大型样本理论基础

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"Elements of Large Sample Theory" 是一本由E.L. Lehmann编写的关于高等概率论和大样本理论的专业书籍，被广泛认为是该领域的经典著作。这本书属于Springer Texts in Statistics系列，由知名统计学家George Casella、Stephen Fienberg和Ingram Olkin担任顾问。该书以PDF文档的形式提供，适合对统计学有深入研究的读者。在统计学领域，大样本理论是研究当样本容量趋向于无穷大时，统计量的行为和性质的分支。这本书涵盖了以下几个核心知识点： 1. **中心极限定理**：在独立同分布的随机变量序列中，当样本量足够大时，样本均值的分布趋于正态分布，即使原数据分布不是正态的。这是理解和应用大样本理论的基础。 2. **一致性和强一致性**：统计估计的收敛性是大样本理论中的关键概念。一致估计是指随着样本量增加，估计量趋于真实参数的极限；而强一致性则意味着几乎可以确定地随样本量增加，估计量将趋近于真实值。 3. **大数定律**：它描述了独立同分布随机变量的平均值在大量观测下趋于其期望值的现象。分为弱大数定律（概率收敛）和强大数定律（几乎必然收敛）。 4. **误差分析和效率**：书中可能讨论了估计方法的误差特性，包括标准误差、置信区间的构建以及不同估计方法的效率比较。 5. **假设检验**：大样本理论为构建和分析假设检验提供了理论基础，包括似然比检验、wald检验和score检验等。 6. **协方差和相关性**：在大样本情况下，协方差和相关性的性质对于理解数据的结构和进行回归分析至关重要。 7. **矩估计和最大似然估计**：这两种常用的参数估计方法在大样本下有着良好的理论保证。 8. **极限定理**：如Donsker定理和 Functional Central Limit Theorem，这些定理在描述过程或随机函数的极限行为时非常有用。 9. **Bootstrap方法**：这是一种现代统计技术，用于在没有知道确切分布的情况下进行抽样分布的估计。 10. **非参数统计**：书中可能还会涉及非参数方法，这些方法不依赖于特定的概率模型，而是利用数据的顺序或排列信息。 E.L. Lehmann的这本书不仅介绍了这些理论，而且很可能还包含了实际应用案例和问题，帮助读者将理论知识应用于解决实际的统计问题。对于生命科学和社会科学的研究者，以及希望深化统计理论理解的读者来说，这本书是一个宝贵的资源。

1.1 The concept of limit 3

For this sequence also, the numbers get arbitrarily close to a as n gets

large. However, they do so without each member being closer to a than

the preceding one. For a sequence a

,n =1, 2,..., to tend to a limit a as

n →∞, it is not necessary for each a

to be closer to a than its predecessor

n−1

, but only for a

to get arbitrarily close to a as n gets arbitrarily large.

Let us now formalize the statement that the members of a sequence

,n =1, 2,..., get arbitrarily close to a as n gets large. This means

that for any interval about a, no matter how small, the members of the

sequence will eventually, i.e., from some point on, lie in the interval. If

such an interval is denoted by (a − , a + ) the statement says that from

some point on, i.e., for all n exceeding some n

, the numbers a

will satisfy

a − <a

<a+  or equivalently

− a| <for all n>n

.(1.1.5)

The value of n

will of course depend on , so that we will sometimes write

it as n

(); the smaller  is, the larger is the required value of n

().

Deﬁnition 1.1.1 The sequence a

,n =1, 2,..., is said to tend (or con-

verge) to a limit a; in symbols:

→ a as n →∞ or lim

n→∞

=a(1.1.6)

if, given any >0, no matter how small, there exists n

= n

() such that

(1.1.5) holds.

For a formal proof of a limit statement (1.1.6) for a particular sequence

, it is only necessary to produce a value n

= n

() for which (1.1.5) holds.

As an example consider the sequence (1.1.1). Here a = 1 and a

−a = −1/n.

For any given , (1.1.5) will therefore hold as soon as

<or n>



.For

 =1/10,n

= 10 will do; for  =1/100,n

= 100; and, in general, for any

, we can take for n

the smallest integer, which is ≥



In examples (1.1.1)–(1.1.4), the numbers a

approach their limit from

one side (in fact, in all these examples, a

<afor all n). This need not be

the case, as is shown by the sequence







1 −

if n is odd

if n is even







=1+(−1)

.(1.1.7)

It may be helpful to give an example of a sequence which does not tend

to a limit. Consider the sequence

0, 1, 0, 1, 0, 1,...

given by a

=0or1asn is odd or even. Since for arbitrarily large n, a

takes on the values 0 and 1, it cannot get arbitrarily close to any a for all

suﬃciently large n.

The following is an important example which we state without proof.

4 1. Mathematical Background

Example 1.1.1 The exponential limit. For any ﬁnite number c,





→ e

as n →∞.(1.1.8)

To give an idea of the speed of the convergence of a





to its limit

e, here are the values of a

for a number of values of n, and the limiting

value e(n = ∞) to the nearest 1/100.

TABLE 1.1.1.





to the nearest 1/100

n 1 3 5 10 30 50 100 500 ∞

2.00 2.37 2.49 2.59 2.67 2.69 2.70 2.72 2.72

To the closest 1/1000, one has a

500

=2.716 and e =2.718. 

The idea of limit underlies all of large-sample theory. Its usefulness stems

from the fact that complicated sequences {a

} often have fairly simple

limits which can then be used to approximate the actual a

at hand. Table

1.1.1 provides an illustration (although here the sequence is fairly simple).

It suggests that the limit value a =2.72 shown in Table 1.1.1 provides a

good approximation for n ≥ 30 and gives a reasonable ballpark ﬁgure even

for n as small as 5.

Contemplation of the table may raise a concern. There is no guarantee

that the progress of the sequence toward its limit is as steady as the tabu-

lated values suggest. The limit statement guarantees only that eventually

the members of the sequence will be arbitrarily close to the limit value, not

that each member will be closer than its predecessor. This is illustrated by

the sequence (1.1.4). As another example, let







√

n if n is the square of an integer (n =1, 4, 9,...)

1/n otherwise.

(1.1.9)

Then a

→ 0 (Problem 1.7) but does so in a somewhat irregular fashion.

For example, for n =90, 91,... ,99, we see a

getting steadily closer to the

limit value 0 only to again be substantially further away at n = 100. In

sequences encountered in practice, such irregular behavior is rare. (For a

statistical example in which it does occur, see Hodges (1957)). A table such

as Table 1.1.1 provides a fairly reliable indication of smooth convergence

to the limit.

Limits satisfy simple relationships such as: if a

→ a, b

→ b, then

+ b

→ a + b and a

− b

→ a − b,(1.1.10)

· b

→ a · b(1.1.11)

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