Interval Estimates for Probabilities of Non-Perforation 

Derived From a Generalized Pivotal Quantity 

by David W. Webb 

ARL-TR-3596 September 2005 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3596 September 2005 

Interval Estimates for Probabilities of Non-Perforation 

Derived From a Generalized Pivotal Quantity 

by David W. Webb 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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4. TITLE AND SUBTITLE 

 Interval Estimates for Probabilities of Non-Perforation Derived From a 

 Generalized Pivotal Quantity

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6. AUTHOR(S) 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

 

A generalized pivotal quantity is developed that yields exact confidence intervals for the cumulative distribution function 

(CDF) at a specific value when the underlying distribution is assumed to be normal. This problem is similar to the 

development of a tolerance interval, and unsurprisingly, its solution involves the non-central “t” distribution. Confidence 

bands for a normal CDF follow easily. Military applications include vulnerability and lethality assessment (e.g., interval 

estimation for the probability of non-perforation against homogeneous armored targets).

15. SUBJECT TERMS

 confidence interval; confidence limit; generalized pivotal quantity; non-central t distribution; perforation

16. SECURITY CLASSIFICATION OF:

19a. NAME OF RESPONSIBLE PERSON

 David W. Webb

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iii

Contents 

List of Figures iv

List of Tables iv

Acknowledgments v

1. Introduction 1

2. General Solution 2

3. Lower Confidence Bound (LCB) 6

4. Upper Confidence Bound (UCB) 7

5. Two-Sided Confidence Interval 8

6. Iterative Solutions for the Confidence Bounds and Confidence Limits 8

7. Confidence Bands 10

8. An Application 10

9. Summary 12

10. References 13

Appendix A. Proof of the Monotonicity of the Function H 15

Appendix B. MATLAB Code for Probability of Non-perforation LCB 17

Appendix C. Mathematica Code for Probability of Non-perforation LCB 19

Appendix D. MATLAB Code for Probability of Non-perforation UCB 21

Appendix E. Mathematica Code for Probability of Non-perforation UCB 23
iv

Appendix F. MATLAB Code for Probability of Perforation Two-Sided Confidence 

 Interval 25

Appendix G. Mathematica Code for Probability of Perforation Two-Sided Confidence 

 Interval 27

Appendix H. MATLAB Code for Probability of Non-perforation CDF and Lower 

 Confidence Band 29

List of Symbols Used 31 

Distribution List 33 

List of Figures 

Figure 1. Two non-central t density functions, 1f and 2f , with respective non-centrality 

parameters λ1 and λ2 .............................................................................................................. 9 

Figure 2. Estimated probability of non-perforation and 95% lower confidence boundary 

as a function of armor thickness ........................................................................................... 11 

List of Tables 

Table 1. Penetration depths into armor (no units specified). ....................................................... 10 
v

Acknowledgments 

The author wishes to thank the following people: 

 Messrs. Thomas Havel and Michael Zoltoski of the U.S. Army Research Laboratory 

(ARL), who brought this problem to my attention; 

 Professor Thomas Mathew, University of Maryland, Baltimore County, for pointing out 

Weerahandi’s generalized pivotal quantity for 

σ

μ ; 

 Major William H. Kaczynski and Commandant Michael A. Faries, United States Military 

Academy, who are responsible for the Mathematica code that appears in appendix B; and 

 Dr. Gene Cooper, ARL, for his many hours spent fine tuning the Mathematica programs 

and his review of the original manuscript. 
vi

INTENTIONALLY LEFT BLANK
1

1. Introduction 

The estimation of non-perforation probabilities is of significant interest to the developers of 

armor systems, whose objective is to provide protection for vehicles and personnel against 

enemy threats. In particular, the following questions may arise: 

 1. What is the probability that homogenous armor of a given thickness will not be 

perforated by a specific enemy threat? 

 2. How thick should the armor be if one is to be highly confident that the probability of 

non-perforation with this threat is high? 

Conversely, these probabilities are of interest to the developers of armor-piercing projectiles, 

whose objective is to perforate enemy armor. From this point of view, one might ask 

 3. What is the probability that a projectile will be able to perforate an enemy armor of a 

given thickness? 

For a particular projectile, non-perforation1 occurs if its penetration depth, denoted by x, is less 

than the thickness of the armor, denoted by 0 x . Now consider the random variable X, which 

represents the penetration depth of a randomly selected projectile. The probability of nonperforation is an unknown, fixed constant within the interval [0, 1] and is written as ( ) 0 P X ≤ x . 

An estimate for the true non-perforation probability will be based on sample data and as such, is 

subject to sampling variation. However, a point estimate for the non-perforation probability is a 

single value and therefore does not give any sense of the uncertainty associated with it. 

Preferable to a point estimate is an interval estimate that contains the non-perforation probability 

with high probability, i.e., a confidence interval. The width of the interval provides a 

quantitative measure of the estimation error. 

The problem of estimating ( ) 0 P X ≤ x has its origins in acceptance sampling plans for statistical 

quality control in the 1950’s. Lieberman and Resnikoff (1955) and Barton (1961) yielded 

expressions for the minimum variance estimator. It was not until the 1970’s that confidence 

interval estimators surfaced. In their 1977 paper, Owen and Hua derived lower confidence limits 

for ( ) 0 P X ≥ x and described how to obtain one-sided and two-sided limits for ( ) 0 P X ≤ x . 

However, their derivation contains several errors in notation and as a result, is quite difficult to 

follow. Owen and Hua’s paper also provided tables that assist in the calculation of lower 

confidence limits for ( ) 0 P X ≥ x ; these tables were dramatically expanded in Odeh and Owen 

 1 For the remainder of this report, we only discuss non-perforation events, their probabilities of occurrence, and 

interval estimates for these probabilities. Analogous results for the probability of perforation, ( ) 0 P X > x , are easily 

derived if we use the axiomatic property ()() 0 1 0 P X > x = − P X ≤ x .
2

(1980, tables 7.1 through 7.7). Odeh and Owen’s tables are cited in the more recent engineering 

statistics text by Hahn and Meeker (1991, section 4.5). 

