Bootstrap Confidence Intervals for the Parameter of the
Poisson-Prakaamy Distribution with Their Applications
WARARIT PANICHKITKOSOLKUL
Department of Mathematics and Statistics, Faculty of Science and Technology,
Thammasat University,
Phahonyothin Road, Khlong Nung, Khlong Luang, Pathumthani 12120,
THAILAND
Abstract: - Poisson-Prakaamy distribution has been proposed for count data, which is of primary interest in
several fields, such as biological science, medical science, demography, ecology, and genetics. However,
estimating the bootstrap confidence intervals for its parameter has not yet been examined. In this study,
bootstrap confidence interval estimation based on the percentile, basic, biased-corrected, and accelerated
bootstrap methods were examined in terms of their coverage probabilities and average lengths via Monte Carlo
simulation. The results indicate that attaining the nominal confidence level using the bootstrap confidence
intervals was not possible for small sample sizes regardless of the other settings. Moreover, when the sample
size was large, the performances of the bootstrap confidence intervals were not substantially different. Overall,
the bias-corrected and accelerated bootstrap confidence interval outperformed the others for all of the cases
studied. Lastly, the efficacies of the bootstrap confidence intervals were illustrated by applying them to two real
data sets, the results of which match those from the simulation study.
Key-Words: - Interval estimation, Poisson distribution, count data, mixture distribution, bootstrap method,
simulation
Received: October 8, 2022. Revised: April 18, 2023. Accepted: May 9, 2023. Published: May 22, 2023.
1 Introduction
It is well-known that that the number of events
happening in a specific region of time and/or space
has a Poisson distribution, [1]. Data such as the
number of thunderstorms occurring in a particular
locality over a given amount of time, the number of
orders a firm will receive tomorrow, the number of
calls attended per hour at a call center, the number
of defects in a finished product, etc., [2], follow a
Poisson distribution. Although the Poisson
distribution is a basic model for the analysis of
count data, its use is restricted due to the equality of
its mean and variance (equi-dispersion). Count data
often express over-dispersion, with a variance larger
than the mean, compared to the Poisson distribution,
[3], The application of the Poisson distribution to
over-dispersion data can lead to incorrect analyses
and mistaken conclusions, [4]. A popular alternative
when the count data exhibit over-dispersion is to use
a mixed Poisson distribution, [5], in which it is
assumed that the Poisson parameter is a random
variable that has a single parameterized distribution,
[6]. In the literature review, numerous mixed
Poisson distributions had been proposed for the
over-dispersed count data. For example, [7],
proposed the Poisson-Lindley (PL) distribution by
compounding the Poisson distribution with the
Lindley distribution. This distribution comes from
the Poisson distribution when its parameter follows,
[8], distribution. Moreover, the PL distribution was
applied to two real data sets and it can be used as an
approximation to the negative binomial distribution,
[9], and the Hermite distribution, [10]. Later, the
Poisson and Akash, [11], distributions were
combined by [12], to introduce the Poisson-Akash
(PA) distribution. The chi-squared goodness-of-fit
test of the PA distribution based on the maximum
likelihood (ML) estimation had been applied for
five count data sets. The PA distribution fits over
Poisson and PL distributions in almost all data sets.
Recently, [13], combined the Poisson and
Prakaamy distributions to produce the Poisson-
Prakaamy (PP) distribution and investigated its
mathematical and statistical properties including
moment generating function, moments, skewness,
kurtosis, hazard rate function, mean residual life
function, stochastic orderings, mean deviations,
Bonferroni and Lorenz curves. The PP distribution
arises from the Poisson distribution when the
Poisson parameter
(the mean number of events)
follows a Prakaamy distribution, [14]. The method
of moments and maximum likelihood estimation
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were both used to estimate the parameter of the PP
distribution and when it was applied to two real data
sets, it was more suitable than either the Poisson,
PL, [7], PA, [12], and Poisson-Ishita, [15],
distributions.
