DELTD:
An R Package for Kernel Density Estimation using Lifetime Distributions
JAVARIA AHMAD KHAN1, ATIF AKBAR1, B. M. GOLAM KIBRIA2
1Department of Statistics,
Bahuddin Zakariya University, Multan,
PAKISTAN
2Department of Mathematics and Statistics,
Florida International University, Miami, FL 33199,
USA
Abstract: - The R package DELTD is for estimating densities by asymmetrical kernels and calculating MSE. This
package is to estimate densities that are free of boundary bias. The major concern of the package is to enhance its
usefulness in performing inference regarding stated kernels. For this purpose, some lifetime distributions, i.e. Beta,
Birnbaum-Saunders, Erlang, Gamma, and Lognormal are considered here due to their usefulness in life data
analysis, where their estimated values for density estimation can also be observed. Tuna data is also presented in
this package. By using these kernels, densities will be free of boundary problems. This package is a collection of
asymmetrical kernels which belong to the lifetime distribution.
Key-Words: - Asymmetrical kernel, Beta kernel, Birnbaum-Saunders kernel, Erlang kernel, Gamma kernel,
Lognormal kernel, Tuna data
Received: May 28, 2023. Revised: August 29, 2023. Accepted: October 1, 2023. Published: October 20, 2023.
1 Introduction
Density estimation is a process of constructing the
probability density function using underlying data.
Applications of density estimation can be found in
many fields of daily life, which may be:
In Statistics: [1], stated that density estimates can
be applied in the construction of smooth
distribution function estimates via integration,
which then can be used to generate bootstrap
samples from a smooth estimate of the
cumulative distribution function rather than from
the empirical distribution. Other statistical
applications include identifying the
nonparametric part in semiparametric models,
finding optimal scores for nonparametric tests,
and empirical Bayes methods.
In Engineering: The detection of abnormal or
unexpected conditions from measured response
data is an important issue, especially where a
clear and early warning of an abnormal condition
is required. For this purpose, [2], proposed a
method that is based upon the probability density
function (PDF) estimated using a kernel method.
Other examples can be found in, [3], [4], and
references therein.
In Hydrology: The estimated density function of
rainfall, river discharge data, modeling of
precipitation, and other hydroclimatic variables
analyzed with a probability distribution, are used
to gain insight into their behavior and frequency
of occurrence, [5], [6].
In Medicine: [7], sought the study to investigate
whether individuals who live near destinations
(service facilities, etc.) are more intensely
distributed rather than dispersed. They used the
kernel density estimation technique to examine
how much they are active and engage in more
frequent walking for transport and recreation. For
other applications in medicine see, [8], [9], [10].
In Physics: [11], kernel density estimate in the
Lamb wave-based damage detection. They
showed that the distribution of data is based on
the intensity of the noise. In the case of weak
noise, the pdf of measured data could be
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Javaria Ahmad Khan, Atif Akbar, B. M. Golam Kibria
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considered as the normal distribution. However,
in the case of strong noise, the pdf was complex
and did not belong to any type of common
distribution function.
In Bioinformatics and Genetics: in the last few
decades, the importance of the development of
computational systems for automated analysis of
large amounts of data (high-throughput) has
risen. The study, [12], discussed such problems
and their solution based on LOWESS and
running median. Additionally, they measured a
rodent's distance from the arena's wall. They
examined the density of distances from the
boundary when the algorithm to estimate the
boundary is being used and when it is not.
Further application can be observed in, [13],
[14], [15].
In Finance and Economics: [16], suggested using
the sequential method for the estimation of the
size distribution of U.S. family income.
Similarly, [17], [18], provide a healthy literature
to enhance the importance of density estimation
in this field.
One could think of other several applications in
archaeology, [19], climatology, [20], physiology,
[21], astronomy, [22], [23], geoscience, [24],
[25], and continues to be relevant in new areas of
mathematics and information science, [26], [27],
[28].
