Hypothesis testing for families of ergodic processes

General sufficient conditions for the discernibility of two families of stationary ergodic processes are derived. The conditions involve the weak topology for stationary processes. They are analogous in several respects to existing conditions for the discernibility of families of independent and identically distributed (i.i.d.) processes, but require a more refined type of topological separation in the general case. As a first application of the conditions, it is shown how existing discernibility results for i.i.d. processes may be extended to a countable union of uniformly ergodic families. In addition, it is shown how one may use hypothesis testing to study polynomial decay rates for covariance-based mixing conditions.

[1]  W. W. Daniel Applied Nonparametric Statistics , 1979 .

[2]  Benjamin Weiss,et al.  How Sampling Reveals a Process , 1990 .

[3]  R. C. Bradley Basic Properties of Strong Mixing Conditions , 1985 .

[4]  C. Kraft Some conditions for consistency and uniform consistency of statistical procedures , 1955 .

[5]  Karl Petersen Ergodic Theory , 1983 .

[6]  Terrence M. Adams,et al.  On density estimation from ergodic processes , 1998 .

[7]  A. Nobel Limits to classification and regression estimation from ergodic processes , 1999 .

[8]  A. Berger On Uniformly Consistent Tests , 1951 .

[9]  S. Kulkarni,et al.  A general classification rule for probability measures , 1995 .

[10]  P. C. Shields,et al.  The d-Recognition of Processes , 1994 .

[11]  T. Cover On Determining the Irrationality of the Mean of a Random Variable , 1973 .

[12]  L. Schwartz,et al.  A Necessary and Sufficient Condition for the Existence of Consistent Estimates , 1960 .

[13]  G. Lugosi,et al.  Almost sure classification of densities , 2002 .

[14]  ON ORTHOGONAL PROBABILITY MEASURES , 1953 .

[15]  M. Rosenblatt A CENTRAL LIMIT THEOREM AND A STRONG MIXING CONDITION. , 1956, Proceedings of the National Academy of Sciences of the United States of America.

[16]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

[17]  A. Dembo,et al.  A Topological Criterion for Hypothesis Testing , 1994 .

[18]  P. Doukhan,et al.  A new weak dependence condition and applications to moment inequalities , 1999 .

[19]  W. Hoeffding,et al.  Distinguishability of Sets of Distributions , 1958 .

[20]  Peter Whittle,et al.  Hypothesis Testing in Time Series Analysis. , 1951 .

[21]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[22]  J. Doob Stochastic processes , 1953 .