Der Artikel präsentiert einen bahnbrechenden Ansatz zur Modellierung der Wahrscheinlichkeitsverteilung mit der Einführung der Klasse Exponentiated Odd Lindley-X Power Series (EOL-XPS). Diese innovative Familie von Verteilungen wurde entwickelt, um größere Flexibilität und Anpassungsfähigkeit zu bieten und die Beschränkungen bestehender Modelle im Umgang mit komplexen und stark verzerrten Datensätzen zu beheben. Der Artikel leitet akribisch die wichtigsten statistischen Eigenschaften ab, einschließlich der Funktion der kumulativen Verteilung (CDF), der Wahrscheinlichkeitsdichtefunktion (PDF), der Quantilfunktion, der Momente und der Funktion der Gefahrenrate, wodurch ein gründliches Verständnis der EOL-XPS-Klasse entsteht. Ein spezieller Fall, die Exponentiated Odd Lindley-Weibull Poisson (EOL-WP) -Verteilung, wird detailliert untersucht und zeigt ihre Fähigkeit, sowohl monotone als auch nichtmonotone Ausfallraten aufzunehmen. Die praktische Anwendbarkeit der EOL-XPS-Klasse wird anhand von realen Datensätzen demonstriert, darunter COVID-19-Sterblichkeitsraten in Kanada und Ausfallzeiten von Kevlar 49 / Epoxid-Strängen. Der Artikel hebt die überlegene Leistung des Modells bei der Anpassung verschiedener Datensätze hervor und macht es zu einem wertvollen Werkzeug für Zuverlässigkeit und Überlebensanalyse. Darüber hinaus diskutiert der Artikel potenzielle Erweiterungen und zukünftige Forschungsrichtungen und betont die Vielseitigkeit und breite Anwendbarkeit der EOL-XPS-Klasse in verschiedenen Bereichen.
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Abstract
In this article, we introduce a new family of probability distributions, the Exponentiated Odd Lindley-X Power Series (EOL-XPS) class, which is derived by integrating the power series distribution with the exponentiated odd Lindley-X family. We establish several statistical properties of this new class, including moments, the moment-generating function, the quantile function, mean deviations, order statistics, and Rényi entropy. As a special case, we derive the probability density function and cumulative distribution function of the Exponentiated Odd Lindley-Weibull Poisson (EOL-WP) distribution, using the Weibull-Poisson distribution as the baseline. To assess the robustness of the proposed model, we conduct a Monte Carlo simulation study to evaluate the performance of maximum likelihood estimation for parameter estimation. Furthermore, we apply the EOL-WP model to COVID-19 and Kevlar datasets, demonstrating its flexibility and practical relevance in modeling complex data.
Hinweise
All authors contributed equally to this work.
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1 Introduction
The generalization of probability distributions has gained significant attention in recent years, driven by the need to develop more flexible models that provide better fits for diverse applications. A key motivation behind these advancements is the necessity to model various types of data and accurately characterize their probabilistic structures.
In the literature, numerous new classes of distributions have been developed by compounding existing models with power series distributions. Notable contributions include the odd power generalized Weibull-G power series class by [1], the odd Weibull Topp-Leone-G power series family by [2], the Topp-Leone-G power series class by [3], the generalized Gompertz-power series distributions by [4], the exponentiated generalized power series by [5], the Lindley-Burr XII power series distribution by [6], the Burr-Weibull power series class by [7], and the complementary extended Weibull power series class by [8], among others.
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[9] made a significant contribution by introducing a new method for generating families of continuous distributions, known as the T-X family of distributions, which is closely connected to hazard rate functions. Later, in 2013, [10] extended this work by introducing the exponentiated T-X family of distributions using the method proposed by [9].
Building on this foundation, we propose a new family of distributions, the Exponentiated Odd Lindley-X Power Series (EOL-XPS) class, by combining the Exponentiated Odd Lindley-X (EOL-X) family introduced by [11] with the power series distribution.