There are a variety of approaches to estimating
the density; Kernel density estimation, histogram,
data clustering, semi-parametric methods, etc. Kernel
density estimation is one of the very famous
techniques. The kernel estimator proposed by, [29],
[30], is given as;
(1)
where is the kernel function and represents the
bandwidth or smoothing parameter. Due to this, a
serious problem of boundary bias or boundary effect
arises. If reflected in the results variance and bias
showed a sharp increase when estimating them at
points near the boundary region. In other words, it
affects the performance of the estimator at the
boundary points due to boundary effects, then from
the interior points. Such a problem occurs when
variables represent some sort of physical measure
such as time or length. These variables thus, have a
naturally lower boundary, e.g. time of birth or zero
point on a scale. So, when smoothing is carried out
near the boundary and fixed the symmetric kernel is
used, those kernels allocate weights outside the
density support, [31]. To remove those boundary
effects in kernel density estimation, a variety of
methods have been developed in the literature. Some
well-known methods are summarized below:
1.1 Reflection Method
The reflection method was first introduced by, [32],
and then studied by, [33]. The main idea is to reflect
the data points. This not only yields a twice as large
sample size but most importantly yields a sample
drawn from a density with unbounded support. Then
kernel estimator is applied to data of size and then
the new estimate is symmetrical around the origin.
1.2 Transformation Method
To control boundary bias, [34], suggested
transforming the data at both sides to a density that
has its first derivative equal to 0. They suggested
different transformations from a parametric family, in
general, and compared with Rice's adjusted kernel
method. They claimed that their proposed method
produced non-negative estimates and outperformed
Rice's adjustment.
1.3 Pseudo-data Method
The study, [35], presented this method, which is
based on pseudo data, which is beyond the limits of
density support. They claimed that their method is
more adaptive because the pseudo method is more
appropriate for kernels of order 2 and more. The
estimators produced by this method may gain optimal
orders of bias, variance, and lower mean squared
error at . They suggested using the plug-in and
least square cross-validation method for bandwidth
selection.
1.4 Local Linear Method
The study, [36], introduced estimators that utilize
density derivative estimators obtained from local
polynomial fitting. He compared his proposed
estimator and its asymptotically optimal bandwidth
with Sheather and Jones's bandwidth. However, he
showed that former bandwidth overcomes the
boundary problems and later does not. A similar
technique was further used by, [37], in which they
the local polynomial smoothing technique as a
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possible alternative method for the problem. It was
observed that such an estimator possesses desirable
properties such as automatic adaptability for
boundary effects near endpoints. They also obtained
an optimal kernel to estimate the density at endpoints
as a solution to a variation problem.
1.5 Rice's Boundary Modification
The study, [38], adapted Rice's method to the context
of density estimation. The study, [39], proposed a
method based on boundary modification of kernel
regression estimators. In this method, a linear
combination of two kernel regression estimators is
used with two different bandwidths in the boundary
area, as the same is used in the interior. The idea is
similar to the bias reduction technique discussed in,
[40].
To handle this problem, [41], suggested the
solution of this problem by replacing the symmetric
kernels with the asymmetric Beta kernel which never
assigns weight outside the support. Many others used
Chen's idea and proposed other kernels, i.e.
lognormal, Weibull, Inverse Gaussian, etc. All of the
methods perform well with different bandwidths.
However, no package directly estimates the densities
according to these asymmetrical kernels and also
calculates its mean squared error. Due to the reasons
stated above, we have developed a package DELTD,
in R language, [42]. In which, we have used some
asymmetrical kernels for which parent distribution
belongs to the family of lifetime distributions to
estimate the density and to calculate their mean
squared error. To the best of our knowledge, no
package is available that plots the density using a
variety of asymmetrical kernels and calculates their
MSE. There are lots of packages that are frequently
used for density estimation, but almost all of them
use symmetrical kernels. The problem of boundary
bias occurs using the symmetrical kernel as
mentioned above. The package is available from the
Comprehensive R Archive Network (CRAN), [43].
This paper aims to describe this package, and also, to
summarize and conveniently present the functions.
This may help interested readers to apply this kind of
technique to real situations. The structure of this
paper is as follows. Section 2 introduces the lifetime
distributions and their relevant kernel. Section 3
introduces the utility of Mean Squared Error (MSE).
