the PI k ,,'s enter into the computation (weighted andi unweighted) of target ~k wil be treated as two cases and discussed separately. 4Thissdhmac wiov~dad by ...
lAD
TECHNICAL REPORT ARBRL-TR-02570
CONSTRUCTION OF APPROXIMATE CONFIDENCE INTERVALS FOR PROBABILITY-OF-KILL VIA THE BOOTSTRAP CD Malcolm S. Taylor Barry A. Bodt
JJuly 1984
LEVE US ARMY ARMAMENT RESEARCH AND
OPMENT CENTER BALLISTIC RESEARCH LABORATORY
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CONSTRUCTION OF APPROXIMATE CONFIDENCE INTERVALS FOR PROBABILITY-OF-KILL VIA THE BOOTSTRAP______________ 7. AUTHOR(s)
6.
PERFORMING ORG. REPORT NUMBER
8.
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Malcolm S. Taylor
Barry A. Bodt PROGRAM ELEMENT. PROJECT. TASK
10.
9D. PERFORMING ORGANIZATION NAME AND ADDRESS
US Arm)' Ballistic Research LaboratoryARAWOKUINMER ATTN: DRXBR-SECAD RDT&E 1L,161102AH43 Aberdeen Proving Ground, MID 21005-5066 REPORT DATE
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Bootstrap Confidence Interval Nonparametric statistics 2&. ADST, ACT (Coatniue so .vroots
of
H neceosse anzd ldifatif7 by block number),
rhe bootstrap, a computer-intensive pro' edure for data analysis, was'applied to an estimation problem to enable a stateme t4 0 be made about the variability inherent in a probability-fkl esiate ~.The bootstrap was applid ta stratified sample rather than a simple randof~ sample and 'its performance oivauated in this framework, first in an abstract situation whe're the parameter- Pk -, was known and then in three situations where the parameter (P was unknown,"b-utan estimate provided by current vulnerability analysis proc'ures was available. (Cont'd)A JAI 73
1473
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NO11V 65 IS OBSOLETE
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Item 20, Continued.
9iihe investigation was carried out for several grid sizes, or alternatively,. for seve 1 levels of detail, to study the effect of grid size on the estimation of Pk•and the reliability of the confidence intervals constructed.
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TABLE OF CONTENTS Page
LIST OF TABLES ...................
5
.......................
LIST OF ILLUSTRATIONS ..............
7
..........................
INTRODUCTION............................
2.
ESTIMATION OF VULNERABLE AREA AND PROBABILITY OF KILL ...........
3.
THE PERCENTILE METHOD FOR CONFIDENCE INTERVAL CONSTRUCTION .
4.
APPLICATIONS ..............................................
9
1.
5.
11 .
......................
4.1.
Surrogate Target ............
4.2.
Armored Personnel Carrier .......
4.3.
Tank ...............
..................
... 21 ................
Unweighted Procedure ......
4.3.2.
Weighted Procedure ............................
DISTRIBUTION LIST ...........
... 23
......................
. ......
13
... 19
4.3.1.
ACKNOWLEDGEMENT .............
13
... 13
............................
SU1MARY AND CONCLUSIONS .........
..
24 ... 28
................
.........................
..
28
...
29
For
--
Accessinn is IS A&I DTI N,
T•B
justificatiozi
Distr-lbutionl!
Availability Codes lAvail and/or Dist
II 3
kL
C*Sp-cial
I
LIST OF ILLUSTRATIONS Figure
Page 12
...................
1. Surrogate target description ....... ... 2.
Flow chart for the bootstrap computation .................
..
3.
One hundred bootstrapped confidence intervals .... ..........
...
16
4.
Data summary for the surrogate target .....
...
18
5.
Armored personnel carrier target description ...............
6.
Data summary for an armored personnel carrier ....
7.
Tank target description ..........
8.
Data summary for the tank (uniform case) .....
.............
9.
Data summary for the tank (Normal case, a = 1')
.........
......
25
10.
Data stummary for the tank (Normal case, a = 3')
.........
......
26
11.
Data summary for the tank (Normal case, a
.........
27
N:5
CA
..............
... 20 ..........
.....................
=
... 21 ...
91) .....
S15
22
... 24
LIST OF TABLES Page
Table 1.
Number of Confidence Intervals Covering the Parameter ......................... Surrogate Target .......... Mean Width of Confidence Intervals. M.
3.
4. S.
...
16 7.......1
Surrogate Target
Number of Confidence Intervals Covering the Parameter .................... Armored Personnel Carrier ...... ...
...
19
Armored Personnel Carrier. Mean Width of Confidence Intervals. Number of Confidence Intervals Covering the Parameter Unweighted Procedure ........
19
23
....................... Unweighted Procedure .
.
23
6.
Mean Width of Confidence Intervals.
7.
Number of Confidence Intervals Covering the Parameter ... .................... Weighted Procedure, a = 1 ........
