How should we compute confidence bands about an empirical distribution using an assumption about the shape of the underlying distribution such as normality?

There can be uncertainty about the shape of a probability distribution because the sample size of empirical data characterizing it is small. Several methods have been proposed to account for the sampling uncertainty about the distribution shape, including Kolmogorov-Smirnov (KS) confidence bands (Crow et al. 1960) and similar confidence bands (Owen 1995), which are distribution-free in the sense that they make no assumption about the shape family of the underlying distribution. There are analogous confidence-band methods that do make assumptions about the shape or family of the underlying distribution, which can often result in tighter confidence bands (Cheng and Iles 1983; Cheng et al. 1988; Murphy 1995).

Constructing confidence bands requires one to select the probability defining the confidence level, which usually must be less than 100% for the result to be non-vacuous. Confidence bands at the (1−α)% confidence level are defined such that, (1−α)% of the time they are constructed, they will completely enclose the distribution from which the data were randomly sampled. A confidence band about a distribution function is sometimes used as a p-box even though it represents statistical rather than rigorous or sure bounds. This use implicitly assumes that the true distribution, whatever it is, is inside the p-box.

We have implemented KS confidence bands, and we have straightforwardly generalized them for interval sample data. We have begun but not completed the implementation of the methods of Cheng and Iles for normal and other distribution shapes. (Code in R is available.) We have heard however that Cheng and Iles bands are obsolete because newer, better methods based on log-likelihood ratios can now be computed. We need to understand the state of the art for computing confidence bands for empirical distribution functions, both with and without distributional shape assumptions. And we want to extend that state of the art to handle interval data.

The graphs above depict an empirical distribution function (in red) for some data set of sample size 30, and the 95% KS confidence band (on the left) and an approximate 95% confidence band assuming normality of the underlying distribution from which the 30 samples were drawn (on the right).

Confidence intervals at a given confidence level are not uniquely defined, and confidence bands have still greater potential variations. So there are many ways to compute confidence bands. Vicky Montgomery (http://maths.dur.ac.uk/stats/people/fc/thesis-VM.pdf) derived a direct way to compute confidence bands about a normal distribution from sample data. She asserts that it extends to a few other distributions such as exponential and lognormal. It may be worthwhile to understand the variety of approaches, so, whatever the state of the art on the matter, we may want to implement the methods of Montgomery, and possibly those of Chen and Iles. And we also want to generalize them to handle interval sample data.

The confidence band in the right graph above is an approximate one that we stumbled upon. We created a p-box as the envelope of the four normal distributions whose means and standard deviations were the extreme values of univariate confidence intervals (both at the same intended confidence level) for the mean and the standard deviation for normal data. Although one might have expected such a p-box to be too tight, we have observed in Monte Carlo simulations that this p-box covers the true distribution with surprisingly good performance. The fidelity to the nominal coverage rate improves as the confidence level increases. The approach seems to generalize to other distribution shapes pretty well, even though it is of course a complete hack.

A similar hack has been shown by simulation to allow us to compute nontrivial confidence bands for a normal distribution with only a single random data point. Carlos Rodríguez, who by the way used to be at Stony Brook, has an interesting on-line manuscript (http://arxiv.org/abs/bayes-an/9504001v1) that describes how to compute 95% confidence intervals for the mean and standard deviation for the normal distribution from which the sample came. He says [x - 9.679|x|, x + 9.679|x| ] is a 95% confidence interval for the mean, and [ 0, 17|x|] is a 95% confidence interval for the standard deviation. The R script below gives an example of Rodríguez’s one-point confidence band. The function in blue creates the confidence band from the sample value x and a specified confidence level.

I also did a simulation checking the confidence p-box described on the previous page. The R script and its output are below. There is actually a lot of other code available for review from previous simulation studies, some of which involves animations.

onepointnormal = function(x, c=0.95) {

cc = c(0.9, 0.95, 0.99)

tm = c(4.83952, 9.678851, 48.39413)

ts = c(8,17,70)

k = which(cc==c)

return(normal(interval(x - tm[[k]]*abs(x), x + tm[[k]]*abs(x)), interval(0, ts[[k]]*abs(x))))

}

outside = function(m,s,m1,m2,s1,s2) (m < pmin(m1,m2)) | (pmax(m1,m2) < m) | (pmax(s1,s2) < s) | (s < pmin(s1,s2))

many = 1000000

mm = -10:10

vv = c(10.0, 5.0, 2.5, 1, 0.5, 0.25, 0.1, 0.01)

cc = c(0.9, 0.95, 0.99) # confidence levels

ttm = c(4.83952, 9.678851, 48.39413)

tts = c(8,17,70)

mmm <<- sss <<- ccc <<- ppp <<- NULL

for (m in mm) for (v in vv) for (k in 1:length(cc)) {

s = abs(v*m)

x = rnorm(many,m,s)

t = ttm[[k]]

mL = x - t*abs(x)

mR = x + t*abs(x)

t = tts[[k]]

sL = 0

sR = t*abs(x)

o = outside(m,s,mL,mR,sL,sR)

p = length(o[o==FALSE])/many

cat('mean =',m,'\t',

'std =',s,'\t',

'level =',cc[[k]],'\t',

'coverage =', p,'\n')

mmm <<- c(mmm,m)

sss <<- c(sss,s)

ccc <<- c(ccc,cc[[k]])

ppp <<- c(ppp,p)