Section 4 presents the implementation of functions in
package DELTD, with examples and argument
details and lastly, Section 5 is devoted to conclusions
with some suggestions for future work.
2 Life Time Distributions
Distributions that tend to better represent life data are
known as lifetime distributions, [44]. Like lognormal
distribution is found in environmental studies, milk
production of cows, amount of rainfall, the volume of
gas in a petroleum reserve, etc., [45], [46],
Birnbaum-Saunders distribution describes the fatigue
life studies, [47], and Beta distribution is used for
percentages, proportion, rates, and fractions, [48].
Similarly, applications of gamma distribution are
found in neuroscience, in bacterial gene expression,
[49], [50]. The gamma distribution is widely used as
a conjugate prior in Bayesian statistics, etc. Erlang
distribution is a specified case of Gamma
distribution, [51], and is used in queuing theory, in
the mathematical study of waiting in lines. It is also
used in stochastic processes mathematical biology,
etc. In this paper, we are interested in only those
distributions that belong to the lifetime distribution
family and their asymmetrical kernels are available in
the literature, e.g. Beta, Birnbaum-Saunders, Erlang,
Gamma, and Lognormal.
Let be a random sample from a
distribution with an unknown probability density
function which has bounded support on ,
with and representing the bandwidth. In the
following subsections kernels are presented for the
above-stated distributions that we are going to use in
the package and were developed to handle the
problem of boundary bias.
2.1 Beta Kernel
The study, [41], proposed a Beta kernel for
estimating curves with compact support by using the
Beta distribution of the first kind. The beta kernel
smoother is free of boundary bias, achieving the
optimal convergence rate of
for mean integrated
squared error and always allocate non-negative
weights. Further, they compared the beta smoothers
and the local linear smoothers. Beta Kernel is
(2)
2.2 Birnbaum-Saunders Kernel
The study, [52], extended the class of non-negative,
asymmetric kernel density estimators and proposed
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Birnbaum-Saunders (BS) kernel density function.
The density function has bounded support on .
They applied a BS kernel density estimator to high-
frequency intraday time duration data. The
comparisons are made on several nonparametric
kernel density estimators. BS kernel performs better
near the boundary in terms of bias reduction.
(3)
2.3 Erlang Kernel
Erlang kernel is proposed by, [53]. They suggested
using it for non-parametrically estimation of the
probability density function (pdf). Moreover, they
investigated the asymptotic normality of the
proposed estimator.
(4)
2.4 Gamma Kernel
The study, [54], developed the Gamma kernel. As it
is stated above, the reason behind this development is
to handle boundary bias, which arises in symmetrical
kernels. He showed that kernels are nonnegative, has
naturally varying shape, and achieve the optimal rate
of convergence for the mean integrated squared error.
The Gamma kernel which we considered in our
package is as follows:
(5)
2.5 Lognormal Kernel
This kernel is also proposed by, [52]. They showed
that this kernel also performs equally well with the
Birnbaum-Saunders (BS) kernel.
(6)
3 Mean Squared Error
The mean squared error described the squared
difference between the actual and estimated values. It
measures the average of the squares of the errors.
Mathematically, we can express this as
(7)
is the number of data points, represents actual
values and represents estimated value. MSE is a
risk function, corresponding to the expected value of
the squared error loss. It is always non-negative, and
values closer to zero are better, [55].
4 The Package Overview
The package DELTD contains functions for density
estimation by using asymmetrical kernels named
Beta, Birnbaum-Saunders, Erlang, Gamma, and
Lognormal. For these kernels, densities are
calculated and represented graphically. The mean
squared error (MSE) for each kernel can also be
calculated. The following section demonstrates the
use of the DELTD package with simulated examples.
The functions within DELTD are briefly described in
Table 1.
For density estimation, five functions (Table 3)
are presented that are used to estimate the density by
using Beta (plot.Beta), Birnbaum-Saunders (plot.BS),
Erlang (plot.Erlang), Gamma (plot.Gamma) and
lognormal (plot. LogN) kernels. For all kernels,
estimated values for density estimation can also be
analyzed by using Beta, BS, Erlang, Gamma, and
LogN, for details, see Table 2. In our examples for
observing estimated values of density, we generated
a sample by using different distributions with
different sample sizes. Function(s) related to density
estimation depends on the grid size and .
