A new univariate continuous distribution with applications in reliability

Marzouk, Waleed; Shafiq, Shakaiba; Naz, Sidra; Jamal, Farrukh; Sapkota, Laxmi Prasad; Nagy, M.; Mansi, A. H.; Hussam, Eslam; Gemeay, Ahmed M.

doi:10.1063/5.0179914

In this article, the odd Lomax Gompertz distribution has been introduced, which is derived by modifying the Gompertz distribution to serve as a baseline model in the odd generalized Lomax distribution. The newly proposed model offers enhanced flexibility and provides a promising alternative for modeling lifetime data. This study seeks to establish a solid theoretical foundation for its application through the exploration of several properties, such as non-central moments, stochastic orderings quantile function, and entropy measure, for the new model. Additionally, by conducting simulation analysis, the performance of the various estimation methods is being assessed, which enables the identification of the most reliable approach for estimating the unknown parameters of the newly developed model. The simulation analysis of the two-risk metrics, namely, value at risk and expected shortfall, revealed the ability of the distribution to capture diverse failure rate patterns, which makes it particularly relevant for assessing financial risks. Finally, the suggested model is practiced to two real-life datasets to provide the compelling evidence of superior flexibility and practical versatility compared to existing models in the literature.

I. INTRODUCTION

Traditional distributions have long been employed for analyzing and modeling data. However, there are instances where these distributions may fall short in meeting the required level of flexibility and accuracy. In response, researchers have introduced various techniques to enhance the performance of these distributions.¹ One significant advancement in recent decades is making the generalized (G) classes of these traditional distributions. The G classes of the distributions involve extending and expanding the scope of traditional distributions to accommodate a wider range of data patterns and characteristics.² By making G classes of the distributions, the researchers aim to create more flexible and versatile models that better capture the diversity and complexity observed in real-life.³ In addition, the approach of adding parameters has emerged as a widely utilized method for developing more flexible and adaptable distributions. Tahir et al. (2015)⁴ provided a detailed explanation about the formation of new distributions using additional parameters. Another approach to G classes involves combining or transforming existing distributions to create new distributions. This technique allows for the synthesis of novel models that inherit the desirable properties of multiple distributions. These G classes of distributions offered enhanced flexibility and can effectively capture a broader range of data patterns, tailoring to various real-life scenarios. Researchers have developed numerous G class of distributions, including the generalized beta distribution by McDonald and Xu (1995),⁵ generalized exponential distribution by Gupta and Kundu (1999),⁶ the gamma G distribution by Khodabina and Ahmadabadib (2010),⁷ the Marshall–Olkin G by Yousof et al. (2018),⁸ the odd generalized exponential distribution by Tahir and Nadarajah (2015),⁹ the Marshall–Olkin odd Lindley G family of distributions by Jamal et al. (2020),¹⁰ the exponentiated odd Chen G family by Eliwa et al. (2021),¹¹ the generalized odd linear exponential family by Jamal et al. (2022),¹² and the odd exponential Logarithmic family by Chesneau et al. (2022).¹³

Cordeiro et al. (2019)¹⁴ proposed the odd Lomax-G (OLG) family of distributions based on the Lomax generator with the cumulative density function (CDF) and probability density function (PDF), respectively, as¹⁵

Φ (x; θ, β, ψ) = 1 - β^{θ} {[β + \frac{F (x : ψ)}{1 - F (x : ψ)}]}^{- θ}

(1)

and

φ (x; θ, β, ψ) = θ β^{θ} f (x; ψ) {[1 - F (x : ψ)]}^{- 2} {[β + \frac{F (x : ψ)}{1 - F (x : ψ)}]}^{- (θ + 1)},

(2)

where β > 0, θ > 0, and ψ is a positive parameter vector of any continuous distribution.

