Generalized P-Values and Confidence Intervals: A Novel Approach for Analyzing Lognormally Distributed Exposure Data

ArticleinJournal of Occupational and Environmental Hygiene 3(11):642-50 · December 2006with91 Reads
Impact Factor: 1.17 · DOI: 10.1080/15459620600961196 · Source: PubMed
Abstract

The problem of assessing occupational exposure using the mean of a lognormal distribution is addressed. The novel concepts of generalized p-values and generalized confidence intervals are applied for testing hypotheses and computing confidence intervals for a lognormal mean. The proposed methods perform well, they are applicable to small sample sizes, and they are easy to implement. Power studies and sample size calculation are also discussed. Computational details and a source for the computer program are given. The procedures are also extended to compare two lognormal means and to make inference about a lognormal variance. In fact, our approach based on generalized p-values and generalized confidence intervals is easily adapted to deal with any parametric function involving one or two lognormal distributions. Several examples involving industrial exposure data are used to illustrate the methods. An added advantage of the generalized variables approach is the ease of computation and implementation. In fact, the procedures can be easily coded in a programming language for implementation. Furthermore, extensive numerical computations by the authors show that the results based on the generalized p-value approach are essentially equivalent to those based on the Land's method. We want to draw the attention of the industrial hygiene community to this accurate and unified methodology to deal with any parameter associated with the lognormal distribution.

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Journal of Occupational and Environmental Hygiene,3:642–650
ISSN: 1545-9624 print / 1545-9632 online
Copyright
c
2006 JOEH, LLC
DOI: 10.1080/15459620600961196
Generalized P-Values and Confidence Intervals:
ANovel Approach for Analyzing Lognormally
Distributed Exposure Data
K. Krishnamoorthy,
1
Thomas Mathew,
2
and Gurumurthy Ramachandran
3
1
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana
2
Department of Mathematics and Statistics, University of Maryland, Baltimore, Maryland
3
Division of Environmental Health Sciences, School of Public Health, University of Minnesota,
Minneapolis, Minnesota
The problem of assessing occupational exposure using the
mean of a lognormal distribution is addressed. The novel
concepts of generalized p-values and generalized confidence
intervals are applied for testing hypotheses and computing
confidence intervals for a lognormal mean. The proposed
methods perform well, they are applicable to small sample
sizes, and they are easy to implement. Power studies and
sample size calculation are also discussed. Computational
details and a source for the computer program are given.
The procedures are also extended to compare two lognormal
means and to make inference about a lognormal variance.
In fact, our approach based on generalized p-values and
generalized confidence intervals is easily adapted to deal
with any parametric function involving one or two lognormal
distributions. Several examples involving industrial exposure
data are used to illustrate the methods. An added advantage of
the generalized variables approach is the ease of computation
and implementation. In fact, the procedures can be easily coded
in a programming language for implementation. Furthermore,
extensive numerical computations by the authors show that
the results based on the generalized p-value approach are
essentially equivalent to those based on the Land’s method. We
want to draw the attention of the industrial hygiene community
to this accurate and unified methodology to deal with any
parameter associated with the lognormal distribution.
Keywords confidence interval, hypothesis test, Type 1 error
Address correspondence to: K. Krishnamoorthy, Department
of Mathematics, 217 Maxim D. Doucet Hall, P.O. Box 41010,
University of Louisiana at Lafayette, Lafayette, LA 70504; e-mail:
krishna@louisiana.edu.
INTRODUCTION
I
t has been well established that occupational exposure
data and pollution data very often follow the lognormal
distribution. Since Oldham’s
(1)
1953 report that the distribution
of dust levels in coal mines is approximately lognormal, several
authors have postulated the lognormal model for studying and
analyzing workplace pollutant data.
(28)
The most common
explanation for this phenomenon is as follows: workplace
concentrations are related to rates of contaminant generation
and ventilation rates that are variable. Workers move around in
this nonuniform environment, and their activity patterns also
vary from day to day. The workers’ exposures are related to
the above factors in a multiplicative manner. Irrespective of the
distribution of contaminant generation rates, ventilation rates,
and worker activity patterns, their multiplicative interactions
typically lead to exposure distributions that are right skewed
and described well by the lognormal probability distribution.
The validity of lognormality assumption for a given data
set can be easily tested. The fact that the data y
1
,...,y
n
are
said to follow a lognormal distribution if ln(y
1
),...,ln(y
n
)
follow a normal distribution (where “ln” denotes the natural
logarithm) allows us to adequately validate the assumption of
lognormality of a given data set. Thus, testing for lognormality
is simply a matter of validating the normality assumption for
the logged data, and this can be done using many widely
available software programs such as Minitab, SPSS, and SAS
or using some popular methods such as Shapiro-Wilks test or
Anderson-Darling test.
If we let y denote the lognormally distributed exposure
measurement of an employee, then x = ln(y)isdistributed
normally with mean and standard deviation to be denoted by µ
l
and σ
l
, respectively, and the mean of the lognormal distribution
(say, µ)isgivenby
µ = exp(η), where η = µ
l
+ σ
2
l
/2. (1)
If repeated exposure measurements are available from a
single worker, then µ can be viewed as the mean of the
worker, and our approach can be used to estimate the individual
worker’s mean. Our approach is also applicable to estimate
the mean of a similarly exposed group (SEG) of workers
if only one exposure measurement is obtained per worker
642 Journal of Occupational and Environmental Hygiene November 2006
Page 1
NOMENCLATURE
y
1
,...,y
n
sample from a lognormal distribution
x
1
,...,x
n
logged data; x
i
= ln(y
i
), i = 1,...,n
µ
l
population mean of the logged data
σ
l
population standard deviation of the logged data
µ mean of the lognormal distribution;
µ = exp (µ
l
+σ
2
l
/2)
σ
2
variance of the lognormal distribution;
σ
2
= exp (2µ
l
+ σ
2
l
)[exp(σ
2
l
) 1]
σ
g
geometric standard deviation; σ
g
= exp(σ
l
)
¯
x sample mean of the logged data
s sample standard deviation of the logged data
or to estimate the mean contaminant level in a workplace.
If multiple measurements exist for each worker, and both
between- and within-worker variability are significant and need
to be accounted for, then one should use the random effects
model.