Practically, estimation may be quite slow with small
grid points, but it is important to note that for large
grid points, density is smoother. In nonparametric
estimation, bandwidth plays a very important
role. So, also affects the smoothness of density
along with grid points.
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Table 1. Summary of contents of the package
Functions
Description
Beta
Estimate Density Values by the Beta
kernel
BS
Estimate Density Values by Birnbaum-
Saunders kernel
Erlang
Estimate Density Values by Erlang kernel
Gamma
Estimate Density Values by Gamma
kernel
LogN
Estimate Density Values by Lognormal
Kernel
mse
Calculate Mean Squared Error (MSE) by
using different Kernels
plot.Beta
Density Plot by the Beta kernel
plot.BS
Density Plot by Birnbaum-Saunders
kernel
plot.Erlang
Density Plot by Erlang kernel
plot.Gamma
Density Plot by Gamma kernel
plot.LogN
Density Plot by Lognormal kernel
TUNA
Data on Tuna fish
Functions for observing estimated values provide
grid points and estimated values of density. All such
functions have some default arguments, if the user
does not provide such parameters then the function
proceeds by using those arguments. But the user must
provide at least or . If is missing in the function
then the package generates grid points between
minimum and maximum values of vector (y). Only in
case Beta is used, with missing , then grid points
will be generated by using a uniform distribution
as restricted by the author. Similarly, if is
missing then the function proceeds by setting ,
where is the length of the vector (y). In case, if the
user does not provide the then the function uses
(8)
which is described by, [56], for non-normal data.
5 Estimated Values of Density:
Illustrative Examples
Here we are using BS for illustration, with all
missing situations. In the following example, all
arguments of a function are user-defined. Here we
are using a quite small to present results. It's better
to use the same length of grid points for one
kernel. Although, it proceeds unequal halt the plot
() or generate NA.
Table 2. Summary of arguments of Beta, BS, Erlang,
Gamma and LogN
Arguments
Description
a scheme for generating grid points
a vector of positive values
number of grid points
the bandwidth
> alpha = 10
> theta = 15 / 60
> k <- 10
> y <- rgamma(n = 100, shape = alpha,
> scale = theta)
> xx <- seq(min(y) , max(y), length =k)
> h <- 1.1
> den <- BS(x = xx, y = y, k = k, h = h)
It provides;
> den
$x
[1] 0.8556524 1.3077240 1.7597956 2.2118671
[5] 2.6639387 3.1160103 3.5680819 4.0201535
[9] 4.4722250 4.9242966
$y
[1] 0.1167586 0.1461795 0.1572175 0.1595425
[5] 0.1573851 0.1528002 0.1468850 0.1402692
[9] 0.1333330 0.1263139
If the scheme for generating grid points is
unknown; then the function proceeds with the above-
mentioned scheme. But for Beta, it is restricted by
the author that grid points and vector (either real
or simulated) lie between 0 and 1. Any other scheme
will produce for beta-estimated values.
> y <- rgamma(n = 1000, shape = alpha,
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> scale = theta)
> h <- 3
> BS(y = y, k = 90, h = h)
Similarly, the following example describes the
situation, if the user does not mention the number of
grid points. Further, it is not necessary that must be
fixed; it can be calculated by any other source.
> y <- rgamma(n = 10, shape = alpha,
> scale = theta)
> xx <- seq(0.001, 1000, length = 10)
> #any bandwidth can be used
> require(KernSmooth)
# Direct Plug-In Bandwidth
> h <- dpik(y)
> BS(x = xx, y = y, h = h)
It results, where the length of is adopted by default.
[1]
[6]
[1]
[4]
[7]
[10]
, "class")
[1] "list" "BS"
If both the generating scheme and the number of
grid points are missing then the function is halted and
will not process.