1.1 Motivation and Background
In recent years, the Lindley distribution has garnered significant attention due to its effectiveness in modeling complex real-world lifetime data. [12] demonstrated that the one-parameter Lindley distribution outperforms the exponential distribution in modeling survival data. The generalization of probability distributions is primarily driven by the need to enhance flexibility, which is achieved by embedding a basic distribution within a more adaptable structure [13]. Several researchers, including [14‐16], have explored structural properties of various generalized Lindley distributions.
While Lindley and power series distributions, along with their generalizations, have been widely applied, they may not always be suitable for certain theoretical or practical contexts. To address these limitations, we propose a new generalized Lindley model by integrating a generalized Lindley family with the power series distribution. The power series distributions, as discussed by [17], provide a robust framework for compounding probability models.
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1.2 Contribution of This Work
By compounding the generalized Lindley family introduced by [11] with the power series class of distributions, we develop a new generalized family of power series distributions, referred to as the EOL-XPS class. As a special case, we derive the Exponentiated Odd Lindley-Weibull Poisson (EOL-WP) distribution, using the Poisson distribution as the baseline. This new model introduces a compounding parameter, which simultaneously acts as both a scale and shape parameter, enhancing flexibility. The additional shape parameter regulates skewness and kurtosis, allowing the model to accommodate both monotonic and non-monotonic failure rates.
A key application of the proposed model is its ability to fit real-world datasets. We analyze COVID-19 mortality rates in Canada from April 10, 2020, to May 15, 2020, to evaluate the model’s performance and parameter estimation. Various statistical comparisons highlight the model’s versatility. Additionally, we apply the EOL-WP distribution to failure times of Kevlar 49/epoxy strands subjected to constant sustained pressure at 90\(\%\) stress levels, further demonstrating the practical applicability of the model.
1.3 Outline of the Paper
Section 2 derives the cumulative distribution function (CDF) and probability density function (PDF) of the EOL-XPS family and presents some special cases.
Section 3 explores key structural properties of the EOL-XPS family.
Section 4 discusses maximum likelihood estimation (MLE) for parameter estimation.
Section 5 derives the PDF and CDF of the EOL-WP distribution and presents a simulation study.
Section 6 focuses on parameter estimation and goodness-of-fit for two real datasets.
Section 7 discusses managerial insights derived from the model.
Section 9 outlines the future scope of this research.
2 The Model
In this section, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the Exponentiated Odd Lindley-X Power Series (EOL-XPS) family of distributions.
Let N be a discrete random variable following a power series distribution, truncated at zero, with probability mass function (PMF) given by:
where \(C(\theta )=\sum _{n=1}^{\infty }a_{n}\theta ^{n}\) is finite, \(\theta >0\), and \(\{a_{n}\}_{n\ge 1}\) is a sequence of positive real numbers. Taking \(X_{(1)}=\min (X_{1}, X_{2}, ..., X_{N}),\) the CDF and PDF of \(X_{(1)}|N=n\) are defined by
respectively, where S(x) is the survival function and g(x) is the PDF of the baseline distribution G(x). [17] presented some families of power series distribution that include the binomial, Poisson, geometric, and logarithmic distributions. [11] introduced the Exponentiated Odd Lindley-X (EOL-X) family of distributions, demonstrating its effectiveness in modeling reliability and medical datasets.
The cumulative distribution function (CDF) of the EOL-X family is given by:
respectively, for \(\alpha ,\lambda >0\), where \(\xi \) is a vector of parameters of the baseline distribution \(G(x;\xi )\).
By substituting the survival function of the baseline distribution \(G(x; \xi )\) into equations (1) and (2), the CDF and PDF of the EOL-XPS class of distributions are given by
respectively, where \(\xi \) is a vector of parameters from the baseline distribution \(G(x;\xi )\), and \(\alpha ,\lambda ,\theta >0\). In Table 1 below, we present the expressions for the CDF of the EOL-X Poisson, EOL-X Geometric, EOL-X Logarithmic, and EOL-X Binomial distributions.
