NATIONAL BUREAU OF STANDARDS

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REPORT FOR FISCAL TEAR 1951

on

APPLICATION OP STATISTICAL THEORY OF EXTREME VALUES

TO GUST-LOAD PROBLEMS

(NACA Project S50-39; NBS Project 1103-21-5106)

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

E. U. Condon, Director

THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is en- gaged in specialized research, development, and engineering in the field indicated by its

title. A brief description of the activities, and of the resultant reports and publications,

appears on the inside of the back cover of this report.

1. ELECTRICITY. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Electrochemistry.

2. OPTICS AND METROLOGY. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

3. HEAT AND POWER. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Combustion.

1. ATOMIC AND RADIATION PHYSICS. Spectroscopy. Radiometry. Mass Spectrometry, physical Electronics. Electron Physics. Atomic Physics. Neutron Measurements. Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

5. CHEMISTRY. Organic Coatings. Surrace Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

6. MECHANICS. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters.

7. ORGANIC AND FIBROUS MATERIALS. Rubber. Textiles. Paper. Leather. Testing and

Specifications . Organic Plastics. Dental Research.

8. METALLURGY. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy . Corrosion.

9. MINERAL PRODUCTS. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Building Stone. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral

Products.

10. BUILDING TECHNOLOGY. Structural Engineering. Fire Protection. Heating and Air Conditioning. Exterior and Interior Coverings. Codes and Specifications.

I i. APPLIED MATHEMATICS. Numerical Analysis. Computation. Statistical Engineering. Ma- chine Development.

12. ELEC1 RON ICS. Engineering Electronics. Electron Tubes. Electronic Computers.

13. ORDNANCE DEVELOPMENT. Mechanical Research and Development. Electromechanical

Fuzes. Technical Services. Missile Fuzing Research. Missile Fuzing Development. Projectile Fuzes. Ordnance Components. Ordnance Tests. Ordnance Research.

I4-. RADIO PROPAGATION. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

15. MISSILE DEVELOPMENT. Missile Engineering. Missile Dynamics. Missile Intelligence.

Missile Instrumentation. Technical Services.
REPORT FOR FISCAL YEAR 1951

on

APPLICATION OF STATISTICAL THEORY OF EXTREME VALUES

TO GUST-LOAD PROBLEMS

(NAGA Project S5G-39~ N3S Project 1103-21-5106}

SUMMARY

Work carried out 'under the above project f cr the

period July 1* 1950 to March *31 , 1951 was covered by progress

Report submitted under dace of May 28* 1951 (reference 1)

The present report- includes work done during the final quarter

of FY 1951 o April I to June 30* which terminates the contract

year o This work was largely in connection with a paper

entitled "On certain estimators based on large samples of

extremes*" by Julius Lieblein* submitted separately.

ACTIVITIES FROM JULY 1* 1950 TO MARCH 31* 1951

As stated above* activities during this period were

described in the previous progress report (reference l) e In

recapitulation* thev were , briefly?

r i 1 , u J Covered by Contract W“47±2 (S50-39) — Services for conducting a special investigation of the application of

the Statistical Theory of Extreme Values to gust-load

problems — between the National Advisory Committee for

Aeronautics and the National Bureau of Standards*

C ^ 3 Item (1) was sponsored in part by the NAGA and in part

by the NBS • Item (2) was sponsored In part by the

Office of Air Research * Air Materiel Command, USAF, and

in part by the NACAo Publication in each case is being

undertaken by the NBS*
— 2 —

(1) "Lectures on the Statistical Theory of Extreme

Values and their Practical Applications by

S» J e dumb el , prepared for publication in the

Applied Mathematics Series of the NBS (reference 2)«

(2) "Probability Tables for the Analysis of ExtremeValue Data/1 with an introduction by E. J. Gumbel,

accepted for publication in the Applied Mathematics

Series of the NBS (reference 3)©

(3) "Table of the First Moment of Ranked Extremes,"

for sample sizes n * 1(1)10(5)60(10)100 and ranks

1(1)25 (in descending order), computed in preliminary form for use in studying B. F. Kimball’s

graphical techniques* (Submitted as a separate

report©)

(4) Generalized Poisson theory applied to analysis of

VGH records , and found to be unsuitable©

(5) Various estimators of the parameters of the extremevalue distribution studied for bias and efficiency©

As stated in the previous report, the aim of the NACAsponsored research at the National Bureau of Standards has been

the improved application of the statistical theory of extreme

values to gust-load problems© At the Bureau this research has

been carried out in the Statistical Engineering Laboratory

(Section 11 ©3) of the Bureau’s National Applied Mathematics

Laboratories (Division 11, NBS) as Project 1103-21-5106, with

Mr, Julius Lieblein as Project Leader© The previous report also

discussed the background and purpose of the project, and in

particular, the relation of the lectures by Gumbel and the

extreme-value probability tables (items (1)

respectively) to the aims of the project©

and (2) above.
WORK FROM APRIL 1, 1951 TO JUNE 30, 1951

The final quarter of the contract year was devoted largely

to preparation of a report under item (5) above, and also

included some work under items (1), (2), (3), as follows t

( 1 ) “Lectures on the Statistical Theory of Extreme Values

and their Practical Applications f “ by B* J* Oumb el The

manuscript was reviewed by a technical reader for the National

Bureau of Standards Research Editorial Committee* Discussions

with the reader were held and a start was made toward incorporating the numerous suggestions into the manuscript*

