NATIONAL BUREAU OF STANDARDS REPORT

2^98

AN ILLUSTRATION OF COMPUTATIONAL METHODS FOR THE

DETERMINATION OF THE PARAMETERS OF A CERTAIN

RANDOM PROCESS*

*The preparation of this paper was sponsored by the

Uo So Naval Ordnance Test Station^ Inyokern 0

Eugene Lukacs

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT MBS REPORT

1103=10=5119 May 14 , 1953 2498

AN ILLUSTRATION OF COMPUTATIONAL METHODS FOR THE

DETERMINATION OF THE PARAMETERS OF A CERTAIN

RANDOM PROCESS

by

Eugene Lukacs

<NBS>

The publication, rep Approved for public release by the

„ pat t, is prohibited

unless permission is < Director of the National Institute O ndards, Washington

25,d.c, Such perrr Standards and Technology (NIST) ort has been specifically prepared If thi on October 9, 2015 >ort for Its own use.

An Illustration of Computational Methods for the

Determination of the Parameters of a Certain

Random Process*

by

Eugene Lukacs

National Bureau of Standards

1 o Introduction s Statistical evaluation of the result of

measurements is called for whenever repeated measurements of

the same quantity are taken,, However one encounters quite frequently situations where the physical quantity observed varies

with time, We can then take a sequence of observations, but

due to the variability in time we can not say that they are

measurements of the same quantity,, Moreover the situation can

be sucfe/ that it is impossible to take simultaneously several

independent observations or to repeat the experiment under

identical conditions. As an example we mention the motion of

certain physical bodies which receive an initial impulse and

are then moving freely subject only to random influences.

The observed quantity would be in this case either the position or the velocity or the acceleration of the body. The

motion of a wide variety of missiles belongs j©to this class.

As another example we mention the decrease in thickness (or

weight) of a shoe sole or a tire due to the natural wear over

a period of time,

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

Ordnance Test Station, Inyokern.
- 2-

In situations of this kind, the standard statistical techniques can not be applied in a natural manner,, It seems

therefore desirable to find an appropriate probabilistic

model

o

Although we can take observations only at discrete time

points ? it seems quite natural to suppose that a random variable (for instance the position of a moving body) is given at

each instant of time 0 It is customary to try to make as many

observations as possible, the temporal proximity of the observations as well as the nature of the physical phenomenon

will in most cases prevent the observations from being stochastically independent,, This is a situation which is rarely

studied in the theory of the statistical evaluation of measurements o However these considerations suggest strongly that a stochastic process, depending on a continuous time parameter

would be the suitable probabilistic model for such a sequence

of observations,,

A mathematically simple model is obtained if we assume that

the actual sequence of observations comes from a Wiener process with a certain mean value function,, Such a model contains several parameters 9 the variance constant of the process

and the parameters introduced by the mean value function,,

Procedures have been developed [2] [ 3 ] [*+] for the estimation

of these parameters „ The purpose of this paper is to give a
-3”

brief survey of the estimation procedures , the numerical aspects of obtaining these estimates will be emphasized. Finally an example of a Wiener process with mean value function

is constructed by means of random numbers. The main purpose

of the present paper is to illustrate the available statistical techniques by means of this artificial process.

2 . The Wiener process

A stochastic process x(t), depending on a continuous time

parameter, is said to be a Wiener process if

(i) x(t) is a process with independent increments

and initial value x(0) = 0,

(ii) the increment x(t+r) - x(t) is normally distributed with mean zero and variance ct , (where

c > 0).

The constant c is called the va

The value x(t) of the proces

(i) as an increment, x(t) = x(t

(ii) that the mean value functi

is identically zero. It is qui

process which has the essential

process but possesses a non van

We give therefore the following

A stochastic process y(t) is

riance constant of the proces

s can be written according to

) - x(0) so that we see from

on Ex(t) of a Wiener process

te often desirable to study a simplicity of the Wiener

ishing mean value function,

definitions

said to be a Wiener process

s 0

with mean value function f(t) if
-4.

