X1 , . . . , Xn .

• Finally, suppose that one wanted to estimate

• The random variables X1 , . . . , Xn are identi-

the climatological mean temperature at a

cally distributed.

location such as Hamburg (Germany), or Vic-

toria (Canada), without consulting historical However, the independence assumption can not be

temperature observations. The concept of the made when making inferences about time series

simple random sample does not serve us or stochastic processes (Chapter 12). Then models

particularly well here. Our observations are are required that account for the dependence

necessarily con¬ned to an interval of time between observations. One way to do this is to

near the present. Temperatures in the past assume that the sample comes from a stationary

and in the distant future cannot be sampled; and ergodic process. Some types of analysis (e.g.,

only a ¬nite number of observations will be extreme value analysis, see Section 2.9) are able to

taken so temperatures realized after the last cope with dependence quite well; others, such as

observation will not be sampled. To treat hypothesis testing about the mean of a sample (see

the sample as a random sample, we must Section 6.6), cope with dependence very poorly.

make some assumptions about the proper- In general, models are either parametric or

ties of the temperature process. In particular, non-parametric.

we assume that the process is stationary or

• Parametric models require a distributional

cyclo-stationary (meaning that its statistical

properties are time invariant) and that the assumption: that is, the assumption that the

distribution of Xi , i = 1, . . . , n, belongs to

9 Shen et al., [349] have given careful thought to the problem

a certain family of probability distributions

of estimating the sampling error in the global mean temperature

(such as Xi is normal). The model is

that arises from the density and distribution of the observing

parametric because it speci¬es everything

network (including the random network discussed above).

4: Concepts in Statistical Inference

76

there is dependence are addressed in Chapters 6, 8,

about the distribution function except for

9, and 10“12.

a few free parameters (for instance, the

mean and variance in the case of a normal We have seen that a random variable is

distribution). Provided that the distribution a function de¬ned on a sample space and

assumption is correct, the parametric model that it inherits a probability distribution from

leads to very ef¬cient statistical inference the probabilities assigned to the sample space

because it brings a substantial amount of elements. In the same way, a statistic is a function

information into the procedure in addition to de¬ned on a sample, and it inherits its probability

that contained in the data. distribution from those of the random variables

that represent the sample. Thus, a statistic is

• Non-parametric approaches to statistical in- a random variable. Every time we replicate the

ference are distinguished from parametric ˜experiment™ that generates the sample, we get a

methods in that the distributional assumption different set of realizations of the random variables

is replaced by something more general. For that constitute the sample, and thus a different

example, instead of assuming that data come realization of the statistic computed from the

from a distribution having a speci¬c form, sample.

such as the normal distribution, it might be We describe here some basic statistics and

assumed that the distribution is unimodal their probability distributions under the standard

and symmetric. This includes not only the normal conditions. That is, we assume that the

random variables {X1 , . . . , Xn } that represent a

normal distribution, but many other families

of distributions as well. sample are independent and identically distributed

normal random variables with mean µ and

Non-parametric methods are advantageous

variance σ 2 .

when it is not possible to make speci¬c

distributional assumptions. Frequently, non-

parametric methods are only slightly less ef- 4.3.1 The Sample Mean. An example of a

¬cient than methods that use the correct para- simple statistic is the sample mean,

metric model, and generally more ef¬cient

n

1

compared with methods that use the incorrect ¯

X= Xi , (4.2)

parametric model. Non-parametric statistical n i=1

inference is therefore relatively cheap insur-

expressed here as a random variable. Once an

ance against moderate departures from the

experiment has been conducted and a particular

distributional assumptions. We will discuss

sample {x1 , . . . , xn } has been observed, we write

a few non-parametric inference techniques

in Chapter 6. A complete treatment of the

n

1

subject can be found in Conover [88].

¯

x= xi

n

While they allow us to relax the distributional i=1

assumption needed for parametric statistical ¯

inference, these procedures rely more heavily to represent the corresponding realized value of X.

upon the sampling assumptions than do para- By applying (2.16) we see that the random variable

¯

metric procedures. Non-parametric models X has mean and variance

¯

are heavily impacted by departures from the EX =µ (4.3)

sampling assumptions (see Zwiers [442]), so

¯

their use is not advised when there may be Var X = σ /n.

2

(4.4)

dependence within a sample.

Thus, it is apparent that the sample mean can be

regarded as an estimator10 of the true mean and

¯

that the spread of the distribution of X, as well

4.3 Statistics and Sampling

as the uncertainty of the estimator, decreases with

Distributions increasing sample size.

The sample mean has a normal distribution

4.3.0 Introduction. In the rest of this chapter we

when random variables Xi are normally dis-

will make the standard assumptions that a sample

tributed. When observations are not normally

can be represented by a collection of independent

and identically distributed (iid) random variables. 10 The concept of an estimator is discussed with more

The effects of dependence and methods used when precision in Chapter 5.

4.3: Statistics and Sampling Distributions 77

When random variables Xi are normally dis-

tributed, it can be shown that (n ’ 1)S2 /σ 2 ∼

χ 2 (n ’ 1).12 Consequently,

2σ 4

= .

S2

Var (4.7)

n’1

Equation (4.7) shows that we can think of S2 as

an estimator of σ 2 that has decreasing uncertainty

with increasing sample size n. The uncertainty

goes to zero in the limit as the sample becomes

in¬nitely large.

¯

It can also be shown that S2 is independent of X.

Figure 4.1: The probability density function of

the sample mean when the sample consists of

4.3.3 The t Statistic. It is natural to interpret

n = 10 and n = 40 independent and identically

the sample mean as a measure of the location of

distributed random variables. The distribution of

the sample. This measure is often expressed as a

the individual observations is labelled n = 1.

distance from some ¬xed point µ0 . This distance

should be stated in dimensionless units so that the

same inference can be made regardless of the scale

distributed, the Central Limit Theorem11 [2.7.5]

of observation.13

assures us, under quite general conditions, that

Suppose, for now, that µ0 = E(Xi ). When the

the sample mean will have a distribution that ¯