(9,10,11)
The sample mean exposure can be used as an estimate of the
long-term average exposure or the average exposure for a SEG
of workers over an extended period of time. For substances
that cause health effects due to chronic exposures, day-to-day
variability in long-term exposures is less health relevant than
the long-term mean. For such exposures, the arithmetic mean
is the best measure of cumulative exposure over a biologically
relevant time period, since the body would have integrated
exposures over this time period.
(9)
The long-term mean is of
relevance in occupational epidemiology where the estimated
value of the long-term mean is assigned to all workers in a SEG.
Once lognormality has been verified for an exposure sample,
inferences on the parameters of the lognormal distribution
can be made. Whereas there are currently only a few legal
standards and threshold values based on long-term averages,
some researchers have explored the statistics of exposures
exceeding long-term limits.
(3)
To show that the mean exposure
does not exceed the long-term average exposure limit (LTA-
OEL), we may want to test the hypotheses
H
0
: µ LTA-OEL vs. H
a
: µ<LTA-OEL (2)
Note that the null and alternative hypotheses in Eq. 2 are
set up to look for evidence in favor of H
a
. Rejection of the
null hypothesis in Eq. 2 implies that the exposure level is
acceptable.
Another method of assessing workplace exposure, sug-
gested by some investigators,
(2,5,8)
is based on the proportion
of exposure data in excess of the LTA-OEL. Because the
proportion of the measurements that are above the LTA-OEL
is equal to the proportion of the logged measurements that
are above ln(LTA-OEL), this approach reduces to the problem
of hypothesis testing about an upper quantile of a normal
distribution. This hypothesis testing can be carried out using an
appropriate tolerance limit of the normal distribution, and it has
been well addressed in the context of assessing occupational
exposure by Tuggle,
(2)
Selvin et al.,
(5)
and Lyles and Kupper.
(8)
In the context of exposure assessment, the problem of
comparing two lognormal means will arise when we want to
compare exposure levels of two similarly exposed groups of
workers, or when we want to compare two exposure assessment
methods or two different sampling devices. Thus, let y
1
and y
2
be lognormally distributed random variables denoting
exposure levels at two different sites or measurements obtained
by two different methods, and let µ
l1
, µ
l2
and σ
2
l1
, σ
2
l2
denote
the respective means and variances of the normally distributed
random variables ln(y
1
) and ln(y
2
). Then the means of x
1
and
x
2
, say µ
1
and µ
2
, respectively, are given by
µ
1
= exp(η
1
), and µ
2
= exp(η
2
),
where η
1
= µ
l1
+ σ
2
l1
/2 and η
2
= µ
l2
+ σ
2
l2
/2.
(3)
For comparing the exposure levels at the two sites, it is of
interest to test the hypotheses
H
0
: µ
1
µ
2
vs. H
a
: µ
1
2
. (4)
Land
(12)
has proposed exact methods for constructing
confidence intervals and hypothesis tests for the lognormal
mean. His methods, however, are computationally intensive
and depend on the standard deviation of the logged data, which
makes the necessary tabulation difficult. For this reason, Rap-
paport and Selvin
(3)
proposed a simple approximate method
that is satisfactory as long as σ
2
l
3 and n > 5. Zhou and
Gao
(13)
reviewed and compared several approximate methods
and concluded that all the approximate methods are either too
conservative or liberal, except for large samples, in which case,
a method developed by Cox
(12)
is satisfactory. Armstrong
(14)
compared four approximate methods for estimating the confi-
dence intervals (CI) with Land’s
(12)
exact interval. These were
the (a) “simple t-interval, (b) the “lognormal t-interval, (c)
the Cox interval proposed by Land,
(15)
and (d) a variation
of the Cox interval. Armstrong
(14)
found that whereas some
of these approximate intervals were adequate for large sample
sizes (n 25) or small geometric standard deviations (σ
g
=
1.5), none of them were accurate for small sample sizes
and large σ
g
—precisely the situations that are commonly
encountered in occupational exposure assessment. Hewett
and Ganser
(16)
have developed procedures that considerably
simplify the calculation of Land’s exact confidence interval.
In a recent article, Taylor and colleagues
(17)
evaluated several
approximate confidence intervals in terms of their coverage
probabilities and also suggested an improved approximation.
Very little work is available on the problem of comparing
two lognormal means. A large sample test is derived in Zhou
et al.
(18)
for testing the equality of two lognormal means.
Journal of Occupational and Environmental Hygiene November 2006 643
Page 2
The purpose of this article is to illustrate the application
of a novel approach for carrying out tests and confidence
intervals for a single lognormal mean, for the ratio of two
lognormal means, for a single lognormal variance, and for the
ratio of two lognormal variances. The approach is based on the
concepts of generalized p-values and generalized confidence
intervals, collectively referred to as the generalized variables
method. The generalized variables methodology is already
described in Krishnamoorthy and Mathew
(19)
for obtaining
tests and confidence intervals for a single lognormal mean, and
for comparing two lognormal means; however, the lognormal
variance is not considered in that article. In this article, we
extend this approach for obtaining confidence intervals for
the lognormal variance. Even though the lognormal mean
is addressed by Krishnamoorthy and Mathew, we shall first
briefly review the generalized variables procedure for the
lognormal means described in their article and then apply
it to the lognormal variance. We want to draw the attention
of the industrial hygiene community to an accurate and
unified methodology to deal with any parameter associated
with the lognormal distribution. An added advantage of the
generalized variables approach is the ease of computation and
implementation. In fact the procedures can be easily coded
in a programming language for implementation. Furthermore,
extensive numerical results by the authors
(19)
show that for one-
sided tests concerning a single lognormal mean, the results
based on the generalized p-value approach are essentially
equivalent to those based on the Land’s
(12)
method.
The concept of generalized p-value was originally intro-
duced by Tsui and Weerahandi,
(20)
and the concept of gener-
alized confidence intervals was introduced by Weerahandi.
(21)
A later book by Weerahandi
(22)
illustrates several nonstandard
statistical problems where the generalized variable approach
produced remarkably useful results. Because the concepts are
not well known, we have presented them in a brief outline
in Appendix 1. In this article, we first present generalized
variables for making inferences about a normal mean and
variance. We then outline the hypothesis testing and interval
estimation procedures for a single lognormal mean and then for
the difference between two lognormal means. The necessary
algorithms and Fortran and SAS programs to carry out our pro-
cedures are posted at http://www.ucs.louisiana.edu/kxk4695
and are available as an appendix to the online version of
this article on the JOEH website. In a later section, we also
address the problem of obtaining tests and confidence intervals
concerning a single lognormal variance, or the ratio of two
lognormal variances. A confidence interval for the lognormal
variance should be of interest to assess the variability among
exposure measurements.