Table 1
Some Special Cases of EOL-XPS Class of Distributions
2.1 Expansion of Probability Density and Cumulative Distribution Functions
In this section, we derive the expressions for the expansions of the CDF (3) and PDF (4) for the EOL-XPS class of distributions. Substituting the derivative \(C^{\prime }\left( \theta \right) =\sum _{n=1}^{\infty }na_{n}\theta ^{n-1}\) of \(C\left( \theta \right) \) into equation \((4)\), we obtain
where \(\overline{G^\alpha } (x;\xi ) = 1 - G^\alpha (x;\xi )\). The power series expansion of the term \(\exp \left( -\frac{n\lambda G^{\alpha }(x;\xi )}{\overline{G^{\alpha }}\left( x;\xi \right) }\right) \) is given by
We note that \(g_{\alpha \left( 1+p+q\right) }\left( x;\xi \right) \) is the density function for the exponentiated-G (Exp-G) distribution with power parameter \(\alpha \left( 1+p+q\right) \).
Next, we obtain the expression for the CDF of the EOL-XPS random variable. Using the expression for \(C\left( \theta \right) \) and the power series expansion from equation (6), the CDF (3) can be written as
Here, \(G^{\alpha \left( p+q\right) }\left( x;\xi \right) \) is the distribution function for the Exp-G distribution with power parameter \((\alpha \left( p+q\right) +1)\).
3 Some Statistical Properties
In this section, we derive the expressions for the quantile function, moments, moment generating function, mean deviation, Lorenz and Bonferroni curves, distribution of order statistics, and Rényi entropy.
3.1 Quantile Function
The quantile function, \(Q\left( u\right) \), \(0<u<1\), for the EOL-XPS class of distributions was obtained by finding the inverse of the CDF (3). It is given by
where \(G^{-1}\left( .\right) \) is the quantile function of baseline distribution \(G\left( .\right) \), \(W\left( .\right) \) is the Lambert W function, and \(C^{-1}\left( .\right) \) is the inverse function of C(.).
3.2 Moments and Moment Generating Function
By definition, the \(r^{th}\) raw moment about the origin of X is given by
where \(g_{\alpha \left( 1+p+q\right) }\left( x;\xi \right) \) and \(\gamma _{p,q}\) are defined in equations (12) and (11), respectively. We assume that \(W_{\alpha (1+p+q)}\sim Exp-G\left( \alpha \left( 1+p+q\right) \right) \) and \(X\sim EOL-XPS(\alpha ,\lambda ,\theta ,\xi )\). Then the \(r^{th}\) moment for the EOL-XPS random variable is given by
where \(E\left( W_{\alpha \left( 1+p+q\right) }^{r}\right) \) denotes the \(r^{th}\) moment of \(W_{\alpha (1+p+q)}\) which follows an Exp-G distribution with parameters \(\alpha \left( 1+p+q\right) \).
where \(E\left( e^{tW_{\alpha \left( 1+p+q\right) }}\right) \) is the mgf of the Exp-G distribution. Furthermore, the characteristic function is given by \(\phi (t)=E\left( e^{itx}\right) \) where \(i=\sqrt{-1}\). That is,
where \(\Phi _{\alpha \left( 1+p+q\right) }\left( t\right) \) is the characteristic function of the Exp-G distribution.
3.3 Mean Deviation, Bonferroni and Lorenz Curves
If \(X\sim \)\(\hbox {EOL-XPS}(\alpha ,\lambda ,\theta ,\xi )\), and \(\mu \) and M represent the mean and median of X, respectively, then the mean deviation about the mean and about the median can be written, respectively, as
where \(I_{\alpha \left( 1+p+q\right) }\left( t\right) =\int _{0}^{t}xg_{\alpha \left( 1+p+q\right) }\left( x;\xi \right) dx\) is the first incomplete moment of the EOL-XPS distribution and \(\gamma _{p,q}\) is as given in (11).
