NISTIR 7376

Nonparametric Statistical Data 

Analysis of Fingerprint Minutiae 

Exchange with Two-Finger Fusion

Jin Chu Wu

Michael D. Garris

 
NISTIR 7376

Nonparametric Statistical Data 

Analysis of Fingerprint Minutiae

Exchange with Two-Finger Fusion

Jin Chu Wu

Michael D. Garris

Information Technology Laboratory

Information Access Division

November 2006

U.S. DEPARTMENT OF COMMERCE

Carlos M. Gutierrez, Secretary

TECHNOLOGY ADMINISTRATION

Michelle O’Neill, Acting Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

William A. Jeffrey, Director

 
Nonparametric Statistical Data Analysis of Fingerprint 

Minutiae Exchange with Two-Finger Fusion 

Jin Chu Wu and Michael D. Garris 

Image Group, Information Access Division, Information Technology Laboratory 

National Institute of Standards and Technology, Gaithersburg, MD 20899 

Abstract 

A nonparametric inferential statistical data analysis is presented. The utility of this method is 

demonstrated through analyzing results from minutiae exchange with two-finger fusion. The 

analysis focused on high-accuracy vendors and two modes of matching standard fingerprint 

templates: 1) Native Matching - where the same vendor generates the templates and the matcher, 

and 2) Scenario 1 Interoperability - where vendor A’s enrollment template is matched to vendor 

B’s authentication template using vendor B’s matcher. The purpose of this analysis is to make 

inferences about the underlying population from sample data, which provide insights at an 

aggregate level. This is very different from the data analysis presented in the MINEX04 report 

in which vendors are individually ranked and compared. Using the nonparametric bootstrap 

bias-corrected and accelerated (BCa) method, 95 % confidence intervals are computed for each 

mean error rate. Nonparametric significance tests are then applied to further determine if the 

difference between two underlying populations is real or by chance with a certain probability. 

Results from this method show that at a greater-than-95 % confidence level there is a significant 

degradation in accuracy of Scenario 1 Interoperability with respect to Native Matching. The 

difference of error rates can reach on average a two-fold increase in False Non-Match Rate. 

Additionally, it is proved why two-finger fusion using the sum rule is more accurate than singlefinger matching under the same conditions. Results of a simulation are also presented to show 

the significance of the confidence intervals derived from the small size of samples, such as six 

error rates in some of our cases. 

Keywords: bootstrap, fingerprint matching, inferential, interoperability, minutiae exchange, 

nonparametric, significance test, standard templates, statistical data analysis

1. Introduction 

The purpose of this paper is to demonstrate the utility of applying nonparametric inferential 

statistics to biometric test results. There are significant advantages of this approach. Since there 

is no underlying distribution model for fingerprint data, the statistical data analysis must be 

model independent [1][2]. This nonparametric method is applicable on small sizes of samples 

where the Central Limit Theorem cannot be applied. This is particularly useful when the 

availability of samples is limited or the cost of generating more samples is prohibitively high. 

Additionally, the statistics invoked in this article are inferential rather than descriptive. In this 

way, properties of the population are inferred from the sample, which provides potentially deep 

insights rather than analyses that focus only on individuals in the sample. 

1
This study was conducted on two-finger fusion results derived from the Minutiae Interoperability 

Exchange Test 2004 (MINEX04), which was organized and administered by the National 

Institute of Standards and Technology (NIST), and a complete description is found at [3]

including an executive summary and report [4]. The two-finger fusion was conducted by using 

the sum rule. A proof is provided in Appendix A that shows why the two-finger sum rule is 

more accurate than single-finger matching results. The MINEX04 report conducted extensive 

analyses intended to rank participants and to determine a group of participants that demonstrate 

interoperability in different modes (or categories) above a minimum level of accuracy. The 

purpose of the nonparametric data analysis presented in this paper is different in that it is not 

primarily intended to rank participants, but rather, make statistically robust observations 

regarding the collective capabilities of the participants (the population) by analyzing their 

aggregate performance. 

A brief overview of MINEX04 is provided in Section 2. Section 3 describes the steps taken to 

conduct the nonparametric data analysis and presents the statistical results. This includes 

selection of a representative sample of participants, comparison of mean performances, 

computing 95 % confidence intervals, and then conducting statistical significance tests. 

Conclusions are drawn in the final section. 