> y <- rgamma(n = 1000,
> shape = alpha, scale = theta)
band
band
If bandwidth is missing then density points can be
calculated as;
> y <- rgamma(n = 1000,
> shape = alpha, scale = theta)
> xx <- seq(0.001, 100, length = 1000)
> BS(x = xx, y = y, k = 900)
Similarly, Beta, Erlang, Gamma, and LogN can be
used for their respective kernels. For details and
examples see, [43].
6 Density Plot: Illustrative Example
To plot density, any kernel plot() can be used, for
details see Table 3 For continuity, the BS kernel is
used in Figure 1.
Table 3. Summary of arguments of plot.Beta,
plot.BS, plot.Erlang, plot.Gamma and plot.LogN
Arguments
Description
An object of class "Beta", "BS",
"Erlang", "Gamma" or "LogN"
Not presently used in this
implementation
## other details can also be added
plot (den, type "Density
Function", lty lab "Time")
## To add true density along with estimated
, type "p", col = "red")
legend("topright", c("Real Density", "Density by
Birnbaum-Saunders Kernel"), col=c("red", "black"),
lty
Fig. 1: Density Estimation by Using BS Kernel.
Further, the Tuna fish dataset, [57], is used to
enhance the usefulness of these kernels. The data is
about Tuna, which is saltwater fish. Its seasonal
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migration is between waters off the coast of Australia
and the Indian Ocean. The data represents a line
transect aerial survey of Southern Bluefin Tuna in the
Great Australian Bight in summer when the tuna
tends to stay on the surface. The abundance is
measured by
, where is the total number of
surface schools in the Bight and is the survey area.
To estimate , an aircraft with two spotters on board
is used to fly randomly allocated transect lines to
detect tuna schools. Each school sighted from the
transect is counted, and its perpendicular distance to
the transect is measured.
7 Mean Squared Error (MSE):
Illustrative Example
This function helps to examine the accuracy of
different considered estimation methods, in terms of
mean squared error (MSE). Table 4 presents
argument details related to this function. These
functions can be utilized only when data follows
exponential, Gamma, or Weibull distribution.
Similarly, Figure 2 presents the density estimation by
using BS Kernel for Tuna data.
Table 4. Summary of argument of mse
Arguments
Description
kernel
type of kernel which is to be used
type
mention the distribution of vectors. If
exponential distribution then use "Exp". If
use gamma distribution then use
"Gamma". If Weibull distribution then use
"Weibull".
Fig. 2: Density Estimation by Using BS Kernel for
Tuna data
kernel den, type "Exp")
[1] 0.002491046
If a distribution other than above mentioned type is
used then NaN will be produced.
mse (kernel = den, type ="Beta")
[1] NaN
8 Summary
In this paper, we have illustrated the functions of the
package DELTD. The package is about the density
estimation through asymmetrical kernels when parent
distribution belongs to lifetime distributions, e.g.
Beta, BS, Exponential, Erlang, Gamma, Logistic, and
Lognormal distribution. Additionally, their mean
squared error (MSE) and plot are constructed through
simulation data. Till that time, the package was the
first publicly available software for the estimation of
density by using asymmetrical kernel(s).
Density estimation is a powerful tool to collect
information about its unknown distribution from
given data. Due to this, kernel density estimation is
very popular. But typically, symmetrical kernels are
considered for estimation, which are sensitive to
boundary bias. To overcome this problem, [41],
proposed to use asymmetrical kernel which is non-
negative and free of boundary bias. In this paper and
the package DELTD, we combined major
asymmetrical kernels that are based on lifetime
distribution, i.e. Beta, BS, Erlang, Gamma, and
Lognormal. The MSE criteria may be used to
examine the accuracy of estimated kernels with real
data.
Extensions towards the software package with
more lifetime distribution kernels can be added in the
package and other distributions can be introduced
which can help to calculate MSE. This package can
also be combined with Artificial Intelligence. In
which, the function automatically identifies the
suitable kernel; which has minimum MSE with
estimated density. This package can be of interest to
all those practitioners of different scientific fields
who use any lifetime distribution(s) and those who
estimate the densities for different purposes.
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References:
[1] Silverman, B. & Young, G., The bootstrap: to
smooth or not to smooth?, Biometrika, Vol.74,
1987, pp. 469-479.