In this report, the recent concept of generalized inference is used to develop one- and two-sided 

confidence intervals for the probability of non-perforation when it can be assumed that the 

penetrations follow a normal distribution. Although this may be seen as an attempt to “reinvent 

the wheel,” it is meant to show the wide applicability of the generalized approach to confidence 

interval construction. As pointed out by Weerahandi (2004, section 1.7), generalized inference 

may be used to obtain confidence intervals for any function of the normal distribution 

parameters. 

Solutions for the limits of the intervals require iterative calculation of percentiles of non-central t 

distributions. As an alternative to the tables by Odeh and Owen and to avoid the interpolation 

errors inherent in their usage, software code is provided in the appendices so that the practitioner 

can easily obtain exact solutions for the interval limits. 

2. General Solution 

Suppose that n depths of penetration n x , x , , x 1 2 K are collected in an experiment for which the 

observations are assumed to come from a normal distribution with mean μ and standard deviation 

σ. Historically, this assumption has been found to be tenable when the projectiles are of the same 

ammunition type with (approximately) equal muzzle velocities, obliquities, etc., and the target is a 

homogeneous armor. For the purpose of this report, we assume that the armor is thick enough to 

guarantee that no projectiles will perforate. From this sample, an estimate is desired for the 

probability of non-perforation, i.e., the probability that a future penetration, X, from the same 

population of projectiles will be less than or equal to some specified depth, 0 x . This probability is 

written as ( ) 0 P X ≤ x and is known as the cumulative distribution function (CDF) evaluated at 0 x , 

or ( ) 0 F x X . Notice that 

( ) 



 



 −  = Φ



 



 −  = ≤



 



 − ≤ − ≤ = σ

μ

σ

μ

σ

μ

σ

μ 0 0 0

0

x x

P Z X xP X x P , (1) 

in which Z is a standard normal random variable and Φ( )⋅ is the standard normal cumulative 

distribution function (CDF) commonly tabulated in statistics texts and available in many 

software packages. Therefore, calculating a simple point estimate for ( ) 0 P X ≤ x is 

straightforward: one computes the standard normal CDF at an estimate of 

σ

x0 − μ . We obtain 

this estimate by substituting the sample mean ( ∑=

=

n

i

i x

n

x

1

1 ) and the sample standard deviation 
3

( ∑( ) =− − =

n

i

i x x

n

s

1

2

1

1 ) for the population mean ( μ ) and the population standard deviation (σ ), 

respectively2. That is, 

() ( ) . ˆ ˆ ˆ 0 0 0

0 0 



 



 −  = Φ



 



 −  ≈ ≤



 



 − = ≤ = ≤

s

x x

s

x x

P Z

x

FX x P X x P Z

σ

μ (2) 

However, any simple point estimate for ( ) 0 F x X does not give information about its precision. 

What we would prefer to construct is a confidence interval or perhaps a confidence bound for 

( ) 0 F x X that by its width gives some sense of the degree of uncertainty associated with it.3

We begin by recalling from equation 1 that ( ) 



 



 − = Φ

σ

0 μ

0

x

F x X . Since Φ( )⋅ is a strictly 

monotonic function, we can easily attain a confidence interval for 



 



 − Φ

σ

x0 μ by deriving a 

confidence interval for 

σ

x0 − μ and then applying the function Φ( )⋅ to the resulting confidence 

limits (see Mood, Graybill, and Boes, 1974, page 378). 

The following derivation of a confidence interval for 

σ

μ θ − = 0 x is based on a method by 

Weerahandi (1995) in which he uses a generalized pivotal quantity to obtain a confidence 

interval for the coefficient of variation, 

μ

σ , for a normal population with mean μ and standard 

deviation σ. The construction of a generalized pivotal quantity requires one to find a function R

with arguments 

1. X and 2 S (independent and sufficient statistics for the random sample X X n , , 1 K ), 

2. x and 2

s (the observed values of X and 2 S ), 

3. θ , the parameter of interest, and 

4. perhaps a vector, ξ , of additional unknown (nuisance) parameters. 

 2Although this estimate is easily obtained, it does not have the desirable property of being unbiased. An unbiased 

estimator with minimum variance is presented in either Lieberman and Resnikoff (1955) or Barton (1961). 

3Note the distinction between the problems of solving for a confidence interval for 



 



 − Φ

σ

x0 μ and solving for a 

tolerance interval. With the former, the percentile value ( 0 x ) is specified and an interval for its associated CDF 

value, 



 



 − Φ

σ

x0 μ , is desired. On the other hand, with tolerance interval construction, the CDF value (p) is 

specified while an interval for the associated percentile, μ + z pσ , is desired.
4

This function, denoted in full by ( , ; , , θ , ξ ) 2 2 R X S x s , must satisfy the following two conditions: 

Condition 1: ( , ; , ,θ ,ξ ) 2 2 R X S x s has a distribution that is free of any unknown parameters. 

Condition 2: The observed value of R, i.e., ( , ; , ,θ,ξ ) 2 2

r = R x s x s , is equal to θ . 

If such a function can be found, then it is a generalized pivotal quantity for θ , and its percentiles 

can be used to obtain confidence intervals for θ . Since their introduction in the 1980’s, 

generalized pivotal quantities have been regarded as quite challenging to derive. Even Weerahandi 

(1993) states “... the construction of pivotals requires some intuition.” In essence, one obtains the 

pivotal quantity by working backwards from the expression for θ . The mathematical “tricks” of 

adding 0 and/or multiplying by 1 are adroitly employed to link all unknown parameters in θ with 

functions of the sufficient statistics whose distributions are parameter free. Once the linking is 

complete, all remaining random variables are converted to their observed values. 

Step 1: The sufficient statistics for a normally distributed random sample are X and 2 S . 