}

plot(ccc,ppp,ylim=c(min(0.9,ppp),max(1,ppp)))

lines(c(0.9,1),c(0.9,1),col='red')

min(ppp[ccc==.9])

min(ppp[ccc==.95])

min(ppp[ccc==.99])

mean = -10 std = 100 level = 0.9 coverage = 0.90146

mean = -10 std = 100 level = 0.95 coverage = 0.953174

mean = -10 std = 100 level = 0.99 coverage = 0.988459

mean = -10 std = 50 level = 0.9 coverage = 0.902545

mean = -10 std = 50 level = 0.95 coverage = 0.953973

mean = -10 std = 50 level = 0.99 coverage = 0.988754

mean = -10 std = 25 level = 0.9 coverage = 0.908344

mean = -10 std = 25 level = 0.95 coverage = 0.957161

mean = -10 std = 25 level = 0.99 coverage = 0.989588

mean = -10 std = 10 level = 0.9 coverage = 0.90005

mean = -10 std = 10 level = 0.95 coverage = 0.950137

mean = -10 std = 10 level = 0.99 coverage = 0.990114

mean = -10 std = 5 level = 0.9 coverage = 0.957454

mean = -10 std = 5 level = 0.95 coverage = 0.977944

mean = -10 std = 5 level = 0.99 coverage = 0.995543

mean = -10 std = 2.5 level = 0.9 coverage = 0.99954

mean = -10 std = 2.5 level = 0.95 coverage = 0.999838

mean = -10 std = 2.5 level = 0.99 coverage = 0.999975

mean = -10 std = 1 level = 0.9 coverage = 1

mean = -10 std = 1 level = 0.95 coverage = 1

mean = -10 std = 1 level = 0.99 coverage = 1

mean = -10 std = 0.1 level = 0.9 coverage = 1

mean = -10 std = 0.1 level = 0.95 coverage = 1

mean = -10 std = 0.1 level = 0.99 coverage = 1

mean = -9 std = 90 level = 0.9 coverage = 0.901297

mean = -9 std = 90 level = 0.95 coverage = 0.953176

mean = -9 std = 90 level = 0.99 coverage = 0.988683

mean = -9 std = 45 level = 0.9 coverage = 0.902647

mean = -9 std = 45 level = 0.95 coverage = 0.954288

mean = -9 std = 45 level = 0.99 coverage = 0.988759

mean = -9 std = 22.5 level = 0.9 coverage = 0.908553

mean = -9 std = 22.5 level = 0.95 coverage = 0.956828

mean = -9 std = 22.5 level = 0.99 coverage = 0.989618

mean = -9 std = 9 level = 0.9 coverage = 0.900507

mean = -9 std = 9 level = 0.95 coverage = 0.949882

mean = -9 std = 9 level = 0.99 coverage = 0.989885

mean = -9 std = 4.5 level = 0.9 coverage = 0.957159

mean = -9 std = 4.5 level = 0.95 coverage = 0.97774

mean = -9 std = 4.5 level = 0.99 coverage = 0.99551

mean = -9 std = 2.25 level = 0.9 coverage = 0.999513

mean = -9 std = 2.25 level = 0.95 coverage = 0.999864

mean = -9 std = 2.25 level = 0.99 coverage = 0.999976

mean = -9 std = 0.9 level = 0.9 coverage = 1

mean = -9 std = 0.9 level = 0.95 coverage = 1

mean = -9 std = 0.9 level = 0.99 coverage = 1

mean = -9 std = 0.09 level = 0.9 coverage = 1

mean = -9 std = 0.09 level = 0.95 coverage = 1

mean = -9 std = 0.09 level = 0.99 coverage = 1

mean = -8 std = 80 level = 0.9 coverage = 0.900901

mean = -8 std = 80 level = 0.95 coverage = 0.95332

mean = -8 std = 80 level = 0.99 coverage = 0.988584

mean = -8 std = 40 level = 0.9 coverage = 0.902165

mean = -8 std = 40 level = 0.95 coverage = 0.953905

mean = -8 std = 40 level = 0.99 coverage = 0.988829

mean = -8 std = 20 level = 0.9 coverage = 0.907827

mean = -8 std = 20 level = 0.95 coverage = 0.95675

mean = -8 std = 20 level = 0.99 coverage = 0.989531

mean = -8 std = 8 level = 0.9 coverage = 0.900036

mean = -8 std = 8 level = 0.95 coverage = 0.950029

mean = -8 std = 8 level = 0.99 coverage = 0.989964

mean = -8 std = 4 level = 0.9 coverage = 0.957251

mean = -8 std = 4 level = 0.95 coverage = 0.977877

mean = -8 std = 4 level = 0.99 coverage = 0.995571

mean = -8 std = 2 level = 0.9 coverage = 0.999512

mean = -8 std = 2 level = 0.95 coverage = 0.999862

mean = -8 std = 2 level = 0.99 coverage = 0.999983

mean = -8 std = 0.8 level = 0.9 coverage = 1

mean = -8 std = 0.8 level = 0.95 coverage = 1

mean = -8 std = 0.8 level = 0.99 coverage = 1

mean = -8 std = 0.08 level = 0.9 coverage = 1

mean = -8 std = 0.08 level = 0.95 coverage = 1

mean = -8 std = 0.08 level = 0.99 coverage = 1

mean = -7 std = 70 level = 0.9 coverage = 0.900827

mean = -7 std = 70 level = 0.95 coverage = 0.953687

mean = -7 std = 70 level = 0.99 coverage = 0.988885

mean = -7 std = 35 level = 0.9 coverage = 0.902713

mean = -7 std = 35 level = 0.95 coverage = 0.954298

mean = -7 std = 35 level = 0.99 coverage = 0.988857