Using the OLG family of distributions, the authors of Ref. 16 defined a four-parameter flexible model and applied it to model COVID-19 and engineering datasets. In this study, we have also used the OLG family of distributions based on Gompertz distribution. The Gompertz model is a flexible model that can be used in many applied areas of sciences, specifically in the analysis of lifetime data (Karamikabir et al., 2021).¹⁷ This distribution was considered for the survival analysis in some areas, such as biology, computer, gerontology, and marketing science. It has a monotone increasing hazard rate function (HRF), but, in practice, one desires also to consider the situation with non-monotonic HRF. Hence, in the past few years, many generalizations of the Gompertz distribution were proposed,

the generalized Gompertz model presented by El-Gohary et al. (2013),¹⁸ which was better than the classical Gompertz distribution in survival analysis,
the Beta-Gompertz model introduced by Ali et al. (2014)¹⁹ with decreasing, increasing, and bathtub shape HRF,
the odd generalized exponential Gompertz distribution proposed by El-Damcese et al. (2015),²⁰
the Kumaraswamy Gompertz distribution proposed by Silva et al. (2021),²¹
the McDonald Gompertz distribution introduced by Roozegar et al. (2015),²² and
the modified beta Gompertz distribution, which has the bathtub hazard rate function introduced by Elbatal et al. (2018).²³

The generalization of the exponential distribution is the Gompertz distribution, whose CDF is given as

F (x) = 1 - \exp [- \frac{a}{b} (e^{b x} - 1)]

(3)

and PDF is

f (x) = a e^{b x} \exp [- \frac{a}{b} (e^{b x} - 1)], x \geq 0 a n d a, b > 0,

(4)

where a and b are the scale parameters.

By Laplace transform, Lenart (2014)²⁴ provided the rth moment of the random variable X with PDF in Ref. 14 in the following form:

E (X^{r}) = \frac{r! e^{\frac{a}{b}}}{b^{r}} E_{1}^{r - 1} (\frac{a}{b})

(5)

such that

E_{k}^{r} (u) = \frac{1}{Γ (r + 1)} \int_{1}^{\infty} {(l n z)}^{r} z^{- k} e^{- u z} d z

is the generalized integro-exponential function for

\in C

⁠, where

E_{1}^{0} (u) = E_{1} (u) = \int_{1}^{\infty} \frac{e^{- u z}}{z} d z

(see Milgram, 1985).

The generalized hypergeometric function is the Meijer G-function and defined by the contour integral as

\begin{array}{l} G_{p, q}^{m, n} (z| \begin{matrix} a_{1}, \dots, a_{p} \\ b_{1}, \dots, b_{q} \end{matrix}) \\ = \frac{1}{2 π i} \int_{C} \frac{\prod_{j = 1}^{m} Γ (b_{j} - s) \prod_{j = 1}^{n} Γ (1 - a_{j} + s)}{\prod_{j = m + 1}^{q} Γ (1 - b_{j} + s) \prod_{j = n + 1}^{p} Γ (a_{j} - s)} z^{s} d s, \end{array}

where

= \sqrt{- 1}

⁠.

In addition, an alternative representation of the rth moment of the Gompertz random variable X presented by Lenart (2014)²⁴ using the Meijer G-function is as

E (X^{r}) = \frac{r! e^{\frac{a}{b}}}{b^{r}} G_{r, r + 1}^{r + 1, 0} (\frac{a}{b} | \begin{matrix} 1, \dots, 1 \\ 0, \dots, 0 \end{matrix})

(6)

[for more details, see Milgram (1985) and Lenart (2014)].²⁴

The rest of this article is presented as follows: Sec. II demonstrates the odd Lomax Gompertz (OLGo) distribution. In Sec. III, main properties are investigated. Section IV shows some stochastic orderings. In Sec. V, we obtained the mean residual life and mean inactivity time functions. The entropy measure is introduced in Sec. VI. Section VII contains the PDF of the order statistics and its moments. The maximum likelihood estimation and simulation study are presented in Sec. VIII. In Sec. IX, two applications to real datasets are given to prove the flexibility for the proposed new model. This article is concluded in Sec. IX.