Random variables based on these statistics whose distributions are free of unknown parameters 

include 

1. 

σ

− μ = X Y , distributed as a normal random variable with mean zero and variance 

n

1 ; and 

2. ( ) 2

2

1

σ

S V = n − , distributed as a chi-square random variable with n −1 degrees of freedom. 

Step 2: We attempt to construct a random variable from θ that involves Y and/or V. First, 

notice that by adding zero in the form of ( ) − X + X to the numerator, θ can be rewritten in the 

following manner: 

. 0 0 0

σ

μ

σ σ

μ

σ

μ θ − + − = − + − = − = x x X X x X X (3) 

Now the parameter μ is linked to the random variable 

σ

X − μ which was previously noted as 

having a parameter-free distribution. When we make the substitution 

σ

− μ = X Y , the parameter 

μ is removed from θ :

. 0 Y x X+ − = σθ (4) 
5

It remains to manipulate the left addend of equation 4 so that the unknown parameter σ is linked 

to a random variable whose distribution is free of unknown parameters. We can achieve this by 

multiplying the left addend by 1, in the form of ( )

( ) 2

2

1

1

n S

n S

−

− : 

( )

( )

( ) . 1

1 1

1

2

2

0

2

2

0 0 Y n S

S n

x X

Y

n S

x X n S Yx X

+ −

−

− + = −− − + = − = σ σ σ

θ (5) 

Now σ is linked to ( )

2

2 1

σ

n − S which was noted in step 1 as having a parameter-free distribution. 

After ( ) 2

2

1

σ

S V = n − is substituted in equation 5, all unknown parameters are removed from θ : 

( ) . 1

1

1

0

2

2

0 V Y

S n

x X

Y n S

S n

x X

+ −

− + = −

−

− = σ

θ (6) 

Step 3: At this point, we replace the remaining sufficient statistics in equation 6 with their 

observed values; that is, V Y

s n

x x

+ −

−

1

0 . 

Step 4: Finally, we rewrite the random variables V and Y in their original form (as functions of 

the sufficient statistics) to obtain the generalized pivotal quantity: 

( ) ( )

σ

μ

σ

μ σ − + −

−

− = n S X

s n

x x R X S x s 2

2

2 2 0 1

1 , ; , , , . (7) 

As was demonstrated in the development of R, its distribution is free of any unknown parameters. 

Thus, condition 1 is satisfied. Furthermore, the observed value of R is 

( ) , ; , , , , 2

2

2 2 0 θ

σ

μ

σ

μ σ = − + − = = s x

s

x x

r R x s x s

thus satisfying condition 2. Therefore, R is a generalized pivotal quantity whose percentiles can 

be used to obtain a confidence interval for θ . Using the normal cumulative distribution, Φ( )⋅ , 

we then ultimately obtain a desired confidence interval for the probability of non-perforation. 
6

3. Lower Confidence Bound (LCB) 

Since effective armor offers a high probability of non-perforation, engineers and management will 

most often be interested in a lower confidence bound, since this would represent a “pessimistic” 

bound on the true probability of non-perforation, 



 



 − Φ

σ

x0 μ . 

A ( ) 1−α 100% LCB for 

σ

μ θ − = 0 x is that value BL for which 

1−α = P( ) BL ≤ R (8) 

is satisfied.4 We can estimate a generalized LCB for θ by randomly generating a large number 

of observations of R and selecting a value for which equation 5 is empirically satisfied. For 

instance, with α = .05 , one could start by generating 100,000 observations of R, denoted by 

1 2 100000 r,r ,K,r . If the ordered observations are written as () ( ) ( ) 1 2 100000 r < r < K < r , then a 95% 

LCB for θ would be any real value BL so that () () 5000 5001 r B r < L < , such as ( )()

2

5000 5001 r r

BL

+ = . 

However, equation 8 can be expanded to get 

. 11 0

2

2

0

















+ −

− = ≤ 













 − + − − = ≤ Y

n

V

s

x x P B S X

s

x x P BL L

σ

μ

σ

α (9) 

Notice that the random variable Y in equation 9 is equal in probability to 

n

Z , in which Z is a 

standard normal random variable (having mean 0 and variance 1). By using this fact and 

rearranging the terms in the last probability statement, we have 

.

1

1

1

1

0

1,

0

0

0



















 − = ≥



















 − ≥

−

− =

















+ ≥ −

− =

















+ ≥ −

− − =

− −

n

s

x x P T

n

s

x x

n

V

Z nB P

V Z nB

n

n

s

x x P

B

n

Z

n

V

s

x x P

n nBL

L

L

α L

 (10) 

 4It is implicit that the upper bound equals 1 in the probability statement of equation 8, i.e., 1 - α = P(BL ≤ R ≤ 1). 
7

In equation 10, n nBLT −1,− is a non-central t random variable with n −1 degrees of freedom and 

non-centrality parameter − nBL (see for example, Casella and Berger, 1990). However, a noncentral t random variable with non-centrality parameter BL − n is the mirror image of a noncentral t random variable with non-centrality parameter nBL (Johnson and Kotz, 1970). 

Therefore, 



















 − = ≤



















 − − = ≥ − − −

n

s

x x P T

n

s

x x P Tn nBL n nBL

0

1,

0

1, 1 α . (11) 

The final probability expression of equation 11 is the CDF of a non-central t random variable 

with n −1 degrees of freedom and non-centrality parameter BL n , i.e., 

1 . 0

1,



















 − − = −

n

s

x x Gn nBL

α (12) 

A lower confidence bound for 

σ

μ θ − = 0 x , is BL , the solution to equation 12. When we exploit the 

strict monotonicity of the normal distribution function, a ( ) 1−α 100% LCB for the probability of 

non-perforation, 



 



 − Φ

σ

x0 μ , is ( ) Φ BL . 

4. Upper Confidence Bound (UCB) 

A UCB would represent an “optimistic” bound on the true probability of non-perforation. As such, 

it is of little practical value and rarely calculated. However, its derivation is briefly discussed here 

for completeness and as a precursor to the development of a two-sided confidence interval. 