mean = -7 std = 17.5 level = 0.9 coverage = 0.90826

mean = -7 std = 17.5 level = 0.95 coverage = 0.956197

mean = -7 std = 17.5 level = 0.99 coverage = 0.989414

mean = -7 std = 7 level = 0.9 coverage = 0.900508

mean = -7 std = 7 level = 0.95 coverage = 0.950176

mean = -7 std = 7 level = 0.99 coverage = 0.990109

mean = -7 std = 3.5 level = 0.9 coverage = 0.957162

mean = -7 std = 3.5 level = 0.95 coverage = 0.977689

mean = -7 std = 3.5 level = 0.99 coverage = 0.995516

mean = -7 std = 1.75 level = 0.9 coverage = 0.999558

mean = -7 std = 1.75 level = 0.95 coverage = 0.999845

mean = -7 std = 1.75 level = 0.99 coverage = 0.999976

mean = -7 std = 0.7 level = 0.9 coverage = 1

mean = -7 std = 0.7 level = 0.95 coverage = 1

mean = -7 std = 0.7 level = 0.99 coverage = 1

mean = -7 std = 0.07 level = 0.9 coverage = 1

mean = -7 std = 0.07 level = 0.95 coverage = 1

mean = -7 std = 0.07 level = 0.99 coverage = 1

mean = -6 std = 60 level = 0.9 coverage = 0.901119

mean = -6 std = 60 level = 0.95 coverage = 0.9535

mean = -6 std = 60 level = 0.99 coverage = 0.988682

mean = -6 std = 30 level = 0.9 coverage = 0.902424

mean = -6 std = 30 level = 0.95 coverage = 0.953854

mean = -6 std = 30 level = 0.99 coverage = 0.988731

mean = -6 std = 15 level = 0.9 coverage = 0.908304

mean = -6 std = 15 level = 0.95 coverage = 0.956692

mean = -6 std = 15 level = 0.99 coverage = 0.989581

mean = -6 std = 6 level = 0.9 coverage = 0.900298

mean = -6 std = 6 level = 0.95 coverage = 0.950179

mean = -6 std = 6 level = 0.99 coverage = 0.990242

mean = -6 std = 3 level = 0.9 coverage = 0.956803

mean = -6 std = 3 level = 0.95 coverage = 0.977834

mean = -6 std = 3 level = 0.99 coverage = 0.995568

mean = -6 std = 1.5 level = 0.9 coverage = 0.999595

mean = -6 std = 1.5 level = 0.95 coverage = 0.999835

mean = -6 std = 1.5 level = 0.99 coverage = 0.999974

mean = -6 std = 0.6 level = 0.9 coverage = 1

mean = -6 std = 0.6 level = 0.95 coverage = 1

mean = -6 std = 0.6 level = 0.99 coverage = 1

mean = -6 std = 0.06 level = 0.9 coverage = 1

mean = -6 std = 0.06 level = 0.95 coverage = 1

mean = -6 std = 0.06 level = 0.99 coverage = 1

mean = -5 std = 50 level = 0.9 coverage = 0.901328

mean = -5 std = 50 level = 0.95 coverage = 0.953457

mean = -5 std = 50 level = 0.99 coverage = 0.988611

mean = -5 std = 25 level = 0.9 coverage = 0.901662

mean = -5 std = 25 level = 0.95 coverage = 0.9544

mean = -5 std = 25 level = 0.99 coverage = 0.988727

mean = -5 std = 12.5 level = 0.9 coverage = 0.90851

mean = -5 std = 12.5 level = 0.95 coverage = 0.956684

mean = -5 std = 12.5 level = 0.99 coverage = 0.989538

mean = -5 std = 5 level = 0.9 coverage = 0.899967

mean = -5 std = 5 level = 0.95 coverage = 0.949922

mean = -5 std = 5 level = 0.99 coverage = 0.990181

mean = -5 std = 2.5 level = 0.9 coverage = 0.95688

mean = -5 std = 2.5 level = 0.95 coverage = 0.977816

mean = -5 std = 2.5 level = 0.99 coverage = 0.995537

mean = -5 std = 1.25 level = 0.9 coverage = 0.999524

mean = -5 std = 1.25 level = 0.95 coverage = 0.999854

mean = -5 std = 1.25 level = 0.99 coverage = 0.999979

mean = -5 std = 0.5 level = 0.9 coverage = 1

mean = -5 std = 0.5 level = 0.95 coverage = 1

mean = -5 std = 0.5 level = 0.99 coverage = 1

mean = -5 std = 0.05 level = 0.9 coverage = 1

mean = -5 std = 0.05 level = 0.95 coverage = 1

mean = -5 std = 0.05 level = 0.99 coverage = 1

mean = -4 std = 40 level = 0.9 coverage = 0.900576

mean = -4 std = 40 level = 0.95 coverage = 0.953307

mean = -4 std = 40 level = 0.99 coverage = 0.988697

mean = -4 std = 20 level = 0.9 coverage = 0.902269

mean = -4 std = 20 level = 0.95 coverage = 0.954123

mean = -4 std = 20 level = 0.99 coverage = 0.988713

mean = -4 std = 10 level = 0.9 coverage = 0.907883

mean = -4 std = 10 level = 0.95 coverage = 0.956844

mean = -4 std = 10 level = 0.99 coverage = 0.989483

mean = -4 std = 4 level = 0.9 coverage = 0.900318

mean = -4 std = 4 level = 0.95 coverage = 0.950201

mean = -4 std = 4 level = 0.99 coverage = 0.989906

mean = -4 std = 2 level = 0.9 coverage = 0.957412

mean = -4 std = 2 level = 0.95 coverage = 0.977982

mean = -4 std = 2 level = 0.99 coverage = 0.995659

