NATIONAL BUREAU OF STANDARDS REPORT

2279 Statistical

National Bure

[Washington

TABLES USEFUL IN ESTIMATING THE MEAN VALUE

FUNCTION OF A FUNDAMENTAL RANDOM PROCESS

by

I. R. Savage and E. Lukacs

<nbj£>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT

1103-20-5119 16 February 1953

MBS ili:pORl279

TABLES USEFUL IN ESTIMATING THE MEAN VALUE

FUNCTION OF A FUNDAMENTAL RANDOM PROCESS*

by

I. Richard Savage and Eugene Lukacs

ttThe preparation of this paper was

U. S. Naval Ordn t Station

sponsored by the

, Inyoke rn.

The publication, Approved for public release by the

unless permissio Director of the National Institute of

25,d,c. Such Standards and Technology (NIST)

eally prepared I on October 9, 2015

lor In part, Is prohibited

'Standards, Washington

i report has been speclfir report for Its own use,
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TABLES USEFUL IN ESTIMATING THE MEAN

VALUE FUNCTION OF A FUNDAMENTAL

RANDOM PROCESS*

by

Io Richard Savage and Eugene Lukacs

National Bureau of Standards

1 0 Introduction , Let y(t) be a stochastic process and assume

that

(1.1) y(t) = x(t) + f(t)

(1.2) f(t) = Kiffi(t) + K202(t) '+ ... + Ks0s (t)

where x(t) is a fundamental random process.

In a forthcoming paper [1+] H, B, Mann assumes that the y(t)

process is observed over an interval [0,T] and developes a method

for finding the maximum likelihood estimates for ( the parameters

K^, Kp, , , , , Ks of the mean value function f(t)*'", In a technical

note [5] he gives a bbief exposition of his results and also

explicit formulae for the practically important case where f(t) is a polynomial in t. As in least squares estimates it seems also

here advantageous to use orthogonal polynomials. However, it is

here necessary to assume that the derivatives ^.(t) are the first

s polynomials of a system orthogonal with respect to the weight

function 1 over an interval [0,T], The functions ^j(t) become"

then the integrals of Legendre polynomials adapted to the interval

[0,1], As a consequence the mean value function must have initial

and terminal value zbro, i.e,,

(1-3) f (0) = f(T) = 0 , While ' (113). will apply In certain physical situations its failure

will in general prevent the use of orthogonal polynomials in

estimating a mean value function*’"'''*. It will then be necessary

"The preparation of this paper was sponsored by the U, S. Naval

Ordnance Test Station, Inyokern,

**The 0j(t) are known functions, and are subject to certain restrictions,

stated in [JLj.] and [£].

""'Ordinarily the condition (1,3) could be obtained by a rotation of

axis, but in the situation under consideration such a rotation

would disturb the nature of the random process, and hence can not

be used.
- 2 -

to assume that ^(t) = t*J and to use the methods developed for

this case.

Let us therefore suppose that

(1.4) f(t) = Kit + K2

t

2 + + Ks

t‘ According to the formulae given in [41 and [5] the estimate kof Kj is given by

o OT

A

(1.5)

Here

Kj = 2 r r=lt

1 1

dy(t) ( 0 1 >2 , » o . jS )

jn sJ' r

jrT

of the matrix

j+r -1

1

U o where the matrix is the inverse

j+r -1 The purpose of the

»r l^.o.fS

present note is to simplify the computations necessary for obtaining

the estimates (1.5). This is done by studying first the matrix

i+j -1 and giving formulae for finding its inverse. Tables

of the coefficients S^r are then computed for n = 2(1)10. These

tables will be found at the end of this report. The same matrix

occurs in least square theory when an integral is minimized instead

of a sum. The inversion of this and related matrices was studied

by A. R. Collar [2], [3] it is however believed that the proofs

given here are somewhat more elementary.

2. A theorem due to A, Cauchy . The following theorem is due to

A , Cauchy

- tl 1 • Let aq, 0 ..,an , b]_ , . .

.

,bp be 2n numbers and consider

the determinant whose elements are of tne form l/ta^+b^)

(i,k = l,2,...,n). Then

1 . ,n

( 2 . 1 ) a* +b k

j>k J J

i ,k=l , . . ,n 1 . .n

jj

(a j+bk

) j,lc -

This theorem as well as an indication of its proof may also be

found in [ 7 ], page 98 , problem 3 .

3. Inversion of a certain matrix . In the introduction we proposed

to consider the matrix

1

(3.1) Sn -

i+j -1

i » j i »

,n
- 3

-

and to find its inverse. Clearly the determinant of S n

as well

as all its minors are of the form discussed in Section 2. We

can therefore compute the inverse S" 1 by applying Cauchy's theorem.

We denote by the minor of the element in the i-th row and

and j-th column of the determinant A n of the matrix S . If we

write S lj for the element in the i-th row and j-th column of the

n -l inverse S" x then

(3.2) = (-

1

ij

n n

If we use (2.1) to find and A n

we obtain by an elementary

computation /'n 1 \

TT" kt)

2

££' •

it / 2 i 2 \ n—

k

n

J I (n -k ) k_1

t fn-1

TTki

i j (n+i-1) l(n+ j-1) l lk=l

(3.3) ^n

and

(3.1+) n [ (1-1) l( j-1) i] 2ni(n-i) S(n-j) I SET j + i +l

I I (n+k) ! k=;l

It is easy to show inductively that

n-1

(3.5)

n / 2 , 2vn-k, , n

11 TT ( n -k ) k i JJ, fh+kl I n l k=l ( n+k ) J so that we obtain from (3.2), (3.3) and (3.4)

(3.6) S iJ = (-D i+j (nfi-l) l(n+i 7 l) l n 1+ J-1 [(i-1) U j-1 j I]'fi (n-i) l(n-j) l

1+. Formulae for the numerical computation of the It is v|uite

convenient to compute the Si«J recursively. We obtain immediately

from (3*6)

(4.D gij =

(n+i) (n+j)

g

ij

n+l (n+l-i) (n+l- j ) n

for i , j=l ,2 ,

.