2. MINEX04 Overview 

The purpose of the MINEX04 test was to determine the feasibility of using minutiae data (rather 

than image data) as the interchange medium for fingerprint information between different 

fingerprint matching systems. The results of MINEX04 have implications that affect planning 

decisions for projects such as Personal Identity Verification (PIV). PIV was initiated by 

Homeland Security Presidential Directive 12 [5]. This mandated the establishment of a common 

identification standard for federal employees and contractors. It required interoperable use of 

identity credentials to control physical and logical access to federal government locations and 

systems. 

MINEX04 was designed to evaluate whether various populations and combinations of encoding 

schemes, enrolled templates, probe templates, and fingerprint matchers will produce successful 

matches. There were two categories of encoding schemes; the first were proprietary minutiae 

templates generated by the participants (called vendors); the second were standard minutiae 

templates. These standard templates are based on INCITS (International Committee for 

Information Technology Standards) 378 Finger Minutiae Format for Data Interchange [6]. There 

were two standard template types evaluated in MINEX04, but for the purposes of this study, we 

focus on just the results of using the standard ‘A’ templates nicknamed “MIN:A”, which contain 

only the minutiae attributes {x, y, θ, type, quality}. 

A total of 14 vendors participated in MINEX04. These vendors are identified in the MINEX04 

report and subsequently in this report by assigned letters. The identities of these vendors are not 

germane to the purpose of this report, so identities are not revealed herein, however the vendor 

key is published in the full MINEX04 report. Each vendor had to supply NIST with a software 

development kit (SDK) that 

 creates an INCITS 378 MIN:A template from an image 

 produces a comparison score from two MIN:A templates 

2
In this way, matching accuracy could be computed and compared in combinations of three 

dimensions: a gallery (enrolment) template, matched to a probe (authentication) template, 

matched with a specific vendor’s fingerprint minutiae matcher. A simple nomenclature “XY_Z” 

has been adopted to represent the possible combinations, where X represents the vendor that 

generated the enrollment template, Y represents the vendor that generated the authentication 

template, and Z is the vendor that developed the template matcher. 

Standard template matching within MINEX04 was tested in two modes. The first comprised of 

standard templates being generated and matched by the same vendor, referred to as Native 

Matching and nicknamed “MIN:A.XX_X” The second involved testing the interoperability of 

matching a standard template from one vendor with a standard template generated by a different 

vendor and then matched potentially by yet another different vendor. 

While all the possible combinations of interoperability were studied in MINEX04, there is one 

combination that has greatest operational relevance. This is the interoperable scenario where a 

subject is enrolled in vendor-P’s system, but then attempts to authenticate with a different 

vendor-Q’s system. This is the case when a person enrolled by one agency’s system visits and 

presents his credentials to a different agency. In this scenario, the subject presents his finger to 

vendor-Q’s system and a standard template is generated; this template is then matched to vendorP’s enrolled template with the match being conducted by vendor-Q’s matcher. This is referred to 

as Scenario 1 Interoperability and nicknamed “MIN:A.YX_X”. For the purposes of this study, 

only MIN:A.XX_X and MIN:A.YX_X template-matcher combinations are analyzed. 

MINEX04 used four different and distinct collections of fingerprints (called datasets) named: 

POEBVA, POE, DOS, & DHS2. A description of these datasets and their NIST Fingerprint 

Image Quality (NFIQ) distributions are documented in the MINEX04 report. All datasets used 

were comprised of left and right-index fingers using live-scan plain impressions. The subject 

sample sizes of each dataset were 60 thousand mates and 120 thousand non-mates. The testing 

was performed by using the second instance of the mates as the enrollment image and the first 

instance as the authentication image. So for each dataset there were 60 thousand mate (genuine) 

template comparison scores. The non-mate scores were generated by comparing the non-mate 

authentication samples to the same enrollment images used with the mates. This generated 120 

thousand non-mate (impostor) template comparison scores. 

One and two-finger authentication was evaluated in the MINEX04 test. The two-finger 

comparison scores were produced in a score-level fusion process by summing a subject’s left and 

right-index finger comparison scores. Given a set of genuine and a set of impostor two-finger 

template comparison scores, performance measures of False Non-Match Rate (FNMR) and False 

Match Rate (FMR) were computed and Detection Error Tradeoff (DET) characteristic curves 

compared. The analysis in this paper focuses only on two-finger fusion results. Note the proof 

in Appendix A. 