3.4 Hazard Rate Function
The survival function or the reliability function R(t) is the probability of an item not failing prior to some time t, and the survival function is defined by
$$\begin{aligned} R(t) = 1-F(t), \end{aligned}$$
where F(.) is the CDF of the EOL-XPS family of distributions. Using equations (10) and (16), the hazard function can be written as
The probability density function of the ith order statistics, say \(X_{i:n}\), for a random sample \(X_{1},X_{2}\ldots ,X_{n}\) from the EOL-XPS family is given by
where equation (21) is the density function for the Exp-G distribution with the power parameter \(\alpha \left( 1+p+q+h+z\right) \).
3.6 Rényi and Shannon Entropy
In this section, we derive measures of uncertainty for the proposed model. There are two common measures of entropy, namely, Shannon entropy by [19] and Rényi entropy by [20]. Rényi entropy generalizes Shannon entropy. Therefore, we present the Rényi entropy of the EOL-XPS distribution. By definition, the Rényi entropy is given by
is the Rényi entropy for the Exp-G distribution with power parameter \(\alpha \left( \frac{p+z}{\nu }-1\right) \).
4 Maximum Likelihood Estimation
This section discusses the maximum likelihood estimation method for estimating EOL-XPS distribution parameters. If \(X_1, X_2, \ldots , X_n\) is a random sample from the EOL-XPS family of distributions where \(x_1, x_2, \ldots , x_n\) are observed values with parameters \(\alpha ,\lambda ,\theta >0\), and \(\xi \) is the parameter vector for the baseline distribution G(.), then the log-likelihood function for the EOL-XPS family is given, from equation (4), by
The maximum likelihood estimates (MLEs) of the parameters are obtained by maximizing the log-likelihood function. This involves solving a system of non-linear equations, which are derived by computing the partial derivatives of the likelihood function with respect to each parameter and setting them equal to zero. If A is the function of \(\xi ,\alpha ,\lambda ,\theta \), and B is the function of \(x_{i},\alpha \), and \({\xi }\), then taking partial derivatives of equation (25) with respect to \(\alpha ,\lambda ,\theta \), and \(\xi \) gives
The maximum likelihood estimators \({\hat{\alpha }}\), \({\hat{\lambda }}\), \({\hat{\theta }}\), and \({\hat{\xi }}\) of the unknown parameters \(\alpha \), \(\lambda \), \(\theta \), and \(\xi \), respectively can be obtained by solving the non-linear equations \(\frac{\partial l}{\partial \alpha }=0\), \(\frac{\partial l}{\partial \lambda }=0\), \(\frac{\partial l}{\partial \theta }=0\), and \(\frac{\partial l}{\partial \xi }=0.\) Since the non-linear system of equations do not have closed-form solutions, we use numerical method such as the quasi-Newton algorithm to numerically optimize the likelihood function given by equation (25).
5 Special Case: Exponentiated Odd Lindley-Weibull Poisson (EOL-WP) Distribution
We demonstrate the utility of the proposed model by an illustration that involves the Weibull distribution as a baseline distribution and the Poisson distribution for the power series. Other models can be derived from the proposed family of distributions by changing the baseline and the power series distributions. The CDF and PDF of the special case, simulation study based on new model, and applications to real data are presented in this section.
The CDF and PDF of the EOL-WP distribution are given by
The EOL-WP distribution takes various shapes for the PDF as shown in Figure 1 (left). The graph of the hazard rate function (as shown in Figure 1 (right)) exhibits both monotonic and non-monotonic hazard rate functions.
5.1 Simulation Study
In this section, we present the Monte Carlo Simulation results from the EOL-WP distribution. Four sets of parameter values were chosen for sample sizes 25, 50, 100, and 200. Root mean square error (RMSE) and average bias (ABias) were used to assess the consistency of the model parameters. The results of this simulation study are shown in Table 2 and they indicate that the model produces consistent parameter estimates based on the decay in both RMSE and ABias values.