Following a similar progression to that of the previous section, a ( ) 1−α 100% UCB for 

σ

μ θ − = 0 x is that value BU satisfying 

( ) 

















 − − = ≤ = ≥ −

n

s

x x P R B P Tn nBU U

0

1, 1 α . 

Therefore, 



















 − = −

n

s

x x Gn nBU

0

1, α , (13) 
8

and a ( ) 1−α 100% UCB for the probability of non-perforation, 



 



 − Φ

σ

x0 μ , is ( ) Φ BU . 

5. Two-Sided Confidence Interval 

A two-sided confidence interval might be of interest to the researcher simultaneously wanting 

lower and upper limits for the probability of non-perforation, neither of which is automatically 

set to their extreme value of 0 or 1, respectively. 

A ( ) 1−α 100% two-sided confidence interval for 

σ

μ θ − = 0 x is given by values CL and CU

satisfying 

( ) = P R ≤ CL 2

α and ( ) = P R ≥ CU 2α

Following steps similar to those used in determining one-sided confidence bounds, two-sided 

confidence limits for 

σ

μ θ − = 0 x are solutions to 



















 − − = −

n

s

x x Gn nCL

0

1, 2

1 α , (14) 

and 



















 − = −

n

s

x x Gn nCU

0

1, 2

α . (15) 

Thus, a ( ) 1−α 100% two-sided confidence interval for the probability of non-perforation, 





 



 − Φ

σ

x0 μ , is ( ) ()() Φ CL Φ CU , . 

6. Iterative Solutions for the Confidence Bounds and Confidence Limits 

As an example, consider solving equation 12 for the LCB of 

σ

μ θ − = 0 x . We start by rewriting 

the equation so that the right side equals 0: 
9

1 0 0

1, + − =



















 −

− α

n

s

x x Gn nBL

. (16) 

In equation 16, n, 0 x , x , s, and α are fixed constants, and the only unknown is BL . Therefore, 

if we let ( ) ; , , , , 1 0

0 1, + −



















 − = − α α

n

s

x x H B n x x s Gn nBL L , the problem becomes one of solving for 

the root of the function H. We can show H to be a monotonic decreasing function in BL by 

recognizing that the non-central t distribution function is monotonic decreasing in the noncentrality parameter (see figure 1) or by formal proof (see appendix A). Therefore, one can use 

the bisection method or other root-finding algorithm to solve for BL . Finally, the LCB for the 

probability of non-perforation is ( ) Φ BL . 

The bisection method can be easily programmed into most mathematical software packages. 

MATLAB5

 and Mathematica6 programs appear in appendices B through G for calculating an 

LCB, a UCB, and a two-sided confidence interval for 



 



 − Φ

σ

x0 μ . 

Figure 1. Two non-central t density functions, 1f and 2f , with 

respective non-centrality parameters λ1 and λ2 . (As the noncentrality parameter increases, the area to the right of 0 x

under the non-central t curve decreases.) 

 5

MATLAB is a registered trademark of The MathWorks.

6Mathematica is a registered trademark of Wolfram Research, Inc.

λ1 λ2

x0

f1 f2

λ1 λ2

x0

f1 f2
10

7. Confidence Bands 

By plotting the function 



 



 − Φ

s

x x for all real numbers x, one obtains a smooth, monotonically 

increasing function that is a simple estimate for the entire CDF. A lower confidence band for the 

CDF is a plot of the function ( ) B ( ) x Φ L . This band lies below the estimated CDF and enables one 

to see the region above the band in which the entire actual CDF lies (with the stated level of 

confidence). If one fixes the probability level on the vertical axis, then the corresponding x-axis 

value from the lower confidence band is an upper tolerance bound. In the context of armor 

design, the upper tolerance bound is very important as it states with a specified level of 

confidence what armor thickness is needed to stop a (high) percentage of projectiles from 

perforating the materiel. A MATLAB program for generating a lower confidence band appears in 

appendix H. 

Upper confidence bands and two-sided confidence bands are constructed in a similar fashion; 

however, they are likely to be of much less interest to the armor designer. Although MATLAB 

programs for their construction are not included here, the interested reader could easily tailor the 

program in appendix H to accomplish this. 

8. An Application 

Table 1 gives the penetrations of 14 projectiles into an extended armor pack. Engineers plan to use 

115 units as the armor thickness. To determine if this thickness will provide enough protection, an 

estimate is desired for the probability that the next projectile fired will not penetrate deeper than 

115 units. Letting X be the penetration of this next projectile, we seek an estimate for 

P( ) X ≤ 115 . 

Table 1. Penetration depths into armor (no units specified). 

47 81 89 99 114 

59 86 90 100 118 

80 88 90 113 

The data are first checked for normality by Lilliefors Test (see Conover, 1980). The value of the 

test statistic is 0.1713, which corresponds to a P-value greater than 0.20. Therefore, the 

assumption of normality is not rejected. 

Summary statistics for the sample data are x = 89.571 and 390.725 2

s = . Equation 2 is used to 

obtain a point estimate for P( ) X ≤ 115 . The solution is 
11

( ) ( ) 1.286 0.901

390.725

115 89.571 115 ˆ = Φ = 













 − P X ≤ = Φ . 

To achieve a 95% LCB for P( ) X ≤ 115 , we start by using equation 12: 

.95 ( ) 4.813 13, 14BL= G . 

Using the bisection method, we obtain a solution of = 0.6672 BL . Therefore, a 95% LCB for 

P( ) X ≤ 115 is Φ( ) 0.6672 = .7477 . That is, for an armor of thickness 115 units, one can be 95% 

confident that the probability of non-perforation is at least 74.77%. 