The scope of the MINEX04 test was large and varied as fully documented in Reference [3]. 

Only a subset of these results is used in the data analysis herein. This includes results from all 

four datasets, with standard MIN:A templates, on two-finger fusion scores, and with only two 

template-matchers combinations (Native Matching and Scenario 1 Interoperability) as described 

in the following section. 

3
3. Nonparametric Inferential Statistical Data Analysis 

The purpose of the nonparametric inferential statistical data analysis presented in this paper is to 

make statistical inferences regarding the collective capabilities of the fingerprint matching 

community (the population) based on analyzing the aggregate performance of fingerprint 

matching vendors (the sample). The utility of this statistical method is therefore greatest when 

addressing operational decisions requiring an assessment of aggregate performance. 

In the case of the various combinations of template generators and matchers, one can think of an 

information technology manager who is responsible for procuring and deploying standard 

template fingerprint authentication systems within his enterprise. Furthermore, he/she may be 

required to procure standard template technology from a pool of competent providers and not 

just from a single source. The category MIN:A.XX_X represents this case. This manager must 

also consider visitors gaining access to his enterprise via credentials enrolled and issued by 

various vendors not within his control. What is the expected impact on the enterprise when 

dealing with standard templates interoperably? The category MIN:A.YX_X represents this case. 

What might this manager infer from the test results regarding the pool of such potential vendors? 

These types of questions can be addressed by a nonparametric inferential statistical approach. 

The properties of the population of MIN:A.XX_X and the population of MIN:A.YX_X will be 

analyzed and compared. 

3.1 The Method 

The statistical approach to the data analysis in this paper is as follows. A group of high-accuracy 

vendors is selected from those who participated in the test. The mean of the sample is computed. 

To compute a confidence interval about the mean, the nonparametric bootstrap is used by 

making no assumption about the distribution of population. The sample, which can be relatively 

small in size, is replicated through a process of resampling with replacement and a confidence 

interval is calculated from the replicated set. Next, to determine if two samples are significantly 

different and to make the test results more convincible, nonparametric significance tests are 

applied. The details and results of each of these steps as applied to MIN:A.XX_X and 

MIN:A.YX_X are described in the following sections. 

3.2 Sample Selection 

In our analysis we desire to gain understanding and insight into the capabilities of vendors who 

demonstrate a relatively high level of matching accuracy. Not all the vendors participating in 

MINEX04 demonstrated such a desirable level. It was necessary therefore to determine a pool of 

sufficiently accurate vendors for us to conduct our analysis. The participation in the test is 

voluntary. Thus, such a sample selection focusing on high-accuracy vendors is random. 

The Ongoing MINEX tests [7] were initiated while conducting the analyses in this paper. A 

primary purpose to the Ongoing MINEX tests is to evaluate and publish a certified list of 

vendors that exhibit a level of template matching interoperability above a minimum level of 

accuracy. This level of accuracy is set such that all vendors in the interoperable group achieve, 

within the context of Scenario 1 Interoperability, a FNMR less than or equal to 0.01 at a fixed 

FMR of 0.01. The details of this process and the current list of compliant vendors are posted on 

the Ongoing MINEX website. For the purposes of this report, we chose to analyze the six 

vendors {A,B,C,D,F,G} whose matcher performance in MINEX04 was determined compliant 

4
using the Ongoing MINEX criteria. It is noted that an ad hoc sample selection of vendors was 

conducted independent of the Ongoing MINEX testing, and the results were complementary. 

3.3 Comparison of Means 

Given the selected set of six vendors {A,B,C,D,F,G}, error rates from each vendor performing 

Native Matching on each of the four datasets were calculated. This generated six error measures 

per dataset, and the mean Native Matching (MIN:A.XX_X) error rate was computed for each 

dataset. Error rates were also calculated from every combination of two vendors performing 

Scenario 1 Interoperability. This generated thirty error measures per dataset (each of six vendors 

used interoperably with the remaining five vendors), and the mean Scenario 1 Interoperability 

(MIN:A.YX_X) error rate was computed for each dataset. 

