A Study on Alpha Power Lomax Distribution
Rana A. Bakoban
1 , E. A. Farag 1,2,
Najwa S. Alsulami 1
1 Department of Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabi
2 Faculty of Science, Mathematics Development, Helwan University, Egypt
|
ABSTRACT |
||
In this paper,
we refer to the new distribution an alpha power Lomax distribution. Various
properties of the proposed distribution are obtained including mode,
quantiles, entropies, and order statistics are obtained. Parameters of the
proposed distribution are estimated using maximum likelihood, ordinary least
squares and weighted least squares. Simulation study is conducted to compare
between estimators. |
|||
Received 15 September 2022 Accepted 16 October 2022 Published 31 October 2022 Corresponding Author Rana A. Bakoban, rabakoban@uj.edu.sa
DOI 10.29121/IJOEST.v6.i5.2022.393 Funding: This research
received no specific grant from any funding agency in the public, commercial,
or not-for-profit sectors. Copyright: © 2022 The
Author(s). This work is licensed under a Creative Commons
Attribution 4.0 International License. With the
license CC-BY, authors retain the copyright, allowing anyone to download,
reuse, re-print, modify, distribute, and/or copy their contribution. The work
must be properly attributed to its author. |
|||
Keywords: Alpha Power
Lomax Distribution, Maximum Likelihood Estimation, Renyi
Entropies, Stress–Strength Reliability and Order Statistics |
1. INTRODUCTION
Improvement over standard distributions has gained popularity in statistical theory over the past few years. Typically, new distributions are created by combining existing distributions or adding a new parameter using generators. Alpha power transformation (APT) a novel approach for integrating an additional parameter in continuous distribution, was introduced by Mahdavi and Kundu (2017). Basically, the concept was put forth to add skewness to the baseline distribution. Then: Let g(x) be the probability density function (PDF) of a continuous random variable X.:
Equation 1
The corresponding cumulative distribution function (CDF)
Equation
2
Where α is the shape parameter and G(x) and g(x) are the CDF and the PDF of the baseline distribution.
Mahdavi and Kundu (2017) developed a brand-new method for creating distributions by adding a second shape parameter to a well-known baseline distribution. They used the exponential baseline distribution to introduce the alpha power exponential distribution, and they researched the fundamental characteristics as well as parameter estimation for the suggested distribution. Alpha power Weibull distribution (APWD) was introduced by Nassar et al. (2017). The capacity of APWD to simulate monotone and non-monotone failure rate functions, which are often used in reliability studies, is what gives its significance. Moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength were among the attributes of APWD that were discovered. The parameters were estimated using the maximum likelihood method. The significance of the APWD was demonstrated using of actual data sets. Devi et al. (2017) introduced the entropy of the Lomax probability distribution. They derived entropy of a Lomax probability distribution as well as their order statistics which is used in business, economics, actuarial modelling, waiting problems, and biological sciences.
This study’s objective is to provide a new more adaptable distribution. The structure of this article is as follows: We go over the distribution of Alpha power Lomax (APL) in Section 2. In Section 3, new statistical properties are investigated. The Renyi entropy is derived in Section 4. In Section 5, the stress-strength reliability is described. Section 6 presents the order statistics from the Alpha power Lomax. Section 7 examines estimate techniques. Section 8 provides a simulation study. In Section 9, application is offered.
2. ALPHA POWER LOMAX DISTRIBUTION
In this section, we will define a new model is called alpha power Lomax distribution, denoted (APL), with three parameters (α, λ, β).
According to El-Houssainy et al. (2016) cumulative’s distribution function of Lomax distribution with scale parameter λ > 0 is
Equation 3
where, β > 0 and λ > 0.
The probability density function corresponding to (3) reduces to
Equation 4
The CDF of APL is generated by renaming Equation 3 as a baseline cumulative function in Equation 2.
Equation 5
and corresponding PDF of APL is defined as follows
Equation 6
where λ is the scale parameter and α is the shape parameter.
The PDF curves for the APL distribution are shown in Figure 1, Figure 2, Figure 3 with various parameter values.
Figure 1
Figure 1 The APL’s PDF curves with λ = 2, β = 3 |
Figure 2
Figure 2 The APL’s PDF curves with α = 3, λ = 2 |
Figure 3
Figure 3 The APL’s PDF curves with α = 3, β = 2 |
Figure 1, Figure 2, Figure 3 show that for various values of the parameters, the APLD density curve is uni- modal and positively skewed.