Table 2
Monte Carlo Simulation Results
\(\omega =0.10\)
\(\lambda =0.02\)
\(\theta =0.02\)
\(\omega =0.15\)
\(\lambda =0.10\)
\(\theta =0.01\)
Parameter
n
Mean
RMSE
ABias
Mean
RMSE
ABias
\(\omega \)
25
0.0921
0.0087
−0.0079
0.1416
0.0103
−0.0084
50
0.0951
0.0054
−0.0049
0.1440
0.0080
−0.0060
100
0.0963
0.0041
−0.0037
0.1451
0.0058
−0.0049
200
0.0974
0.0028
−0.0026
0.1468
0.0041
−0.0032
\(\lambda \)
25
0.0369
0.0191
0.0169
0.1070
0.0278
0.0070
50
0.0292
0.0107
0.0092
0.1083
0.0302
0.0083
100
0.0267
0.0075
0.0067
0.1040
0.0231
0.0040
200
0.0244
0.0048
0.0044
0.1037
0.0187
0.0037
\(\theta \)
25
0.0640
0.1033
0.0440
1.1214
1.6169
1.1114
50
0.0836
0.3303
0.0636
0.7395
1.3125
0.7295
100
0.0347
0.0341
0.0147
0.6525
1.1514
0.6425
200
0.0317
0.0303
0.0117
0.3799
0.8718
0.3699
\(\omega =0.10\)
\(\lambda =0.01\)
\(\theta =0.10\)
\(\omega =0.15\)
\(\lambda =0.15\)
\(\theta =0.01\)
Parameter
n
Mean
RMSE
ABias
Mean
RMSE
ABias
\(\omega \)
25
0.0938
0.0076
−0.0062
0.1430
0.0091
−0.0070
50
0.0967
0.0043
−0.0033
0.1451
0.0073
−0.0049
100
0.0985
0.0025
−0.0015
0.1462
0.0058
−0.0038
200
0.0990
0.0018
−0.0010
0.1476
0.0040
−0.0024
\(\lambda \)
25
0.0186
0.0111
0.0086
0.1465
0.0438
−0.0035
50
0.0138
0.0052
0.0038
0.1403
0.0453
−0.0097
100
0.0116
0.0027
0.0016
0.1410
0.0400
−0.0090
200
0.0110
0.0018
0.0010
0.1419
0.0321
−0.0081
\(\theta \)
25
0.1283
0.0852
0.0283
1.4731
2.0808
1.4631
50
0.1194
0.0440
0.0194
1.3152
1.8700
1.3052
100
0.1156
0.0355
0.0156
1.0385
1.5975
1.0285
200
0.1132
0.0296
0.0132
0.7380
1.2746
0.7280
6 Applications
In this section, we present real data to illustrate the applicability of the proposed model. We compared the EOL-WP distribution to the non-nested models: odd power generalized Weibull-Weibull Poisson(OPGWWP) by [1], odd Weibull Topp-Leone-log logistic Poisson (OWTLLLP) by [2], Topp-Leone-Weibull Poisson (TLWP) by [3], generalized Gompertz power series (GGP) by [4], exponentiated generalized exponential logarithmic (EGEL) by [5], Topp-Leone-Gompertz Weibull (TLGW) by [21], Topp-Leone-Weibull Lomax (TLWLx) by [22], and beta odd Lindley-exponential (BOLE) by [23] distributions.
Model performance was evaluated using various goodness-of-fit (GoF) statistics which include; -2log-likelihood statistic \((-2\ln (L))\), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (AICC), Cramér-von Mises (\(W^{*}\)), Anderson-Darling (\(A^{*}\)), Kolmogorov-Smirnov (K-S) statistic, as well as its associated p-value. The best-fitting model is the one with the lowest goodness-of-fit statistics values and the highest p-value for the K-S statistic. Furthermore, GoF plots are also considered which include histogram of data and fitted PDF, probability plot, fitted CDF, Kaplan-Meier curves, total time on test (TTT), and the fitted HRF function to demonstrate how the proposed model fit the selected data.