The Prakaamy distribution is a lifetime
continuous distribution introduced by [14], with a
probability density function (pdf) defined as
65
5
(;) 1 ,
120
x
f
xxe
0, 0.x
(1)
It is a mixture of the exponential distribution
having scale parameter
and gamma distribution
having shape parameter 4 and scale parameter
with their proportions 55
/ ( 120)
and
5
120 / ( 120),
respectively. This distribution has
been applied to model lifetime data from
engineering and biomedical science. Furthermore,
[14], also showed that the Prakaamy distribution is a
better model than either the exponential, [8], [11],
[16], distributions. The important statistical
properties of the Prakaamy distribution had been
discussed by [14], such as mean deviations,
Bonferroni and Lorenz curves, order statistics,
Renyi entropy measure, and stress-strength
reliability. Plots of the pdf of the Prakaamy
distribution with some specified values of the
parameter
are shown in Fig.1.
In statistics, the confidence interval (or the
interval estimation) is a range of values that is likely
to contain the true value of the population parameter
of interest, which is a key output for many statistical
analyses and has a critical role to play in the
interpretation of parameter estimations, [17]. The
reviewed literature does not contain any efforts
aimed at calculating bootstrap confidence intervals
for the parameter of the PP distribution. Bootstrap
confidence intervals for estimating the parameter
provide a way of quantifying the uncertainty in
statistical inference based on a sample of data. The
concept is to run a simulation study based on the
actual data for estimating the likely extent of
sampling error, [18]. The objective of the current
study is to assess the efficiencies of three bootstrap
confidence intervals, namely, the percentile
bootstrap (PB), the basic bootstrap (BB), and the
bias-corrected and accelerated (BCa) bootstrap to
estimate the parameter of the PP distribution.
Additionally, none of the bootstrap confidence
intervals will be exact (i.e., the actual confidence
level is exactly equal to the nominal confidence
level 1
) but they will all be consistent, meaning
that the confidence level approaches 1
as the
sample size gets large, [19]. In light of the
impossibility of a theoretical comparison of these
bootstrap confidence intervals, we conduct a
simulated study to evaluate their relative merits.
Moreover, the bootstrap confidence intervals had
been compared via a simulation study in several
studies (see, [20], [21], [22]). In this study, a Monte
Carlo simulation study is conducted to compare
their performance, and used the results to determine
the best performing method based on the coverage
probability and the average length.
2 Theoretical Background
A discrete random variable Y is said to have a
Poisson distribution, with a parameter 0,
if it
has a probability mass function (pmf) given by
(;) ,
!
y
e
py y
0,1,2,...,y (2)
where e is an Euler’s number approximately equal
to 2.718282. The value of
is equal to the
expected value of Y and also to its variance. Hence,
the Poisson parameter
follows a Prakaamy
distribution. The construction of mixed Poisson
distributions showed the details in [5], [6], [13].
Let X be a random variable that follows the
Poisson-Prakaamy (PP) distribution with the
parameter ,
it is denoted as ~PP( ).X
In [13],
the authors defined the pmf of the PP distribution as
5
6543
2
56
(1)
15 85
225 274 120
(;) ,
(120)(1)
x
xxx
xx
px
where 0,1,2,..., 0.x
Plots of the pmf of the PP distribution with some
specified parameter values
are shown in Fig.2.
The expected value and variance of X are
respectively as follows:
5
5
720
() 120
EX
and
11 10 6
5
22
25
840
3840 86400 86400
var( ) .
120
X
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Fig.1: Plots of the pdf of the Prakaamy distribution for
= 0.5, 1, 1.5 and 2
The point estimator of
is obtained by
maximizing the log-likelihood function
log ( ; )
i
Lx
or the logarithm of joint pmf of
1,..., .
n
XX
Therefore, the ML estimator for
of
the PP distribution is derived by the following
processes:
6
51
543
5
2
1
log ( ; )
log ( 6)log( 1)
120
15 85
log ( 1) 225 274 120
i
n
i
i
niii
iii
Lx
nx
xxx
xx
5
5
4
543
15
2
( 720) ( 6)
( 120) 1
5( 1) .