mean = -4 std = 1 level = 0.9 coverage = 0.999554

mean = -4 std = 1 level = 0.95 coverage = 0.999861

mean = -4 std = 1 level = 0.99 coverage = 0.99997

mean = -4 std = 0.4 level = 0.9 coverage = 1

mean = -4 std = 0.4 level = 0.95 coverage = 1

mean = -4 std = 0.4 level = 0.99 coverage = 1

mean = -4 std = 0.04 level = 0.9 coverage = 1

mean = -4 std = 0.04 level = 0.95 coverage = 1

mean = -4 std = 0.04 level = 0.99 coverage = 1

mean = -3 std = 30 level = 0.9 coverage = 0.90094

mean = -3 std = 30 level = 0.95 coverage = 0.953578

mean = -3 std = 30 level = 0.99 coverage = 0.988356

mean = -3 std = 15 level = 0.9 coverage = 0.902728

mean = -3 std = 15 level = 0.95 coverage = 0.95394

mean = -3 std = 15 level = 0.99 coverage = 0.98867

mean = -3 std = 7.5 level = 0.9 coverage = 0.908238

mean = -3 std = 7.5 level = 0.95 coverage = 0.956665

mean = -3 std = 7.5 level = 0.99 coverage = 0.989543

mean = -3 std = 3 level = 0.9 coverage = 0.900023

mean = -3 std = 3 level = 0.95 coverage = 0.949909

mean = -3 std = 3 level = 0.99 coverage = 0.990035

mean = -3 std = 1.5 level = 0.9 coverage = 0.957263

mean = -3 std = 1.5 level = 0.95 coverage = 0.977906

mean = -3 std = 1.5 level = 0.99 coverage = 0.995605

mean = -3 std = 0.75 level = 0.9 coverage = 0.999507

mean = -3 std = 0.75 level = 0.95 coverage = 0.999843

mean = -3 std = 0.75 level = 0.99 coverage = 0.999982

mean = -3 std = 0.3 level = 0.9 coverage = 1

mean = -3 std = 0.3 level = 0.95 coverage = 1

mean = -3 std = 0.3 level = 0.99 coverage = 1

mean = -3 std = 0.03 level = 0.9 coverage = 1

mean = -3 std = 0.03 level = 0.95 coverage = 1

mean = -3 std = 0.03 level = 0.99 coverage = 1

mean = -2 std = 20 level = 0.9 coverage = 0.900875

mean = -2 std = 20 level = 0.95 coverage = 0.953152

mean = -2 std = 20 level = 0.99 coverage = 0.988782

mean = -2 std = 10 level = 0.9 coverage = 0.902647

mean = -2 std = 10 level = 0.95 coverage = 0.953945

mean = -2 std = 10 level = 0.99 coverage = 0.988827

mean = -2 std = 5 level = 0.9 coverage = 0.907894

mean = -2 std = 5 level = 0.95 coverage = 0.956392

mean = -2 std = 5 level = 0.99 coverage = 0.989503

mean = -2 std = 2 level = 0.9 coverage = 0.8999

mean = -2 std = 2 level = 0.95 coverage = 0.950107

mean = -2 std = 2 level = 0.99 coverage = 0.989982

mean = -2 std = 1 level = 0.9 coverage = 0.957324

mean = -2 std = 1 level = 0.95 coverage = 0.977835

mean = -2 std = 1 level = 0.99 coverage = 0.995475

mean = -2 std = 0.5 level = 0.9 coverage = 0.99955

mean = -2 std = 0.5 level = 0.95 coverage = 0.999868

mean = -2 std = 0.5 level = 0.99 coverage = 0.999976

mean = -2 std = 0.2 level = 0.9 coverage = 1

mean = -2 std = 0.2 level = 0.95 coverage = 1

mean = -2 std = 0.2 level = 0.99 coverage = 1

mean = -2 std = 0.02 level = 0.9 coverage = 1

mean = -2 std = 0.02 level = 0.95 coverage = 1

mean = -2 std = 0.02 level = 0.99 coverage = 1

mean = -1 std = 10 level = 0.9 coverage = 0.900994

mean = -1 std = 10 level = 0.95 coverage = 0.953378

mean = -1 std = 10 level = 0.99 coverage = 0.988584

mean = -1 std = 5 level = 0.9 coverage = 0.902692

mean = -1 std = 5 level = 0.95 coverage = 0.954426

mean = -1 std = 5 level = 0.99 coverage = 0.988806

mean = -1 std = 2.5 level = 0.9 coverage = 0.908552

mean = -1 std = 2.5 level = 0.95 coverage = 0.956521

mean = -1 std = 2.5 level = 0.99 coverage = 0.989274

mean = -1 std = 1 level = 0.9 coverage = 0.899838

mean = -1 std = 1 level = 0.95 coverage = 0.95021

mean = -1 std = 1 level = 0.99 coverage = 0.990094

mean = -1 std = 0.5 level = 0.9 coverage = 0.957085

mean = -1 std = 0.5 level = 0.95 coverage = 0.978178

mean = -1 std = 0.5 level = 0.99 coverage = 0.995688

mean = -1 std = 0.25 level = 0.9 coverage = 0.999517

mean = -1 std = 0.25 level = 0.95 coverage = 0.99987

mean = -1 std = 0.25 level = 0.99 coverage = 0.999975

mean = -1 std = 0.1 level = 0.9 coverage = 1

mean = -1 std = 0.1 level = 0.95 coverage = 1

mean = -1 std = 0.1 level = 0.99 coverage = 1

mean = -1 std = 0.01 level = 0.9 coverage = 1

mean = -1 std = 0.01 level = 0.95 coverage = 1

mean = -1 std = 0.01 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 0 std = 0 level = 0.9 coverage = 1