2.1. The survival and the hazard functions
The following forms could be used to represent the survival and hazard rate functions for the APL distribution, respectively.
Equation 7
and
Equation
8
Figure 4, Figure 5 display the survival function of APLD and the hazard function for different values of parameters.
Figure 4
Figure 4 The APL’s survival function curves with λ = 2, β = 2 |
Figure 5
Figure 5 The The APL’s hazard function curves
show with λ = 2 and |
3. Statistical Properties
3.1. Quantile function
Let u = F (x) where U follows uniform (0, 1). By using the transformation method, we consider the random variable X of APL as follows
Equation
9
3.1.1. THE MEDIAN
The median of the APL distribution can be obtained by putting u = 0.5 in Equation 9 as follows
Equation 10
3.2. The mode
The mode of the distribution can be found by solving the following equation
By taking the derivative of Equation 6 and equating it to zero and solving for x, mode becomes
Equation
11
This is equation is not linear. The Newton-Raphson method can be used to solve it numerically.
3.2.1. Skewness and kurtosis
Because the APL’s moments are not exist, we’ll apply an alternate forms using quartiles. Bowley skewness was considered as an alternate measure to determine asymmetry (Kenney and Keeping (1962)) of a distribution that takes the form
where Q represents the quantile function from Equation 9. Moors (1988) proposed a different method to determine the kurtosis of the distribution based on octiles, and it has the following form
Table 1
Table 1 The Mode, Median, Skewness and Kurtosis of APL Distribution for Different Values of the Parameters. |
||||||
α |
λ |
β |
Mode |
Median |
Skewness |
Kurtosis |
4.5 |
0.5 |
2 |
0.000679 |
0.373801 |
0.341431 |
1.58388 |
8.5 |
0.5 |
2 |
0.097225 |
0.458852 |
0.327832 |
1.57484 |
12.5 |
0.5 |
2 |
0.148810 |
0.512296 |
0.321915 |
1.57231 |
4.5 |
1 |
2 |
0.001358 |
1.024560 |
0.341431 |
1.58388 |
4.5 |
3 |
2 |
0.004074 |
3.07377- |
0.341431 |
1.58388 |
4.5 |
6 |
2 |
0.008149 |
6.147550 |
0.341431 |
1.58388 |
4.5 |
0.5 |
2.5 |
0.014549 |
0.281491 |
0.309599 |
1.50100 |
4.5 |
0.5 |
3 |
0.020491 |
0.225435 |
0.288054 |
1.45078 |
4.5 |
0.5 |
3.5 |
0.022919 |
0.187873 |
0.272518 |
1.41727 |
Table 1 the following:
1) For constant values of λ and β, the median, mode increasing as α increases, while the kurtosis and skewness decrease.
2) For constant values of α and β, the median and mode increase while λ is increasing but the kurtosis and skewness remain stable.
3) For constant values of λ and α, the mode increasing as β increases, while the median, kurtosis, and skewness decrease.
4. Renyi Entropy of APL
Renyi entropy was invented by Renyi (1959). The variation of the uncertainty is measured by a random variable’s entropy X. Additionally, Song (2001) has effectively used the theory of entropy to a variety of applications, including information theory, engineering, and physics.
Theorem 1: The Renyi entropy of APL
Equation 12
Proof. For the density function f(x), the Renyi entropy is defined as:
Equation 13
Using Equation 6, we obtain
by substituting z = (1 + x) −β and using the series representation,
Equation 14
Then we have
Hence, the theorem is proved.
5. The Stress-Strength Reliability
In literature related to engineering, reliability, and bio statistics, the stress-strength model has attracted a lot of interest. Let X1 represents stress and X2 represents strength. The stress-strength model’s most basic version states that a system failure occurs when the stress exceeds the strength. Kotz and Pensky (2003). The stress-strength reliability for the APL is given by the ensuing theorem.
Theorem 5.1. The form of APL’s stress-strength reliability
Equation 15
Proof. The stress-strength reliability can be defined as
Equation 16
using the Equation 5 and Equation 6 of APL distribution, stress strength R, can be obtained as
by substituting z = (1 + x) −β and using the series representation
Equation
17
Hence, the theorem is proved.