At this point, a reasonable question might be “With 95% confidence, what armor thickness 

would offer at least 90% protection against perforation?” The lower confidence band for ( ) Φ BL , 

shown in figure 2, will help determine this. We begin by extending a horizontal (dashed) line 

from the “y”-axis at P( ) non − perforation = .90 to the lower confidence band and then dropping 

to the x-axis to find the desired armor thickness. This thickness of 131.25 is the 90% upper 

tolerance bound. 

20 40 60 80 100 120 140 160 180

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ARMOR THICKNESS

P(NON-PERFORATION)

LC

Band

Estimated

90% Upper Tolerance Bound = 131.25

Figure 2. Estimated probability of non-perforation and 95% LCB as a function of armor 

thickness. (The line formed by the bounds is referred to as the lower confidence 

band. From the band, one is able to determine what armor thickness yields a certain 

minimum probability of non-perforation, e.g., with 95% confidence, one can state 

that a thickness of 131.25 will stop at least 90% of projectiles from perforating the 

material.) 
12

A 95% two-sided confidence interval for P( ) X ≤ 115 is calculated from equations 12 and 13. It 

has a lower limit given by ( ) Φ CL in which CL is the solution to 



















 − − = −

n

s

x x Gn nCL

0

1, 2

1 α

, or 

.975 ( ) 4.813 13, 14CL= G . The upper limit is given by ( ) Φ CU in which CU is the solution to 



















 − = −

n

s

x x Gn nCU

0

1, 2

α

, or .025 ( ) 4.813 13, 14CU= G . Using the bisection method again, we obtain 

= .5566 CL and 9901 =1. CU . Therefore, the lower confidence limit is Φ( ) .5566 = .7111, and the 

upper confidence limit is Φ( ) 1.9901 = .9767 . We can be 95% confident that the true probability 

of non-perforation is between 71.11% and 97.67%. 

9. Summary 

Reporting an interval estimate along with a point estimate gives designers a sense of the error of 

estimation which is a function of the sample size used in the study and the inherent variability of 

each projectile-armor interaction. Using a generalized pivotal quantity, we have achieved an 

interval estimate for the probability that a homogeneous armor plate of specified thickness will 

successfully stop a projectile from perforating. In the derivation of this interval estimate, we 

have assumed that the penetration depths are random and that they follow a normal distribution 

of unknown mean and variance. 

The interval limit (or limits if a two-sided interval is desired) is a function of the sample mean 

and sample standard deviation of penetration depths, the number of shots fired, the thickness of 

the armor and percentiles of a non-central t distribution. However, since the non-centrality 

parameter associated with this distribution is a function of the interval limit, numerical methods 

are required to obtain the final solution. MATLAB and Mathematica codes to perform these 

calculations are provided in appendices B through G. 

Confidence bands that graphically display the entire the relationship between armor width and 

bounds on the probability of non-perforation follow naturally. These bands may be of help to 

armor designers in selecting an armor width that will provide a high degree of protection against 

specific enemy threats. 
13

10. References 

Barton, D.E. Unbiased estimation of a set of probabilities. Biometrika 1961, 48, 227–229. 

Casella, G.; Berger, R.L. Statistical Inference, 2nd Ed., Duxbury Press: Belmont, CA,1990. 

Conover, W. J. Practical Nonparametric Statistics, 2nd ed.; John Wiley & Sons: New York, 

1980. 

Hahn G. J.; Meeker, W. Q. Statistical Intervals: A Guide for Practitioners, John Wiley & Sons: 

New York, 1991. 

Johnson, N. L.; Kotz, S. Distributions in Statistics --- Continuous Univariate Distributions - 2, 

John Wiley & Sons: New York, 1970. 

Lieberman, G. J.; Resnikoff, G. J. Sampling plans for inspection by variables. Journal of the 

American Statistical Association 1955, 50, 457–516. 

Mood, A.; Graybill, F.; Boes, D. Introduction to the Theory of Statistics, 3rd ed.; McGraw-Hill: 

New York, 1974. 

Odeh, R.E.; Owen, D.B. Tables for Normal Tolerance Limits, Sampling Plans, and Screening, 

Marcel Dekker, Inc.: New York, 1980 

Owen, D. B.; Hua, T. A. Tables of Confidence Limits on the Tail Area of the Normal 

Distribution, Communications in Statistics – Simulation and Computation B 1977, 6, 285–

311. 

Weerahandi, S. Generalized Confidence Intervals, Journal of the American Statistical 

Association 1993, 88, 899–905. 

Weerahandi, S. Exact Statistical Methods for Data Analysis, Springer: New York, 1995. 

Weerahandi, S. General Inference in Repeated Measures, John Wiley & Sons: New York, 

2004. 
14

INTENTIONALLY LEFT BLANK
15

Appendix A. Proof of the Monotonicity of the Function H 

Show that H( )⋅ is a monotonic decreasing function. That is, show that if λ1 < λ2 , then 

() ( ) H λ1 > H λ2 . 

Proof:

Let n, 0 x , x , s, α , λ1 , and λ2 be fixed constants with λ1 < λ2 . Let X1 and Y be independent 

random variables, X1 being normally distributed with mean λ1 and variance 1, and Y following a 

chi-squared distribution with n −1 degrees of freedom. The support of X1 is the set of all real 

numbers, and the support of Y is the set of all positive reals. 

Now define 

s

x x

n

nY W −

− = 01 . The support of W is either the set of all positive real numbers (if 

x ≥ x 0 ) or the set of all non-positive real numbers (if x < x 0 ). However, in general, we denote 

the support of W by Θ . 

