of spatial variability computed from the analyses

upon the unknown binomial parameter and the

and AMIP simulations performed with two climate

remaining eight of the random variables have a

models.

joint distribution that depends only upon the value

Spatial variance of DJF mean φ500 of S. If such a transformation exists, then S is

in m2 said to be a suf¬cient statistic for the unknown

parameter because it contains all the information

NMC

that can be found in the sample about the unknown

Year analyses Model A Model B

parameter. Suf¬cient statistics are therefore very

79/80 451 471 205 good test statistics.

80/81 837 209 221

81/82 598 521 373

6.3 Monte Carlo Simulation

82/83 979 988 419

83/84 555 234 334

6.3.1 General. The analytical procedures men-

84/85 713 331 265

tioned above, as well as other theoretical methods

85/86 598 217 291

used to derive the distributions of test statistics,

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often result in intractable mathematical problems.

87/88 270 448 582

6.3: Monte Carlo Simulation 105

The Monte Carlo method is often used when this

happens.2 The idea is to simulate the statistical

Cummulative frequency (%)

Relative frequency (%)

model on a computer under the assumption that

H0 is true. The computer is used to generate a large

number of realizations of the test statistic, say S,

which in turn are used to construct an empirical

estimate of the distribution of S under H0 . Finally,

the estimated distribution is used to determine the

critical value κp just as its analytical counterpart

˜

would be used if it were available.

1000 10 000 100 000 100 000 cummulative

The Monte Carlo method is a powerful tool

because it substantially increases the range of

problems that will yield to statistical reasoning. As Figure 6.6: Monte Carlo simulation of the

probability function f S ( j) of (6.7) with n =

with all powerful tools, there are also a number

of pitfalls to be avoided. Although the Monte 51 cases. The functions are derived from 1000,

Carlo approach can be applied to any statistic, 10 000, and 100 000 trials. The distribution

function FS ( j), estimated from 100 000 trials is

heuristically derived statistics may not be ef¬cient

and can result in misleading inferences. For also shown.

example, the invariance principle [4.3.3], which

requires that the same inference be made under all

with the convention ζk = ζk’8 if k > 8. This

linear transformations of the data, may be violated.

alternative was chosen because it was anticipated

that the ζk will vary smoothly with k if H0 is false

6.3.2 Example. The Monte Carlo method in such a way that phases on one half of the circle

was used to study the relationship between the are preferred over those in opposite sectors. A

appearance of tropical storms in the Southwest natural test statistic for this setup is

Paci¬c and the phase of the tropical Madden-and-

j+3

Julian Oscillation (MJO) [399]. The latter is a

stochastic oscillation that affects the intensity of S = max (Fk ’ Fk+4 ) . (6.7)

j

k= j

convection in the tropical West Paci¬c. Intensi¬ed

convection may, in turn, be associated with S is a discrete random variable that takes

increased tropical cyclogenesis and vice versa. values between zero and n, the total number of

We therefore consider the null hypothesis: ˜H0 : storms observed. In this example, 51 storms were

the frequency of tropical storms in the West Paci¬c observed in a ¬ve year period.

is independent of the phase of the Madden-and- To make an inference about (6.6) we need to

Julian Oscillation.™ To test this hypothesis we need determine the probability distribution f S ( j) of S

an objective measure of the phase of the MJO. given that H0 is true. This was done with the

One such measure is given by the oscillation™s Monte Carlo method by repeatedly:

˜POP index™ [15.2.3]. The observed phases can

• generating n independent realizations

then be classi¬ed into one of eight 45—¦ sectors.

x1 , . . . , xn from the discrete uniform

Each tropical cyclone is assigned to the sector

distribution on the set of integers {1, . . . , 8}

corresponding to the phase of the MJO on the

[2.4.4],

day of genesis. Then, if Fk , k = 1, . . . , 8, is the

frequency of storms in sector k, the null hypothesis • computing the frequencies f1 , . . . , f8 , and

may be re-expressed as

• ¬nally obtaining a realization of S by

H0 : ζk = 1/8, (6.6) substituting the realized frequencies into

(6.7).

where ζk = E(Fk ).

By doing this often, the probabilities P (S = j) for

A reasonable alternative hypothesis Ha is

j = 1, . . . , n can be estimated.

j+3

Estimates based on 1000, 10 000, and 100 000

(ζk ’ ζk+4 ) > 0,

Ha : max

samples are shown in Figure 6.6. The three

j

k= j

estimates are very similar. The differences

2 The ideas discussed here are closely related to the arise from sampling variations: slightly different

estimates of the true probability function are

bootstrapping ideas discussed in Section 5.5.

6: The Statistical Test of a Hypothesis

106

information with which to test this hypothesis we

might

1 randomly select a (large) sample of rock

formations that have not been altered by

humans, and

2 count the number of rock formations arranged

as the Mexican Hat.

Let us assume that no other Mexican Hat-like

formations are found. Humans have traversed most

of the rocky desert of the world at one time or

another and it would appear that the Mexican

Hat is unique in the collective experience of

these travellers. Therefore, the chances of ¬nding

another Mexican Hat among, say, one million

randomly selected rocks, are nil. Thus we may

reject the null hypothesis at a small signi¬cance

level, and give credence to the explanation given

in Figure 6.8.

Obviously we can generalize this example to

include many different null hypotheses of the type

˜rare event is common.™

The problem with these null hypotheses is that

they were derived from the same data used to

Figure 6.7: The Mexican Hat at the border between

conduct the test. We already know from previous

Utah and Arizona”is this rock naturally formed?

exploration that the Mexican Hat is unique, and

[3]

its rarity leads us to conjecture that it is unnatural.

Unfortunately, statistical methodology can not take

obtained each time the Monte Carlo procedure is us any farther in this instance unless we are willing

repeated. The estimate obtained from the 100 000 to wait a very long time so that tectonic processes

trial sample, of course, has less uncertainty than can generate a new independent realization of the