We have used two examples to illustrate our methods.
The first example involves the sample of air lead levels data
collected from a lab by the National Institute of Occupational
Safety and Health (NIOSH) health hazard evaluation staffs.
The problem is to assess the contaminant level within the
facility based on a sample. We also illustrate the generalized
variable method for testing the equality of the means of
measurements obtained by two different methods. For this
purpose we used the data presented in O’Brien et al.
(23)
Generalized Variables for the Mean and Variance of
a Normal Distribution
As the mean of a lognormal distribution is a function of
the mean and variance of a normal distribution, we present
the generalized variables for the mean and variance of a
normal population. The details of construction of generalized
variables can be found in Krishnamoorthy and Mathew
(19)
or
in Weerahandi,
(22)
and for easy reference they are provided in
Appendix 1. Let X
1
, ..., X
n
be a sample from a normal popu-
lation with mean µ
l
and variance σ
2
l
, N (µ
l
2
l
). The sample
mean and the variance of the X
i
s are respectively given by
¯
X =
1
n
n
i=1
X
i
and S
2
=
1
n 1
n
i=1
(X
i
¯
X)
2
. (5)
Let Z and V be independent random variables with
Z =
n(
¯
X µ
l
)
σ
l
N (0, 1), and V
2
=
(n 1)S
2
σ
2
l
χ
2
n1
,
(6)
where χ
2
r
denotes the central chi-square distribution with r
degrees of freedom. Let
¯
x and s be the observed values of
¯
Xand S, respectively. Following the procedure outlined in the
appendix, a generalized variable for making inferences on µ
l
is given by
G
µ
l
=
¯
x
¯
X µ
l
σ
l
/
n
σ
l
n
s
S
µ
l
=
¯
x
Z
V /
n 1
s
n
µ
l
(7)
= T
µ
l
µ
l
,
where
T
µ
l
=
¯
x
Z
V /
n 1
s
n
, (8)
and Z and V are as defined in (Eq. 6). In the above, G
µ
l
denotes the generalized test variable for µ
l
, and T
µ
l
denotes
the generalized pivot statistic (the statistic that can be used for
making inference about the unknown parameter) for µ
l
.We
shall now show that G
µ
l
satisfies the three conditions given in
(Eq. A3) of Appendix 1: (1) For a given
¯
x and s, the distribution
of G
µ
l
does not depend on the nuisance parameter σ
2
l
; (2) it
follows from Step 1 of Eq. 7 that the value of G
µ
l
at (
¯
X, S) =
(
¯
x, s)isµ
l
; (3) it follows from Step 3 of Eq. 7 that, for a given
¯
x ands, the generalized variable is stochastically decreasing
with respect to µ
l
and hence the generalized p-value for testing
H
0
: µ
l
µ
l0
vs. H
a
:µ
l
l0
is given by
sup
H
0
P(G
µ
l
0) = P(G
µ
l
0|µ
l
= µ
l0
)
= P(T
µ
l
µ
l0
)
= P
t
n1
<
¯
x µ
l0
s/
n
,
644 Journal of Occupational and Environmental Hygiene November 2006
Page 3
which is the p-value based on the usual t-test. To get the
last equality, we used the fact that Z /(V /
n 1) follows a
Student’s t distribution with degrees of freedom n 1, t
n1
.
For a given
¯
x and s, the lower α/2 quantile T
µ
l
,α/2
of T
µ
l
and
the upper α/2 quantile T
µ
l
,1α/2
of T
µ
l
form a 1 α generalized
confidence interval for µ
l
. This generalized CI is indeed
equal to the usual t-interval; that is, (T
µ
l
,α/2
, T
µ
l
,1α/2
) =
(
¯
x t
n1,1α/2
s
n
,
¯
x + t
n1,1α/2
s
n
), where t
m, p
denotes the
100 pth percentile of the Student’s t distribution with m degrees
of freedom.
The generalized test variable for the variance σ
2
l
is given by
G
σ
2
l
=
s
2
V
2
/(n 1)
σ
2
l
= T
σ
2
l
σ
2
l
, (9)
where
T
σ
2
l
=
s
2
V
2
/(n 1)
(10)
is the generalized pivot statistic, and V is as defined in Eq. 6.
Again, for a given s
2
, the generalized 1 α CI for σ
2
l
is formed
by the lower and upper α/2 quantiles of T
σ
2
l
and is equal to the
usual CI based on a chi-square distribution with n 1degrees
of freedom.
Even though the generalized variable method produced
exact inferential procedures for the normal parameters, in
general, the generalized variable method is not necessarily
exact. In other words, the generalized p-value may not satisfy
the conventional properties of the usual p-value. In such cases,
the properties (such as Type I error rates of the generalized
variable test and coverage probability of the generalized
confidence limits) of the generalized variable method should
be evaluated numerically.
Suppose we are interested in making inference about a
function of µ
l
and σ
2
l
, say, q(µ
l
2
l
). Then, the generalized
test variable for q(µ
l
, σ
2
l
)isgivenbyq(T
µ
l
, T
σ
2
l
) q(µ
l
2
l
),
and the generalized pivot statistic is given by q(T
µ
l
, T
σ
2
l
). For a
given
¯
x and s, the variable q(µ
l
, σ
2
l
) depends only on the ran-
dom variables Z and V whose distributions do not depend on
any unknown parameters. Therefore, Monte Carlo simulation
can be used to find a generalized CI for q(µ
l
2
l
). This will be
illustrated for the lognormal case in the following section.
Inference about a Lognormal Mean
Let y
1
,...,y
n
be a sample of exposure measurements and
let x
i
= ln(y
i
), i = 1,...,n. Then, x
1
,...,x
n
is a random
sample from a N (µ
l
,σ
2
l
) distribution. Since the lognormal
mean exp(µ
l
+ σ
2
l
/2) is a function of µ
l
and σ
2
l
, the results
of the preceding section can be readily applied to construct
a generalized test variable and a generalized pivot statistic
for the lognormal mean. From the preceding section, we
have the generalized test variable for making inference on
η = (µ
l
+ σ
2
l
/2) as
G
η
= T
µ
l
+
T
σ
2
l
2
η
=
¯
x
Z
V /
n 1
s
n
+
s
2
2V
2
/(n 1)
η (11)
= T
η
η,
where
T
η
=
¯
x
Z
V /
n 1
s
n
+
s
2
2V
2
/(n 1)
(12)
and Z and V are as defined in Eq. 6. For given sample statistics
¯
x and s,wenote that G
η
is stochastically decreasing in η,
and hence the generalized p-value for testing (Eq. 2) is given
by
P(G
η
0|η = ln(LTA-OEL))
= P(T
η
ln(LTA-OEL)). (13)
The null hypothesis in Eq. 2 will be rejected whenever the
probability in Eq. 13 is less than the nominal level α.