mean = 0 std = 0 level = 0.95 coverage = 1

mean = 0 std = 0 level = 0.99 coverage = 1

mean = 1 std = 10 level = 0.9 coverage = 0.900367

mean = 1 std = 10 level = 0.95 coverage = 0.953059

mean = 1 std = 10 level = 0.99 coverage = 0.988591

mean = 1 std = 5 level = 0.9 coverage = 0.902865

mean = 1 std = 5 level = 0.95 coverage = 0.954519

mean = 1 std = 5 level = 0.99 coverage = 0.988937

mean = 1 std = 2.5 level = 0.9 coverage = 0.9085

mean = 1 std = 2.5 level = 0.95 coverage = 0.956769

mean = 1 std = 2.5 level = 0.99 coverage = 0.989432

mean = 1 std = 1 level = 0.9 coverage = 0.899753

mean = 1 std = 1 level = 0.95 coverage = 0.949857

mean = 1 std = 1 level = 0.99 coverage = 0.99003

mean = 1 std = 0.5 level = 0.9 coverage = 0.957081

mean = 1 std = 0.5 level = 0.95 coverage = 0.977958

mean = 1 std = 0.5 level = 0.99 coverage = 0.995578

mean = 1 std = 0.25 level = 0.9 coverage = 0.999517

mean = 1 std = 0.25 level = 0.95 coverage = 0.999876

mean = 1 std = 0.25 level = 0.99 coverage = 0.999977

mean = 1 std = 0.1 level = 0.9 coverage = 1

mean = 1 std = 0.1 level = 0.95 coverage = 1

mean = 1 std = 0.1 level = 0.99 coverage = 1

mean = 1 std = 0.01 level = 0.9 coverage = 1

mean = 1 std = 0.01 level = 0.95 coverage = 1

mean = 1 std = 0.01 level = 0.99 coverage = 1

mean = 2 std = 20 level = 0.9 coverage = 0.900946

mean = 2 std = 20 level = 0.95 coverage = 0.953264

mean = 2 std = 20 level = 0.99 coverage = 0.988755

mean = 2 std = 10 level = 0.9 coverage = 0.902506

mean = 2 std = 10 level = 0.95 coverage = 0.953974

mean = 2 std = 10 level = 0.99 coverage = 0.988749

mean = 2 std = 5 level = 0.9 coverage = 0.908199

mean = 2 std = 5 level = 0.95 coverage = 0.9568

mean = 2 std = 5 level = 0.99 coverage = 0.989395

mean = 2 std = 2 level = 0.9 coverage = 0.899994

mean = 2 std = 2 level = 0.95 coverage = 0.949793

mean = 2 std = 2 level = 0.99 coverage = 0.99012

mean = 2 std = 1 level = 0.9 coverage = 0.957363

mean = 2 std = 1 level = 0.95 coverage = 0.977966

mean = 2 std = 1 level = 0.99 coverage = 0.995662

mean = 2 std = 0.5 level = 0.9 coverage = 0.99954

mean = 2 std = 0.5 level = 0.95 coverage = 0.999848

mean = 2 std = 0.5 level = 0.99 coverage = 0.999978

mean = 2 std = 0.2 level = 0.9 coverage = 1

mean = 2 std = 0.2 level = 0.95 coverage = 1

mean = 2 std = 0.2 level = 0.99 coverage = 1

mean = 2 std = 0.02 level = 0.9 coverage = 1

mean = 2 std = 0.02 level = 0.95 coverage = 1

mean = 2 std = 0.02 level = 0.99 coverage = 1

mean = 3 std = 30 level = 0.9 coverage = 0.900915

mean = 3 std = 30 level = 0.95 coverage = 0.953466

mean = 3 std = 30 level = 0.99 coverage = 0.98853

mean = 3 std = 15 level = 0.9 coverage = 0.902884

mean = 3 std = 15 level = 0.95 coverage = 0.954171

mean = 3 std = 15 level = 0.99 coverage = 0.989005

mean = 3 std = 7.5 level = 0.9 coverage = 0.908001

mean = 3 std = 7.5 level = 0.95 coverage = 0.956728

mean = 3 std = 7.5 level = 0.99 coverage = 0.989599

mean = 3 std = 3 level = 0.9 coverage = 0.900134

mean = 3 std = 3 level = 0.95 coverage = 0.95011

mean = 3 std = 3 level = 0.99 coverage = 0.989815

mean = 3 std = 1.5 level = 0.9 coverage = 0.957343

mean = 3 std = 1.5 level = 0.95 coverage = 0.978127

mean = 3 std = 1.5 level = 0.99 coverage = 0.995378

mean = 3 std = 0.75 level = 0.9 coverage = 0.99954

mean = 3 std = 0.75 level = 0.95 coverage = 0.999865

mean = 3 std = 0.75 level = 0.99 coverage = 0.999977

mean = 3 std = 0.3 level = 0.9 coverage = 1

mean = 3 std = 0.3 level = 0.95 coverage = 1

mean = 3 std = 0.3 level = 0.99 coverage = 1

mean = 3 std = 0.03 level = 0.9 coverage = 1

mean = 3 std = 0.03 level = 0.95 coverage = 1

mean = 3 std = 0.03 level = 0.99 coverage = 1

mean = 4 std = 40 level = 0.9 coverage = 0.901514

mean = 4 std = 40 level = 0.95 coverage = 0.953471

mean = 4 std = 40 level = 0.99 coverage = 0.988667

mean = 4 std = 20 level = 0.9 coverage = 0.902503

mean = 4 std = 20 level = 0.95 coverage = 0.954104

mean = 4 std = 20 level = 0.99 coverage = 0.988893