6.1 COVID-19 Data Canada: April 10- May 15, 2020
The first data representing the mortality rate from COVID-19 in Canada for the period April 10, 2020 to May 15, 2020, [24]. The observations are as follows: 3.1091, 3.3825, 3.1444, 3.2135, 2.4946, 3.5146, 4.9274, 3.3769, 6.8686, 3.0914, 4.9378, 3.1091, 3.2823, 3.8594, 4.0480, 4.1685, 3.6426, 3.2110, 2.8636, 3.2218, 2.9078, 3.6346, 2.7957, 4.2781, 4.2202, 1.5157, 2.6029, 3.3592, 2.8349, 3.1348, 2.5261, 1.5806, 2.7704, 2.1901, 2.4141 and 1.9048.
Table 3
Parameter estimates and goodness-of-fit statistics for various models fitted for COVID-19 data from Canada
Estimates
Statistics
Model
\(\theta \)
\(\lambda \)
\(\omega \)
\(\alpha \)
\(-2\log \, L\)
AIC
AICC
BIC
\(W^*\)
\(A^*\)
K-S
p-value
EOL-WP
4.4894
13.0960
0.5286
24.9928
94.7679
102.7679
104.0582
109.102
0.0742
0.4245
0.0999
0.8647
(2.6271)
(41.1178)
(0.2516)
(5.5072)
\(\alpha \)
\(\beta \)
\(\lambda \)
\(\theta \)
OPGWWP
0.7730
0.0652
1.9407
\(6.3697\times 10^{-9}\)
114.6469
122.6469
123.9372
128.9810
0.2712
1.5834
0.2017
0.1067
(0.2526)
(0.0223)
(0.2033)
(0.0110)
\(\alpha \)
\(\lambda \)
\(\gamma \)
\(\theta \)
OWTLLLP
0.2344
1.2393
0.9017
\(2.8047\times 10^{-10}\)
237.8660
245.8650
247.1553
252.1990
0.0973
0.5607
0.7922
\(2.2\times 10^{-16}\)
(0.0778)
(0.3294)
(0.4566)
(\(2.0494\times 10^{-3}\))
\(\alpha \)
\(\beta \)
b
\(\theta \)
TLWP
0.2265
1.4964
8.4369
7.8349\(\times 10^{-8}\)
96.1882
104.1883
105.4786
110.5224
0.0959
0.5611
0.1070
0.8037
(0.2921 )
(0.6801)
(10.8870)
(0.0394)
\(\theta \)
\(\alpha \)
\(\beta \)
\(\gamma \)
GGP
3.7041
19.8230
1.4537
\(9.5700\times 10^{-9}\)
94.8978
102.8978
104.1881
109.2319
0.0740
0.4304
0.1029
0.8404
(2.4248)
(16.1050)
(0.2273)
(0.0306)
p
\(\alpha \)
\(\beta \)
\(\lambda \)
EGEL
0.7411
2.0230
34.9405
0.7013
96.6285
104.6286
105.9189
110.9626
0.1063
0.6266
0.1206
0.6711
( 0.5611)
(0.0647)
(20.7874)
(0.1869)
\(\theta \)
\(\gamma \)
b
\(\lambda \)
TLGW
0.0209
4.6206
16.3142
0.2066
96.2382
104.2382
105.5286
110.5723
0.0964
0.56368
0.1061
0.8118
(0.0661)
(3.8975)
(0.1150 )
( 0.1405 )
a
b
\(\alpha \)
\(\theta \)
TLWLx
0.5861
0.7649
2.0682
6.1765
96.1577
104.1577
105.4480
110.4918
0.0953
0.5571
0.1056
0.8169
(1.5081)
(3.1331)
(3.4471)
(9.2453)
a
b
\(\lambda \)
\(\theta \)
BOLE
16.5972
2.2109
41.2948
0.0172
96.2139
104.2139
105.5042
110.5480
0.0968
0.5661
0.1064
0.8089
(0.3385)
(1.4392 )
(0.0073)
(0.0050)
Based on the results presented in Table 3, we conclude that the EOL-WP distribution exhibits greater flexibility compared to the selected models when applied to COVID-19 mortality data in Canada. Figures 2, 3, and 4 visually illustrate the effectiveness of the proposed model in fitting the dataset.
