We propose a class of estimators for the parameters of a GARCH(p, q) sequence. We show that our estimators are consistent and asymptotically normal under mild conditions. The quasi-maximum likelihood and the likelihood estimators are discussed in detail. We show that the maximum likelihood estimator is optimal. If the tail of the distribution of the innovations is polynomial, even a quasi-maxim...

In finite mixtures of location-scale distributions, if there is no constraint or penalty on the parameters, then the maximum likelihood estimator does not exist because the likelihood is unbounded. To avoid this problem, we consider a penalized likelihood, where the penalty is a function of the minimum of the ratios of the scale parameters and the sample size. It is shown that the penalized max...

when a change occurs in a process, one expects to receive a signal from a control chart as quickly as possible. upon the receipt of signal from the control chart a search for identifying the source of disturbance begins. however, searching for assignable cause around the signal time, due to the fact that the disturbance may have manifested itself into the rocess sometimes back, may not always l...

The consistency of the quasi maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non-degenerate random variable. In this paper we propose empirical likelihood methods based on weighted score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic an...

In a regular full exponential family, the maximum likelihood estimator (MLE) need not exist in traditional sense. However, MLE may completion of family. Existing algorithms for finding solve many linear programs; they are slow small problems and too large problems. We provide new, fast, scalable methodology This is based on conventional computations which come close, sense, to These construct m...