ABSTRACT
The foundations of large sample theory are laid down by the concepts of s to ch as tic co n v e rg en c e and weak co n v e rg en ce . Consistency is a mini mal requirement in large sample theory, and the basic relationship between consistency and stochastic convergence has been thoroughly discussed in Chapter 2. Consistency (whether of an estim ator or a test statistic) may, however, fail to convey the full statistical information contained in the data set at hand, and, hence, by itself, it may not provide an efficient statistical conclusion. To illustrate this point suppose tha t we have n independent ran dom variables X \ , . . . , X n drawn from an unknown distribution with mean H and finite variance a 2 (which are both unknown). The sample mean X n is a natural estim ator of //, and from the results of Chapter 2, we may conclude that as n increases, X n converges to // in a well-defined manner (viz., in probability/alm ost surely/second mean). This certainly endows us with an increasing confidence on the estim ator (X n) with increasing sample sizes. Suppose now th a t we desire to set a confidence in te rv a l (Ln ,Un) for fi with a prescribed coverage p ro b a b ili ty or con fidence coeffic ien t 1 — a (for some a , 0 < a < 1), i.e., we intend to determine suitable statistics Ln and Un (Ln < Un) based on a sample X \ , . . . , X n , such th a t
(3.1.1)
If a were known, we could have used the Chebyshev Inequality (2.3.1) to obtain that, for every n > 1,
(3.1.2)
so that, on setting we have
We may even get sharper bounds (for Ln and Un) by using the Markov Inequality (2.3.4) (assuming higher order moments) or the Bernstein In-
equality (2.3.7) (assuming a finite moment generating function) instead of the Chebyshev Inequality. On the other hand, if X has a normal distri bution with mean /i and variance <r2, we have, for every real x and every n > 1,
(3.1.4)
Let us define r€ by ^ (rO = 1 — £, 0 < £ < 1 . Then, we have from (3.1.4) that, for every n > 1 and a (0 < a < 1),
(3.1.5)
Let us compare the width of the two bounds in (3.1.3) and (3.1.5). Note that
(3.1.6) The rhs of (3.1.6) is strictly less than 1 for every a (0 < a < 1). For a=0.01, 0.025, 0.05 and 0.10, it is equal to 0.258, 0.354,0.438 and 0.519, respectively. A similar picture holds for the case of the Markov or Bernstein Inequality based bounds. This simple example illustrates that the knowledge of the actual sampling distribution of a statistic generally leads to more precise confidence bounds than those obtained by using some of the probability inequalities considered in Chapter 2. For a statistic Tn (viz., an estimator of a param eter 0), more precise studies of its properties may be made when its sampling distribution is known. A similar situation arises when one wants to test for the null hypothesis (Ho) that pi is equal to some specified value fio (against one-or two-sided alternatives). In order that the test has a specified margin (say, a , 0 < a < 1) for the Type I error, one may use the probability inequalities from Chapter 2 for the demarcation of the critical region . However, this generally entails some loss of the power of the test. Knowledge of the distribution of the test statistic (under Ho) provides a test which has generally better power properties.