II. THE OLGo DISTRIBUTION

In this section, we propose the new model called the odd Lomax Gompertz (OLGo) distribution. It is derived by combining the Lomax generator having CDF in Eq. with the modified Gompertz distribution having CDF in Eq. by considering that a = 1. The CDF of the OLGo model can be written as

Φ (x; θ, β, b) = 1 - {\{1 + \frac{1 - \exp [- b^{- 1} (e^{b x} - 1)]}{β \exp [- b^{- 1} (e^{b x} - 1)]}\}}^{- θ},

(7)

and the PDF corresponding to CDF in Eq. is

\begin{align} φ (x; θ, β, b) & = (\frac{θ}{β}) \exp \{b x + b^{- 1} (e^{b x} - 1)\} \\ \times {\{1 + \frac{1 - \exp [- b^{- 1} (e^{b x} - 1)]}{β \exp [- b^{- 1} (e^{b x} - 1)]}\}}^{- θ - 1} . \end{align}

(8)

In addition, the HRF of the OLGo model is given as

\begin{align} h (x) & = (\frac{θ}{β}) \exp \{b x + b^{- 1} (e^{b x} - 1)\} \\ \times {\{1 + β^{- 1} (\exp [b^{- 1} (e^{b x} - 1)] - 1)\}}^{- 1} . \end{align}

(9)

The OLGo distribution as expressed in Eq. (7) encompasses several notable special cases, which are as follows:

If θ = 1, then we obtain the two parameter Marshall–Olkin Gompertz (MOGo) distribution.
If b tends to zero, then we get the Marshall–Olkin generalized exponential (MOGEx) distribution.
If θ = 1 and b tends to zero, then we get the Marshall–Olkin exponential (MOEx) distribution.
If θ = 1and β = 1, then we obtain the one parameter Gompertz (Go) distribution.
If β = 1 and b tends to zero, then we get the generalized exponential (GEx) distribution.
If θ = 1, β = 1, and b tends to zero, then we get the exponential (Ex) distribution.

Figure 1 depicts various configurations of the OLGo model based on specific parameter values. Meanwhile, Fig. 2 demonstrates the versatility of the OLGo model’s HRF, which can exhibit decreasing, increasing, and bathtub-like shapes. Clearly, the OLGo model showcases a significantly greater degree of flexibility compared to the Go model. Different plots of proposed model CDF and survival function are presented in Fig. 3.

FIG. 1.

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Graphs of the OLGo PDF for some parameters values.

FIG. 2.

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Graphs of the OLGo HRF function for some parameters values.

III. MAIN PROPERTIES

A. Density function expansion

Based on the binomial expansion, we can rewrite the PDF expressed in Eq.

\begin{array}{l} φ (x) & = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} {(- 1)}^{m} (\frac{θ (\begin{matrix} k \\ m \end{matrix}) (\begin{matrix} - θ - 1 \\ k \end{matrix})}{β^{k + 1}}) e^{b x} \\ \times \exp [- b^{- 1} (m - k - 1) (e^{b x} - 1)] . \end{array}

Hence, the PDF of the OLGo model can be written as

φ (x) = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} ω_{k, m} (β, θ) f_{G o} (m - k - 1, b),

(10)

where

f_{G o} (m - k - 1, b)

is the Gompertz (Go) PDF with parameters m − k − 1 and b and

ω_{k, m} (β, θ) = \frac{{(- 1)}^{m} θ (\begin{matrix} - θ - 1 \\ k \end{matrix}) (\begin{matrix} k \\ m \end{matrix})}{β^{k + 1} (m - k - 1)}

⁠.

FIG. 3.

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Graphs of survival and CDF function for some parameters values.