Error rates were computed as the resulting FNMR at a specified level of FMR. Two levels of 

FNMR were computed and compared in this analysis. The first was FNMR at an FMR of 0.01, 

and results are reported in Table 1. The second level was FNMR at an FMR of 0.001, and these 

results are reported in Table 2. Comparing the means between MIN:A.XX_X and MIN:A.YX_X 

within each table, there is a consistent degradation in values when switching from Native 

Matching to Scenario 1 Interoperability. How accurate are these means? 

FMR = 0.01 

POEBVA POE DOS DHS2 

Mean 

FNM

R 

Conf. 

Interval 

Mean 

FNMR

Conf. 

Interval

Mean 

FNMR

Conf. 

Interval 

Mean 

FNMR 

Conf. 

Interval

MIN:A.XX_

X 0.0022 (0.0013, 

0.0028) 0.0024 (0.0013,0.0031) 0.0063 (0.0043, 0.0081) 0.0133 (0.0083,0.0256) 

MIN:A.YX_

X 0.0048 (0.0041, 

0.0057) 0.0049 (0.0041,0.0057) 0.0117 (0.0102, 0.0134) 0.0183 (0.0146,0.0237) 

Table 1. Means and 95 % confidence intervals of FNMR with FMR at 0.01 by dataset. 

FMR = 0.001 

POEBVA POE DOS DHS2 

Mean 

FNM

R 

Conf. 

Interval 

Mean 

FNMR

Conf. 

Interval

Mean 

FNMR

Conf. 

Interval 

Mean 

FNMR 

Conf. 

Interval

MIN:A.XX_

X 0.0045 (0.0028, 

0.0061) 0.0043 (0.0026,0.0054) 0.0120 (0.0083, 0.0150) 0.0195 (0.0138,0.0315) 

MIN:A.YX_

X 0.0099 (0.0081, 

0.0120) 0.0095 (0.0080,0.0113) 0.0204 (0.0176, 0.0236) 0.0295 (0.0242,0.0372) 

Table 2. Means and 95 % confidence intervals of FNMR with FMR at 0.001 by dataset. 

5
3.4 95 % Confidence Intervals 

To answer this question, the 95 % confidence intervals in different cases were computed using 

the nonparametric bootstrap bias-corrected and accelerated (BCa) method [8]. The statistical 

computing program R (Version 2.0.1) was used with the package “boot.”1

 For the 

MIN:A.XX_X category from each dataset, the six error measurements were re-sampled with 

replacement up to 100000 bootstrap replications, and the limits of the nonsymmetrical 95 % 

confidence interval were computed. The same process was followed for the MIN:A.YX_X 

category. Only in each case, thirty error measurements were re-sampled with replacement. The 

resulting confidence intervals are reported along side their corresponding means in Table 1 & 

Table 2. A simulation is provided in Appendix B that shows the significance of the confidence 

intervals computed from a sample of six error rates for the MIN:A.XX_X category. 

The means and their associated confidence intervals for the MIN:A.XX_X and MIN:A.YX_X 

categories are plotted side by side for each dataset in Figure 1 & Figure 2 for two different values 

of FMR, respectively. Looking at the results from the first three datasets in both figures, a clear 

pattern is observed. The 95 % confidence intervals for MIN:A.XX_X are completely separated 

with no vertical overlap to the 95 % confidence intervals for the corresponding MIN:A.YX_X 

sample. Comparing the results for the POEBVA dataset, the average error rate for Scenario 1 

Interoperability is more than twice that of Native Matching with standard templates. This is true 

for both FNMR measured at FMR=0.01 and FMR=0.001. 

The results are quite different for the DHS2 dataset. In this case, the mean error rates are 

considerably higher and the 95 % confidence intervals are much larger and the intervals for 

MIN:A.XX_X overlap largely with the corresponding intervals for MIN:A.YX_X. This dataset 

is known to have unique image quality characteristics resulting in much poorer image quality as 

reflected in the MINEX04 Report which lists DHS2 as having the largest percentage of worst 

(NFIQ Quality 5) fingerprints. The differences observed with DHS2 in our analysis are 

attributed to the difference in image quality of this dataset. The confidence intervals are 

particularly large due the existence of an outlier from one vendor’s sample distribution of 

FNMRs. 

The results from the other three datasets are consistent. These datasets are comprised of 

fingerprints captured with more modern sensors and quality control processes, while DHS2 

contains legacy data captured with older sensor technology and with fewer quality controls. 

