, achieving asymptotic optimality.

Abstract of first part:

Nonparametric estimation of a hazard rate from left truncated and right censored data is a typical situation in applications, and a number of consistent and rate-optimal estimators, under the mean integrated squared error (MISE) criterion, have been proposed. It is known that, under a mild assumption, neither truncation nor censoring affects the rate of the MISE convergence. Hence a sharp constant of the MISE convergence is needed to create a benchmark for an estimator.

This work develops the theory of sharp minimax nonparametric estimation of the hazard rate with left truncated and right censored data. It is shown how left truncation and right censoring affect the MISE. The proposed data-driven sharp minimax estimator adapts to smoothness of an underlying hazard rate and it also adapts to unknown distributions of the truncating and censoring random variables. Performance of the proposed estimator is illustrated via analysis of simulated and real data, and for real data nonparametric estimates are complemented by hypotheses testing and confidence bands.

The abstract of second part:

Superefficiency in nonparametric estimation and new rates under a shrinking minimax.

In order to compute Gini index for a particular country or a region at given time, a procedure is needed which will minimize both the error of estimation as well as the cost of sampling without any assuming any income distribution. It is well known that error in estimation decreases when the sample size increases which in turn will increase the overall cost of sampling. In the same way, if one wants to minimize the cost of sampling, then one has to use a smaller sample size which in turn will increase the error of estimation. So, a procedure is required which will act as a trade-off between the estimation accuracy and the sampling cost.