Large Sample Theory

1. Representative textbooks are, in a roughly increasing order of difficulty, Hoel (1971); Freund (1971); Mood, Graybill, and Boes (1974); Cox and Hinkley (1974), and Bickel and Doksum (1977).

2. For a more complete study of the subject, the reader should consult Chung (1974) or Lofeve (1977), the latter being more advanced than the former. Rao (1973), which is an excellent advanced textbook in mathematical statistics, also gives a concise review of the theory of probability and random variables.

3. If a set function satisfies only (i) and (iii) of Definition 3.1.2, it is called a measure. In this case the triplet defined in Definition 3.1.3 is called a measure space. Thus the theory of probability is a special case of measure theory. (A standard textbook for measure theory is Halmos, 1950.)

4. If we have a measure space, then a function that satisfies the condition of Defini­tion 3.1.4 is called a measurable function. Thus a random variable is a function measurable with respect to the probability measure.

5. Lebesgue measure can be defined also for certain non-Borel sets. The Lebesgue measure defined only for Borel sets is called Borel measure.

6. For У to be a random variable, h must satisfy the condition {(oh[X{pi)] < у) Є Л for every y. Such a function is said to be Borel-measurable. A continuous function except for a countable number of discontinuities is Borel-measurable.

7. The following is a simple example in which the Riemann-Stieltjes integral does not exist. Suppose F(x) = 0 for a S x £ c and F(x) = 1 for с < x Ш b and suppose h(x) = F(x). Then, depending on the point x* we choose in an interval that contains c, S„ is equal to either 1 or 0. This is a weakness of the Riemann-Stieltjes integral. In this example the Lebesgue-Stieltjes integral exists and is equal to 0. However, we will not go into this matter further. The reader may consult references cited in Note 2 to Chapter 3.

8. Sometimes we also say X„ converges to X almost everywhere or with probability one. Convergence in probability and convergence almost surely are sometimes re­ferred to as weak convergence and strong convergence, respectively.

9. Between M and a. s., we cannot establish a definite logical relationship without further assumptions.

10. The law of large numbers implied by Theorem 3.2.1 (Chebyshev) can be slightly generalized so as to do away with the requirement of a finite variance. Let {Xj} be independent and suppose ЕХ,1+г < M for some S > 0 and some M < °°. Then X„ — EXn 0. This is called Markov’s law of large numbers.

11. The principal logarithm of a complex number rew is defined as log r + ів.

12. It seems that for most practical purposes the weak consistency of an estimator is all that a researcher would need, and it is not certain how much more practical benefit would result from proving strong consistency in addition.

13. Lai, Robbins, and Wei (1978) proved that if {«,} are assumed to be independent in Model 1 and if the conditions of Theorem 3.5.1 are met, the least squares estimator is strongly consistent. Furthermore, the homoscedasticity assumption can be relaxed, provided the variances of {u,} are uniformly bounded from above.

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