9 n

By means of (4-1) it is possible to compute from the elements of

the elements of S

-

}:-, in the first n rows and columns. We n n+l

still have to determine the elements of the last column. We can
- 4

-

do this by deriving a relation between the elements of and the

coefficients of certain orthogonal . polynomials . Consider a polynomial of degree m

(4.2) Pm<

x ) = 1 + a.m,lx + am,mx

m

( am,rt+0)

*~such

r that

(4*3)

n 1

x

k

Pm

(x)dx = 0 for k = 0 ,1 , . . . , (m-1

)

0.

Equatiom (4«3) determines the polynomials P (x)

,

they are the

Legendre', polynomials adapted to the interval1 0 < x < 1. They are

obtained from the Legendre polynomials formed for the interval

-1 i x i +1 by a linear transformation of the variables. They

are discussed and to a certain extent tabulated in [6], This

system of polynomials is orthogonal and we have

(4.4) pm ( x ) pn( x ) dx = U0

2m+l if m = n

Moreover

,

<*-5>

For the proof of (4*4) and (4*5) the reader is referred to

W. E. Milne's book [6] chapter IX, Section 69 . We write ara q = 1 and substitute (4«2) and (4*3) and obtain,

considering also (4.4)*

m a

2Ll1 _

.m (k—0 ,1 , . . . ,m) , j =0 k+ j +1 (2m+l)art(P u

k,.

where 6^m

is the Kronecker delta. We rewrite these equations as

m+l am i i 1

(4.6) 2 Fvl'r = To rrr

6

k m+i (k=i,...,m+i) . j_l K J-1 ( 2m+l )

a

m jln k,m+i

Considering the obvious relation

m+1 1 0 hk

2 TT—T S n = 6

lc=l j+k-l m+l hk

we solve the system (4*6) and obtain
- 5

-

,m+l ,k

m,k-l (2m+X)a m+1

or

(4-7) C4k = < 2m+i >

. _ ek,m+l lm p k-l bm+l

From (4.7) and (4.5) see then that for k = 1, (m+1)

/i, ox -m+l,k _ Q k,m+1 _ . .m+k-1 , 0 ,/ 2m\ / m \/m+k-l| (4.8) Sm+1

> - sm+1 -(-l) ( 2m+1

)( m ic-i )

Formulae (4*1) and (4*8) can t>e used to compute systematically

tables of the elements of S“^ . m

5. An alternate method for computing S . We normalize the polynomials (4.2) and obtain

m

(5.D Qm(x) = ./5m+l Pm(x) = 2 b x

K (m=0,l,2,...)

k=0

It is seen from (4.5) and (4*4) that

(5.2) »> k

= ( -D k vssr(£)(T)

and

(5.3)

0

0

Vx >Vx)dx = 6m>n

We consider the matrix

% " 5 m,k

m,k=0 , . . . , (N-l)

It follows thfen from (5.3) that

(5.4)

' I

Here I denotes the identity matrix and b! the transposed of B^.

From (5*4) we see that

(5.5) = B^j B^j

If tables of the coefficients of the polynomials P (x) are known

it is possible to find S^" from (5.5). On the other hand one

could also derive (3.0) 'from (5«5) and (5.2).
- 6 -

6. Description of Tables and their preparation . Tables are given

for the inverse of the m.mx l n [see equation (3.1)] for n=2,..,10

That is the quantities [see S ( equation (3.2)] are tabulated for l<i^j<n<10.

Since the matrix Sn is symmetric its inverse is also, hence

in the following tables is given only for j i.

These tables were obtained by the use of equations (4«i)

,

(4*7) and (4.8) . The tables were checked by using the relationships:

( 6 . 1 ) n

Sij/(i+k-l) = 6-j k (j,k = i-1

All of the above relations were checked for j=k, and supplementarily a few others were examined and found correct.

Acknowledgement : The authors wish to thank Mr. Edwin L. Grab who

prepared and checked the tables.

REFERENCES

[1] A. Cauchy, Exercises d» Analyse et de phys . math. Vol . 2,

pp. 151-159, Paris I 8I4.I. Also Oeuvres' completes 2 e serie

XII

.

[2] A. R. Collar, On the Reciprocation of Certain Matrices.

Proceedings of the Royal Society of Edinburgh, Vol. 59.

pp. 195-206, (1939).

[3] A. R. Collar, On the Reciprocal of a Segment of a Generalized

Hilbert Matrix. Proceedings of the Cambridge Philosophical Society . Vol. 47 , pp . 11-17, (1951).

[41 H. B. Mann, On the Estimation of Parameters Determining the

Mean Value Function of a Stochastic Process. To be

published, in Sankhya.

[51 H. B. Mann, Statistical Techniques Applicable to the Analysis

of a Fundamental Random Process. Technical Note WCRR-52-7

Flight Research Laboratory, Wright Air Development Center

Dayton, Ohio, 1952.

[6] W. E. Milne, Numerical Calculus. Princeton University Press,

Princeton, New Jersey, 1949.

[ ( ] G. Polya-G. Szego, Aufgaben und Lehrsatze aus der Analysis,

Vol.' 2. (Grundlehren der math. Wiss. Vol. 20) J. Springer

Berlin, 1924* Reprinted by Dover Publications, New York,

1945.
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