[2] Desforges, M., Jacob, P. & Cooper, J.,
Applications of probability density estimation
to the detection of abnormal conditions in
engineering, Proceedings of The Institution of
Mechanical Engineers, Part C: Journal of
Mechanical Engineering Science, Vol.212,
1998, pp. 687-703.
[3] Girolami, M. & He, C., Probability density
estimation from optimally condensed data
samples, IEEE Transactions on Pattern
Analysis and Machine Intelligence, Vol.25,
2003, pp. 1253-1264.
[4] Hollands, K. & Suehrcke, H., A three-state
model for the probability distribution of
instantaneous solar radiation, with applications,
Solar Energy, Vol.96, 2013 pp. 103-112.
[5] Rajagopalan, B., Lall, U. & Tarboton, D.,
Evaluation of kernel density estimation
methods for daily precipitation resampling,
Stochastic Hydrology and Hydraulics, Vol.11,
1997, pp. 523- 547.
[6] Kim, K. & Heo, J., Comparative study of flood
quantiles estimation by nonparametric models,
Journal of Hydrology, Vol.260, 2002, pp. 176-
193.
[7]
King, T., Thornton, L., Bentley, R. & Kavanagh,
A., The use of kernel density estimation to ex-
amine associations between neighborhood
destination intensity and walking and physical
activity, PLoS One, Vol.10, No.e0137402,
2015.
[8] Shankar, P., The use of the compound
probability density function in ultrasonic tissue
characterization, Physics in Medicine &
Biology, Vol.49, 2004, pp. 1007.
[9] Rosado-Mendez, I., Drehfal, L., Zagzebski, J. &
Hall, T., Analysis of coherent and diffuse
scatter
ing using a reference phantom, IEEE
Transactions on Ultrasonics, Ferroelectrics,
And Frequency Control, Vol.63, 2016, pp.
1306-1320.
[10] Kang, E., Lee, E., Jang, M., Kim, S., Kim, Y.,
Chun, M., Tai, J., Han, W., Kim, S. & Kim, J.,
Reliability of computer-assisted breast density
estimation: comparison of interactive
thresholding, semiautomated, and fully
automated methods, American Journal of
Roentgenology, Vol.207, 2016, pp. 126-134.
[11] Yu, L. & Su, Z., Application of kernel density
estimation in lamb wave-based damage
detection, Mathematical Problems in
Engineering, Vol.2012, 2012.
[12]
Sakov, A., Golani, I., Lipkind, D., Benjamini, Y.
& Others., High-throughput data analysis in
behavior genetics, The Annals of Applied
Statistics, Vol.4, 2010, pp. 743-763.
[13] Ewens, W. & Grant, G., Statistical methods in
bioinformatics: an introduction, Springer
Science & Business Media, 2006.
[14] Knapp, B., Frantal, S., Cibena, M., Schreiner,
W. & Bauer, P., Is an intuitive convergence
definition of molecular dynamics simulations
solely based on the root mean square deviation
possible?, Journal of Computational Biology,
Vol.18, 2011, pp. 997-1005.
[15]
Sawle, L. & Ghosh, K., Convergence of
molecu
lar dynamics simulation of protein native
states: Feasibility vs self-consistency dilemma,
Journal of Chemical Theory and Computation,
Vol.12, 2016, pp. 861-869.
[16] Wu, X., Calculation of maximum entropy
densities with application to income
distribution, Journal of Econometrics, Vol.115,
2003, pp.347-354.
[17] Tortosa-Ausina, E., Cost efficiency and product
mix clusters across the Spanish banking
industry, Review of Industrial Organization,
Vol.20, 2002, pp. 163-181.
[18] Alemany, R., Bolancé, C. & Guillén, M., A
nonparametric approach to calculating value-at-
risk, Insurance: Mathematics and Economics,
Vol.52, 2013, pp. 255-262.
[19] Baxter, M., Beardah, C. & Westwood, S.,
Sample size and related issues in the analysis of
lead isotope data, Journal of Archaeological
Science, Vol.27, 2000, pp. 973-980.