6. Order Statistics of APL
Consider denotes the order statistic of a random sample
of size n from APL distribution with CDF, F(x), and PDF, f(x), the PDF of Xr is given by Balakrishnan
and Cohen (2014)
Equation
18
By using Equation 5, Equation 6 the PDF the rth order statistic from APLD is given by:
Equation 19
The PDF of the smallest order statistic, X1, is as follows
Equation 20
And the PDF of the largest order statistic, Xn, is given by
Equation 21
7. METHODS OF ESTIMATION
In the section, we will study classical estimation methods to estimate unknown parameters of the APL distribution. Estimation methods that used are: (1) maximum likelihood method, (2) Ordinary least squares, (3) Weighted least squares.
7.1. MAXIMUM LIKELIHOOD ESTIMATION OF APL
Let X1, X2, ...Xn be a random sample of size n from the APL distribution. By using (6), we obtain the likelihood function L (x, α, λ, β) as
Equation 22
and the log-likelihood function is
Equation 23
Equation 24
Then by deriving the log-likelihood function with respect to α, λ and β we get
Equation 25
Equation 26
Equation 27
After solving Equation 25, Equation 26 and Equation 27 simultaneously using the Newton-Raphson method in Mathematica 12.0, the MLE of α, β and λ could befound.
7.2. Ordinary Least Squares Estimation of APL
Swain et al. (1988) First suggested the least-squares estimators and
weighted least-squares estimators. Assuming a random sample of size n from the APL distribution, with the observations
being in the following order: x1:n <
x2:n <
…. < xn:n we may obtain the ordinary least
squares (OLS) estimates α , λ and β as follows.
Equation
28
Then by deriving the S (α, λ, β) function with respect to α, λ, β, we get
Equation 29
Equation 30
Equation
31
After solving Equation 29, Equation 30, and Equation 31 simultaneously using the Newton-Raphson method in Mathematica 12.0, the OLS of α, β and λ could be obtained.
7.3. Weighted Last Squares Estimation of APL
Weighted least squares (WLS) estimates are represented using the following α, β and λ form
Equation 32
Then by deriving the W (α, λ, β) function with respect to α, λ, β, we get
Equation 33
Equation 34
Equation
35
After solving Equation 33, Equation 34 and Equation 35 simultaneously using the Newton-Raphson method in Mathematica 12.0, the WLS of, and could be obtained.
8. SIMULATION STUDY
To make sense of the theoretical findings, simulation using Mathematica 12.0 have been carried out, of the estimating issue presented in the earlier parts. Simulated research has been carried out for mean square error (MSE) and average MLEs.
The estimator computation algorithm is
Step 1: Create a random sample of size n using Equation (9) for the starting parameter values (α, β, λ).
Step 2: Use the Newton-Raphson method to solve the equations in to estimate the MLE, LSE and
WLSE of the parameter α to solve the equations given in (25), (29) and (33), respectively.
Step 3: Compute the estimator of entropy from (12), using the estimates in the previous step.
Step 4: 1000 times, repeat the first three steps.
Step 5: Calculate the MSE.
