Sequential point estimation of parameters in a threshold AR(1) model

AbstractWe show that if an appropriate stopping rule is used to determine the sample size when estimating the parameters in a stationary and ergodic threshold AR(1) model, then the sequential least-squares estimator is asymptotically risk efficient. The stopping rule is also shown to be asymptotically efficient. Furthermore, non-linear renewal theory is used to obtain the limit distribution of appropriately normalized stopping rule and a second-order expansion for the expected sample size. A central result here is the rate of decay of lower-tail probability of average of stationary, geometrically β-mixing sequences

Rates of convergence of an adaptive kernel density estimator for finite mixture models

Kernel smoothing methods are widely used in many areas of statistics with great success. In particular, minimum distance procedures heavily depend on kernel density estimators. It has been argued that when estimating mixture parameters in finite mixture models, adaptive kernel density estimators are preferable over nonadaptive kernel density estimators. Cutler and Cordero-Braña [1996, J. Amer. Statist. Assoc. 91, 1716-1721] introduced such an adaptive kernel density estimator for the minimum Hellinger distance estimation in finite mixture models. In this paper, we investigate the convergence properties of a practical version of their adaptive estimator under some regularity conditions, and compare them with those of a nonadaptive estimator. The rates of convergence of the bias and variance of the proposed estimator are established.Minimum Hellinger distance estimation Adaptive kernel density estimator Mean squared error

Contemporary Developments in Statistical TheoryA Festschrift for Hira Lal Koul /

XI, 396 p. 160 illus., 60 illus. in color.online

Estimation of the offspring mean in a controlled branching process with a random control function

AbstractControlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773] for branching processes with immigration and provide a unified limit theory of estimation

Estimation of the offspring mean in a controlled branching process with a random control function

Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757-1773], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757-1773] for branching processes with immigration and provide a unified limit theory of estimation.Branching processes Random control function Weighted conditional least squares estimator Weak convergence Diffusion approximation