[20]
Hannachi, A., Quantifying changes and their
un
certainties in probability distribution of
climate variables using robust statistics, Climate
Dynamics, Vol.27, 2006, pp. 301-317.
[21] Paulsen, O. & Heggelund, P., The quantal size
at retinogeniculate synapses determined from
spontaneous and evoked EPSCs in guineapig
thalamic slices, The Journal of Physiology,
Vol.480, 1994, pp. 505-511.
[22] Rau, M., Seitz, S., Brimioulle, F., Frank, E.,
Friedrich, O., Gruen, D. & Hoyle, B., Accurate
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photometric redshift probability density
estimation–method comparison and
application, Monthly Notices of The Royal
Astronomical Society, Vol.452, 2015, pp. 3710-
3725.
[23] Cavuoti, S., Amaro, V., Brescia, M., Vellucci,
C., Tortora, C. & Longo, G., METAPHOR: a
machine-learning-based method for the
probability density estimation of photometric
redshifts, Monthly Notices of The Royal
Astronomical Society, Vol.465, 2016, pp. 1959-
1973.
[24] Li, X. & Gong, F., A method for fitting
probability distributions to engineering
properties of rock masses using Legendre
orthogonal polynomials, Structural Safety,
Vol.31, 2009, pp.335- 343.
[25] Woodbury, A., A FORTRAN program to
produce minimum relative entropy
distributions, Computers & Geosciences,
Vol.30, 2004, pp. 131-138.
[26] Lu, N., Wang, L., Jiang, B., Lu, J. & Chen, X.,
Fault prognosis for process industry based on
information synchronization, IFAC Proceedings
Volumes, Vol.44, 2011, pp. 4296-301.
[27] Hajihosseini, P., Salahshoor, K. & Moshiri, B.,
Process fault isolation based on transfer entropy
algorithm, ISA Transactions, Vol.53, 2014, pp.
230-240.
[28] Xu, S., Baldea, M., Edgar, T., Wojsznis, W.,
Blevins, T. & Nixon, M., Root cause diagnosis
of plant-wide oscillations based on information
transfer in the frequency domain, Industrial &
Engineering Chemistry Research, Vol.55, 2016,
pp. 1623-1629.
[29] Nadaraya, E., On estimating regression, Theory
of Probability & Its Applications, Vol.9, 1964,
pp. 141-142.
[30] Watson, G., Smooth regression analysis,
Sankhyā: The Indian Journal of Statistics,
Series A, 1964, pp. 359-372.
[31] Jou, P., Akhoond-Ali, A., Behnia, A. & Chini-
pardaz, R., Parametric and nonparametric
fre
quency analysis of monthly precipitation in
Iran,
Journal of Applied Sciences, Vol.8, 2008,
pp. 3242-3248.
[32] Schuster, E., Incorporating support constraints
into nonparametric estimators of densities,
Communications in Statistics-Theory and
Meth
ods, Vol.14, 1985, pp. 1123-1136.
[33] Cline, D. & Hart, J., Kernel estimation of
densi
ties with discontinuities or discontinuous
deriva
tives, Statistics: A Journal of Theoretical
and Applied Statistics, Vol.22, 1991, pp. 69-84.
[34]
Marron, J. & Ruppert, D., Transformations to
re
duce boundary bias in kernel density
estimation, Journal of The Royal Statistical
Society: Series B (Methodological), Vol.56,
1994, pp. 653-671.
[35]
Cowling, A. & Hall, P., On pseudo data methods
for removing boundary effects in kernel density
estimation, Journal of The Royal Statistical
Society: Series B (Methodological), Vol.58,
1996, pp. 551-563.
[36] Cheng, M., Boundary aware estimators of inte
grated density derivative products, Journal of
The Royal Statistical Society: Series B
(Statistical Methodology), Vol.59, 1997, pp.
191-203.
[37] Zhang, S. & Karunamuni, R., On kernel density
estimation near endpoints, Journal of Statistical
Planning and Inference, Vol.70, 1999, pp. 301-
316.
[38] Cheng, M. & Others, Choice of the bandwidth
ratio in Rice’s boundary modification, Journal
of The Chinese Statistical Association, Vol.44,
2006, pp. 235-251.