mean = 4 std = 10 level = 0.9 coverage = 0.907939

mean = 4 std = 10 level = 0.95 coverage = 0.956518

mean = 4 std = 10 level = 0.99 coverage = 0.989474

mean = 4 std = 4 level = 0.9 coverage = 0.900175

mean = 4 std = 4 level = 0.95 coverage = 0.950077

mean = 4 std = 4 level = 0.99 coverage = 0.989954

mean = 4 std = 2 level = 0.9 coverage = 0.957297

mean = 4 std = 2 level = 0.95 coverage = 0.977673

mean = 4 std = 2 level = 0.99 coverage = 0.995519

mean = 4 std = 1 level = 0.9 coverage = 0.99957

mean = 4 std = 1 level = 0.95 coverage = 0.999843

mean = 4 std = 1 level = 0.99 coverage = 0.99998

mean = 4 std = 0.4 level = 0.9 coverage = 1

mean = 4 std = 0.4 level = 0.95 coverage = 1

mean = 4 std = 0.4 level = 0.99 coverage = 1

mean = 4 std = 0.04 level = 0.9 coverage = 1

mean = 4 std = 0.04 level = 0.95 coverage = 1

mean = 4 std = 0.04 level = 0.99 coverage = 1

mean = 5 std = 50 level = 0.9 coverage = 0.900845

mean = 5 std = 50 level = 0.95 coverage = 0.953293

mean = 5 std = 50 level = 0.99 coverage = 0.988742

mean = 5 std = 25 level = 0.9 coverage = 0.902493

mean = 5 std = 25 level = 0.95 coverage = 0.954137

mean = 5 std = 25 level = 0.99 coverage = 0.988701

mean = 5 std = 12.5 level = 0.9 coverage = 0.907606

mean = 5 std = 12.5 level = 0.95 coverage = 0.956733

mean = 5 std = 12.5 level = 0.99 coverage = 0.989564

mean = 5 std = 5 level = 0.9 coverage = 0.900381

mean = 5 std = 5 level = 0.95 coverage = 0.949969

mean = 5 std = 5 level = 0.99 coverage = 0.989995

mean = 5 std = 2.5 level = 0.9 coverage = 0.957277

mean = 5 std = 2.5 level = 0.95 coverage = 0.977822

mean = 5 std = 2.5 level = 0.99 coverage = 0.99554

mean = 5 std = 1.25 level = 0.9 coverage = 0.999565

mean = 5 std = 1.25 level = 0.95 coverage = 0.999855

mean = 5 std = 1.25 level = 0.99 coverage = 0.999978

mean = 5 std = 0.5 level = 0.9 coverage = 1

mean = 5 std = 0.5 level = 0.95 coverage = 1

mean = 5 std = 0.5 level = 0.99 coverage = 1

mean = 5 std = 0.05 level = 0.9 coverage = 1

mean = 5 std = 0.05 level = 0.95 coverage = 1

mean = 5 std = 0.05 level = 0.99 coverage = 1

mean = 6 std = 60 level = 0.9 coverage = 0.901056

mean = 6 std = 60 level = 0.95 coverage = 0.953628

mean = 6 std = 60 level = 0.99 coverage = 0.988577

mean = 6 std = 30 level = 0.9 coverage = 0.902501

mean = 6 std = 30 level = 0.95 coverage = 0.954166

mean = 6 std = 30 level = 0.99 coverage = 0.98892

mean = 6 std = 15 level = 0.9 coverage = 0.907952

mean = 6 std = 15 level = 0.95 coverage = 0.956429

mean = 6 std = 15 level = 0.99 coverage = 0.989522

mean = 6 std = 6 level = 0.9 coverage = 0.90021

mean = 6 std = 6 level = 0.95 coverage = 0.949929

mean = 6 std = 6 level = 0.99 coverage = 0.990035

mean = 6 std = 3 level = 0.9 coverage = 0.957086

mean = 6 std = 3 level = 0.95 coverage = 0.97789

mean = 6 std = 3 level = 0.99 coverage = 0.995579

mean = 6 std = 1.5 level = 0.9 coverage = 0.999526

mean = 6 std = 1.5 level = 0.95 coverage = 0.999861

mean = 6 std = 1.5 level = 0.99 coverage = 0.999971

mean = 6 std = 0.6 level = 0.9 coverage = 1

mean = 6 std = 0.6 level = 0.95 coverage = 1

mean = 6 std = 0.6 level = 0.99 coverage = 1

mean = 6 std = 0.06 level = 0.9 coverage = 1

mean = 6 std = 0.06 level = 0.95 coverage = 1

mean = 6 std = 0.06 level = 0.99 coverage = 1

mean = 7 std = 70 level = 0.9 coverage = 0.900829

mean = 7 std = 70 level = 0.95 coverage = 0.953498

mean = 7 std = 70 level = 0.99 coverage = 0.988666

mean = 7 std = 35 level = 0.9 coverage = 0.902375

mean = 7 std = 35 level = 0.95 coverage = 0.954199

mean = 7 std = 35 level = 0.99 coverage = 0.988835

mean = 7 std = 17.5 level = 0.9 coverage = 0.908072

mean = 7 std = 17.5 level = 0.95 coverage = 0.956938

mean = 7 std = 17.5 level = 0.99 coverage = 0.98945

mean = 7 std = 7 level = 0.9 coverage = 0.900605

mean = 7 std = 7 level = 0.95 coverage = 0.949936

mean = 7 std = 7 level = 0.99 coverage = 0.989944

mean = 7 std = 3.5 level = 0.9 coverage = 0.957014