P( ) X1 < W can be expressed as a double integral of the joint density of X1 and W, namely, 

( ) () ( ) ∫ ∫

Θ ∞−

< =

w

X W P X W f x f w dx dw 1 1 . (A-1) 

The double integral of equation A-1, can be rewritten as the sum of two double integrals: 

( ) () ( )

( )

() ( )

( )

∫ ∫ ∫ ∫

Θ Θ − −

− −

−∞

< = +

w

w

X W

w

P X W f X x fW w dx dw f x f w dx dw

2 1

1

2 1

1 1

λ λ

λ λ

. (A-2) 

Each of these double integrals is positive, so by dropping the latter of the two, we obtain the 

inequality, 

( ) () ( )

( )

∫ ∫

Θ

− −

−∞

< >

2 1

1 1

w λ λ

P X W f X x fW w dx dw. (A-3) 

However, the double integral in equation A-3, represents a probability, namely, 

( ) ( ) P X1 < W − λ 2 − λ1 , which in turn equals P( ) X1 + λ 2 − λ1 < W . So, 

P( )( ) X1 < W > P X1 + λ 2 − λ1 < W . (A-4) 

Now define ( ) X2 = X1 + λ2 − λ1 ; then X 2 is normally distributed with mean λ 2 and variance 1. 

Therefore, 

P( )( ) X1 < W > P X2 < W , (A-5) 
16

Consider the standard normal random variable, Z, having mean 0 and variance 1. Employing the 

equalities X1 = Z + λ1 and X2 = Z + λ2 and the definition of W, we expand the above result to 

obtain 















 −

− > + < 













 −

− + <s

x x

n

nY P Z

s

x x

n

nY P Z 0

2

0

1 1 1

λ λ . (A-6) 

Rearranging the expressions inside each of the probability statements, 



















 − <

−

+

>



















 − <

−

+

n

s

x x

n

Y

Z

P

n

s

x x

n

Y

Z

P 1 0 2 0

1 1

λ λ . (A-7) 

By the definition of a non-central t random variable, 



















 − > <



















 − − < −

n

s

x x P T

n

s

x x P Tn n

0

1,

0

1,λ1 λ2 . (A-8) 

Each of these probabilities is the cumulative non-central t distribution function evaluated at 

n

s

x − x 0 : 



















 − >



















 − − −

n

s

x x G

n

s

x x Gn n 0

1, 01,λ1 λ2 . (A-9) 

Next, we add the constant α −1 to both sides of the inequality: 

1 1 0

1, 01, 1 2 + −



















 − + − >



















 − − λ α − λ α

n

s

x x G

n

s

x x Gn n . (A-10) 

Using the definition of the function H( )⋅ , we obtain the desired result. 

() ( ) H λ1 > H λ2 . (A-11) 
17

Appendix B. MATLAB Code for Probability of Non-perforation LCB 

MATLAB Release 14, Version 7, code for calculating an LCB for the probability of nonperforation, 



 



 − Φ

σ

x0 μ . The user may supply his or her own data and values for alpha, x0, and 

epsilon. 

% Define variables % 

alpha=.05; x0=115; epsilon=.000001; 

% Declare data % 

data=[47 59 80 81 86 88 89 90 90 99 100 113 114 118]'; 

% Calculate summary statistics and define K % 

n=length(data); xbar=mean(data); sigma=std(data); 

K=sqrt(n)*(x0- xbar)/sigma; 

% Initialize flag and bounds on non-centrality parameter % 

flag=0; lononcent=xbar-10*sigma; hinoncent=xbar+10*sigma; 

% Execute bisection method % 

noncent=(lononcent+hinoncent)/2; 

while flag==0 

 if hinoncent-lononcent < epsilon 

 flag=1; 

 elseif nctcdf(K,n-1,noncent)+alpha-1 < 0 

 hinoncent=noncent; noncent=(noncent+lononcent)/2; 

 else 

 lononcent=noncent; noncent=(noncent+hinoncent)/2; 

 end 

end 

%Print lower confidence bound for probability of non-perforation % 

fprintf('%3.0f%% LCB for P(non-perforation) = %6.4f\n',... 

 100*(1-alpha),normcdf(noncent/sqrt(n))) 
18

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19

Appendix C. Mathematica Code for Probability of Non-perforation LCB 

Mathematica Version 5.1 code for calculating an LCB for the probability of non-perforation, 





 



 − Φ

σ

x0 μ . The user may supply his or her own data and values for alpha, x0, and epsilon. 

(* Remove warning messages that do not affect results but clutter output *) 

Off[General::spell]; Off[General::spell1]; 

Off[NIntegrate::slwcon]; Off[NIntegrate::ncvb]; 

(* Enable use of statistical tools *) 

<<Statistics`ContinuousDistributions` 

<<Statistics`NormalDistribution` 

(* Clear variables for use *) 

Clear[data,n,noncent,x0,a,epsilon,lononcent,hinoncent,flag,K,ndist] 

(* Define variables *) 

alpha=.05; x0=115; epsilon =.000001; 

(* Declare data *) 

data={47,59,80,81,86,88,89,90,90,99,100,113,114,118}; 

(* Calculate summary statistics and define K *) 

n=Length[data]; xbar=Mean[data]//N; sigma=StandardDeviation[data]//N; 

K=(x0-xbar)/(sigma/Sqrt[n])//N; 

(* Initialize flag and bounds on non-centrality parameter *) 

flag=0; lononcent=xbar-10sigma//N; hinoncent=xbar+10sigma//N; 

(* Execute bisection method *) 

H:=CDF[NoncentralStudentTDistribution[n-1,noncent],K]+alpha-1 

While[flag==0,noncent=Mean[{lononcent,hinoncent}]; 

 If[hinoncent-lononcent<epsilon,flag=1, 

 If[H<0,hinoncent=noncent,lononcent=noncent]]] 

(* Print lower confidence bound for probability of non-perforation *) 

Print[TraditionalForm[StringForm["``0% LCB for P(non-perforation) = ``", 

 100 (1-alpha),CDF[NormalDistribution[0,1],noncent/Sqrt[n]]]]] 
20

INTENTIONALLY LEFT BLANK
21

Appendix D. MATLAB Code for Probability of Non-perforation UCB 

MATLAB Release 14, Version 7, code for calculating a UCB for the probability of nonperforation, 



 



 − Φ

σ

x0 μ . The user may supply his or her own data and values for alpha, x0, and 

epsilon. 