The generalized pivot statistic for interval estimation of η is
given by T
η
. Appropriate quantiles of T
η
can be used to obtain
confidence intervals for η or for the lognormal mean exp(η).
Specifically, if T
η,p
,0< p < 1, denotes the pth quantile of T
η
,
then (T
η,α/2
, T
η,1α/2
)isa1α generalized confidence interval
for η, and (exp(T
η,α/2
), exp(T
η,1α/2
)) is a 1 α generalized
confidence interval for the lognormal mean exp(η). One-sided
limits for η and exp(η) can be similarly obtained. In particular,
a1 α lower limit for exp(η)isgivenby exp(T
η,α
).
Through numerical results, Krishnamoorthy and
Mathew
(17)
noted that the confidence limits based on
Land’s
(12)
approach and the generalized confidence interval
are practically the same. However, computationally, our
approach is very easy to implement. The simple algorithm
presented in Appendix 2 of Krishnamoorthy and Mathew
(19)
can be used for computing the generalized p-value and the
generalized confidence interval.
Power Studies and Sample Size Calculation for
Testing a Lognormal Mean
We shall now discuss the power of the test based on the
generalized p-value in Eq. 13. For a given sample size n and
for a given value of µ
l
and σ
l
such that H
a
in Eq. 2 holds
(i.e., η = µ
l
+ σ
2
l
/2 < ln(LTA-OEL)), the power of the
test can be estimated by Monte Carlo simulation. In practice,
however, practitioners are mainly interested in finding the
required sample size to have a specified power at a given
level of significance. The sample size can be calculated using
an iterative method. For power calculation, an algorithm and
Fortran and SAS programs based on the algorithm are posted
at http://www.ucs.louisiana.edu/kxk4695 and are available
as an appendix to the online version of this article. Using
this program, we computed sample sizes that are required to
have a power of 0.90 at the level of significance α = 0.05 for
various parameter configurations, and these are presented in
Table I. As an example, if an employer speculates that the mean
exposure level is 40% (the value R in Table I) of the LTA-OEL,
and the geometric standard deviation is 2.0, then the required
sample size to have a power of at least 0.90 at the level 0.05
is 13.
We observe from Table I that the power of the test increases
as the ratio R decreases, which is a natural requirement for a
test. We also note that the power decreases as σ
g
increases and,
Journal of Occupational and Environmental Hygiene November 2006 645
Page 4
TABLE I. Sample Size for Testing Equation 2 to
Attain a Power of 0.90 at the Level of 0.05, Using the
Generalized P-Value Test
σ
g
R 1.5 2.0 2.5 3.0 3.5
0.1 4 (.96) 6 (.94) 8 (.90) 11 (.90) 13 (.91)
0.2 4 (.90) 7 (.91) 11 (.91) 16 (.90) 21 (.91)
0.3 5 (.93) 10 (.91) 16 (.90) 21 (.91) 30 (.90)
0.4 6 (.91) 13 (.90) 23 (.91) 35 (.91) 45 (.90)
0.5 8 (.93) 18 (.91) 33 (.90) 52 (.90) 68 (.90)
0.7 18 (.91) 56 (.90) 99 (.90) 162 (.90) 235 (.90)
0.8 37 (.90) 120 (.90) 241 (.90) 363 (.90) 563 (.90)
Note: R =
µ
l
LTA
-OEL
; σ
g
= exp(σ
l
) = geometric standard deviation; the
numbers in parenthesis represent actual attained powers; LTA-OEL =1.0; the
lognormal mean.
hence, large samples are required to make correct decisions
when σ
g
is expected to be large.
Comparison of Two Lognormal Means
Consider the independent lognormal random variables y
1
and y
2
so that x
1
= ln(y
1
) N (µ
l1
2
l1
) and x
2
= ln(y
2
)
N (µ
l2
2
l2
). Then the lognormal means are given by E(y
1
) =
exp(η
1
) and E(y
2
) = exp(η
2
), where
η
1
= exp
µ
l1
+ σ
2
l1
/2
and η
2
= exp
µ
l2
+ σ
2
l2
/2
. (14)
Thus, hypothesis tests and confidence intervals for the ratio
of the two lognormal means are respectively equivalent to
those for the difference η
1
η
2
.Weshall now develop gen-
eralized p-values and generalized confidence intervals for this
problem.
We shall first consider the testing problem
H
0
: η
1
η
2
vs. H
a
: η
1
2
. (15)
Let y
1 j
, j = 1,...,n
1
, and y
2 j
, j = 1,...,n
2
, denote
random samples from the lognormal distributions of y
1
and
y
2
, respectively. Let x
1 j
= ln(y
1 j
), j = 1,...,n
1
, and x
2 j
=
ln(y
2 j
), j = 1,...,n
2
. The sample means
¯
x
1
and
¯
x
2
and the
sample variances s
2
1
and s
2
2
are then given by
¯
x
i
=
1
n
i
n
i
j=1
x
ij
and s
2
i
=
1
n
i
1
n
i
j=1
(x
ij
¯
x
i
)
2
, i = 1, 2.
(16)
It follows from Eq. 12 that the generalized variable for η
i
can be expressed as
T
η
i
=
¯
x
i
Z
i
V
i
/
n
i
1
s
i
n
i
+
s
2
i
2V
2
i
/(n
i
1)
, i = 1, 2,
(17)
where Z
i
N (0, 1) and V
2
i
χ
2
n
i
1
, for i = 1, 2, and all
these random variables are independent. The generalized test
variable for testing (Eq. 15) is given by
G
η
1
η
2
= T
η
1
T
η
2
(η
1
η
2
) (18)
and the generalized pivot statistic to construct CI for η
1
η
2
for is given by
T
η
1
η
2
= T
η
1
T
η
2
. (19)
Forgiven sample statistics, G
η
1
η
2
is stochastically decreas-
ing in η
1
η
2
. Thus the generalized p-value for testing the
hypotheses in Eq. 15 is given by
sup
H
0
P(G
η
1
η
2
0) = P(G
η
1
η
2
0|η
1
η
2
= 0)
= P(T
η
1
η
2
0). (20)
Forgiven sample statistics, the confidence intervals for
η
1
η
2
can be computed using the percentiles of T
η
1
η
2
.