...
25
Mean Width of Confidence Intervals Weighted Procedure, a = 1 ..........
...
25
Number of Confidence Intervals Covering the Parameter Weighted Procedure, a = 3 ...... ..... ....................
...
26
Mean Width of Confidence Intervals Weighted Procedure, a = 3 ...... ..... ....................
...
26
Number of Confidence Intervals Covering the Parameter ... .................... Weighted Procedure, a = 9 ......
...
27
Mean Width of Confidence Intervals Weighted Procedure, a = 9 ...... ..... ....................
...
27
8.
9.
10.
11.
12.
.
....................
7
.
PREVIOUS PAG
ISBLN
V
1. INTRODUCTION The bootstrap, so named by Efron to convey its self-help attributes, is a conceptually simple technique. It is one of a number of procedures known as resampling plans whose goal is to extract information from a set of data through repeated inspection. Although the theoretical foundation is incomplete and many of the available results are empirical, this is an area of active research that will grow in importance as computation becomes faster and cheaper. The bootstrap attempts to address an important statistical problem: having computed an estimate of some parameter, say a mean or a probability or a correlation, what accuracy can be attached to the estimate? Accuracy refers to the "-±- something" that often accompanies statistical estimates and is commonly conveyed tirough such devices as variance, standard error, confidence interval, etc.. n
Before detailing the mechanics of the bootstrap procedure, it is instructive to study the following simple example. Consider a data set consisting of a random sample of size n from an unknown probability _oI
distribution
F
X 1 =Xl, X 2 --'x
on 2 ,...,
the
real
line,
X 1, X
2
- F.
,
Xn =Xn, the sample mean.x"' n
Having
observed
x, is computed for use as
an estimate of the distribution mean 14. A crucial fact is that the sample values X 1 , X2
,...,
Xn contain more information
than the estimate Z. They also provide an estimate for the accuracy of ., namely
the standard error of Z Unfortunately, equation (1) has no obvious extension to estimators other than Z The bootstrap is one method of making this extension. Let P be the empirical distribution of the data with probability mass I assigned to each
n
xi and letXl*,X 2 ,... Xn• be a random sample fromF. •,nother words, eachX i is drawn independently with replacement from the set (Xl,X 2 , ... ,Xn). The bootstrap estimate of standard error for an estimator 8(Y 1 , X 2 , ... , Xn) is
I
B = (Var [
X 2 ,.
., X,,()]}
2)
R Fron,Toostwrap method" anotherlook at the Jackknife,"Ann. Siatist., 7(1979), pjx 1-26.
9
PREVIOUS PAGE GLANK___j& ~~~IS
error. An application of the bootstrap procedure to confidence interval construction xill be the
sub'rct of Sec-ion 3 of this report.
2. ESTIMATION OF VULNERABLE AREA AND PROBABILITY OF KILL Consider the conceptual item of military hardware illustrated in Figure 1. Without loss of generality, the component edges are assumed to be aligned with the superimposed 30x30 rectangular grid. The color in the ith cell represents the probability Pk1h, that the target will be killed should a prescribed round of ammunition fired from a fixed range and orientation impact within that cell. The vulnerable area A . for the target is calculated by Av •
.Pk h, A i - A i
_ Pk 1h,
(S)
i
where i is indexed over the cells of the grid. In this example, the cell areas are identical, i.e., A1 A2 ... A. The overall probability of kill Pk is
Pk
A
(6)
where A p is the total presented area of the target. For the conceptual target shown in Figure 1, the exact values of vulnerable area and probability of kill are A Pk == 0.21.
=-- 188.1 and
In practice, the values Pk 1h, are unknown, aid estimates Pk h, are required to obtain the approximations
Pk -
A
(8)
The method by which the values Pk 1h: are produced will not be considered here. Determining valid Pklh, estimates is a persistent problem because of the diffrulty in modeling the underlying damage mechanisms and constitutes an additional source of uncertainty. The procedure used here assumes the individual kjh, 's are accurately represented and doe' not attempt to assign a component of variation due to this factor.
i
11
0
o.1 o.2 0.3
o. 7
o.9
-tt-
I.itr
.S roae ag td s rp in
3. THE PERCENTILE METHOD FOR CONFIDENCE INTERVAL CONSTRUCTION The bootstrap procedure was applied to construct an approximate confidence interval fur the parameter Pk. A percentile method 2 was used which allows an approximate confidence interval to be assigned to any real-valued parameter 0 distribution of --- (F).
O(F) based on the bootstrap
Let
~)
:
•Pr, (0^ < t }(9)
be the probability distribution of 0
.
0,low(a) =
For 0 < a < 0.5 , defirne
(a), 0p(a)
(
(10)
An approximate 1 -2a central confidence interval for 0 may then be chosen as
which is the central I -- 2a portion of the distribution of bootstrapped 0". Under Monte Carlo sampling,
NP