15 85
(1) 225 274 120
n
iiii
ii
nnx
xxx
xx
Solving the equation log ( ; ) 0
i
Lx
for ,
we
have the non-linear equation
5
5
4
543
15
2
( 720) ( 6)
(120) 1
5( 1) 0,
15 85
(1) 225 274 120
n
iiii
ii
nnx
xxx
xx
where
x
denotes the sample mean. Since the ML
estimator for
does not provide the closed-form
solution, the non-linear equation can be solved by
the numerical iteration methods such as Newton-
Raphson, bisection, and Ragula-Falsi methods. In
this paper, the maxLik package, [23], with the
Newton-Raphson method was used for ML
estimation in the statistical software R, [24].
3 Bootstrap Confidence Intervals
Confidence intervals are obtained from a
parametric estimator of the standard errors of a
quantity of interest .
Then, the (1 )100%
confidence interval for
is obtained by adding or
subtracting the standard error multiplied by a
critical value (for example, 1( /2)
ˆˆ
()).zSE
This
calculation assumes that the distribution of the
estimator of
is approximately normal, [21].
However, there are several situations in which the
assumption of normality is incorrect. In these cases,
or when the standard error is very difficult to
estimate, an alternative is to use techniques based
on the bootstrap method, [25]. In this paper, we
focus on three bootstrap confidence intervals for
the parameter of the PP distribution. The bootstrap
confidence intervals, that are most popular in
practice, are the percentile bootstrap, basic
bootstrap, and bias-corrected and accelerated
bootstrap confidence intervals, [19]. The computer-
intensive bootstrap methods described in this study
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provide an alternative for constructing approximate
confidence intervals without having to assume the
underlying distribution, [26]. In this study, a boot
package, [27], was used for estimating the
bootstrap confidence intervals in the statistical
software R. See the details of some bootstrap
methods in [28], [29]. Furthermore, the bootstrap
methods and applications are in [30], [31].
3.1 Percentile Bootstrap (PB) Confidence
Interval
The PB confidence interval is the interval between
the ( / 2) 100
and (1 ( / 2)) 100
percentiles
of the distribution of
estimates obtained from
resampling or the distribution of *
ˆ,
where
represents a parameter of interest and
is the
significance level (e.g.,
= 0.05 for 95%
confidence intervals), [32]. The PB confidence
interval for
can be obtained as follows:
1) B random bootstrap samples of the mother
distribution are generated with a replacement where
B is the number of bootstrap replications,
2) a parameter estimate *
ˆ
is computed from
each bootstrap sample,
3) all B bootstrap parameter estimates are
ordered from lowest to highest, and
4) the (1 )100%
percentile bootstrap
confidence interval is constructed in the following
form:
**
() ()
ˆˆ
,,
PB r s
CI
(4)
where *
()
ˆr
represents the th
r quantile of the set of
ordered quantiles from lowest to highest, *
()
ˆ
s
represents the th
s
quantile of the same set,
(/2) ,rB
(1 ( / 2)) ;
s
B
and
is the
significance level. This study used B= 2,000 and
= 0.05; the quantile corresponding to the lower
limit of the confidence interval was **
() (50)
ˆˆ
r
(the 50th quantile) and that corresponding to the
upper limit was **
() (1950)
ˆˆ
s
(the 1950th quantile).
3.2 Basic Bootstrap (BB) Confidence
Interval
The BB confidence interval is sometimes called the
simple bootstrap confidence interval and is a
method as easy to apply as the PB confidence
interval. Suppose that the quantity of interest is
and that the estimator of
is ˆ.
The BB
confidence interval assumes that the distributions
of ˆ
and *
ˆˆ
are approximately the same,
[26]. The (1 )100%
basic bootstrap confidence
interval for
is
**
() ()
ˆˆ ˆˆ
2,2 ,
BB s r
CI
(5)
where *
()
ˆr
and *
()
ˆ
s
are the same quantile of the
empirical distribution of bootstrap estimates *
ˆ
used in (4) for the PB confidence interval.
