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Page 2

Lecture Notes in Statistics
Edited by S. Fienberg, J. Gani, K. Krickeberg,
1. Oikin, and N. Wermuth

94

Page 134

Extreme-value Analysis 129
of Canadian Wind Speeds

1.00

0.90

0.80

0.70

en
0 0.60
a:
Il.

a
w 0.50
~
:E
t= 0.40 en
w

0.30

0.20

0.10

0.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

EXPECTED PROS

FIGURE 4: A Pop plot of the Toronto monthly maxima of the transformed wind speed, illustrating the
goodness of fit of the Gumbel distribution. The diagonal lines represent the upper and lower
10% and 5% critical values for the KS statistic.

goodness-of-fit statistics determined separately for each month. The two month-by-
month analyses of goodness of fit revealed very few differences.

When the parameters were adjusted month by month, the statistics and plots did
reveal a slightly closer fit than in the case when the parameters were held fixed for all
months. However, even in the latter case, little evidence was found that the chosen
model is inappropriate. The results ofthe KS and AD tests were very consistent with
each other, and neither test rejected the null hypothesis more frequently than would
be expected in a set of 60 independent tests (12 tests at each station). The results ofthe
AD test are summarized in Table 4. The actual significance level of the goodness-of-
fit tests should be very close to the nominal level, because the model specified in the
tests is determined from a sample 12 times larger than the sample used in the tests.

In conclusion, it was evident that the Gumbel distribution is a suitable model for
monthly maximum wind speed after the data have been transformed, and that the
same model is appropriate for all months of the year.

Page 135

130 Case Studies in Data Analysis
Case Study No.5, Pages 119-144

TABLE 4: Anderson-Darling goodness-of-fit statistic indicating the difference between the
Gumbel distribution fitted to the monthly maxima of transformed wind speed (all months
combined) and corresponding empirical cumulative distribution functions determined
separately for each month.

Goodness-of-fit

Month Vancouver Regina Toronto Montreal Shearwater

Jan. 1.782 2.121" 0.297 0.534 0.898
Feb. 0.861 1.165 1.599 0.628 0.283
Mar. 0.502 2.904 b 0.817 0.605 0.770
Apr. 0.483 0.452 1.224 0.455 0.437
May 0.364 0.750 0.827 0.597 2.238"
Jun. 1.168 0.540 0.923 0.878 0.856
Jul. 0.795 0.967 0.457 0.382 1.977'
Aug. 0.885 0.746 0.879 0.567 0.511
Sep. 0.533 2.787 b 0.692 0.250 2.411 •
Oct. 0.931 0.361 0.684 0.823 0.749
Nov. 0.647 1.044 0.947 3.971 ' 0.758
Dec. 1.095 0.487 1.295 1.392 0.310

·Significant at the 10% level.
bgignificant at the 5% level.
'Significant at the 1% level.

4. ESTIMATES OF RETURN VALUES

The "raw" data for this analysis, the filtered wind speeds, may be thought of as
fluctuations about a stochastic process varying on long time scales. The extreme-
value analysis must also take the variability of this underlying process, the 365-day
moving average of wind speed, into account. The best way in which to accomplish
this would be to conduct an extreme-value analysis with it as well. However, there
are several difficulties with such an analysis. The 365-day moving averages contain
trends and display the effects of interventions. They also display fluctuations on long
time scales, and thus the record of 38 years (35 years at Shearwater) simply does not
contain enough information to do a reliable extreme-value analysis even if the
nonstationary components of the record could somehow be taken into account.

The fact that the data exhibit trends and interventions due to human activities
implies than an estimate of the variability of the long-time-scale component of the
wind speed (the 365-day moving average) must be more of a statement of policy than
anything else. It is therefore left to policy planners working in concert with climatol-
ogists to make some statement regarding the future annual mean maximum daily
wind speeds.

Lacking a "policy" statement regarding the future effects of urbanization and
interventions on wind speed, the following rather conservative approach was taken.
First it was noted that the trend at all five stations has been decreasing with time.
Secondly, it was assumed that:

(I:) there will be continuing urbanization of the observing sites and hence the
downward trends will not be reversed;

(2) management and standardization of observing sites will continue to improve
(meaning that it is much less likely today than, say, 20 years ago that an instrument
will be placed in an inappropriate location); and

Page 268

Vol. 52: P.K. Goel, T. Ramalingam, The Matching Methodology:
Some Statistical Properties. vili, 152 pages, 1989.

Vol. 53: B.C. Arnold, N. Balakrishnan, Relations, Bounds and
Approximations for Order Statistics. ix, 173 pages, 1989.

Vol. 54: K.R. Shah, B.K. Sinha, Theory of Optimal Designs. viii,
171 pages, 1989.

