The application of the statistical theory of extreme values to gust-load problems Page: 4 of 16
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REPORT 991-NATIONAL ADVISORY COMMITTEl FOR AERONAUTICS
more complex, method of obtaining confidence limits for the
distribution of extreme values. In the interest of simplicity,
the present discussion will be restricted to the method given
by Gumbel.
The method of obtaining control curves derived by Gumbel
depends essentially on the determination of control intervals
at discrete points along the distribution by the utilization
of the properties of the distribution of the mth, values.
Within the range 0.1< W*(x) <0.9, the distribution of
the mth values of a continuous variable having moderate
skew distribution and possessing a mode is asymptotically
normal around the most probable mth value (reference 5)
with a standard error (a/~, given by:-(13)
(q r) _ /[W*(x)J [1- W*(x)]
The function (oa,)m is hereinafter called the "reduced
standard error" and is shown as a function of F*(x,) and yin figure 3. A horizontal interval around the true mth
value given by, - Sm<m< + Sm
where Sm is defined by
(14)
(15)gives a probability of about 0.68 that the mth value in a
sample of n observed maximum values will lie in the enclosed
interval. Similarly, the interval E, +2Sm has a probability
of about 0.95 of enclosing the mth value from a sample of n
observed maximum values.
In order to obtain a control interval at the larger values, a
fundamental property of the distribution of maximum values
may be utilized. The most probable largest value of n
maximum values has been shown (reference 4) to have a
distribution of the same form as the distribution of the
maximum values with the origin shifted to the right by a.68
.8/
.?7
.63
.54 .46
F'rm)
.38.3/
.25
.20
.16 .13 J0
3..
.0
fayf
I s - -- - - - - - - - - - - --/
0 ~-- - - - - - - - --1.0
-.5
0
.5
/1.0
1.5
2.0
FIGURE 3.-Reduced standard error for rth values.
2.5
S =
S cr1
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Press, Harry. The application of the statistical theory of extreme values to gust-load problems, report, July 22, 1949; (https://digital.library.unt.edu/ark:/67531/metadc65484/m1/4/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.