3.3 Bias-corrected and Accelerated (BCa)
Bootstrap Confidence Interval
The BCa bootstrap confidence interval incorporates
a bias-correction factor and an acceleration factor
to explain for both bias and skewness of the
bootstrap parameter estimates to resolve the over
coverage probability problems in the PB
confidence interval, [32], [33]. The mathematical
details of the BCa adjustment were provided in
[19], [34]. The estimator of a bias-correction
factor 0
ˆ
z is computed as the proportion of the
bootstrap estimates less than the original parameter
estimate ˆ,
1*
0ˆˆ
ˆ#/,zB
where 1
is
the inverse function of a standard normal
cumulative distribution function (e.g.,
1(0.975) 1.96).
The estimator of an
acceleration factor ˆ
a is calculated through
jackknife resampling (i.e., “leave one out”
resampling), which involves generating n
replicates of the original sample, where n is the
number of observations in the sample. The first
jackknife replicate is obtained by leaving out the
first case (1)i
of the original sample, the second
by leaving out the second case (2),i , and so on,
until n samples of size 1n are obtained. For each
of the jackknife resamples, ()
ˆi
is obtained. The
average of these estimates is () ( )
1
ˆˆ
/.
n
i
i
n
Then, the acceleration factor ˆ
a is estimated as
follows,
3
() ( )
13/2
2
() ( )
1
ˆˆ
ˆ.
ˆˆ
6
n
i
i
n
i
i
a
With the values of 0
ˆ
z and ˆ,a the values 1
and
2
are calculated,
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Fig. 2: Plots of the pmf of the PP distribution for
= 0.5, 1, 1.5 and 2
0/2
10
0/2
ˆ
ˆˆˆ
1
zz
zaz z
and
01(/2)
20
01(/2)
ˆ
ˆ,
ˆˆ
1
zz
zaz z
where 1( /2)
z
is the percentile (1 ( / 2))100
of a
standard normal distribution (e.g. 1 (0.05/2) 1.96z).
Then, the (1 )100%
BCa bootstrap confidence
interval for
is as follows
**
() ()
ˆˆ
,,
BCa j k
CI
(7)
where
1
j
B
and
2.kB
When 0
ˆ0z
and
ˆ0,a the BCa confidence interval is equivalent to
the PB confidence interval.
4 Simulation Study
The interval estimation for the parameter of the PP
distribution estimated via several bootstrap
confidence intervals was considered in this study.
Due to the unavailability of a direct theoretical
comparison, a Monte Carlo simulation study was
designed using R, [24], version 4.2.3 to cover cases
with different sample sizes ( n = 10, 30, 50, 100,
and 500). To observe the effect of small and large
variances, the true value of the parameter (
) was
set as 0.1, 0.3, 0.5, 0.8, 1, and 1.5. The number of
bootstrap replications ( B) was set as 2,000
because, [35], claimed that 2,000 bootstrap samples
are sufficient to estimate the coverage probability
for the 95% confidence intervals with a standard
error of just under 0.5%. The practical problem of
choosing B was discussed in [36], [37]. Bootstrap
samples of size n were generated from the original
pseudo-random sample which has a PP distribution
and each simulation was repeated 1,000 times.
Without loss of generality, the nominal confidence
level 1
was set at 0.95. The proportion of
samples for which the known parameter was
contained in the bootstrap confidence interval. This
proportion was an empirical coverage probability.
The performances of the bootstrap confidence
intervals were compared in terms of their coverage
probabilities and average lengths; the one with a
coverage probability greater than or close to the
nominal confidence level (meaning that it contains
the true value) and the shortest average length can
be used to most accurately estimate the bootstrap
confidence interval for the parameter of interest for
a particular scenario. Hence, it is obvious that when
coverage probability is the same, a smaller average
length indicates that the bootstrap confidence
interval is appropriate for the specific case.
The results of the study were tabulated in Table
1. For n=10, the coverage probabilities of all three
bootstrap confidence intervals tended to be less
than 0.90 and so had not reached the nominal
confidence level. Nevertheless, the BCa bootstrap
confidence interval outperformed the others in
these scenarios. For n=30, all of the bootstrap
confidence intervals once again provided coverage
probabilities that were less than the nominal
confidence level of 0.95. For n50, all of the
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bootstrap confidence intervals attained coverage
probabilities close to the nominal confidence level
and provided similarly average lengths. However,
the BCa bootstrap confidence interval tended to
have a coverage probability that was closer to the
nominal confidence level of 0.95. As the sample
size was increased, the coverage probabilities of the
bootstrap confidence intervals tended to increase
and approach the nominal confidence level of 0.95.