Different sample sizes of the ML estimate for α, entropy and MSE for the real parameter values are shown in Table 2. The LSE estimations for α, entropy and MSE for true parameter values are presented in Table 3. Additionally, the WLSE estimates for α, entropy, and MSE for the genuine parameter val- ues are shown in Table 4
Table 2
Table 2 The MLE and MSE for the Parameters α. |
|||||
Parameters |
n |
Mean (α) |
MSE (α) |
MLE Entropy |
MSE Entropy |
α = 2 |
30 |
2.10996 |
0.055305 |
0.794999 |
0.001414 |
β = 4 |
50 |
2.10507 |
0.050296 |
0.794381 |
0.001292 |
λ = 3 |
200 |
2.09217 |
0.049327 |
0.792177 |
0.001273 |
ρ = 2 |
500 |
2.07664 |
0.041060 |
0.789856 |
0.001079 |
|
1000 |
2.04933 |
0.031904 |
0.785526 |
0.000860 |
α = 3.5 |
30 |
3.49168 |
0.081989 |
0.901258 |
0.000717 |
β = 3 |
50 |
3.47555 |
0.080602 |
0.899818 |
0.000716 |
λ = 2 |
200 |
3.47064 |
0.078377 |
0.899322 |
0.000695 |
ρ = 2 |
500 |
3.47054 |
0.074389 |
0.893239 |
0.000648 |
|
1000 |
3.44995 |
0.065321 |
0.893229 |
0.000566 |
α = 5.5 |
30 |
5.49820 |
0.084452 |
1.815170 |
0.000363 |
β = 3 |
50 |
5.49050 |
0.083072 |
1.814670 |
0.000359 |
λ = 2 |
200 |
5.48660 |
0.082376 |
1.814400 |
0.000352 |
ρ = 2 |
500 |
5.47801 |
0.079926 |
1.813870 |
0.000345 |
|
1000 |
5.47643 |
0.075352 |
1.813790 |
0.000326 |
Table 3
Table 3 The LSE and MSE for the parameters α. |
|||||
Parameters |
n |
Mean (α) |
MSE (α) |
MLE Entropy |
MSE Entropy |
α = 2 |
30 |
2.167800 |
0.059513 |
0.804842 |
0.001468 |
β = 4 |
50 |
2.166970 |
0.057549 |
0.804195 |
0.001427 |
λ = 3 |
200 |
2.133420 |
0.045056 |
0.799589 |
0.001126 |
ρ = 2 |
500 |
2.087780 |
0.029733 |
0.792361 |
0.000760 |
|
1000 |
2.008250 |
0.010686 |
0.779427 |
0.000294 |
α = 3.5 |
30 |
3.464680 |
0.087933 |
0.898712 |
0.000789 |
β = 3 |
50 |
3.453120 |
0.083220 |
0.897577 |
0.000744 |
λ = 2 |
200 |
3.385090 |
0.082102 |
0.891249 |
0.000713 |
ρ = 2 |
500 |
3.277550 |
0.079814 |
0.881001 |
0.000697 |
|
1000 |
3.171800 |
0.051935 |
0.870616 |
0.000321 |
α = 5.5 |
30 |
5.482420 |
0.084139 |
0.885941 |
0.000360 |
β = 2 |
50 |
5.457720 |
0.083646 |
0.884451 |
0.000351 |
λ = 1 |
200 |
5.412630 |
0.083646 |
0.881737 |
0.000321 |
ρ = 2 |
500 |
5.307800 |
0.073992 |
0.874050 |
0.000316 |
|
1000 |
5.192590 |
0.073594 |
0.868265 |
0.000207 |
Table 4
Table 4 The WLSE and MSE for the Parameters α |
|||||
Parameters |
n |
Mean (α) |
MSE (α) |
MLE Entropy |
MSE Entropy |
α = 2 |
30 |
2.176210 |
0.062294 |
0.806236 |
0.001535 |
β = 4 |
50 |
2.172040 |
0.060366 |
0.805593 |
0.001495 |
λ = 3 |
200 |
2.171150 |
0.059024 |
0.805502 |
0.001487 |
ρ = 2 |
500 |
2.167290 |
0.059968 |
0.804785 |
0.001462 |
|
1000 |
2.164850 |
0.056621 |
0.804519 |
0.001402 |
α = 3.5 |
30 |
3.476270 |
0.089437 |
0.899785 |
0.000800 |
β = 3 |
50 |
3.463550 |
0.087462 |
0.898549 |
0.000778 |
λ = 2 |
200 |
3.46067 |
0.084138 |
0.898370 |
0.000747 |
ρ = 2 |
500 |
3.453780 |
0.083912 |
0.897605 |
0.000754 |
|
1000 |
3.451360 |
0.082047 |
0.897472 |
0.000734 |
α = 5.5 |
30 |
5.483850 |
0.086180 |
0.886005 |
0.000312 |
β = 2 |
50 |
5.483810 |
0.085978 |
0,886004 |
0.000311 |
λ = 1 |
200 |
5.480090 |
0.084704 |
0.885813 |
0.000308 |
ρ = 2 |
500 |
5.474100 |
0.082359 |
0.885431 |
0.000301 |
|
1000 |
5.458950 |
0.081502 |
0.884541 |
0.000295 |
From Table 2, Table 3, Table 4, we observed that the MSE of the estimate α and entropy decreases as the sample size increase. Also, the mean of the estimate α and the MLE of entropy decreases as the sample size increase.
9. Applications
In this section. The inverse Lomax distribution (ILD)
introduced by Hassan and Mohamed (2019) and the exponential lomax
distribution (ELD) introduced by Abdo
et al. (2015) will be used to
compare the fit of the APLD with other lifetime models that are well-known.