mean = 7 std = 3.5 level = 0.95 coverage = 0.977733

mean = 7 std = 3.5 level = 0.99 coverage = 0.995592

mean = 7 std = 1.75 level = 0.9 coverage = 0.999567

mean = 7 std = 1.75 level = 0.95 coverage = 0.999852

mean = 7 std = 1.75 level = 0.99 coverage = 0.999977

mean = 7 std = 0.7 level = 0.9 coverage = 1

mean = 7 std = 0.7 level = 0.95 coverage = 1

mean = 7 std = 0.7 level = 0.99 coverage = 1

mean = 7 std = 0.07 level = 0.9 coverage = 1

mean = 7 std = 0.07 level = 0.95 coverage = 1

mean = 7 std = 0.07 level = 0.99 coverage = 1

mean = 8 std = 80 level = 0.9 coverage = 0.901081

mean = 8 std = 80 level = 0.95 coverage = 0.953371

mean = 8 std = 80 level = 0.99 coverage = 0.9887

mean = 8 std = 40 level = 0.9 coverage = 0.902631

mean = 8 std = 40 level = 0.95 coverage = 0.954115

mean = 8 std = 40 level = 0.99 coverage = 0.988826

mean = 8 std = 20 level = 0.9 coverage = 0.908033

mean = 8 std = 20 level = 0.95 coverage = 0.956805

mean = 8 std = 20 level = 0.99 coverage = 0.98942

mean = 8 std = 8 level = 0.9 coverage = 0.89986

mean = 8 std = 8 level = 0.95 coverage = 0.950246

mean = 8 std = 8 level = 0.99 coverage = 0.989974

mean = 8 std = 4 level = 0.9 coverage = 0.957185

mean = 8 std = 4 level = 0.95 coverage = 0.977921

mean = 8 std = 4 level = 0.99 coverage = 0.995536

mean = 8 std = 2 level = 0.9 coverage = 0.999538

mean = 8 std = 2 level = 0.95 coverage = 0.999855

mean = 8 std = 2 level = 0.99 coverage = 0.999983

mean = 8 std = 0.8 level = 0.9 coverage = 1

mean = 8 std = 0.8 level = 0.95 coverage = 1

mean = 8 std = 0.8 level = 0.99 coverage = 1

mean = 8 std = 0.08 level = 0.9 coverage = 1

mean = 8 std = 0.08 level = 0.95 coverage = 1

mean = 8 std = 0.08 level = 0.99 coverage = 1

mean = 9 std = 90 level = 0.9 coverage = 0.90081

mean = 9 std = 90 level = 0.95 coverage = 0.953361

mean = 9 std = 90 level = 0.99 coverage = 0.988647

mean = 9 std = 45 level = 0.9 coverage = 0.903148

mean = 9 std = 45 level = 0.95 coverage = 0.953831

mean = 9 std = 45 level = 0.99 coverage = 0.988847

mean = 9 std = 22.5 level = 0.9 coverage = 0.908524

mean = 9 std = 22.5 level = 0.95 coverage = 0.956632

mean = 9 std = 22.5 level = 0.99 coverage = 0.989471

mean = 9 std = 9 level = 0.9 coverage = 0.899952

mean = 9 std = 9 level = 0.95 coverage = 0.94963

mean = 9 std = 9 level = 0.99 coverage = 0.990026

mean = 9 std = 4.5 level = 0.9 coverage = 0.957099

mean = 9 std = 4.5 level = 0.95 coverage = 0.9777

mean = 9 std = 4.5 level = 0.99 coverage = 0.995621

mean = 9 std = 2.25 level = 0.9 coverage = 0.999526

mean = 9 std = 2.25 level = 0.95 coverage = 0.999855

mean = 9 std = 2.25 level = 0.99 coverage = 0.999981

mean = 9 std = 0.9 level = 0.9 coverage = 1

mean = 9 std = 0.9 level = 0.95 coverage = 1

mean = 9 std = 0.9 level = 0.99 coverage = 1

mean = 9 std = 0.09 level = 0.9 coverage = 1

mean = 9 std = 0.09 level = 0.95 coverage = 1

mean = 9 std = 0.09 level = 0.99 coverage = 1

mean = 10 std = 100 level = 0.9 coverage = 0.900966

mean = 10 std = 100 level = 0.95 coverage = 0.953358

mean = 10 std = 100 level = 0.99 coverage = 0.988468

mean = 10 std = 50 level = 0.9 coverage = 0.902639

mean = 10 std = 50 level = 0.95 coverage = 0.954242

mean = 10 std = 50 level = 0.99 coverage = 0.988912

mean = 10 std = 25 level = 0.9 coverage = 0.908217

mean = 10 std = 25 level = 0.95 coverage = 0.956788

mean = 10 std = 25 level = 0.99 coverage = 0.989347

mean = 10 std = 10 level = 0.9 coverage = 0.900262

mean = 10 std = 10 level = 0.95 coverage = 0.949955

mean = 10 std = 10 level = 0.99 coverage = 0.989977

mean = 10 std = 5 level = 0.9 coverage = 0.95688

mean = 10 std = 5 level = 0.95 coverage = 0.977909

mean = 10 std = 5 level = 0.99 coverage = 0.995468

mean = 10 std = 2.5 level = 0.9 coverage = 0.999543

mean = 10 std = 2.5 level = 0.95 coverage = 0.99982

mean = 10 std = 2.5 level = 0.99 coverage = 0.999982

mean = 10 std = 1 level = 0.9 coverage = 1

mean = 10 std = 1 level = 0.95 coverage = 1