Because, given
¯
x
1
,
¯
x
2
, s
2
1
and s
2
2
, the distribution of T
η
1
η
2
is free of any unknown parameters, the percentiles of T
η
1
η
2
can be estimated using Monte Carlo simulation. We can also
construct confidence intervals for the difference between the
lognormal means, that is, exp(η
1
) exp(η
2
). For this, we
can use the percentiles of exp(T
η
1
) exp(T
η
2
), where T
η
1
and T
η
2
are given in Eq. 17. Note that algorithms similar to
Algorithm 1 can be easily developed for computing the above
generalized p-values and confidence intervals. An algorithm
and Fortran and SAS programs for computing the generalized
p-value test and the CI for exp(T
η
1
) exp(T
η
2
) are posted at
http://www.ucs.louisiana.edu/kxk4695 and are available as
an appendix to the online version of this article.
Power Properties of the Generalized Test for the
Two-Sample Case
For given sample sizes n
1
and n
2
, parameters µ
l1
l2
l1
and σ
l2
the powers of the generalized test based on Eq.
20 can be estimated using Monte Carlo method. A Fortran
program and SAS codes for computing the power (along
with a help file) are posted at http://www.ucs.louisiana.edu/
kxk4695 and are available as an appendix to the online
version of this article. The help file also contains an algo-
rithm that can be coded in any desired computing language.
Krishnamoorthy and Mathew
(19)
computed powers for several
sample sizes and parameter combinations. It is observed in this
article that the generalized test possesses all natural properties.
However, the power of the test depends on µ
l1
µ
l2
l1
and
σ
l2
. Therefore, to compute the required sample sizes to attain a
specified power, the practitioner should have knowledge about
µ
l1
µ
l2
l1
, and σ
l2
.
Inference about a Lognormal Variance and
Geometric Standard Deviation
For the assessment of the extent of variability among
the exposure measurements, confidence intervals, or tests
concerning the variance becomes necessary. If y denotes the
lognormally distributed exposure measurements, then x =
ln(y)isdistributed normally with mean µ
l
and variance σ
2
l
.
646 Journal of Occupational and Environmental Hygiene November 2006
Page 5
TABLE II. Monte Carlo Estimates of the Sizes of the Generalized P-Value Test Based on Equation 24 for
Lognormal Variance in Equation 21; Nominal Level = 0.05
σ
l
= 0.5 σ
l
= 1.0 σ
l
= 1.5
µ
l
n = 10 n = 15 n = 20 n = 10 n = 15 n = 20 n = 10 n = 15 n = 20
0.00 .050 .041 .047 .047 .048 .046 .050 .044 .049
0.30 .046 .056 .052 .053 .044 .049 .043 .046 .056
0.70 .047 .050 .052 .045 .050 .048 .049 .044 .049
1.00 .050 .050 .054 .048 .049 .050 .048 .044 .049
1.30 .054 .046 .043 .052 .049 .045 .045 .051 .050
1.50 .048 .060 .048 .053 .048 .049 .048 .050 .045
1.70 .053 .053 .051 .043 .045 .048 .052 .048 .042
2.00 .055 .053 .054 .046 .054 .054 .054 .048 .047
The variance of y,tobedenoted by σ
2
,isgivenby
σ
2
= exp
2µ
l
+ σ
2
l

exp
σ
2
l
1
. (21)
As far as we are aware, no procedures (except obvious large
sample procedures) are known for computing a confidence
interval or for testing hypotheses concerning σ
2
.Itturns out
that the ideas of generalized p-values and generalized confi-
dence intervals provide solutions to this problem, regardless
of the sample size. We shall now construct a generalized pivot
statistic that can be used to compute a confidence interval for
σ
2
, and a generalized test variable that can be used for testing
the hypotheses
H
0
: σ
2
σ
2
0
vs. H
a
: σ
2
2
0
, (22)
where σ
2
0
is a known constant. Note that it is by rejectingH
0
that we conclude that the variability is small, that is, below the
bound σ
2
0
.
Using earlier notations, the generalized test variable for σ
2
is given by
G
σ
2
= exp
2T
µ
l
+ T
σ
2
l

exp
σ
2
l
1
σ
2
= exp
2
¯
x
Z
V /
n 1
s
n
+
s
2
V
2
/(n 1)
×
exp
s
2
V
2
/(n 1)
1
σ
2
, (23)
where Z and V are as defined in Eq. 6. The generalized pivot
statistic for constructing CI for σ
2
l
is given by
T
σ
2
= exp
2
¯
x
Z
V /
n 1
s
n
+
s
2
V
2
/(n 1)
exp
s
2
V
2
/(n 1)
1
. (24)
Arguing as in previous sections, the generalized p-value for
testing the hypotheses in Eq. 22 is given by
P
G
σ
2
0
σ
2
= σ
2
0
= P
T
σ
2
σ
2
0
. (25)
Furthermore, the percentiles of T
σ
2
can be used for com-
puting a generalized confidence interval for σ
2
.Analgorithm
(similar to Algorithm 1 in Appendix 2) can be easily developed
for computing the above generalized p-value and confidence
interval. We also note that the above procedure can be
easily extended for the purpose of comparing two lognormal
variances.
To understand the validity of the generalized test based
on Eq. 25, we estimated its sizes (Type I error rates) using
Monte Carlo method for various values of µ
l
l
and n =
10,15, and 20. The sizes are estimated for testing hypotheses
in Eq. 22 at the nominal level 0.05, and they are given in Table
II. For a good test, the estimated sizes should be close to
the nominal level. We see from Table II that the estimated
sizes are very close to the nominal level for all the cases
considered.
The generalized variable for a geometric standard deviation
σ
g
= exp(σ
l
)isgivenby
G
σ
g
= exp
G
σ
2
l
, (26)
where the generalized variable G
σ
2
l
for σ
2
l
is given in Eq. 9.