(2.1 ) y(t) = x(t) + f (t) „

Here f(t) is a fixed real valued function of t such that

f (0) = 0.' and x(t) is a Wiener process.

The process y(t) is completely determined by its variance

parameter and its mean value function i(t). In order to be

able to apply estimation procedures to the function f(t) it is

desirable to give it in such a form that it contains certain

parameters. A fairly general assumption is that

(2.2) f (t) = k

1 01 (t) + k

202 (t)+... k

s 0s (t).

Here the functions 0.(0...0 (t) are s arbitrary but completely

I s specified functions while the coefficients k. ...k are the

parameters of the mean value function. The functions 0^ ft ) »

*0S (t

are only subject to certain res t ric tions *[ 3 ] which are satisfied in all cases of practical interest.

3 o The estimation Procedure . We assume in the following that the y(t) process - as described by (2.1) and (2.2) - is observed over a finite interval [o,T] so that one sample curve is known over this interval.

We compute the following quantities

\

*These assumptions a re§’°~XaT~All the functions 0 (t) are twice

differentiable. (b) For any set a l9 ».. ? a of real numbers,

I s not all zero, we have a^0. 1 ( t)+ . . .a

g

0

g

( t) 0 0 for some tset of positive measure which is contained in [o ? T].
-5-

(3

i

•1) f’ij = J

0'iCt) 0> ,(t)dt (i,j = 1,2,...,s)

and then the quantities which are determined by the matrix

relation

(3.2) (( $

ij

>) = (( f.pr1 x J

Next one has to compute the integrals

(3.3)

|

0

s i (t)dy(t) = y(T)0' i (T) - J

y( t)0 ,, i

( t)d t (i=1,,„, ? s)

o ° A

The estimates k. of the parameters k. are then

J J s J

(3.*0 k. = y~! 0' (t)dy(t) ( j=1 ,2, „ . . ,s)

J v— 1 J0

A

The estimates k^ ( j=1 ,2, , , , ,s) are limits of maximum likelihood

estimates computed from observations taken over a finite set of

points. It can be shown that under rather general conditions

A

the estimates k. are unbiased estimates of k. and that the

A

J J

'\ A /v A T j j covariance of k. and k. is given by o(k.k.) = c

J

, J

-t J It should be emphasized that the estimates k. are not maxJ irnum likelihood estimates and that the estimated mean value

(3.5) ?(t) = k

1 0^(t) + t 0

s (t)

is not a least square fit of the data. Moreover it can be

A

shown that the mean value curve f(t) has another optimal

A

property,. We say that f(t) is a best linear estimate of

f(t) if
-6-

(a) E £(t) = f(t) (i 0 e 0 f(t) is an unbiased estimate

j. T ^

(b) E

j

[f (t) - £(t)] 2dt <_ E [

[f (t) - 7(t)] 2dt

o o s where f(t) is an estimate, f(t) = > , k 0 (t) such that _ v=1

V J ^

! A)

k v =

I v^(t)dy(t) and also E ( k^) = k v

„ 0

The estimate (3»5) is a best linear estimate in this sense„

We finally give an estimate for the variance constant,.

We divide the interval [0, T] into N equal parts of length

V -

jq

and denote the subdivision points by t^ = i ?T (i = 0,1,

2, 0 »o,N)o The quantity

N

(3.6) ?= t~ I] [y(t

n

) - y(t„_i)-f ( t ) + f(t

n.,)]

2

n=1

can be used as an estimate for the variance constant c, This

estimate is slightly biased but its bias converges to zero

as N increases.

We still have to choose the functions 0^ ( t

)

9 02

( O » ° • »0

g

(t)

In this report we consider only the case in which the 0j(t)

are polynomials. However other choices are possible. For

instance it might be advantageous to choose the 0.(t) as trigJ onometric functions when phenomena with a definite periodicity

are studied.