Notably, Figure 2 demonstrates how the model adapts its tails to capture outlying values, highlighting the impact of the additional parameter in regulating tail weight. Moreover, the model successfully identifies the increasing hazard rate followed by a stabilization phase, which aligns with the observed characteristics of the COVID-19 pandemic. This suggests that the EOL-WP distribution is well-suited for modeling real-world mortality patterns with complex hazard rate behavior.
Fig. 2
Fitted Densities and Probability Plots for Covid Data
Parameter estimates and goodness-of-fit statistics for various models fitted for Kevlar dataset
Estimates
Statistics
Model
\(\theta \)
\(\lambda \)
\(\omega \)
\(\alpha \)
\(-2\log \, L\)
AIC
AICC
BIC
\(W^*\)
\(A^*\)
K-S
p-value
EOL-WP
5.8998
0.4780
0.5106
1.6060
203.2952
211.2952
211.7119
221.7557
0.1214
0.7430
0.0729
0.6563
(2.2462)
(0.1987)
(0.0555)
(0.5239)
\(\alpha \)
\(\beta \)
\(\lambda \)
\(\theta \)
OPGWWP
1.5938
0.5588
0.4533
7.1433\(\times 10^{-8}\)
206.7744
214.7744
215.1911
225.2349
0.1544
0.9266
0.0827
0.4938
(0.4322)
(0.0732)
(0.0985)
(0.1210)
\(\alpha \)
\(\lambda \)
\(\gamma \)
\(\theta \)
OWTLLLP
0.4112
1.3698
1.4732
\(2.2795\times 10^{-8}\)
215.2848
223.2836
223.7002
233.7440
0.1770
1.0209
0.2062
0.0003
(0.2351)
(0.8066)
(0.3808)
(0.0111)
\(\alpha \)
\(\beta \)
b
\(\theta \)
TLWP
0.4056
1.0604
0.7929
\(1.2711\times 10^{-7}\)
205.5743
213.5743
213.9910
224.0348
0.1652
0.9586
0.0844
0.4680
(0.1755)
(0.2665)
(0.3215)
(0.0291)
\(\theta \)
\(\alpha \)
\(\beta \)
\(\gamma \)
GGP
1.0673
1.0630
1.1189
\(4.3201\times 10^{-9}\)
215.3074
223.3080
223.7246
233.7685
0.1271
0.7777
0.1367
0.0458
(0.9297)
(0.3012)
(0.1550)
(0.0165)
p
\(\alpha \)
\(\beta \)
\(\lambda \)
EGEL
\(3.7308\times 10^{-8}\)
0.6013
0.8574
1.3988
205.8439
213.8439
214.2606
224.3044
0.1814
1.0312
0.0815
0.5129
(0.0237)
(0.2141)
(0.1105)
(0.4981)
\(\theta \)
\(\gamma \)
b
\(\lambda \)
TLGW
0.3734
1.5051
2.4138
0.3145
205.9311
213.9311
214.3478
224.3916
0.1681
0.9760
0.0842
0.4706
( 0.4297)
(2.0038)
(0.9797)
(0.2598)
a
b
\(\alpha \)
\(\theta \)
TLWLx
0.6616
0.7096
1.3632
0.6078
205.4314
213.4314
213.8480
223.8918
0.1624
0.9447
0.0834
0.4824
( 0.6595)
( 0.9639)
(0.9469)
(0.4455 )
a
b
\(\lambda \)
\(\theta \)
BOLE
0.87089
30.3780
20.7710
0.0014
205.6593
213.6593
214.0760
224.1198
0.1813
1.0308
0.0891
0.3978
(0.1064)
(\(4.9971\times 10^{-6}\))
(\(7.8244\times 10^{-6}\))
(\(2.2665\times 10^{-4}\))
Furthermore, the results presented in Table 4 confirm that the EOL-WP distribution demonstrates greater flexibility compared to the selected models when applied to the Kevlar dataset. Figures 5, 6, and 7 illustrate the effectiveness of the proposed model in fitting the data.