( 2 ) “Probability Tables for the Analysis of Extreme - Value Data *” The manuscript was reviewed by a technical reader 9 several minor changes were made, and the manuscript was accepted

for publication in the Applied Mathematics Series of the NBS*

( 3 ) “Table of the First Moment of Ranked Extremes •

Several entries were corrected and an explanatory section added

to make the table suitable for submitting as a separate report*

(5) Report on large- sample estimators * As above Indicated

the chief outcome of the work of the final quarter is a report

entitled “On Certain Estimators Based on Large Samples of

Extremes,” by Julius Lieblein, which is submitted separately*

This paper provides several useful and efficient methods for

analyzing large samples of extreme-value data that have advantages over certain methods suggested by other workers in the
field and previously used in an NACA research report (reference

4). Certain technical aspects of the forthcoming paper are

indicated in the abstract comprising the Appendix to the present

report. It is planned to present this paper at the Eastern

Sectional Meeting of the Institute of Mathematical Statistics

in Washington, D. C., October 26-27, 1951, subject to the

necessary clearance by the NACA*

INDICATED FURTHER RESEARCH

The above-mentioned progress report for the first three

quarters of FY 1951 was accompanied by a discussion of proposals

for further research (reference 5) that concerned five items

warranting attention if the project is continued through a

second year* These are briefly, (i) estimators, (ii) prediction

(iii) graphical methods, (iv) testing a single sample, and (v)

testing two samples.

Of the above items, one (estimators) depends upon the

extension of previous work under the project. The work on

estimators is intended to concentrate on methods of analyzing

very small samples of data. The availability of item (3)

previously mentioned, "Table of the First Moment of Ranked

Extremes/1 makes it possible for the first time to find methods

applicable to very small samples which are unbiased, that is,

the methods will give answers which will tend to cluster about
a 5 “

the correct value „ But to find among these the most efficient

methods 9 i,e 8 those whose !l clustering” is the most concentrated

about the true value , so that fewer observations are necessary

for the same precision, it is necessary to have available tables

of the second moments (variances and covariances) of ranked

extremes in small samples • Computation of these tables is

expected to absorb about one-third of the estimated total

project cost of $15,000©
6

REFERENCES

1* National Bureau of Standards: Progress Report on Application of Theory of Extreme Values to Gust-Load

Problems, for the period July 1, 1950 - March 31,

1951 (NACA Project S50-39; NBS Project 1103-21-5106).

Enclosure to letter from E. TJ. Condon to Ho Lo Dryden

dated 28 May 1951.

2. Gumbel, E. J. : Lectures on the Statistical Theory of

Extreme Values and their Practical Applications. To

be submitted for publication in the Applied Mathematics

Series of the NBS.

3. Probability Tables for the Analysis of Extreme -Value Lata.

To be published in the Applied Mathematics Series of

the NBS.

4. Press, Harry: The Application of the Statistical Theory

of Extreme Values to Gust-Load Problems. NACA TN 1926,

November 1949, pp 0 10-11.

5. National Bureau of Standards: Proposal for Renewal of

NACA Contract with NBS on Application of Extreme Values

to Gust-Load Problems. Enclosure to letter from

E. U. Condon to H. L. Dryden dated 28 May 1951.
APPENDIX

(Abstract of paper for presentation at the Eastern

Sectional Meeting of the Institute of MathematicalStatistics in Washington, D®C®, October 26-27 , X95X«)

On Certain Estimators Based on large Samples of Extremes ® Julius Lieblein, Statistical Engineering Laboratory, National

Bureau of Standards®

E® J® Gumbel and Bo F® Kimball have given estimators for the

parameters of the asymptotic distribution of largest values,

/ \ Prob {X <x} = exp(-e“ which are especially adapted

to the analysis of data in large samples® The present paper

applies the theory of order statistics to the problem of seeking

more efficient estimators which are at the same time simpler to

compute® Several large -sample estimators are found which, with

one exception, appear yto have greater efficiency than those

derived from the methods of Gumbel and Kimball, yet require

much less effort in computation® If punch cards are used the

work can be handled by a mechanical sorter which ranks the

observations in order of size and then selects a small number

of them with predetermined ranks®

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The National Bureau of Standards is the principal agency of

the Federal Government for fundamental and appi ied research in physics, mathematics, chemistry, and engineering. Its activities range

from the determination of physical constants and properties of ma- terials, the development and maintenance of the national standards

of measurement in the physical sciences, and the development of

methods and instruments of measurement, to the development of special

devices for the military and civilian agencies of the Government.

The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, cal ibration services,

and various scientific and technical advisory services. A major

portion of tne NBS work is performed for other government agencies,

particularly the Department of Defense and the Atomic Energy Commission. The functions of the National Bureau of Standards are set

forth in the Act of Congress, March 3, IQ 0 1 , as amended by Congress

in Public Law 619, 1950. The scope of activities is suggested in

the listing of divisions and sections on the inside of the front

cover

Reports and Pub Heat i ons

The results of the Bureau’s work take the form of either actual

equipment and devices or published papers and reports. Deports are

issued to the sponsoring agency of a particular project or program.

Published papers appear either in the Bureau’s own series of publications or in the journals of professional and scientific societies.

The Bureau itself publishes three monthly periodicals, available

from the Government Printing Of f ice : the Journal of Research, which

presents complete papers reporting technical investigations; the

Technical News Bulletin, which presents summary and preliminary re- ports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for

radio communications throughout the world. There are also five

series of nonperiodical publications: the Applied Mathematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can be found in NBS

Circular 460, Publications of the National Bureau of Standards (75

cents). Information on calibration services and fees can be found

in NBS Circular 483, Testing by the National Bureau of Standards

(25 cents). Both are available from the Government Printing Of fice

Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications, National Bureau

of Standards, Washington 25, D. C.