possibilities, which we call α and β risk: the probability of “good” materiel failing to meet the criteria (false reject), and the probability of “bad” material passing the criteria (false acceptance).
These risks may increase or decrease, depending on the test quantities and acceptance criteria used in LAT. This is what is called “discrimination.”
MONTE CARLO SIMULATION
Monte Carlo Simulation of a target with 50 points, CoI = (0.31, 0.51), σx
and σy = 4. = 3,
EXAMPLE: SMALL- CALIBER AMMUNITION For example, some recent 7.62mm M80 ball LAT data provide an idea of the past performance of this ammunition rela- tive to its accuracy requirement. Using a mathematical relationship between the different measures of accuracy, we could select an average ES and maximum ES requirement comparable to the current LAT requirement of 7.5-inch average MR (AMR). This can be verified graphically where the inflection points (50th per- centiles of P[a]) of the OC curves of each should be approximately aligned.
BIVARIATE DISTRIBUTION Bivariate Normal Distribution with µx
ρ = 0, σx = 3, and σy = 4. , µy , and
Keeping the sample sizes and number of targets in the LAT constant, we can perform numerous Monte Carlo simu- lations across a range of σ’s, which will allow us to compare the discrimination of the proposed LAT requirement with the current requirement.
OBLIQUE PROJECTION Oblique projection of probability density of above distribution.
Comparing the OC curves of the disper- sion metrics illustrates the increased risk associated with switching to a maximum ES requirement vs. averaging the targets. The risk of falsely rejecting good ammu- nition (α-risk) increases to ~25%, and the risk of falsely accepting bad ammu- nition (β-risk) increases to ~20% at the same points where AMR α = β = ~10%. This result can be deduced intuitively by understanding the nature of these different measures: Whereas ES uses only informa- tion from two rounds per target, MR uses
information from all rounds and therefore is much more statistically efficient.
Over the course of numerous LATs, this increase in risk translates to excessive non- value-added production costs, reduced performance, and schedule impact. In this example, these are all due to using a differ- ent method to calculate dispersion from the same data.
CONCLUSION When determining weapon or muni- tion accuracy requirements or how to test for accuracy, it is important to understand not only how sample size, target and group breakout, and accep- tance criteria affect the discrimination and risks inherent in any test, but also how selecting the right method to cal- culate target impact dispersion can affect discrimination and risk as well. This dis- cussion has merely scratched the surface of these considerations. Armaments engi- neers and scientists need to address these issues in test and evaluation so that the warfighter doesn’t have to deal with them in combat.
SSG DOUGLAS RAY (USA Ret.) is the lead Mathematical Statistician at the U.S. Army Research, Development, and Engi- neering Center, in the Statistical Methods and Analysis Group, part of the Quality Engineering and System Assurance Direc- torate’s Reliability Management Branch at Picatinny Arsenal, NJ. He retired in 2010 from the Army National Guard, where he was an Airborne Infantryman. He holds a B.S. in applied mathematics from the Uni- versity of Rhode Island and an M.S.E. in engineering science with a concentration in applied statistics from the New Jersey Institute of Technology. Ray is produc- tion, quality, and manufacturing Level III certified and is a U.S. Army Acquisition Corps member.
A S C . A RMY.MI L 71
SCIENCE & TECHNOLOGY
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