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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT 10 Ju±y 19 ° 2 NBS REPORT

1103-21-5119

Progress Report for April - June 1S52

on

Applications of the Theory of Stochastic

Processes to the Study of Trajectories

(NBS Project 1103=21-5119)

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, reprinting, or reprodu AnorOVed for DllbliC release by the

from the Office of the Director, Nation . . r

which a report has been specifically Director of the National Institute OT Standards and Technology (NIST) on October 9, 2015

d unless permission is obtained In writing

s not needed, however, by an agency for that particular report for Its own use.
This is a special progress report to the U.S. Naval

Ordnance Test Station which sponsors NBS Project 1103-21-51.19,

Application of the theory of stochastic processes to the

study of trajectories

.

J. H. Curtiss

Chief, National Applied

Mathematics Laboratories
PROGRESS REPORT FOR APRIL-JUNE 1952

ON

APPLICATION OP THE THEORY OF STOCHASTIC PROCESSES

TO THE STUDY OF TRAJECTORIES

(NBS Project 1103-21-5119)

I o Summary

This report contains a summary of the work done luring

the quarter. Results of this work are briefly stated Technical

reports written in connection with this project are mentioned

but are transmitted separately.

II. Discussion of work done during the quarter . The first task under the project is the development

of a test for the hypothesis that a sequence of observations

comes from a fundamental random process (Wiener process).

In this connection a theorem was derived during the preceding

quarter. A statement of this theorem was already givep in

the progress report for the quarter January - March 1952

.

[NBS report 1598] , A detailed treatment may be found in a paper by E, Lukacs entitled "A property of strongly continuous

processes” (NBS report let's) which was transmitted to the

U.S. Naval Ordnance Test Station at the close of the quarter.

This work provides an analysis of the various assumptions underlying the proposed stochastic model of a fundamental random process (F.R.P.). On the basis of this

analysis one can conclude that standard statistical techniques

can be used to test the hypothesis that observed trajectory

data come from a F.R.P. One has to form the deviations

x(t) from the mean value curve. If N equidistant observations

are taken during the fewae interval (0,T)-'one has to form the

increments A n - x(n*r) ~ x(n-lT) for T = T/N and n = 1,2,,.. N,

and test the hypothesis that the A n are independently distributed. A rather wide variety of such tests is known, rejection of the hypothesis of independence of the D n would

imply rejection of the hypothesis of a F.R.P.

Another alternative approach was also studied. This

approach does not make use of the fact that the normality of

the process is — under certain assumptions — equivalent to

a continuity property of the process. If one wants to test

the hypothesis that a sequence of observations comes from

a Wiener process, one slight be willing to assume a priori

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the continuity of the variance function but should like to

test whether the increments are normally and independently

distributed. One has therefore to test simultaneously two

hypotheses: the hypothesis of normality and the hypothesis

of randomness (identical and independent distribution of the

random variables). This establishes a connection between

this project and a study currently carried under Task 1103-

11 - 1X07/52-2. Mr, I, R. Savage proved the following theorem:

Let x, ...x, be a sample of n identically and independently

distributed random variables. Then any rank order statistic

and any symmetric statistic are independently distributed.

This theorem can be applied to construct a test for the_

hypothesis mentioned above in which the evidence from a X

test of normality and from a rank-order test of randomness

can be combined. Mr, Savage's results are summarized in

NBS Report No. 1694 entitled "Simultaneous Tests of Randomness

and other Hypotheses" which will be issued at the beginning

of the next quarter

.

E. Lukacs and Professor H. B. Mann visited the Naval

Ordnance Test Station in June 1952, and discussed possible

applications and problems to be solved under this project.

Professor Mann also gave lectures on his recent work on the

estimation of mean value functions of stochastic processes.

/
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