Additionally, we take into account the model selection
parameters, such as the Akaike information criterion (AIC), log-likelihood (), Bayesian information criterion
(BIC), consistent Akaike information criterion (CAIC), and Hannan-Quinn
information criterion (HQIC). The lowest values of AIC, BIC, CAIC, HQIC and
greatest value of,
all point to a better fit of the data, Where
k is the number of parameters; n is the sample size and (θˆ)
is the log-likelihood function evaluated at the highest likelihood estimates.
Data I: COVID-19 cases every day in Saudi Arabia
From 1 April to 2 May, this information reveals the daily COVID-19 instances in Saudi Arabia. These statistics were obtained from the Saudi Ministry of Health’s website, located at https://covid19.moh.gov.sa. These findings are
110, 157, 165, 154, 140, 206, 138, 272, 137, 355, 364, 382, 429, 472,435, 493, 518, 762, 1132, 1088, 1122,
1147, 1141, 1158, 1172, 1197,1223, 1289, 1266, 1325, 1351, 1344, 1362.
Table 5 displays descriptive statistics for this data.
Table 5
Table 5 Descriptive Statistics for the Number of Daily COVID-19 Cases in Saudi Arabia |
|||
Measure |
Value |
Measure |
Value |
N |
33 |
Minimum |
110 |
Maximum |
1362 |
Mean |
727.455 |
Q1 |
255.5 |
Q3 |
1178.25 |
Median |
518. |
Skewness |
0.0350687 |
Kurtosis |
1.26721 |
Variance |
230684.0 |
Standard deviation |
480.295 |
|
The performance of the APLD in comparison to other models is shown in Table 6 along with the MLEs for the model parameters.
Table 6
Table 6 MLEs of the Model Parameters and the Statistics of the AIC, BIC, CAIC, HQIC and for the Number of Daily COVID-19 Cases in Saudi Arabia |
|||||||
Distributions |
MLE α |
MLE entropy |
|
AIC |
BIC |
CAIC |
HQIC |
APLD |
34.7454 |
2.73066 |
-243.261 |
492.532 |
497.012 |
493.350 |
494.033 |
ELD |
0.04329 |
0.39915 |
-425.628 |
857.257 |
861.746 |
858.085 |
858.768 |
ILD |
182.650 |
2.95628 |
-253.035 |
512.070 |
516.560 |
512.898 |
513.581 |
Table 6 shows that, among the models taken into consideration, the APLD has the smallest AIC, CAIC, BIC, and HQIC values. Which shows that the APLD seems to be a model that might work well with the data set.
Data II: (Natural phenomena) Badr (2019) presented the consist of 30 successive March precipitation (in inches) observations.
77, 52, 90, 174, 162, 205, 81, 131, 120, 32, 195, 95, 120, 81, 47,281, 143, 187, 337, 118, 220, 135, 300,
475, 309, 248, 151, 96, 210,189.
Table 7 displays descriptive statistics for this data.
Table 7
Table 7 Descriptive Statistics for the of 30 Successive March Precipitation (in inches) |
|||
Measure |
Value |
Measure |
Value |
N |
30 |
Minimum |
32 |
Maximum |
475 |
Mean |
168.7 |
Q1 |
95. |
Q3 |
210. |
Median |
147. |
Skewness |
1.12002 |
Kurtosis |
4.30607 |
Variance |
9786.15 |
Standard deviation |
98.925 |
|
The performance of the APLD in comparison to other models is shown in Table 8 along with the MLEs for the model parameters.
Table 8
Table 8 MLEs of the Model Parameters and the Statistics of the AIC, BIC, CAIC, HQIC and for the of 30 Successive March Precipitation (In Inches) |
|||||||
Distributions |
MLE α |
MLE entropy |
,e |
AIC |
BIC |
CAIC |
HQIC |
APLD |
66.161 |
2.83245 |
-200.811 |
401.622 |
412.201 |
408.422 |
409.183 |
ELD |
0.0370132 |
0.351725 |
-476.589 |
959.178 |
963.757 |
959.978 |
960.739 |
ILD |
440.78 |
3.040180 |
-296.266 |
598.532 |
603.111 |
599.322 |
600.093 |
Table 8 is results show that while, e has the largest value and the APLD has the smallest AIC, CAIC, BIC, and HQIC values. As a result, the APLD performs better than other models.