However, it was pointed out earlier that the generalized
variable approach gives the same confidence interval for σ
2
l
as
the conventional chi-square interval. From this, a confidence
interval for the geometric standard deviation is easily obtained
as
exp
s
(n 1)
χ
2
n1,1α/2
, exp
s
(n 1)
χ
2
n1,α/2

, (27)
where χ
2
m, p
denotes the 100 pth percentile of the central chi-
square distribution with df = m. The expression in (Eq. 27) is
an exact 1 α confidence interval for σ
g
.
Similarly, a test for
H
0
: σ
g
c vs. H
a
: σ
g
> c, (28)
is essentially a test concerning the variance σ
2
l
, and the usual
chi-square test for the variance can be applied.
Journal of Occupational and Environmental Hygiene November 2006 647
Page 6
Illustrative Examples
Example 1
The data represent air lead levels collected by NIOSH at the
Alma American Labs, Fairplay, Colorado, for health hazard
evaluation purpose (HETA 89-052) on February 23, 1989. The
air lead levels were collected from 15 different areas within
the facility.
Air Lead Levels (µg/m
3
): 200, 120, 15, 7, 8, 6, 48, 61, 380,
80, 29, 1000, 350, 1400, 110
For this data, the mean (=254) is much larger than the
median (=80), which is an indication that the distribution
is right skewed. The normal probability plots (Minitab 14.0,
default method) were constructed for the actual lead levels
(Figure 1A) and for the logged lead levels (Figure 1B). It
is clear from Figures 1A and 1B that the distribution of the
data is far away from a normal distribution (p-value < 0.05),
butalognormal model adequately describes the data (p-value
0.871). The p-values are based on the Anderson-Darling test.
Therefore, we apply the methods of this paper to make valid
inferences about the mean lead level. Based on the logged data,
we have the observed values
¯
x = 4.333 and s = 1.739. Using
these numbers in Algorithm 1, we computed the 95% upper
limit for exp(η)as2405. We also computed the 95% lower limit
for the lognormal mean as 141. That is, the mean air lead level
within the facility exceeds 141 µg/m
3
with 95% confidence.
Suppose we want to test whether the mean is greater than
some arbitrary value (e.g., 120 µg/m
3
) that could be a limit
value
H
0
: µ 120 vs. H
a
: µ<120,
where µ = exp(η) (with η = µ
l
+ σ
2
l
/2) denotes the actual
unknown mean air lead levels within the lab facility. Using
again Algorithm 1, we computed the generalized p-value as
FIGURE 1. Normal probability plots of (A) actual lead levels, and (B) logged air lead levels
648 Journal of Occupational and Environmental Hygiene November 2006
Page 7
TABLE III. Summary Statistics for Airborne Con-
centration of Metalworking Fluids (MWF) in 23 Plants
Method Sample Size ¯xs
Thoracic MWF aerosol 23 1.277 0.835
(gravimetric analysis)
Closed-face MWF analysis 23 0.979 0.917
Note:
¯
x = mean of the logged data; s =standard deviation of the logged data.
0.97, and so we conclude that the data do not provide enough
evidence to indicate that the mean air lead levels within the
facility is less than 120 µg/m
3
.
Regarding the lognormal variance, we computed the maxi-
mum likelihood estimate as 2337098 µg/m
3
. This estimate is
obtained by replacing µ
l
and σ
2
l
in Eq. 21, respectively, by
¯
x
and ((n 1)s2/n). We also computed a 95% confidence interval
for the lognormal variance, using the generalized pivot statistic
T
σ
2
in (24), as (128538, 2956026772).
Finally, we computed the 95% CI for the geometric standard
deviation using the generalized variable in Eq. 26 as (3.57,
15.49); using the exact formula in Eq. 27, we get (3.57, 15.53).
Example 2
In this example, we shall illustrate the generalized variable
procedures for testing the equality of the means of mea-
surements obtained by two different methods. The data were
reported in Table I of O’Brien et. al.,
(23)
and represent total mass
of metalworking fluids (MWF) obtained by thoracic MWF
aerosol and closed-face MWF aerosol. Normal probability
plots of logged data indicated that the lognormality assumption
about the original data is tenable. The means and the standard
deviations of the logged data are given in Table III. Let µ
t
and µ
c
denote the true means of the airborne concentrations
by thoracic MWF aerosol and closed-face MWF aerosol,
respectively. To test the equality of the means, we consider
H
0
: µ
t
= µ
c
vs. H
a
: µ
t
= µ
c
.
Using the summary statistics in Table III, we simulated
D = exp(T
η
1
)exp(T
η
2
), where T
η
1
and T
η
2
are given in Eq. 17,
100,000 times. The generalized p-value for the above two-tail
test can be estimated by 2 × min{proportion of Ds < 0,
proportion of Ds > 0}. Our simulation yielded the generalized
p-value of 0.244. The lower 2.5 and the upper 2.5 percentiles of
D form a 95% confidence interval for the difference between
the means and is computed as (–0.657, 0.145). Thus, at the
5% level, both generalized p-value and the confidence interval
indicate that there is no significant difference between the
means.
CONCLUSIONS
S
everal attempts have been made in the literature for
drawing inferences concerning the mean of a single
lognormal distribution. To a much lesser extent, attempts have
also been made to draw inferences for the ratio of the means
of two lognormal distributions. These problems have certain
inherent difficulties associated with them, and the available
solutions are either approximate, or are applicable only to
large samples, or are difficult to compute. In this article, we
have explored a novel approach for solving these problems,
based on the concepts of generalized p-values and generalized
confidence intervals. It turns out that these concepts provide
a unified and versatile approach for handling any parametric
function associated with one or two lognormal distributions.
Even though analytic expressions are not available for the
resulting confidence intervals or p-values, their computation is
both easy and straightforward. We have provided the necessary
programs for their computation, and we have also illustrated
our approach using several examples dealing with the analysis
of exposure data. In writing this article, our intention has
been to draw the attention of industrial hygienists to this new
methodology.
ACKNOWLEDGMENT
T
his research was supported by a grant from the National
Institute of Occupational Safety and Health (NIOSH).
REFERENCES
1. Oldham, P.: The nature of the variability of dust concentrations at the
coal face. Br. J. Ind. Med. 10:227–234 (1953).
2. Tuggle, R.M.: Assessment of occupational exposure using one-sided
tolerance limits. Am. Ind. Hyg. Assoc. J. 43:338–346 (1982).
