Four general multivariate stationary extremal Markovian processes

ABSTRACT

Two general multivariate stationary Markovian process with maximization structure (denoted by Max-AR(1) and MaxI-AR(1)) are developed respectively. Max-AR(1) is a subclass of MaxI-AR(1). The characterization of the Max-AR(1) and MaxI-AR(1) to be stationary is studied. Some properties of the two maximization processes are derived. Two more related general multivariate stochastic Markovian process with minification structure are analogously constructed (denoted by Min-AR(1) and MinI-AR(1)). Some well known maximization and minification processes are special cases of these four extermal Markovian processes. Two of them are simulated and some point estimations are provided as an illustration of the wide application of these four processes.

Keywords:

• Extremal Markovian process
• Max-AR(1), MaxI-AR(1), Min-AR(1), MinI-AR(1)
• Stationarity
• Semi-logistic processes
• MSL⁽¹⁾-AR(1), MSL⁽²⁾-AR(1)
• Pareto and semi-Pareto processes

AMS Subject Classifications:

• 60 E 10
• 60 J 05
• 62 M 10

Acknowledgments

The author would like to thank the anonymous referees for many helpful comments which led to the improvement of this paper. The author would also like to express her appreciation to her former research assistant, Mr. An-Pang Yu whose work on computer graphics and simulations enhanced the readability and completeness of this paper.