As in least squares theory it seems also here to be convenient to use orthogonal polynomials. The purpose of introducing orthogonal polynomials is in the present case the
-7-

(3.1) we see that it is then necessary to assume that the debecome the integrals of Legendre polynomials adapted to the

interval [0,1 ]„ As a consequence the mean value function

must have initial and terminal value zero, i„e<,

Condition (3°7) will not be fulfilled in general,, Moreover,

it is not possible to enforce this condition by a rotation of

the axis. In the situation under consideration such a rotation would change the character of the process and prevent it

from being a Wiener process. This means that in general it

will not be possible to use orthogonal polynomials to estimate

the mean value function of a Wiener process. The use of orthogonal polynomials is restricted to certain physical situations where the quantity measured starts at a certain level

which is reached again at the end of the period of observation,.

In the general case (3.7) is not satisfied and it is then

convenient to choose for the 0j(t) consecutive powers of t„

We assume therefore

rivatives 0' .(t) are the first s polynomials of a system

orthogonal with respect to the weight function 1 over an

interval [0 ? T]„ It follows then that the functions 0.(t)

3

(3.7) f(0) = f(T) = 0 o

s)
- 8-

We obtain therefore from (3®1)> (3. 3) and (3*^)

§ =

ij 1 ' r ij i + j - 1

i+j-1

(3.9)

r

( 3 . 10 )

jt v ’dyd) = T

1 ’y(T) - (i-D \yy( t)

1 2dt

for i = 2, . . ,

,

t

v-1 dy(t)

v=1 n

It is easy to show that

(3.11) $

jv = s21

• mj+V~1

j vl J

where the matrix Jj S v

J| is the inverse of the matrix

j+v-1

' j yV— 1 ; . t i )S

The computational work involved in this estimation procedure can be greatly reduced by providing tables of the

matrices |Js^ v

{|

. These are inverses of finite segments of

the Hilbert matrix and were tabulated for segments up to and

including order 10. These tables [ 5 ] should be used whenever

a polynomial of degree not exceeding 10 is the mean value

function. The scope of these tables will be sufficient in

most cases. It should be remarked that with the use of these
-9-

tables the estimation of the mean value function is by no means

more laborious than the computation of a least square fit to

the same data, One could even say that the estimation of the

mean value function is in some respects more convenient than

a least square fit. This is due to the fact that the table to

be used for estimating the mean value curve depends only on

the degree of the polynomials but is independent of the

number of points observed in [0 ? T], The tables for fitting oR^io^onouu

polynomials by least squares depend on the contrary on the degree of the polynomials as well as on the number of point s,

4, Construction of an example 6 We next construct an example to which we will apply

the technique discussed in the preceding section. We use random numbers to construct ’•data" which simulate observations

from a Wiener process with mean value function.

We choose the following polynomial of degree four

(4.1 ) f(t) = 3400 + 31 Ot - 2,7t 2 + 43 10"^ 0 t--2,6x10“

L

'

t

H

This polynomial will represent the mean value function of our

fictitious process. We assume that T = 100 and that we take

observations at all integer values 0 jC n <_ 100, We compute

therefore the values of f(n) ? n = 0(1)100, In order to

obtain the simulated observations we odd random numbers as "errors" to the values f(n). These random numbers were

obtained from H, Wold's table [6], These tables contain random
- 10-

norraal deviates representing a normal population of zero mean

and unit variance. They are arranged in colums of 50 and

the sums X] (x) of each column is also given. We still must

select the variance constant c. In view of the arrangement of

Wold’s tables it was decided to choose

(4.2) c = .