Notably, Figure 5 highlights the model’s ability to accommodate reverse-J shaped data while effectively adjusting its tails to capture outlying values. Additionally, as shown in Figure 7, the model successfully captures the non-increasing hazard rate behavior, further demonstrating its adaptability and robustness in modeling complex lifetime data.
Fig. 5
“Fitted Densities and Probability Plots for Kevlar Data”
The newly proposed EOL-XPS class of distributions introduces a compounding parameter that functions as both a scale and shape parameter. The additional shape parameter enhances the model’s ability to regulate skewness and kurtosis, making it adaptable to a wide range of data distributions. Furthermore, the EOL-XPS class effectively accommodates both monotonic and non-monotonic failure rates, increasing its applicability in diverse reliability and survival analysis contexts.
As a result, the EOL-XPS class offers significant advantages in managerial applications that require modeling complex and highly skewed datasets. Compared to traditional models, which often struggle with such complexities, the EOL-XPS class provides greater flexibility and improved precision, making it a superior choice for analyzing intricate real-world data.
Its practical applications include:
1.
Lifetime data analysis
2.
Reliability analysis
3.
Insurance data analysis for highly skewed insurance claims.
The EOL-XPS class of distributions is a versatile model with potential applications in fields such as finance, engineering, and environmental sciences. It can be extended to multivariate data and enhanced with machine learning to improve predictions in high-dimensional or time-series data. Various estimation techniques, such as Bayesian methods, could enhance parameter estimation, particularly in noisy scenarios. Combining EOL-XPS with other distributions may lead to more adaptable models for specific applications, while its use in stochastic processes, such as survival analysis, provides insights into event-time distributions. Using extreme value models as a baseline could improve the fitting of highly skewed data. Additionally, refining the theoretical properties of the EOL-XPS distribution could further enhance its practical and analytical value across various disciplines.
8 Concluding Remarks
In this work, we propose a new class of distributions called the exponentiated odd Lindley-X power series (EOL-XPS) family of distributions. We derive expressions for various statistical properties, including the quantile function, moments, moment-generating function, mean deviation, hazard rate function, order statistics, and entropy. As a special case, we consider the exponentiated odd Lindley-Weibull Poisson (EOL-WP) distribution and conduct a simulation study to assess the validity of the model parameters. Furthermore, we demonstrate the usefulness and versatility of the proposed model by applying it to a COVID-19 dataset from Canada and failure time data (in hours) of Kevlar 49/epoxy strands under 90\(\%\) pressure.
9 Future Scope of Research
The Exponentiated Odd Lindley-X Power Series (EOL-XPS) class of distributions provides a flexible framework with promising future applications and developments. It can be extended to handle multivariate datasets, making it suitable for complex domains such as finance, engineering, and environmental sciences. Integrating machine learning techniques with the model could enhance predictive accuracy in high-dimensional or time-series data. Bayesian estimation methods offer a robust approach for parameter inference, particularly in scenarios with limited or noisy data. Hybridizing the EOL-XPS class with other distribution families may lead to highly adaptable models for specialized applications, while its extension to stochastic processes such as survival analysis and reliability engineering could yield valuable insights into event-time distributions. Additionally, refining its theoretical properties, including entropy measures and order statistics, would enhance its analytical utility. These advancements would expand the scope and impact of the EOL-XPS class, contributing to both statistical theory and practical problem-solving across various disciplines.
Acknowledgements
The authors would like to acknowledge anonymous reviewers whose contributions greatly improved this work.
Declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
Ethical statements
The paper is not presently under review for publication in any other context or language.
Code availability
Upon a reasonable request, the codes used in this study can be provided.
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where \(C(\theta )=\sum _{n=1}^{\infty }a_{n}\theta ^{n}\) is finite, \(\theta >0\), and \(\{a_{n}\}_{n\ge 1}\) is a sequence of positive real numbers. Taking \(X_{(1)}=\min (X_{1}, X_{2}, ..., X_{N}),\) the CDF of \(X_{(1)}|N=n\) are given by