B. Measures of position and random number generation

Assuming that X has the OLGo model, then the quantile function (QF) of X is given as

Q_{X} (p) = \ln {\{1 + \ln {\{\frac{β - (β - 1) {(1 - p)}^{1 / θ}}{{(1 - p)}^{1 / θ}}\}}^{b}\}}^{\frac{1}{b}},

(11)

where

p \in (0, 1)

⁠. The QF provides us with the capability to utilize Bowley’s skewness coefficient and Moor’s kurtosis coefficient. These coefficients are robust against the influence of outliers and remain applicable even in cases where moments are unavailable. The coefficients for Bowley and Moor are expressed as follows:

S K = \frac{Q_{X} (3 / 4) + Q_{X} (1 / 4) - 2 Q_{X} (1 / 2)}{Q_{X} (3 / 4) - Q_{X} (1 / 4)}

and

K = \frac{Q_{X} (7 / 8) + Q_{X} (3 / 8) - Q_{X} (5 / 8) - Q_{X} (1 / 8)}{Q_{X} (6 / 8) - Q_{X} (2 / 8)},

where

Q_{X} (i / 8)

is the ith octile for X such that i = 1, 2, …, 7. If U is a continuous random variable uniform on the unit interval (0, 1), then we can use Eq. to generate random numbers from the OLGo distribution when the parameters are known.

C. Ordinary moments

This section focuses on providing various forms for the moments of the OLGo distribution and presenting the moment generating function (MGF). It aims to establish a comprehensive understanding of the statistical properties and characteristics of the OLGo model. Suppose X∼OLGo(θ, β, b), then the rth moment of X is

μ_{r}^{'} = E (X^{r}) = \int_{0}^{\infty} x^{r} φ (x) d x .

(12)

Immediately from the expansion of the pdf given in Eq. and using it in Eq. , we get

μ_{r}^{'} = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} ω_{k, m} (β, θ) E (X_{G o}^{r}),

(13)

where

E (X_{G o}^{r})

is rth moment of the Gompertz variable X with parameters

(m - k - 1)

and b. Hence, from Eqs. and and by replacing a with

(m - k - 1)

two, we get

\begin{array}{l} μ_{r}^{'} & = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} ω_{k, m} (β, θ) r! b^{- r} \exp [b^{- 1} (m - k - 1)] E_{1}^{r - 1} \\ \times (b^{- 1} (m - k - 1)) . \end{array}

Thus, the rth moment of the OLGo distribution is

μ_{r}^{'} = \sum_{k = 0}^{\infty} \nabla_{k}^{(r)} E_{1}^{r - 1} (b^{- 1} (m - k - 1)) .

From Eq. , we can get an alternative form for moments of the OLGo distribution in terms of Meijer G-function as

μ_{r}^{'} = \sum_{k = 0}^{\infty} \nabla_{k}^{(r)} G_{r, r + 1}^{r + 1, 0} (b^{- 1} (m - k - 1) |\begin{matrix} 1, \dots, 1 \\ 0, \dots, 0 \end{matrix}),

(14)

where

\nabla_{k}^{(r)} = \sum_{m = 0}^{k} \frac{{(- 1)}^{m} (\begin{matrix} - θ - 1 \\ k \end{matrix}) (\begin{matrix} k \\ m \end{matrix}) r! θ \exp [b^{- 1} (m - k - 1)]}{β^{k + 1} b^{r} (m - k - 1)} .

Table I shows the first four moments, skewness, variance, and kurtosis for selected values of the parameters of the OLGo distribution. Now, we can use the relationship between MGF and non-central moments in order to get a formula of the MGF of OLGo distribution. The relationship between MGF and non-central moments is

M_{X} (t) = \sum_{r = 0}^{\infty} \frac{t^{r}}{r!} μ_{r}^{'} .

(15)

Hence, from Eqs. or , we can get the MGF of OLGo distribution by substituting into Eq. .

TABLE I.

Moments, variance, skewness, and kurtosis of the OLGo distribution.

Parameter (θ, β, b)	μ₁	μ₂	μ₃	μ₄	Variance	Skewness	Kurtosis
0.5, 0.5, 0.5	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
1.0, 0.5, 0.5	0.5248	0.5230	0.7167	1.1938	0.2475	1.4813	12.7506
1.5, 0.5, 0.5	0.3544	0.2492	0.2513	0.3180	0.1235	1.7334	12.8953
2.0, 0.5, 0.5	0.2657	0.1429	0.1129	0.1146	0.0723	1.8814	13.4008
0.5, 1.0, 0.5	1.1926	2.1277	4.6454	11.5560	0.7052	0.7185	27.3966
0.5, 1.5, 0.5	1.3303	2.4834	5.5419	13.9436	0.7136	0.5629	39.7108
0.5, 2.0, 0.5	1.4299	2.7573	6.2571	15.8894	0.7124	0.4598	52.1468
0.5, 0.5, 1.0	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
0.5, 0.5, 1.5	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
0.5, 0.5, 2.0	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331