Random numbers from a normal population with zero mean and

were

variance c = y-y^obtained by dividing the values (x) in

Wold’s table by n/ 3« A set of 100 random numbers (denoted

in the following by w^) was derived in this manner by taking

100 columns in Wold’s table. We started with the first column

and used all consecutive columns with the exception of the

random numbers on page 7» These were skipped in agreement

n with the warning given in the tables. The sum ~> I w.

i=1

1

(n = 0,1,2,o,o,100) with w

q = 0 were computed and added as

’’errors” to the ’’true values”. In this manner

n

(*+.3) g(a) = f(a) + w

i (i = 0,1,, ,,,100)

i=0

was computed. These values of g(«) are the simulated observations, We will apply in the following our estimation procedure to these simulated observations. Since we wish to

use a Wiener process as our model we have to adjust the observations to make the initial value equal to zero. This can
=11 -=

be accomplished by subtracting g(0) from all observations,,

This leaves the increments unchanged and does therefore not

affect the character of the process „ We denote by

y(n) = g(n) - g(0)„

Table 1 gives for each n the values of f(n)

and y (n)

o

w

i? g(n)

i=1 1
-1 2 n f (n) n £

i=0 0 3400.00 0 1 3707.34 +6 2 4009.74 + 7 3 4306.84 + 2 4 4799.49 -0 7 4887.71 "3 6 7171 .77 -1 

7 7471 .82 -14 8 7728.17 -11 9 6000 . 94 -13

10 6270.40 “13 1 

6736.73 -12

12 6800.11 -12

13 7060.77 “13

14 7318.80 - 4

17 7774.46 - 7

16 7827.89 - 6

17 8079.24 - 2

18 8328.68 - 3

19 8776.37 - 0

20 8822.40 - 0

21 9066.96 + 7

22 9310.16 + 9

23 9772.12 + 8

24 9792.97 + 4

27 10032.81 + 7

26 10271 .77 + 6

27 10709.89 + 6

28 10747.33 + 13

29 10984.13 + 1 8

30 11220.40 + 28

31 11456.19 +27

32 11691 .59 + 22

33 11926.65 + 22

34 12161 .42 + 29

37 12395.96 + 30

36 12630.31 +33

37 12864.50 +30

38 13098.56 +31

39 13332.52 + 22

4o 1 3566.40 + 19,

g(n) y(n)

3400.00 0.00

3713o39 31 3»39

4oi6 0 76 6l 6. 76

4309.40 909.40

4599.08 1 1 99o08

4883 . 97 1 483* 97

51 60.49 1760.49

5437.23 2037 o 23

5716.49 2316.49

5987.23 2587.23

6256.70 2856.70

6523.92 3123.92

6787<.85 3387o85

7047.53 3647.53

7314.71 3914.71

7568.79 4168.79

7821.80 4421 .80

8076.61 4676.61

8325.48 4925.48

8576.03 5176.03

8822.19 5422.19

9074.89 5674.89

9319.37 5919.37

9760.81 6160.81

9797.31 6397.31

10038.32 6638.32

10278.08 6878.08

10716.33 7116.33

10761.22 7361.22

11002.20 7602.20

11248.66 7848.66

11483.47 8083.47

11714.13 8314.13

11949.77 8549.77

12190.84 8790.84

12426.60 9026.60

12663.93 9263.93

12894.93 9494.93

13130.09 9730.09

13374.80 9974.80

13787.68 10185.68

w 

1

00

05

02

56

.4i

74

26

59

66

71

70

.81

.26

,22

.09

67

09

63

,20

32

,21

.93

.19

.69

34

51

33

44

.89

07

26

28

54

92

42

64

62

43

73

28

28
13'

n f(n) 23-4 w s i=0 J

g(n)