3. Rappaport, S. M., and S. Selvin: A method for evaluating the mean
exposure from a lognormal distribution. Am. Ind. Hyg. Assoc. J. 48:374–
379 (1987).
4. Selvin, S., and S.M. Rappaport: Note on the estimation of the mean
value from a lognormal distribution. Am. Ind. Hyg. Assoc. J. 50:627–630
(1989).
5. Selvin, S., S. M. Rappaport, R. Spear, J. Schulman, and M. Francis:
A note on the assessment of exposure using one-sided tolerance limits.
Am. Ind. Hyg. Assoc. J. 48:89–93 (1987).
6. Borjanovic, S.S., S.V. Djordjevic, and M.D. Vukovic-Pal: A method
for evaluating exposure to nitrous oxides by application of lognormal
distribution. J. Occup. Health 41:27–32 (1999).
7. Saltzman, B.E.: Health risk assessment of fluctuating concentrations
using lognormal models. J. Air Waste Manag. Assoc. 47:1152–1160
(1997).
8. Lyles, R.H., and L.L. Kupper: On strategies for comparing occupational
exposure data to limits. Am. Ind. Hyg. Assoc. J. 57:6–15 (1996).
9. Rappaport, S.M.: Assessment of long-term exposures to toxic substances
in air. Ann. Occup. Hyg. 35:61–121 (1991).
10. Lyles, R.H., L.L. Kupper, and S.M. Rappaport: Assessing regulatory
compliance of occupational exposures via the balanced one-way random
effects ANOVA model. J. Agric. Biol. Environ. Statist. 2:64–86 (1997).
11. Krishnamoorthy, K., and T. Mathew: One-sided tolerance limits in
balanced and unbalanced one-way random models based on generalized
confidence limits. Technometrics 46:44–52 (2004).
12. Land, C.E.: Hypotheses tests and interval estimates. In Lognormal
Distribution (E.L. Crow and K. Shimizu, eds.). New York: Marcel Dekker,
1988. pp. 87–112.
13. Zhou, X.H., and S. Gao: Confidence intervals for the lognormal mean.
Statist. Med. 16:783–790 (1997).
Journal of Occupational and Environmental Hygiene November 2006 649
Page 8
14. Armstrong, B.G.: Confidence intervals for arithmetic means of lognor-
mally distributed exposures. Am. Ind. Hyg. Assoc. J. 53:481–485 (1992).
15. Land, C.: An evaluation of approximate confidence interval methods for
lognormal means. Technometrics 14:145–158 (1972).
16. Hewett, P., and G.H. Ganser: Simple procedures for calculating
confidence intervals around the sample mean and exceedance fraction
derived from lognormally distributed data. Appl. Occup. Environ. Hyg.
12:132–142 (1997).
17. Taylor, D.J., L.L. Kupper, and K.E. Muller: Improved approximate
confidence intervals for the mean of a log-normal random variable. Statist.
Med. 21:1443–1459 (2002).
18. Zhou, X.H., S. Gao, and S.L. Hui: Methods for comparing the means of
two independent lognormal samples. Biometrics 53:1129–1135 (1997).
19. Krishnamoorthy, K., and T. Mathew: Inferences on the means of lognor-
mal distributions using generalized p-values and generalized confidence
intervals. J. Statist. Plan. Infer. 115:103–121 (2003).
20. Tsui, K.W., and S. Weerahandi: Generalized p-values in significance
testing of hypotheses in the presence of nuisance parameters. J. Am. Statist.
Assoc. 84:602–607 (1989)
21. Weerahandi, S.: Generalized confidence intervals. J. Am. Statist. Assoc.
88:899–905 (1993).
22. Weerahandi, S.: Exact Statistical Methods for Data Analysis.New York:
Springer-Verlag, 1995.
23. O’Brien, D.M., G.M. Piacitelli, W.K. Sieber, R.T. Hughes, and J.D.
Catalano: An evaluation of short-term exposures to metal working fluids
in small machine shops. Am. Ind. Hyg. Assoc. J. 62:342–348 (2001).
APPENDIX 1
The Generalized Confidence Interval
and Generalized P-Value
A general setup where the concepts of generalized con-
fidence intervals and generalized p-values are defined is as
follows. Consider a random variable X whose distribution
depends on a scalar parameter of interest θ and a nuisance
parameter (parameter that is not of direct inferential interest)
η, where η could be a vector. Here X could also be a vector.
Suppose we are interested in computing a confidence interval
for θ. Let x denote the observed value of X , that is, x represents
the data that has been collected. To obtain a generalized
confidence interval for θ ,weneed a generalized pivot statistic
(the pivotal quantity based on which inferential procedures will
be developed) T
1
(X ; x) that is a function of the random
variable X, the observed data x , and the parameters θ and η,
and satisfying the following two conditions:
(i) Given x, the distribution of T
1
(X ; x)isfree of the
unknown parameters θ and η;
(ii) The observed value of T
1
(X ; x), namely,
T
1
(x; x)isequal to θ. (A1)
The percentiles of T
1
(X ; x) can then be used to obtain
confidence intervals for θ. Such confidence intervals are
referred to as generalized confidence intervals. For example,
if T
1α
denotes the 100 (1 α)th percentile of T
1
(X ; x),
then T
1α
is a generalized upper confidence limit for θ .A
lower confidence limit or two-sided confidence limits can be
similarly defined.
Now suppose we are interested in testing the hypothesis
H
0
: θ θ
0
vs. H
a
: θ>θ
0
, (A2)
where θ
0
is a specified quantity. Suppose we can define a gen-
eralized test variable T
2
(X ; x) satisfying the following
conditions:
(i) For a given x, the distribution of T
2
(X ; x)isfree of
the nuisance parameter η;
(ii) The observed value of T
2
(X ; x), namely,
T
2
(x; x)isfree of any unknown parameters;
(iii) For a given x and η, the distribution of T
2
(X ; x)is
stochastically monotone in θ (i.e., stochastically increas-
ing or decreasing in θ). (A3)
In general, for a given x and η, T
2
(X ; x)isstochas-
tically decreasing in θ, and the generalized p-value for
testing Eq. A2 is given byP
(
T
2
(X ; x) t
)
, where t =
T
2
(x; x). On the other hand, if T
2
(X ; x)isstochas-
tically decreasing in θ, then the generalized p-value for
Eq. A2 is defined as P
(
T
2
(X ; x) t
)
.Ingeneral, the
observed value t is equal to θ
0
, and as the distribution
of T
2
(X ; x)isfree of the nuisance parameter η, the
generalized p-value at θ
0
can be computed using Monte Carlo
simulation.