% Define variables % 

alpha=.05; x0=115; epsilon=.000001; 

% Declare data % 

data=[47 59 80 81 86 88 89 90 90 99 100 113 114 118]'; 

% Calculate summary statistics and define K % 

n=length(data); xbar=mean(data); sigma=std(data); 

K=sqrt(n)*(x0- xbar)/sigma; 

% Initialize flag and bounds on non-centrality parameter % 

flag=0; lononcent=xbar-10*sigma; hinoncent=xbar+10*sigma; 

% Execute bisection method % 

noncent=(lononcent+hinoncent)/2; 

while flag==0 

 if hinoncent-lononcent < epsilon 

 flag=1; 

 elseif nctcdf(K,n-1,noncent)-alpha < 0 

 hinoncent=noncent; noncent=(noncent+lononcent)/2; 

 else 

 lononcent=noncent; noncent=(noncent+hinoncent)/2; 

 end 

end 

% Print upper confidence bound for probability of non-perforation % 

fprintf('%3.0f%% UCB for P(non-perforation) = %6.4f\n',... 

 100*(1-alpha),normcdf(noncent/sqrt(n))) 
22

INTENTIONALLY LEFT BLANK
23

Appendix E. Mathematica Code for Probability of Non-perforation UCB 

Mathematica Version 5.1 code for calculating a UCB for the probability of non-perforation, 





 



 − Φ

σ

x0 μ . The user may supply his or her own data and values for alpha, x0, and epsilon. 

(* Remove warning messages that do not affect results but clutter output *) 

Off[General::spell]; Off[General::spell1]; 

Off[NIntegrate::slwcon]; Off[NIntegrate::ncvb]; 

(* Enable use of statistical tools *) 

<<Statistics`ContinuousDistributions` 

<<Statistics`NormalDistribution` 

(* Clear variables for use *) 

Clear[data,n,noncent,x0,a,epsilon,lononcent,hinoncent,flag,K,ndist] 

(* Define variables *) 

alpha=.05; x0=115; epsilon =.000001; 

(* Declare data *) 

data={47,59,80,81,86,88,89,90,90,99,100,113,114,118}; 

(* Calculate summary statistics and define K *) 

n=Length[data]; xbar=Mean[data]//N; sigma=StandardDeviation[data]//N; 

K=(x0-xbar)/(sigma/Sqrt[n])//N; 

(* Initialize flag and bounds on non-centrality parameter *) 

flag=0; lononcent=xbar-10sigma//N; hinoncent=xbar+10sigma//N; 

(* Execute bisection method *) 

H:=CDF[NoncentralStudentTDistribution[n-1,noncent],K]-alpha 

While[flag==0,noncent=Mean[{lononcent,hinoncent}]; 

 If[hinoncent-lononcent<epsilon,flag=1, 

 If[H<0,hinoncent=noncent,lononcent=noncent]]] 

(* Print upper confidence bound for probability of non-perforation *) 

Print[TraditionalForm[StringForm["``0% UCB for P(non-perforation) = ``", 

 100 (1-alpha),CDF[NormalDistribution[0,1],noncent/Sqrt[n]]]]] 
24

INTENTIONALLY LEFT BLANK
25

Appendix F. MATLAB Code for Probability of Perforation Two-Sided 

Confidence Interval 

MATLAB Release 14, Version 7, code for calculating a two-sided confidence interval for the 

probability of non-perforation, 



 



 − Φ

σ

x0 μ . The user may supply his or her own data and values 

for alpha, x0, and epsilon. 

% Define variables % 

alpha=.05; x0=115; epsilon=.000001; 

% Declare data % 

data=[47 59 80 81 86 88 89 90 90 99 100 113 114 118]'; 

% Calculate summary statistics and define K % 

n=length(data); xbar=mean(data); sigma=std(data); 

K=sqrt(n)*(x0- xbar)/sigma; 

% Initialize flag and bounds on non-centrality parameter % 

flag=0; lononcent=xbar-10*sigma; hinoncent= xbar+10*sigma; 

% Execute bisection method for lower limit % 

noncent=(lononcent+hinoncent)/2; 

while flag==0 

 if hinoncent-lononcent < epsilon 

 flag=1; 

 elseif nctcdf(K,n-1,noncent)+alpha/2-1 < 0 

 hinoncent=noncent; noncent=(noncent+lononcent)/2; 

 else 

 lononcent=noncent; noncent=(noncent+hinoncent)/2; 

 end 

end 

% Print lower confidence limit for probability of non-perforation % 

fprintf('%4.0f%% CI for P(non-perforation) = (%6.4f, ',... 

 100*(1-alpha),normcdf(noncent/sqrt(n))) 

% Re-initialize flag and bounds on non-centrality parameter % 

flag=0; lononcent=xbar-10*sigma; hinoncent= xbar+10*sigma; 
26

% Execute bisection method for upper limit % 

noncent=(lononcent+hinoncent)/2; 

while flag==0 

 if hinoncent-lononcent < epsilon 

 flag=1; 

 elseif nctcdf(K,n-1,noncent)-alpha/2 < 0 

 hinoncent=noncent; noncent=(noncent+lononcent)/2; 

 else 

 lononcent=noncent; noncent=(noncent+hinoncent)/2; 

 end 

end 

% Print upper confidence limit for probability of non-perforation % 

fprintf('%6.4f)\n',normcdf(noncent/sqrt(n))) 
27

Appendix G. Mathematica Code for Probability of Perforation Two-Sided 

Confidence Interval 

Mathematica Version 5.1 code for calculating a two-sided confidence interval for the probability 

of non-perforation, 



 



 − Φ

σ

x0 μ . The user may supply his or her own data and values for alpha, 

x0, and epsilon. 