40 13566 0 4-0

41 1 3800 o 20

4-2 1 4033.94

4-3 1 426y 0 6l

44 1 4501 0 21

45 1 4734 o 71

46 l4968o11

47 1 5201 o 37

48 1 5434 o 48

49 15667o38

1 9o28 1 3535.68

24,97 13825.17

24,01 15-057.95

34,91 15302,52

39o 54 15540.75

37o63 15772.35

39o 20 15007.31

38,24 15239.61

41 o34 15575.82

43,4o 1 5710.77

50 1 5900,00 4l 0 94

51 16132.34 39.91

52 16364,32 46 , 44

53 l6595o39 54,57

54 16826,96 51 .10

55 17057o46 44 0 44

56 17287.32 47.63

57 17516,44 48,87

58 17744,73 5i .55

59 17972.07 55.07

1 5941 o 94

161 72.25

1 64 1 0 o 76

1 6650 o 46

168780O6

1 7101 o90

1 7334 o 95

17565 o31

17796o 28

1 8027d4

60 1 8198,40 55.15

61 18423.56 56.79

62 1 8647,46 58,08

63 1 8869.95 64,22

64 19090,92 65,90

65 19310,21 64, 83

66 19527.70 65.97

67 19743.22 65.96

68 19956,62 65.29

69 20167.74 67.54

18253.55

1 8480 o 35

1 870 5 o 54

1 8934 0 1

7

1 91 56 o 81

1 9375.04

19593o67

19809ol8

20021 o9l

20235o28

70 20376 o 40

71 20582 o 44

72 20 785 066

73 20985 089

74 21 1 82o 92

75 21 3760 56

76 21 566 06O

77 21752o83

78 21935.02

79 22112o96

74.56 20450.96

78.49 20660.93

78.65 20864.31

79.59 21 065.48

81 .32 21 264 , 24

83.12 21459.68

84.1 8 21 650 0 78

84.83 21837.66

81 .67 22016.69

86.44 22199.40

22286 0 4o 82 0 1 6 223680 56

y(n)

1 01 85o68

1 O425o 1

7

10-757.95

10902o52

1 1 l4o 0 75

11372,34

1l607o31

1 1 839 061

1 2075o 82

12310.77

12541 .94

12772,25

13010,76

13250,46

134780 06

13701 ,90

13934.95

l4i65o31

143960 28

l4627ol4

l4853o55

1 5080,35

I5305o54

15534.17

1 5756,81

I5975o04

16198067

1 6409o 1

8

16621 ,91

l6835o28

1 7050.96

17260,93

1 7464,31

1 7665448

1 7864,24

1 8059 068

1 8250.78

1 8437,66

1 861 6,69

1 8799

»

4o

80 1 8968 , 56
-14- n

n f (n) 23 -i

i=0

g(n) y(n)

80 22286 .40 82.16 22368.56 18968.56

81 22455.12 81.37 22536.49 19136.49

82 2261 8086 80.44 22699.30 19299.30

83 22777<>38 84.07 22861 .45 19461 .45

84 22930.42 82.34 2301 2.76 19612.76

85 23077.71 82.09 13159.80 19759.80

86 23219.00 78.16 23297.16 19897.16

87 23353.99 73.10 23427.09 20027.09

88 23482.42 74.72 23557.14 201 57. 1 

89 23603o98 75.86 23679.84 20279.84

90 23718.40 84.38 23802.78 20402. 78

91 23825.36 86.21 23911 .57 20511 .57

92 23924.57 86.97 24011 .54 20611 .54

93 2401 5.70 91 .86 24107.56 10707.56

94 24098.44 90.60 24 1 89.04 20789.04

95 241 72.46 81.73 24254.19 20854.19

96 24237.44 78,38 2431 5.82 20915.82

97 24293.03 80.21 24373.24 20973.24

98 24338.88 80.01 2441 8.89 21018.89

99 24374.66 83.05 24457.71 21057.71

100 24400.00 79.30 24479.30 21079.30
-1 5-

5. Computation of the estimated mean value curve and of

the variance constant

We next compute the estimate of the mean value function

using formulae (3.9), (3.10) and (3.11) in order to obtain

the integral (3.10) we have to compute the integrals

i—

2

y(t) t dt for i = These integrals were evaluated by means of Simpson's rule

Xo+hh - hi (5.D y dx = 3Cy0

+1+

(yi

+ ys + *^+ yn _ 1 ) + 2(y2

-fyl++oo 0 +yn __ 2

)