APPENDIX 2
Algorithm for Computing the Generalized P-Value
and the Generalized Confidence Interval
The following algorithm given by Krishnamoorthy and
Mathew
(17)
can be used for computing the generalized p-value
and the generalized confidence interval.
Foragiven logged data set, compute the observed sample
mean and variance, namely,
¯
x ands
2
, respectively.
For i = 1tom
Generate a standard normal variate Z
Generate a chi-square random variate V
2
with degrees of
freedom n 1
Set T
ηi
=
¯
x
Z
V /
n 1
s
n
+
s
2
2V
2
/(n 1)
Set K
i
= 1ifT
ηi
> ln(LTA - PEL), else K
i
= 0
(end i loop)
1
m
m
i=1
K
i
is the generalized p-value for testing the hypothe-
ses in Eq. 2. The 100(1 α)th percentile of T
η
1
,...,T
η
m
,
denoted by T
η,1α
,isthe 1 α generalized upper confidence
limit for η = µ
l
+σ
2
l
/2. Furthermore, exp (T
η,1α
)isthe 1 α
generalized upper limit for the lognormal mean.
Based on our experience, we recommend simulation con-
sisting of at least 100,000 (i.e., the value of m)toget consistent
results regardless of the initial seed used for random number
generation. The above algorithm can be easily programmed
in any programming language. A Fortran program and SAS
codes for computing generalized p-values for one-tail tests
and one-sided confidence limits is posted at http://www.ucs.
louisiana.edu/kxk4695. Interested readers can download
these files from this address.
650 Journal of Occupational and Environmental Hygiene November 2006
Page 9
    • "The computation of confidence limits for some rather complicated parameters comes up in many industrial hygiene applications, and the concept of a generalized confidence interval has proved very fruitful to address such problems. In a series of articles, Krishnamoorthy and Mathew (2002, 2009) and Krishnamoorthy et al., (2006 Krishnamoorthy et al., ( , 2007) have successfully applied the generalized confidence interval idea for the analysis of industrial hygiene data. In particular, Krishnamoorthy and Mathew (2009) have developed an accurate upper confidence limit for the symmetric-range accuracy , using the generalized confidence interval approach, in the context of normally distributed sample measurements. "
    [Show abstract] [Hide abstract] ABSTRACT: The symmetric-range accuracy of a sampler is defined as the fractional range, symmetric about the true concentration, that includes a specified proportion of sampler measurements. In this article, we give an explicit expression for assuming that the sampler measurements follow a one-way random model so as to capture different components of variability, for example, variabilities among and within different laboratories or variabilities among and within exposed workers. We derive an upper confidence limit for based on the concept of a ‘generalized confidence interval’. A convenient approximation is also provided for computing the upper confidence limit. Both balanced and unbalanced data situations are investigated. Monte Carlo evaluation indicates that the proposed upper confidence limit is satisfactory even for small samples. The statistical procedures are illustrated using an example.
    Full-text · Article · Mar 2013 · Annals of Occupational Hygiene
    0Comments 0Citations
    • "A confidence interval for g can be obtained along the lines of the method for the percentiles given in earlier section and using the generalized variable approach. For more details on the generalized variable approach in the present context, see the articles by Krishnamoorthy et al. (2006 Krishnamoorthy et al. ( , 2011) and Krishnamoorthy and Mathew (2009b). Specifically, an approximate 'generalized pivotal quantity (GPQ)' for g, which can be constructed following the lines of Krishnamoorthy et al. (2010), as follows. "
    [Show abstract] [Hide abstract] ABSTRACT: The problem of assessing occupational exposure using the mean or an upper percentile of a lognormal distribution is addressed. Inferential methods for constructing an upper confidence limit for an upper percentile of a lognormal distribution and for finding confidence intervals for a lognormal mean based on samples with multiple detection limits are proposed. The proposed methods are based on the maximum likelihood estimates. They perform well with respect to coverage probabilities as well as power and are applicable to small sample sizes. The proposed approaches are also applicable for finding confidence limits for the percentiles of a gamma distribution. Computational details and a source for the computer programs are given. An advantage of the proposed approach is the ease of computation and implementation. Illustrative examples with real data sets and a simulated data set are given.
    Full-text · Article · Jun 2011 · Annals of Occupational Hygiene
    0Comments 2Citations
    • "Generalized procedures have been successfully applied to several problems of practical importance. The areas of applications include comparison of means, testing and estimation of functions of parameters of normal and related distributions (Weerahandi,2345, Krishnamoorthy and Mathew [6], Johnson and Weerahandi [7], Gamage, Mathew and Weerahandi [8]); testing fixed effects and variance components in repeated measures and mixed effects ANOVA models (Zhou and Mathew [9], Gamage and Weerahandi [10], Chiang [11], Krishnamoorthy and Mathew [6], Weerahandi [5], Mathew and Webb [12], Arendacka [13]); interlaboratory testing (Iyer, Wang and Mathew [14]); bioequivalence (McNally, Iyer and Mathew [15]); growth curve modeling (Weerahandi and Berger [16], Lin and Lee [17]); reliability and system engineering (Roy and Mathew [18], Tian and Cappelleri [19], Mathew, Kurian and Sebastian [20]); process control (Burdick, Borror and Montgomery [21], Mathew, Kurian and Sebastian [22]); environmental health (Krishnamoorthy, Mathew and Ramachandran [23]) and many others. The simulation studies in Johnson and Weerahandi [7], Weerahandi [4,5] Zhou and Mathew [9], Gamage and Weerahandi [10], among others have demonstrated the success of the generalized procedure in many problems where the classical approach fails to yield adequate confidence intervals. "
    [Show abstract] [Hide abstract] ABSTRACT: Generalized confidence intervals provide confidence intervals for complicated parametric functions in many common practical problems. They do not have exact frequentist coverage in general, but often provide coverage close to the nominal value and have the correct asymptotic coverage. However, in many applications generalized confidence intervals do not have satisfactory finite sample performance. We derive expansions of coverage probabilities of one-sided generalized confidence intervals and use the expansions to explain the nonuniform performance of the generalized intervals. We then show how to use these expansions to obtain improved coverage by suitable calibration. The benefits of the proposed modification are illustrated via several examples.
    Full-text · Article · Aug 2009 · Journal of Multivariate Analysis
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