(* Remove warning messages that do not affect results but clutter output *) 

Off[General::spell]; Off[General::spell1]; 

Off[NIntegrate::slwcon]; Off[NIntegrate::ncvb]; 

(* Enable use of statistical tools *) 

<<Statistics`ContinuousDistributions` 

<<Statistics`NormalDistribution` 

(* Clear variables for use *) 

Clear[data,n,noncent,x0,a,epsilon,lononcent,hinoncent,flag,K, 

 ndist,lowerlimit,upperlimit] 

(* Define variables *) 

alpha=.05; x0=115; epsilon =.000001; 

(* Declare data *) 

data={47,59,80,81,86,88,89,90,90,99,100,113,114,118}; 

(* Calculate summary statistics and define K *) 

n=Length[data]; xbar=Mean[data]//N; sigma=StandardDeviation[data]//N; 

K=(x0-xbar)/(sigma/Sqrt[n])//N; 

(* Initialize flag and bounds on non-centrality parameter *) 

flag=0; lononcent=xbar-10sigma//N; hinoncent=xbar+10sigma//N; 

(* Execute bisection method for lower limit *) 

H:=CDF[NoncentralStudentTDistribution[n-1,noncent],K]+alpha/2-1 

While[flag==0,noncent=Mean[{lononcent,hinoncent}]; 

 If[hinoncent-lononcent<epsilon,flag=1, 
28

 If[H<0,hinoncent=noncent,lononcent=noncent]]] 

lowerlimit = N[CDF[NormalDistribution[0,1],noncent/Sqrt[n]]]; 

(* Re-initialize flag and bounds on non-centrality parameter *) 

flag=0; lononcent=xbar-10sigma//N; hinoncent=xbar+10sigma//N; 

(* Execute bisection method for upper limit *) 

H:=CDF[NoncentralStudentTDistribution[n-1,noncent],K]-alpha/2 

While[flag==0,noncent=Mean[{lononcent,hinoncent}]; 

 If[hinoncent-lononcent<epsilon,flag=1, 

 If[H<0,hinoncent=noncent,lononcent=noncent]]] 

upperlimit = N[CDF[NormalDistribution[0,1],noncent/Sqrt[n]]]; 

(* Print confidence interval for probability of non-perforation *) 

Print[TraditionalForm[StringForm["``0% CI for P(non-perforation) = (``,``)", 

 100 (1-alpha),lowerlimit,upperlimit]]] 
29

Appendix H. MATLAB Code for Probability of Non-perforation CDF and 

Lower Confidence Band 

MATLAB Release 14, Version 7, code for plotting an estimated CDF and lower confidence 

band. The user may supply his or her own data and values for alpha, x0, and epsilon. 

% Define variables % 

alpha=.05; epsilon=.0001; 

% Declare data % 

data=[47 59 80 81 86 88 89 90 90 99 100 113 114 118]'; 

% Calculate summary statistics % 

n=length(data); Xbar=mean(data); SDev=std(data); 

% Initialize counter for points along the X-axis % 

i=0; 

% Declare armor thickness % 

for x0=Xbar-3*SDev:SDev/10:Xbar+4*SDev 

 i=i+1; x(i)=x0; 

 

 % Calculate plug-in estimate for probability of non-perforation % 

 pointest(i)=normcdf((x0-Xbar)/SDev); 

 

 % Define K % 

 K=sqrt(n)*(x0-Xbar)/SDev; 

 % Initialize flag and bounds on non-centrality parameter % 

 flag=0; lononcent=-1000; hinoncent=1000; 

 % Execute bisection method for lower confidence bound % 

 noncent=(lononcent+hinoncent)/2; 

 while flag==0 

 if hinoncent-lononcent < epsilon 

 flag=1; 

 elseif nctcdf(K,n-1,noncent)-1+alpha < 0 
30

 hinoncent=noncent; noncent=(noncent+lononcent)/2; 

 else 

 lononcent=noncent; noncent=(noncent+hinoncent)/2; 

 end 

 end 

 lcb(i)=normcdf(noncent/sqrt(n)); 

end 

% as a function of armor thickness, plot non-perforation probability curve % 

% in blue, and lower confidence band for probability of non-perforation in % 

% red % 

plot(x,pointest,'b-',x,lcb,'r-') 

xlabel('ARMOR THICKNESS'); ylabel('P(NON-PERFORATION)') 

text(Xbar-.25*SDev,.5,'Estimated','Color','b','Rotation',69,... 

 'HorizontalAlignment','Center') 

text(Xbar+.75*SDev,.5,'LC Band','Color','r','Rotation',67,... 

 'HorizontalAlignment','Center') 
31

List of Symbols Used 

Scalar Values

n number of observations in sample data 

ir the i

th observed value of the random variable R

( )i r the i

th ordered, observed value of the random variable R

s observed sample standard deviation 

i x the i

th observed value from a data set; in this paper, the observed depth of penetration 

from the i

th projectile 

0 x a specified thickness of armor 

x observed sample mean 

p z the value from the standard normal distribution having an area of p to its right under the 

density curve 

α level of significance 

σ population standard deviation for a normally distributed random variable 

θ parameter for which a confidence interval is constructed; may actually be a function of 

one or more parameters 

μ population mean for a normally distributed random variable 

ξ a set of one or more nuisance parameters 

Random Variables

R( ) generalized test variable; a function of random data, observed data, parameter of interest 

and perhaps other nuisance parameters 

S sample standard deviation of depths of penetration 

T student’s t random variable 

V chi-square random variable 

X depth of penetration for a randomly selected projectile 

X sample mean depth of penetration 
32

Y a normal random variable with mean zero and variance 

n

1

Z a standard normal random variable, with mean zero and variance one 

Functions

F( ) cumulative distribution function 

F( ) ˆ estimated cumulative distribution function 

G( ) student’s t cumulative distribution function 

Φ( ) standard normal cumulative distribution function 

Miscellaneous

BL lower confidence bound 

BU upper confidence bound 

CL lower confidence limit 

CU upper confidence limit 

P( ) probability of the parenthesized expression 

P( ) ˆ estimated probability of the parenthesized expression 

Σ summation 

^ estimate of 
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