+ yJ ] n

J

where a is an even number*

The results were compared in all cases to the approximation by summation and for i = 5 also to the use of a seven

point Lagrangian interpolation formula [1]* It was found

that results obtained by the seven-point formula and Simpson's rule differ only little while straight summation

would be too crude an approximation to the integral. As

an example, we mention that

f y(t)t^dt = ^6 l

, 1 57 , *+ 55,032

by Simpson's rule, while *+6l , 1 60, 1*+1 , 829 was obtained from

the 7-point formula and *+71 , 752 , 1 77,029 by straight surama- 7 i-2 tion* The integrals l y(t)t dt were therefore computed

o by (5.1), the values are given in table 2.
-16-

1

2*

5

6

Table 2

T

{ i-2

j y ( t) t dt

o

1 ,209,256.51

78,807,137.03

5,825,790,539.5^

461 ,157,^55,032.19

38,111 ,464,889,820.59

2—1

We next compute from table 2 the values a^ - iy(T)T

— i ( i— 1 )

j y(t)t*~2dt for i = 2,. ..,6 while = y(T) = }'-|qq.

'0 These are given in table 3*

Table 3

12

\

5

6

a.

1

2.10793 x 10^

1 .79735 x 10§

1.59536 x 10 °

1 .44077 x 10 10

1 .31650 x 1012

1 . 21 4 1 4 x 10 11+ A

We finally compute form (3*10) the estimates k. of the

J coefficients of the mean value function and obtain

"k . =

J i *

v=1

JV for j - 1,...,s. The elements

of the inverse matrix were taken from the tables [ 5 ] menA

tioned above. The coefficients kj were computed for mean

value function of various degrees, using always the same

’•data 1 * to be fitted. The results of these computations
-17”

are given in table 4 0

Table 4

A

Coefficients k , for estimating mean value curves of degree 9

J 9=3

1 257o 267

2 46 „ 962x10 2

3 ~9 o 34x1 0“3

4

5

6

9=4

308„ 773

-2 o 621

1

4.21 66x 1 O

-2

-2„5753xiO"4

S=5

307. *+95

-2o>+932

3.8332x10

-2

- 2 . 1 28x 1 0"4

-1 .79x1 O'7

5=6

300 0 785

-1 0 4868

-8 e 638x1

0~3

70266x10“^

-3 9 63352x10

"6

2 o 8l 82x1

0” 8

Using these coefficients the values f (t) of the estimated

mean value curve of degree s were computed for t=1 and

t=10(10)100 and s = 3 ? 4 9 596 0 In addition the mean value esA

timate f^(t) of degree 4 was computed for t=0(1)100 and a least square fit to the simulated observations was also determined 0 A comparison of these curves given in table 5?

while table 6 permits to compare the 4-th degree estimate with

A

the 4-th degree least squares fit„ We denote by f^(t) the

4-th degree mean value estimate and by f^(t) the 4-th degree

least square fit 0 If we write again y(t) for the (adjusted)

simulated observations then we see from table 6 that

[f^(t)-y(t)] 2 = 3351 0 71

100

t=0

[ f^( t)-y(t) ] = 3l66 0 90

while
- 18-

Table 5

Comparison between the coordinates, the observations and

the true values

"true values" "Simulated Least squa

t f (t)-f(0) observations"

y(t)

1 307o3^ 31 3o39 303,52

10 2870.40 2856,70 2865,12

20 5422.40 5422,19 5423,79

30 7820.40 7848,66 7834,18

40 10166.40 10185,68 10195,57

50 12500.00 12541 ,94 12545-23

60 1 4798.40 14853.55 14858,43

70 1 6976,40 17050,96 1 7048,42

80 1 8886,40 18968,56 1 8966,44

90 30318,40 20402,78 20401 ,75

100 21000,00 21079,30 21081 ,58

Estimates of mean value curve of degree

t 3 4 5 6

1 257,73 306,19. 305.04 299.29

10 2610,26 2865,21 2861 .83 2991 .90

20 5258,32 5423,14 5424.66 5442.31

30 7888,16 7834.08 7839.22 7851 .25

40 10443,74 10196.58 10200.83 10191 .15

50 12869 .OO 12547.08 12547.31 12526.12

60 1 5107,92 14860.69 14856.81 14846.98

70 1 7104,44 17050.36 17045.67 17057.57

80 1 8802.54 18967.36 18966,27 18983.58

90 20146,18 20401 .13 20404 . 87 20399.69

100 21082.70 21079.30 21079.50 21079.30
- ^9 - t y(t)