Moreover, the average lengths of the bootstrap
confidence intervals increased when the value of
was increased because of the relationship between
the variance and .
Unsurprisingly, as the sample
size was increased, the average lengths of all three
bootstrap confidence intervals decreased, with the
BCa bootstrap confidence interval providing the
shortest average length for all situations studied.
Furthermore, the average length of the BCa
bootstrap confidence interval was significantly
different from the others when the sample size was
small (n=10). It was observed that the PB and BB
confidence interval provided the average length
were not substantially different for all sample sizes.
In summary, the BCa bootstrap confidence interval
performed well in terms of coverage probability
and average length for moderate and large sample
sizes ( n50).
Table 1. Coverage probability and the average length of the 95% bootstrap confidence intervals for
the PP
distribution
n
Coverage probability Average length
PB BB BCa PB BB BCa
10 0.1 0.895 0.890 0.90 0.052 0.052 0.051
0.3 0.891 0.882 0.891 0.1717 0.1717 0.1695
0.5 0.900 0.896 0.904 0.3185 0.3188 0.3164
0.8 0.900 0.888 0.89
8
0.5331 0.5339 0.5293
1.0 0.892 0.863 0.897 0.6633 0.6641 0.6653
1.5 0.903 0.879 0.891 0.8859 0.8859 0.8994
30 0.1 0.945 0.930 0.943 0.0301 0.0301 0.0298
0.3 0.927 0.920 0.926 0.0982 0.0984 0.0975
0.5 0.931 0.933 0.937 0.1800 0.1798 0.1780
0.8 0.935 0.928 0.940 0.3091 0.3089 0.3055
1.0 0.938 0.928 0.935 0.4023 0.4020 0.3989
1.5 0.942 0.924 0.937 0.5650 0.5651 0.5653
50 0.1 0.931 0.936 0.939 0.0235 0.0234 0.0233
0.3 0.945 0.944 0.948 0.0771 0.0771 0.0767
0.5 0.938 0.935 0.937 0.1381 0.1380 0.1371
0.8 0.946 0.949 0.949 0.2393 0.2390 0.2372
1.0 0.947 0.934 0.937 0.3117 0.3115 0.3094
1.5 0.939 0.938 0.936 0.4423 0.4428 0.4421
100 0.1 0.944 0.952 0.945 0.0166 0.0166 0.0165
0.3 0.946 0.947 0.949 0.0546 0.0545 0.0544
0.5 0.955 0.946 0.951 0.0976 0.0976 0.0972
0.8 0.940 0.926 0.933 0.1694 0.1693 0.1685
1.0 0.947 0.934 0.951 0.2199 0.2200 0.2195
1.5 0.947 0.941 0.944 0.3192 0.3192 0.3192
500 0.1 0.952 0.958 0.956 0.0074 0.0074 0.0074
0.3 0.943 0.947 0.947 0.0244 0.0245 0.0244
0.5 0.950 0.950 0.954 0.0437 0.0437 0.0436
0.8 0.941 0.939 0.941 0.0760 0.0760 0.0760
1.0 0.956 0.960 0.956 0.0992 0.0991 0.0991
1.5 0.949 0.941 0.946 0.1433 0.1434 0.1433
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5 Applications to Real Data
In this section, we used two real-world data sets to
demonstrate the applicability of the bootstrap
confidence intervals for estimating the parameter of
the PP distribution. Besides these two data sets, the
proposed bootstrap confidence intervals can be
applied to the other count data fitted to the PP
distribution. The application of the PP distribution
was shown in [13]. For the applications of the
mixed Poisson distributions, we refer to the
references, [5], [6], [7], [12], [15], and references
therein for further details
5.1 Application to the Number of Mistakes
in Copying Groups of Random Digits
The number of mistakes in copying groups of
random digits collected by [38], was used for this
application. The data consisting of 60 observations
were summarized in Table 2. For this data set, the
sample mean and the standard deviation were
0.7833 and 1.1213, respectively. This study used
the chi-squared goodness-of-fit test for checking
whether the sample data is likely to be from a
specific theoretical distribution, [39]. The chi-
squared statistic was 0.8074 and the p-value was
0.6678. Thus, a PP distribution with ˆ
= 3.0831 is
suitable for this dataset. Table 3 reported the 95%
bootstrap confidence intervals for the parameter of
the PP distribution. The results corresponded with
the simulation results because the average length of
the BCa bootstrap confidence interval was shorter
than those of the PB and BB confidence intervals.