Therefore, the statistical results with DHS2 are considered not representative and dismissed. The 

DHS2 results are intentionally left in this report as they point out how critical proper test design 

and sample selection are to achieving relevant conclusions using nonparametric inferential 

methods. 

 

1

 Specific hardware and software products identified in this paper were used in order to perform the analyses 

described herein. In no case does identification of any commercial product, trade name, or vendor, imply 

recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the 

products and equipment identified are necessarily the best available for the purpose. 

6
Figure 1. Means FNMR and 95 % confidence intervals with FMR at 0.01 by dataset 

Figure 2. Means FNMR and 95 % confidence intervals with FMR at 0.001 by dataset 

7
3.5 Statistical Significance Tests 

Are all the above observed differences significant? To help answer this question, two 

nonparametric statistical significance tests, the permutation test [9] and the Wilcoxon rank sum 

test [10], are applied. Both test the null hypothesis (that there is no difference in distributions) 

between two populations. Two-sided p-values are generated that indicate the significance level 

whether the two distributions are the same. A p-value less than 5 % then represents that the two 

distributions are likely to be different with a confidence level of greater than 95 %. 

The results of the permutation test are listed in Table 3. The first row of values are from samples 

of FNMR where FMR = 0.01, and the second row are from samples of FNMR where FMR = 

0.001. The statistical computing program S-Plus (Version 7.0.0) with the package “resample” 

was used to compute p-values. For each dataset, the MIN:A.XX_X sample and the 

MIN:A.YX_X sample were input to S-Plus and p-values were computed. Once again a pattern 

for the first three datasets is observed in that all p-values are well less than 0.05 indicating that 

the underlying populations of MIN:A.XX_X are significantly different from their corresponding 

populations of MIN:A.YX_X. The results on dataset DHS2 are strikingly different with p-values 

considerably larger than 0.05. This confirms the large and overlapping confidence intervals 

observed for this dataset in the previous figures for the same reasons. 

Results of the Wilcoxon rank sum test are listed in Table 4 and were computed using R. The 

Wilcoxon rank sum test is used because samples between the MIN:A.XX_X and MIN:A.YX_X 

categories are unpaired. The same pattern of p-values is observed confirming the earlier 

permutation test results and the mean error rates along with confidence intervals. 

Permutation Test Category 1 Category 2 POEBVA POE DOS DHS2 

FMR

MIN:A.XX_X MIN:A.YX_X 0.0022 0.0053 0.0027 0.4214 0.01 

MIN:A.XX_X MIN:A.YX_X 0.0089 0.0034 0.0119 0.1773 0.001

Table 3 Two-sided p-values of the permutation test with FMR at 0.01 and 0.001, by dataset 

Wilcoxon Rank Sum Test Category 1 Category 2 POEBVA POE DOS DHS2 

FMR

MIN:A.XX_X MIN:A.YX_X 0.0024 0.0063 0.0032 0.1463 0.01 

MIN:A.XX_X MIN:A.YX_X 0.0074 0.0063 0.0290 0.1721 0.001

Table 4 Two-sided p-values of Wilcoxon rank sum test with FMR at 0.01 and 0.001, by 

dataset 

8
4. Conclusions 

The application of nonparametric inferential statistics has been demonstrated on fingerprint 

minutiae exchange with two-finger fusion. Advantages include no assumptions of an underlying 

distribution model, the handling of confidence intervals and significance tests, small sample sizes 

can be used, and properties of the population are inferred providing important insights. The 

method begins with careful sample selection. If the sample is not representative, then results will 

be irrelevant and potentially misleading as demonstrated with the results reported on dataset 

DHS2. Given a sample, the 95 % confidence intervals are computed using the nonparametric 

bootstrap BCa method. The underlying populations of two samples are then compared using 

nonparametric significance tests such as the permutation test and the Wilcoxon rank sum test. 

Using this method, MINEX04 results were studied. Six high-accuracy vendors were selected 

and their ability to match standard fingerprint templates natively (MIN:A.XX_X) and 

interoperably (MIN:A.YX_X) were compared using two-finger fusion. Analysis showed that at 

a greater-than-95 % confidence level there is a significant degradation in accuracy of Scenario 1 

Interoperability with respect to Native matching. The difference of error rates can reach on 

average a two-fold increase in FNMR. A proof was also provided to show why two-finger 

fusion using the sum rule is better than single-finger matching results under the same conditions. 