Table 6 A C t ) 0 0 0 1 313.39 306 . 1 9 2 61 6 . 56 607.39 3 909.40 903.85 4 1199.08 1195.79 5 1483.97 1483.45 6 1 760.49 1767.05 7 2037.23 2046.82 8 2316.49 2322.96 9 2587.23 2595.70

10 2856.70 2865.21 1 1

3123.92 3131 .70

12 3387.85 3395.36

13 3647.53 3656.37

14 3914.71 3914.90

15 4i 68.79 4l 71 .1 2 1 6 4421 . 80 4425.20

17 4676 .61 4677.30 1 8 4925.48 4927.56

19 5176.03 5176.13

20 5422.19 5423.14

21 5674.89 5668.74

22 5919.35 5913.05

23 6160.81 6156.18

24 6397.31 6398.27

25 6638.32 6639.40

26 6878.08 6879.67

27 7116.33 7119.20

28 7361 .22 7358.06

29 7602.20 7596.33

30 7848.66 7834.08

31 8083.47 8071 .42

32 8314.13 8308.42

33 8549.57 8545.08

34 8790 . 84 8781 .48

35 9026.60 901 7.66

36 9263.93 9253.68

37 9494.93 9489.55

38 9730.09 9725.32

39 9954.80 9960.99

4o 101 85.68 10196.58

Vt}

-3.11

303.52

605.13

901 .95

1194.23

1482.21 1 766 o 09

2046 o 1 2

2322.50

2595.43

2865.12

3131 .77

3395.57

3656.69

3915.32

4i 71 .62

4425.76

4677.90

4928o19

51 76.77

5423.79

5669.38

5913.66

61 56 . 76

6398.79

6639.86

6880.08

7119.53

7358.31

7596.50

7834.18

8071 .43

8308.30

8544.85

8781 .14

9017.22

9253.12

9488.88

9724.53

9960.09

10195.57
-20- t y(t) VO VO

40 10185.68 10196.58 10195.57

4l 10425.17 10432.11 10430.99

42 10657.95 10667.58 1 0666.34

43 10902.52 10902.98 10901 .63

44 1 1 l40o75 11138.30 11136.84

45 11372.34 11373.52 11371 .96

1+6 11607.31 1 1608. 64 11606.97

47 11839.61 11843.60 1 1 84 1 . 84

48 12075.82 12078.39 12076.54

49 12310.77 12312.97 12311 .02

50 12541 .94 12547.09 12545.23 5 

1 2772.25 1 2781 .06 12779.14

52 13010.76 13014.66 13012.67

53 13250.46 13247.81 13245.77

54 13478.06 13480.45 13478.36

55 13701 .90 13712.50 13710.36

56 13934.95 13943.86 13941 .69

57 l4i 65.31 14174.47 l4i 72.26

58 14396.28 14404.21 14401 .98

59 14627. 14 14632.98 14630.74

60 14853.55 14860.69 14858.43

61 15080.35 1 5087.20 1 50 84. '94

62 1 5305.54 15312.41 15310.20

63 15534.17 1 5536.17 15533.93

64 1 5756.81 15758.37 15756.15

65 15975.04 1 5978.86 15976.67

66 16193.67 16197.50 16195.34

67 1 6409.1 8 1 6414.13 1 6412.01

68 16621 .91 16628.59 1 6626. 52

69 16835.28 16840.73 16838.72

70 17050.96 17050.36 1 7048.42

71 17260.93 17257.32 17255.45

72 17464,31 1 7461 .42 17459.63

73 17665.48 17662.48 17660.77

74 1 7864.24 1 7860 . 29 1 7858.68

75 18059.68 1 8054.66 1 8053.14

76 18250.78 18245.38 18243.97

77 18437.66 18432.23 18430.93

78 1 861 6.69 1 8614.99 18613.82

79 18799.40 18793.45 1 8792.40

80 18968.56 18967.36 1 8966.44
-21

t y(t) *4<0 -d" 80 1 89680 56 18967.36 1 8966.44

81 19136. 49 19136.50 19135.71

82 1 9299 0 30 19300 o 60 19299.96

83 1 9^-61 .45 19459.43 19458.93

84 1 961 2 o 76 1 961 2o73 19612.38

85 19759.80 19760.23 19760.03

86 1 9897o 1

6

19901 .66 19901 .62

87 20027.09 20036.75 20036.93

88 201 57.14 20165.22 201 65. 5l

89 20279.84 20286.78 20287.23

90 20402 0 78 20401 .13 20401 .75