mean = 10 std = 1 level = 0.99 coverage = 1

mean = 10 std = 0.1 level = 0.9 coverage = 1

mean = 10 std = 0.1 level = 0.95 coverage = 1

mean = 10 std = 0.1 level = 0.99 coverage = 1

There were 50 or more warnings (use warnings() to see the first 50)

>

> plot(ccc,ppp,ylim=c(min(0.9,ppp),max(1,ppp)))

> lines(c(0.9,1),c(0.9,1),col='red')

>

> min(ppp[ccc==.9])

[1] 0.899753

> min(ppp[ccc==.95])

[1] 0.94963

> min(ppp[ccc==.99])

[1] 0.988356

It seems clear (but perhaps requires demonstration) that we can apply this construction when the value of the sample is an interval, but we would also like to extend the idea so that we can develop Rodríguezconfidence bands for

• other confidence levels,

• other sample sizes, and

• other distribution shapes.

Although Rodríguez describes the derivation of the confidence interval, it is not quite clear how to find analogous intervals at other confidence levels beyond those he explicitly gives in his paper. Neither is it clear how we could generalize it for two or a handful of points. We could extend it to lognormal distributions with a transformation, but it is not clear how to extend the argument to other important distribution families. We would especially like to extend the method for symmetric, unimodal distributions generally. Rodríguez gives a 95% confidence interval for the mean of a symmetric unimodal distribution as [x – 19|x|, x + 19|x|], but it is not clear whether or how this could be parlayed into a confidence band that only assumes symmetry, unimodality and a single random sample.

Below is a partial bibliography of the subject of confidence bands. I feel sure that it is not up to date.

Arnold, B.C., and R.M. Shavelle. 1998. Joint confidence sets for the mean and variance of a normal distribution. The American Statistician 52: <<>>.

Cheng, R.C.H. 1987. Confidence bands for two-stage design problems. Technometrics 29: 301-309.

Cheng, R.C.H. and T.C. Iles. 1983. Confidence bands for cumulative distribution functions of continuous random variables. Technometrics 25: 77-86.

Cheng, R.C.H. and T.C. Iles. 1988. One-sided confidence bands for cumulative distribution function. Technometrics 30: 155-159.

Cheng, R.C.H., B.E. Evans and J.E. Williams. 1988. Confidence band estimations for distributions used in probabilistic design. Intemational Journal of Mechanical Sciences 30: 835-845.

Cheng, R.C.H., B.G. Williams and M. Cooper. 1971. The treatment of errors in the deconvolution of line profile measurements. Philosophical Magazine 23: 181, 115-133.

Cooper, N.R., J. Margetson, S. Humble. 1986. Probability of failure calculations and confidence band estimates for an annular brittle disc fractured under centrifugal loading. Journal The Journal of Strain Analysis for Engineering Design 21: 121-126.

Escobar, L.A., Y. Hong, and W.Q. Meeker. 2009. Simultaneous confidence bands and regions for log-location-scale distributions with censored data. Journal of Statistical Planning and Inference 139: 3231-3245.

Frey, J., O. Marrero and D. Norton. 2008. Minimum-area confidence sets for a normal distribution. Journal of Statistical Planning and Inference 139: 1023-1032.

Hong, Y., L.A. Escobar and W.Q. Meeker. 2010. Coverage probabilities of simultaneous confidence bands and regions for log-location-scale distributions. Statistics & Probability Letters 80: 733-738.

Jeng, S.-L., and W.Q. Meeker. 2001. Parametric simultaneous confidence bands for cumulative distributions from censored data. Technometrics 43: 450-461.

Kanofsky, P., and R. Srinivasan. 1972. An approach to the construction of parametric confidence bands on cumulative distribution functions. Biometrika <<>>.

Littell, R.C., and K.M. Portier. 1992. Robust parametric confidence bands for distribution functions. Journal of Statistical Computation and Simulation 41: 187-200.

Rosenkrantz, W.A. 2000. Confidence bands for quantile functions: a parametric and graphic alternative for testing goodness of fit. The American Statistician 54: 185-190.