ABSTRACT
In problems of statistical inference (on the basis of a representative sample), statistical decision rules are usually based on suitable estim ators (of param eters of interest), test statistics (for appropriate statistical hypotheses) and other plausible functions of sample observations. These s ta t is t ic s may be real-or vector-valued random functions or may even be more general ran dom functions, such as s to c h a s tic p ro c esses . Being stochastic in nature, these statistics are characterized by chance fluctuations (governed by the underlying model and other factors) which generally exhibit a diminishing trend with the increase in the size of the sample on which they are based. A minimal requirement for a good statistical decision rule is its increasing reliability with increasing sample sizes (termed co n s is ten cy ) . Thus, for an estimator, consistency relates to an increasing closeness to its population counterpart with increasing sample sizes. However, in view of the stochastic nature of the estimator, this closeness needs to be defined and interpreted in a meaningful and precise m anner incorporating the stochastic nature of the fluctuation of the estim ator around the param eter it estimates. Gen erally, a d is ta n c e fu n c tio n (or n o rm ) of this stochastic fluctuation is incorporated in the formulation of this closeness, and consistency refers to the convergence of this norm to 0 in a well-defined manner. Similarly, dealing with a test of significance (for a statistical hypothesis), consistency refers to the increasing degree of confidence in the ability of the test to re ject the null hypothesis when it is not true as the sample size increases. In the framework of a more general statistical decision rule, consistency refers to decreasing risk s of making incorrect statistical decisions with increasing sample sizes. Basically, the consistency of a statistical decision rule rests on some co n v e rg en ce p ro p e r t ie s of an associated sequence of statistics, and this concept is intim ately related to the dual one of s to c h a s tic con v erg en ce in the classical theory of probability.