CONFLICT OF INTERESTS
None.
ACKNOWLEDGMENTS
None.
REFERENCES
Abdo, N. F., El-bassiouny, A. H. and Shahen, H. S. (2015). Exponential Lomax Distribution. International Journal of Computer Applications, 121, 14-29. https://doi.org/10.5120/21602-4713.
Badr, M. M. (2019). Goodness of Fit Tests for the Compound
Rayleigh Distribution with Application to Real Data. Heliyon, 5,1-6. https://doi.org/10.1016/j.heliyon.2019.e02225.
Balakrishnan, N. and Cohen, A. C. (2014). Order Statistics and Inference: Estimation Methods, Academic Press, San Diego, California.
Devi, B., Kumar, P. and Kour,
K. (2017). Entropy of Lomax Probability Distribution And Its Order Statistic. International
Journal of Statistics and Systems,
12(2), 175-181.
Dey, S., Nassar, M. and Kumar, D. (2019). Alpha Power Transformed Inverse Lindley Distribution : A Distribution With an Upside-Down Bathtub-Shaped Hazard Function. Journal of Computational and Applied Mathematics, 130-145. https://doi.org/10.1016/j.cam.2018.03.037.
El-Houssainy, A., Rady, W. and Elhaddad, T. (2016). The Power Lomax Distribution With an Application to Bladder Cancer Data. Springerplus, 1838,1-22. https://doi.org/10.1186/s40064-016-3464-y.
Hassan, A., and Mohamed, R. (2019). Weibull Inverse Lomax Distribution. Pakistan Journal of Statistical and Operation Research, 587-603. https://doi.org/10.18187/pjsor.v15i3.2378.
Ijaz, M., Asim, S. M. And Alamgir, M. (2019). Lomax Exponential
Distribution With an Application to Real-Life Data. Plos One, 12-14. https://doi.org/10.1371/journal.pone.0225827.
Ihtisham, S., Khalil, A., Manzoor, S., Khan, S. A. and Ali, A. (2019). Alpha-Power Pareto Dis- Tribution its Properties and Applications. Plos One, 6-14. https://doi.org/10.1371/journal.pone.0218027.
Lomax, K, S. (1954). Business Failures: Another Example of the Analysis of Failure Data. Joumal of the American Statistical Association, 49, 847-852. https://doi.org/10.1080/01621459.1954.10501239.
Kriver, J. F. and Keeping, E. S. (1962). Mathematics of Statistics, 3rd edition, Part 1, Princeton, New Jersey.
Kotz, S. and Pensky, M. (2003). The Stress-Strength Model and its Generalizations : Theory and Applications, World Scientific, New Jersey. https://doi.org/10.1142/9789812564511.
Mahdavi, A. and Kundu, D. (2017). A New Method for
Generating Distributions with an Application to Exponential Distribution. Communications
in Statistics- Theory and Methods, 46(13), 6543-6557. https://doi.org/10.1080/03610926.2015.1130839.
Moors,
J. (1988). A Quantile Alternative For Kurtosis, Journal of the Royal
Statistical Society : Series D (The Statistician), 37, 25-32. https://doi.org/10.2307/2348376.
Nassar, M. Alzaatreh, A., Mead, M. and Abo-Kasem, O. (2017). Alpha Power Weibull Distribution : Properties and Applications. Communications in Statistics Theory and Methods, 20(46), 10236-10252. https://doi.org/10.1080/03610926.2016.1231816.
R'enyi,
A. (1959). On The Dimension and Entropy of Probability Distributions. Acta
Math. Hung. 10, 193-215.
https://doi.org/10.1007/BF02063299.
Song, K.S. (2001). R'enyi Information,
Loglikelihood and an Intrinsic Distribution Measure. J.Stat. Plan. Inference 93,
51-69. https://doi.org/10.1016/S0378-3758%2800%2900169-5.
Swain, J., Venkatraman, S. and Wilson, J. (1988). Least Squares Estimation of Distribution Function in Johnson's Translation System. Journal of Statistical Computation and Simulation, 29, 271-297. https://doi.org/10.1080/00949658808811068.
This work is licensed under a: Creative Commons Attribution 4.0 International License
© Granthaalayah 2014-2022. All Rights Reserved.