5.2 Application to the Number of
Chromatid Aberrations in Human
Leukocytes
In studies, [40], [41], authors reported the number
of chromatid aberrations in human leukocytes. The
dataset was given in Table 4; the total sample size
is 400. For this data set, the sample mean and the
standard deviation were 0.5475 and 1.0609,
respectively. For the chi-squared goodness-of-fit
test, [39], the chi-squared statistic was 2.4877 and
the p-value was 0.4775. Thus, a PP distribution
with ˆ
= 3.5328 is suitable for this data set. The
95% bootstrap confidence intervals for the
parameter of the PP distribution were tabulated in
Table 5. The results corresponded with the
simulation results because the average length of the
BCa bootstrap confidence interval was shorter than
those of the PB and BB confidence intervals.
6 Conclusions and Discussion
Percentile bootstrap (PB), basic bootstrap (BB),
and bias-corrected and accelerated (BCa) bootstrap
confidence intervals were proposed for estimating
the parameter of the PP distribution. It appears that
the sample size ( n) had a significant effect on the
proposed bootstrap confidence intervals. When the
sample sizes were small ( n=10 and 30), the
coverage probabilities for all three were
substantially lower than 0.95. When the sample
size was sufficiently large ( n50), the coverage
probabilities and average lengths using all three
bootstrap confidence intervals were not markedly
different. According to our findings, the BCa
bootstrap confidence interval performed the best
for almost all of the situations in both the
simulation study and using real data sets. Our
findings provided the simulation results which
correspondent with the study of [21]. They
compared the coverage of four bootstrap
confidence intervals via a Monte Carlo Simulation.
Their results indicated that the coverage
probabilities of the BCa bootstrap confidence
interval were almost always higher than those
obtained with the other confidence intervals.
The limitation of the current study is that none
of the bootstrap confidence intervals were exact but
they would be consistent, meaning that the
coverage probability approaches 0.95 as the sample
sizes get large. In addition, three bootstrap
confidence intervals are not easy to compute and
are computer intensive. However, there are
numerous available packages in R for computing
the bootstrap confidence intervals such as boot
package, [27], bootstrap package, [42], semEff
package, [43], and BootES package, [44]. Since R
is open-source, users are free to download these
packages.
Future research could focus on the other
confidence intervals to compare with the proposed
bootstrap confidence intervals. The constructions of
the interval estimators for the population mean and
the coefficient of variation are interesting.
Moreover, there is no research on the hypothesis
testing for the parameter of the PP distribution.
These issues can be studied in the future.
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Table 2. The number of mistakes in copying groups of random digits
Number of mistakes per group 0 1 2 3
Observed frequency 35 11 8 6
Expected frequency 35.0138 12.6752 6.1166 6.1945
Table 3. The 95% bootstrap confidence intervals and corresponding widths using all intervals for the parameter
in the number of mistakes in copying groups of random digits
Methods Confidence intervals Width
s
PB (2.7683, 3.5497) 0.7814
BB (2.6167, 3.4060) 0.7893
BCa (2.7458, 3.5069) 0.7611
Table 4. The number of chromatid aberrations in human leukocytes
Number of chromatid aberrations 0 1 2 3 4
Observed frequency 268 87 26 9 10
Expected frequency 271.9948 77.7099 28.8613 12.3989 9.0351
Table 5. The 95% bootstrap confidence intervals and corresponding widths using all intervals for the parameter
in the number of chromatid aberrations in human leukocytes
Methods Confidence intervals Width
s
PB (3.3359, 3.7916) 0.4557
BB (3.2778, 3.7318) 0.4540
BCa (3.3295, 3.7666) 0.4371
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