Results of a simulation using the nonparametric bootstrap are also reported that show the 

significance of the confidence intervals derived from the small size of samples in our case. 

9
APPENDIX A. Proof of Two-Finger Fusion Sum Rule 

There are several ways to deal with two-finger fusion from the output of single-finger matching. 

For instance, the sum rule adds up the two similarity scores that are generated from the rightindex finger matching and the left-index finger matching. An alternative is the maximum rule 

where the maximum of the two similarity scores is selected as the fused score. In this paper, the 

sum rule is adopted. 

A qualitative proof is as follows showing why two-finger fusion using the sum rule improves 

results of single-finger matching in terms of the operational criteria under the same conditions. 

Suppose that for right-index finger matching, a score Gr is selected from the distribution of 

genuine comparison scores, and a score Ir is selected from the distribution of impostor 

comparison scores. And for left-index finger matching, the corresponding genuine score and 

impostor score are Gl and Il, respectively. Assume further that Ir<Gr and Il<Gl. Indeed, this 

assumption is valid in most cases. The distances between genuine scores and impostor scores in 

the single-finger case are Dr = Gr – Ir and Dl = Gl – Il, respectively. Using the sum rule for twofinger fusion, the fused genuine score is (Gr + Gl) and the fused impostor score is (Ir + Il). The 

distance between these two fused scores is Df = (Gr + Gl) - (Ir + Il) = Dr + Dl. This indicates that 

Df must be greater than Dr and Dl, respectively. In general, the greater is the distance between 

distributions of genuine and impostor scores, the more accurate the fingerprint matcher [1]. 

Therefore, the accuracy using the fused two-finger scores is increased. 

The effect of the sum rule is to create greater separation between the genuine and impostor score 

distributions and thus reduce the overlapping area of the two distributions. This can also be 

evidenced by examining the discrete probability distribution functions for the genuine and 

impostor comparison scores. In this case, MIN:A.XX_X results from a single vendor on the 

POEBVA dataset are presented. Figure 3 & Figure 4 show the results from matching 

individually the right-index finger and the left-index finger, respectively. Figure 5 shows the 

results of two-finger fusion using the same vendor’s technology. Note that the probabilities in 

these three figures have been cut at 0.001 in order to show clearly the relative positions of two 

distributions of the genuine and impostor scores. Comparing the two-finger fusion results with 

the single-finger results demonstrates the effect of the sum rule in that there is greater separation 

and less overlap of the genuine and impostor distributions in Figure 5. 

As a result, the sum rule of two-finger fusion improves the shape of the receiver operating 

characteristic (ROC) curve as opposed to using one-finger matching under the same conditions 

[1]. An ROC curve is created by applying the operational criteria of a score threshold across the 

genuine and impostor distributions measuring the True Accept Rate (TAR)2 and FMR at each 

threshold. An ROC analysis evaluates vendors’ performance. It is shown in Figure 6 that the 

ROC curve of two-finger fusion invoking the sum rule is higher than the ROC curves using onefinger matching. 

 

2

 Equivalently, (FNMR = 1 – TAR). 

10
Figure 3. The discrete probability distribution functions of the genuine and impostor 

scores for right-index finger matching. 

Figure 4. The discrete probability distribution functions of the genuine and impostor 

scores for left-index finger matching. 

Figure 5. The discrete probability distribution functions of the genuine and impostor 

scores for two-finger fusion using the sum rule. 

11
Figure 6 The three ROC curves of the right-index finger, the left-index finger, and the twofinger fusion. 

APPENDIX B. Significance of Confidence Intervals 

The 95 % confidence intervals for the MIN:A.XX_X category were obtained using the bootstrap 

BCa method on error measurements taken from a set of six vendors. With a sample size of six, 

one might question whether the results are significant. This section addresses this question, first 

in a theoretical manner [8][11], followed by supporting evidence from a simulation. 