Vol. 55:L. McDonald,B. Manly,J. Lockwood,J. Logan (Editors),
Estimation and Analysis of Insect Populations. Proceedings,
1988. xiv, 492 pages, 1989.

Vol. 56: J.K. Lindsey, The Analysis of Categorical Data Using
GUM. v,168 pages,1989.

Vol. 57: A. Decarli, B.J. Francis, R. Gilchrist, G.U.H. Seeber
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lar Inequality Constraints. xii, 291 pages, 1990.

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Vol. 65: A. Janssen, D.M. Mason, Non-Standard Rank Tests. vi,
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Vol. 66: T. Wright, Exact Confidence Bounds when Sampling
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Vol. 67: MA. Tanner, Tools for Statistical Inference: Observed
Data and Data Augmentation Methods. vi, 110 pages, 1991.

Vol. 68: M. Taniguchi,HigherOrder Asymptotic Theory forTime
Series Analysis. viii, 160 pages, 1991.

Vol. 69: NJ.D. Nagelkerke, Maximum Likelihood Estimation of
Functional Relationships. v, 110 pages, 1992.

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pages, 1992.

Vol. 71: E.M.R.A. Engel, A Road to Randomness in Physical
Systems. ix, 155 pages, 1992.

Vol. 72: J.K. Lindsey, The Analysis of Stochastic Processes using
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Vol. 73: B.C. Arnold, E. Castillo, J.-M. Sarabia, Conditionally
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Vol. 74: P. Barone, A. Frigessi, M. Piccioni (Editors), Stochastic
Models, Statistical Methods, and Algorithms in Image Analysis.
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Vol. 75: P.K. Goel, N.S. Iyengar (Editors), Bayesian Analysis in
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Vol. 76: L. Bondesson, Generalized Gamma Convolutions and
Related Classes of Distributions and Densities. viii, 173 pages,
1992

Vol. 77: E. Mammen, When Does Bootstrap Work? Asymptotic
Results and Simulations.vi, 196 pages, 1992.

Vol. 78: L. Fahnneir, B. Francis, R. Gilchrist, G. Tutz (Editors),
Advances in GUM and Statistical Modelling: Proceedings of the
GUM92 Conference and the 7th International Worlcshop on
Statistical Modelling, Munich, 13-17 July 1992. ix, 225 pages,
1992

Vol. 79: N. Schmitz. Optimal Sequentially Planned Decision
Procedures. xii, 209 pages, 1992.

Vol. 80: M. Fligner,J. Verducci (Editors), Probability Models and
Statistical Analyses for Ranking Data. xxii, 306 pages, 1992

Vol. 81: P. Spirtes, C. Glymour, R. Scheines, Causation, Predic-
tion, and Search. xxiii, 526 pages, 1993.

Vol. 82: A. Koroste1ev and A. Tsyhakov, Minimax Theory of
Image Reconstruction. xii, 268 pages, 1993.

Vol. 83:C. Gatsonis,J.Hodges,R. Kass,N. Singpurwalla (Editors),
Case Studies in Bayesian Statistics. xii, 437 pages, 1993.

Vol. 84: S. Yamada, Pivotal Measures in Statistical Experiments
and Sufficiency. vii, 129 pages, 1994.

Vol. 85: P. Doukhan, Mixing: Properties and Examples. xi, 142
pages, 1994.

Vol. 86: W. Vach, Logistic Regression with Missing Values in the
Covariates. xi, 139 pages, 1994.

Vol. 87: J. M0ller, Lectures on Random Voronoi Tessellations.vii,
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Vol. 88: J.E. Kolassa, Series Approximation Methods in
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Vol. 89:P. Cheeseman,R.W. Oldford (Editors), Selecting Models
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Vol. 92: M. Eerola, Probabilistic Causality in Longitudinal
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Vol. 93: Bernard Van Cutsem (Editor), Classification and
Dissimilarity Analysis. xiv, 238 pages, 1994.

Vol. 94: Jane F. Gentleman and G.A. Whitmore (Editors), Case
Studies in Data Analysis. viii, 262 pages, 1994.

Page 269

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Series Editors:

Professor S. Fienberg
Department of Statistics
Carnegie Mellon University
Pittsburgh, Pennsylvania 15213
USA

Professor J. Gani
Stochastic Analysis Group, SMS
Australian National University
Canberra ACT 2601
Australia

Professor K. Krickeberg
3 Rue de L 'Estrapade
75005 Paris
France

Professor 1. Olkin
Department of Statistics
Stanford University
Stanford, California 94305
USA

Professor N. Wermuth
Department of Psychology
Johannes Gutenberg University
Postfach 3980
D-6500 Mainz
Germany

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