[39] John, R., Boundary modification for kernel
regression, Communications in Statistics-
Theory and Methods, Vol.13, 1984, pp. 893-
900.
[40] Schucany, W. & Sommers, J., Improvement of
kernel type density estimators, Journal of The
American Statistical Association, Vol.72, 1977,
pp. 420-423.
[41] Chen, S., Beta kernel smoothers for regression
curves, Statistica Sinica, 2000, pp. 73-91
[42] R Core Team R: A Language and Environment
for Statistical Computing, (R Foundation for
Statistical Computing), 2023, https://www.R-
project.org/, ISBN 3-900051-07-0
[43] DELTD: Kernel Density Estimation using
Lifetime Distributions Manual (Online),
https://CRAN.R-project.org/package=DELTD
(Accessed Date: October 13, 2023)
[44] Bekker, P., A lifetime distribution model of
de
preciable and reproducible capital assets,
Ams
terdam: Vrije Universiteit, 1991.
[45]
Limpert, E., Stahel, W. & Abbt, M., Log-normal
distributions across the sciences: keys and clues:
on the charms of statistics, and how mechanical
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.82
Javaria Ahmad Khan, Atif Akbar, B. M. Golam Kibria
E-ISSN: 2224-2880
754
Volume 22, 2023
models resembling gambling machines offer a
link to a handy way to characterize log-normal
distributions, which can provide deeper insight
into variability and probability—normal or log-
normal: that is the question, BioScience, Vol.51,
2001, pp. 341-352.
[46] Foster, J., Bevis, M. & Raymond, W.,
Precipitable water and the lognormal
distribution, Journal of Geophysical Research:
Atmospheres, Vol.111, 2006.
[47] Birnbaum, Z. & Saunders, S., A new family of
life distributions, Journal of Applied
Probability, Vol.6, 1969, pp. 319-327.
[48] Ferrari, S. & Cribari-Neto, F., Beta regression
for modelling rates and proportions, Journal of
Applied Statistics, Vol.31, 2004, pp. 799-815.
[49] Robson, J. & Troy, J., Nature of the maintained
discharge of Q, X, and Y retinal ganglion cells
of the cat, JOSA A, Vol.4, 1987, pp. 2301-2307.
[50] Wright, M., Winter, I., Forster, J. & Bleeck, S.,
Response to best-frequency tone bursts in the
ventral cochlear nucleus is governed by ordered
inter-spike interval statistics, Hearing Research,
Vol.317, 2014, pp. 23-32.
[51] Jambunathan, M., Some properties of beta and
gamma distributions, The Annals of
Mathematical Statistics, 1954, pp. 401-405.
[52] Jin, X., Kawczak, J. & Others., Birnbaum-
Saunders and lognormal kernel estimators for
modelling durations in high frequency financial
data, Annals of Economics and Finance, Vol.4,
2003, pp. 103-124.
[53] Salha, R., El Shekh Ahmed, H. & Alhoubi, I.,
Hazard rate function estimation using Erlang
kernel, Hazard Rate Function Estimation Using
Erlang Kernel, Vol.3, 2014.
[54] Chen, S., Probability density function
estimation using gamma kernels, Annals of The
Institute of Statistical Mathematics, Vol.52,
2000, pp. 471-480.
[55] Mood, A., Graybill, F. & Boes, D.,
Introduction to the theory of statistics,
McGraw-Hill Inc, New York, 1974.
[56] Silverman, B., Density Estimation, Chapman &
Hall/CRC, 1986.
[57] Buckland, S., Burnham, K., Anderson, D. &
Laake, J., Density estimation using distance
sampling, Chapman Hall, London, 1993.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Javaria Ahmad Khan: conceived the idea,
developed and maintained the package,
programming, graph making, writing, performed
the computations, and final approval of the version
to be published.
- Atif Akbar: editing, proofreading, and final
approval for publication.
- B M Golam Kibria: proofreading, formatting, and
critical review.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en_
US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.82
Javaria Ahmad Khan, Atif Akbar, B. M. Golam Kibria
E-ISSN: 2224-2880
755
Volume 22, 2023