Suppose that a bootstrap is carried out on n distinct values. It implies that the bootstrap space is 

nn

 in the sense that each possible bootstrap sample is selected with equal probability. Further, 

assume that a multinomial-distribution vector of the bootstrap is (k1, k2, ..., kn), where ki stands 

for the number of times the ith value is selected in a bootstrap sample, subject to 

0 ≤ ki ≤ n, i = 1, ..., n; and ∑ . =

=

n

1i

ki n

All bootstrap samples, corresponding to a bootstrap vector, can result in the same bootstrap 

replication of the considered statistic such as mean etc. In this sense, the number of distinct 

bootstrap samples is ⎟. ⎠

⎞ ⎜

⎝

⎛ −

−

1n

12n

A distinct bootstrap sample has the following multinomial probability being selected, 

n

1

!k!k!k

n! )k ,,k,(k Prob n 21 n

21 n = ∗

Λ

Λ . 

12
The original sample of n distinct values is obtained while all ki’s are equal to 1, and thus the 

corresponding probability is n!/nn

. It follows from the above probability formula that this 

probability is the largest one among all others. 

Therefore, while the sample size is six for the MIN:A.XX_X category, the bootstrap space is 66

= 46,656, the number of distinct bootstrap samples is = 462, and the probability of selecting 

the observed statistic is 6!/6

⎟

⎠

⎞ ⎜

⎝

⎛

5

11

6

 = 1.54 %. All these indicate that the confidence intervals of the 

mean derived from a sample of size six using the bootstrap are significant. 

To support this position, a simulation was conducted on the MIN:A.XX_X results that were 

presented in Table 1 for the POEBVA dataset. The associated mean and 95 % confidence 

interval are shown in the top-right of Table 5 in the row labeled Trial 0. To the left of these 

statistics are the underlying FNMR measurements (at an FMR = 0.01) contributed by each of the 

six vendors. 

For each of the subsequent trails (Trials 1-5) reported in Table 5, the underlying genuine and 

impostor scores from each vendor were re-sampled with replacement, respectively, and an 

FNMR was computed at FMR = 0.01 from two new distributions. These simulated FNMR’s are 

reported under each vendor column in the table. Their mean is reported to the right, and the 

bootstrap BCa method was applied to the simulated FNMR’s to compute new 95 % confidence 

intervals, which are also reported. 

Comparing the resulting simulated means and 95 % confidence intervals in the table with the 

original results of Trial 0, it is clear that the means as well as the upper bounds and the lower 

bounds of the confidence intervals fluctuate only by no more than ±0.0002. This indicates that 

the simulation results are very stable. It follows that the means and 95 % confidence intervals 

reported in Table 1 & Table 2 are significant, even though for the MIN:A.XX_X category only 

six error rates were used in the bootstrap. 

MIN:A.XX_X 

Trial 

AA_A BB_B CC_C DD_D FF_F GG_G 

Mean Conf. Interval 

0 0.0024 0.0024 0.0032 0.0013 0.0031 0.0007 0.0022 (0.0013, 0.0028) 

1 0.0026 0.0026 0.0033 0.0013 0.0035 0.0007 0.0023 (0.0014, 0.0030) 

2 0.0023 0.0024 0.0032 0.0011 0.0033 0.0008 0.0022 (0.0013, 0.0029) 

3 0.0024 0.0027 0.0031 0.0013 0.0029 0.0006 0.0021 (0.0012, 0.0028) 

4 0.0024 0.0027 0.0030 0.0014 0.0035 0.0006 0.0023 (0.0013, 0.0029) 

5 0.0019 0.0020 0.0035 0.0013 0.0028 0.0007 0.0020 (0.0013, 0.0028) 

Table 5. Original results (Trial 0) of mean FNMR and 95 % confidence interval compared 

to bootstrap simulation results (Trails 1-5) computed by resampling the underlying genuine 

and impostor scores of Trial 0. (All results are from MIN:A.XX_X on the POEBVA 

dataset with FNMR’s computed at an FMR = 0.01.) 

13
REFERENCES 

[1] J.C. Wu, C.L. Wilson; Nonparametric Analysis of Fingerprint Data on Large Datasets; 

Pattern Recognition, 2007; in press. 

[2] J.C. Wu, C.L. Wilson; An Empirical Study of Sample Size in ROC-Curve Analysis of 

Fingerprint Data; Proceedings of SPIE Volume: 6202; Biometric Technology for Human 

Identification III, Editors: P.J. Flynn, S. Pankanti, April 2006. 

[3] Minutiae Interoperability Exchange Test 2004 (MINEX04); 

http://fingerprint.nist.gov/minex04/index.html. 

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