term™ mean. Then, taking the limit as δ ’ 0, and

That is, the p-quantile is the solution x p of

noting that pkδ tends to δ f X (kδ) as δ ’ 0, we

FX (x p ) = p. obtain

∞ (k+1/2)δ

An example of the annual cycle of the quantiles

lim E(Xδ ) = lim f X (x) d x

kδ

of daily mean temperature at Potsdam, Germany, is δ’0 δ’0 (k’1/2)δ

k=’∞

displayed in Figure 2.1. Note that the distribution ∞

= x f X (x) d x,

is approximately symmetric during the transition

’∞

seasons, but negatively skewed in winter, and

slightly positively skewed in summer. The ˜noise™ thus concluding the argument. A rigorous proof is

evident in these curves is a consequence of obtained by demonstrating that the sample mean

estimating the quantiles from a ¬nite sample of is a consistent estimator of the expectation (see

observations. [5.2.5]).

2: Probability Theory

32

are generally strongly skewed to the right”

2.6.7 The Central Moments: Location, Scale,

and Shape Parameters. The kth moment µ(k) even though small amounts of rainfall occur

considerably more often than large amounts.

of a continuous random variable X is also de¬ned

This occurs because rainfall distributions

as in the discrete case. Speci¬cally

have a wide ˜tail™ that extends far to the right.

∞

(k)

µ =E = x k f X (x) d x.

xk On the other hand, geopotential height tends

’∞

to be somewhat skewed to the left because

The kth central moment µ (k) of a random lows tend to have greater amplitude than

highs.8

variable X is the expectation of (X ’ µ)k , given

by

• The kurtosis, a scaled and shifted version of

∞

(k) the fourth central moment, is given by

µ = (x ’ µ)k f X (x) d x.

’∞

x ’µ 4

γ2 = f X (x) d x ’ 3. (2.19)

Most characteristics of a distribution can be

σ

R

summarized through the use of simple functions

of the ¬rst four moments. These slightly modi¬ed

Kurtosis is a measure of peakedness.

parameters are the mean, variance, skewness, and

Platykurtic distributions, such as the uniform

kurtosis:

distribution, have γ2 < 0 and are less

˜peaked™ than the normal distribution (see

• The mean, also known as the location

[2.7.3]). Distributions with γ2 > 0 are said to

parameter, is given by the ¬rst moment

be leptokurtic, and are more ˜peaked™ than the

µ = µ(1) . normal distribution. The double exponential

distribution, with density f X (x) = 1 e’|x’µ| ,

2

is leptokurtic.

• The variance is given by the second central

moment

The skewness and kurtosis are often referred to

as shape parameters.9

Var(X) = E (X ’ µ)2 (2.17)

Shape parameters can be useful aids in the

2

= E X2 ’ E(X) identi¬cation of appropriate probability models.

This seems to be especially true in extreme value

= µ(2) ’ µ(1) .

2

analysis (Section 2.9) where debate over the

merits of various distributions is often intense.

The properties of the variance, discussed for

However, skewness and kurtosis are often dif¬cult

the discrete case in [2.3.4], extend to the

to estimate well. In practice, it is advisable to use

continuous case, in particular

alternative shape parameters such as L-moments

[2.6.9].

Var(aX + b) = a 2 Var(X). (2.18)

√

The standard deviation σ X = Var(X) is 2.6.8 The Coef¬cient of Variation. When

a random variable, such as precipitation, takes

also often described as a scale parameter.

only positive values a scale parameter called the

• The skewness is a scaled version of the third coef¬cient of variation,

central moment that is given by

c X = σ X /µ X ,

x ’µ 3

γ1 = f X (x) d x.

σ is sometimes used. The standard deviation of such

R

variables is often proportional to the mean and it

Symmetric distributions (i.e., distributions for is therefore useful to describe the scale parameter

which f X (µ ’ x) = f X (µ + x)) have γ1 = 0. relative to the mean.

Distributions for which γ1 < 0 are said to be 8 Holzer [180] shows that this is due to the recti¬cation of

negatively skewed or skewed to the left, and nonlinear interactions in the atmosphere™s dynamics (see also

distributions for which γ1 > 0 are said to be [3.1.8]).

9 The concept of skewness and kurtosis is not limited to

positively skewed or skewed to the right.

continuous random variables. It carries over to discrete random

Daily rainfall distributions, bounded on the variables in the obvious way: by replacing integration with

left by zero and unbounded on the right, summation in the de¬nitions given above.

2.7: Example of Continuous Random Variables 33

X2|2 ≥ X1|2 by de¬nition). The third and fourth

2.6.9 L-Moments. Hosking [183] introduced

an alternative set of scale and shape statistics moments are shape parameters. Standardized L-

called L-moments, which are based on order moments are

statistics. The L-moments play a role similar

• the L-coef¬cient of variation

to that of conventional moments; in particular,

any distribution can be completely speci¬ed by

c X = »(2) /»(1) ,

L

(2.22)

either. The difference is that the higher ( j ≥

3) L-moments can be estimated more reliably

• the L-skewness

than conventional moments such as skewness and

kurtosis. Robust estimators of higher moments are

γ1L = »(3) /»(2) , (2.23)

needed to identify and ¬t distributions such as the

Gumbel, Pareto, or Wakeby distributions used in

• the L-kurtosis

extreme value analysis (see Section 2.9).

To de¬ne the L-moments of a random variable

γ2L = »(4) /»(2) . (2.24)

X we must ¬rst de¬ne related random variables

called order statistics. Let X = (x1 , . . . , xn )T be a

random vector that is made up of n independent, Examples of the application of L-moments

identically distributed random variables, each in climate research include Guttmann [151] and

with the same distribution as X. Suppose x = Zwiers and Kharin [448].

(x1 , . . . , xn )T is a realization of X. Let g(·) be

the function that sorts the elements of an n-

2.7 Example of Continuous Random

dimensional vector in increasing order. That is

Variables

g(x) = (x(1|n) , . . . , x(n|n) )T

2.7.1 The Uniform Distribution. The simplest

where x(i|n) is the ith smallest element of x. The of all continuous distributions is the uniform

random vector that corresponds to g(x) is distribution. A random variable that takes values