91 20511 .57 20507.98 20508.77

92 206l 1 0 54 20607.02 20607.98

93 20707.56 20697.93 20699.06

94 20789.04 20780.40 20781 .70

95 20854 0 1

9

20854.10 20855.57

96 20915.82 20918.71 20920.34

97 20973 0 24 20973.88 20975.68

98 210l8o89 21019.26 21021 .23

99 21057.71 21054.52 21056.65

100 21079.30 21079.30 21081 .58

We still have to estimate the variance constant. This

A

is done for the estimate f^(t) by means of (3.6) and the

data contained in table 6. We obtain the estimate c - l5o05l

as compared with the ?l troe value '

?l c = 50/3.

We also give four figures which permit to visualize

the relative errors of our estimates 0 Figures 1 and 2 compare the estimating polynomials of degrees 3 9 ^ 9 5 and 6.

In figure 1 the graphs of

A f^Ct) - f(t)

and of (t)

74

(t) - f(t)

TMfTt)

are given 9 in figure 2 we find the corresponding errors
=22=

referred to the

f^.(t) - y(t)

observations 9 i 0 e 0 the curves

<v . . %>) - yd)

and r^Ct) = —

A

vh(t) =

In figures 3 and b we corapa

polynomials of degress 3 9 h-

? 5 a

is in figure 3 the curve £(t)

re the error of the estimating

nd 6 9 the ba sis of reference

in figure 4 the curve y(t) 0

6 0 Conclusion ,, lo actual applications we will often encounter situations where more than a single sample curve of

the process are given Q In these cases a number of problems

can a rise „ It might for instance be necessary to test the

hypothesis whether two or more sample curves come from

processes with the same variance constant or with the same

mean value function,. Due to the fact that increments from

a Wiener process behave as if they were independent observations from a normal population many of the questions

which might arise can be treated by standard statistical

techniques o To go into such a discussion would however exceed the scope of this paper 0 The author wishes to acknowledge his thanks to Mr 0 Edwin

Lo Grab who carried out the necessary computations 0
-23-

References

[1] Go Blanchj I n Rhodes 9 Seven-point Lagrangian integration formulas o Jour, Math 0 Phys 0 22 5 204-207

(1943)o

[2] Ho Bo Marm ? The estimation of parameters in certain

stochastic processes 0 Sankhya 11, 97-106

( 1951 ).

[ 3 ] Ho Bo Mann 9 On the estimation of parameter determining

the mean value function of a stochastic process

„

To be published in Sankhya,

[ 4 ] Statistical Techniques Applicable to the analysis of a fundamental random process. Technical Note

WoC oR 0 R=52”7o Flyhdo Res 0 Lab 0

3

Wright Air

Dev, Center 9 Dayton 9 Ohio 9 1952,

[ 5 ] L Ro Savage ? E 0 Lukacs 9 Tables of inverses of finite

segments of the Hilbert Matrix, To be published in the NBS Applied Math, Series,

[6] Ho Wold, Random Normal Deviates ^Tracts for Computers

XXV, Cambridge University Press 9 1948,

FIG. 1

\(t) and 100 Vt}

FIG* 2

100 r., (t) and 100 r. (t)

4 4

0 10 20 90 100

FIG„ 3

100 E.(t)

3

FIG. h

100 r (t)

5

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