Parameter (θ, β, b)	μ₁	μ₂	μ₃	μ₄	Variance	Skewness	Kurtosis
0.5, 0.5, 0.5	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
1.0, 0.5, 0.5	0.5248	0.5230	0.7167	1.1938	0.2475	1.4813	12.7506
1.5, 0.5, 0.5	0.3544	0.2492	0.2513	0.3180	0.1235	1.7334	12.8953
2.0, 0.5, 0.5	0.2657	0.1429	0.1129	0.1146	0.0723	1.8814	13.4008
0.5, 1.0, 0.5	1.1926	2.1277	4.6454	11.5560	0.7052	0.7185	27.3966
0.5, 1.5, 0.5	1.3303	2.4834	5.5419	13.9436	0.7136	0.5629	39.7108
0.5, 2.0, 0.5	1.4299	2.7573	6.2571	15.8894	0.7124	0.4598	52.1468
0.5, 0.5, 1.0	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
0.5, 0.5, 1.5	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331
0.5, 0.5, 2.0	0.9695	1.6044	3.3931	8.3148	0.6643	1.0142	15.5331

D. Probability weighted moments

Probability weighted moments (PWMs) stand out as a pivotal instrument in delineating theoretical probability distributions. They find application in not only estimating parameters but also in conducting hypothesis tests for probability distributions and non-parametrically estimating the distribution underlying an observed sample. The

(n, r) th

PWMs of the random variable X with PDF f(x) and CDF F(x) are defined as

η_{r, n} = E \{X^{r} F {(X)}^{n}\} = \int_{- \infty}^{\infty} x^{r} f (x) F {(x)}^{n} d x .

From Eqs. and and by using the binomial formula, we obtained

\begin{array}{l} f (x) F {(x)}^{n} & = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} \sum_{i = 0}^{n} {(- 1)}^{i + m} (\begin{matrix} n \\ i \end{matrix}) (\begin{matrix} k \\ m \end{matrix}) (\begin{matrix} - (i + 1) θ - 1 \\ k \end{matrix}) \\ \times (\frac{θ}{β^{k + 1}}) e^{\frac{1}{b} (m - k - 1)} \end{array}

e^{- \frac{1}{b} (m - k - 1) e^{b x} + b x} .

Hence, the

(n, r) th

PWMs of the OLGo X are

\begin{array}{l} η_{r, n} & = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} \sum_{i = 0}^{n} {(- 1)}^{i + m} (\begin{matrix} n \\ i \end{matrix}) (\begin{matrix} k \\ m \end{matrix}) (\begin{matrix} - (i + 1) θ - 1 \\ k \end{matrix}) (\frac{θ}{β^{k + 1}}) \\ \times e^{\frac{1}{b} (m - k - 1)} \end{array}

\int_{0}^{\infty} x^{r} e^{b x} e^{- \frac{1}{b} (m - k - 1) e^{b x}} d x .

Using integration by parts, we get

\int_{0}^{\infty} x^{r} e^{b x} e^{- \frac{1}{b} (m - k - 1) e^{b x}} d x = \frac{Γ (r + 1)}{b^{r} (m - k - 1)} E_{1}^{r - 1} (b^{- 1} (m - k - 1)) .

Hence,

η_{r, n} = \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} \sum_{i = 0}^{n} Δ_{k, m, i}^{(r, n)} E_{1}^{r - 1} (b^{- 1} (m - k - 1)),

(16)

where

Δ_{k, m, i}^{(r, n)} = \frac{{(- 1)}^{i + m} (\begin{matrix} n \\ i \end{matrix}) (\begin{matrix} k \\ m \end{matrix}) (\begin{matrix} - (i + 1) θ - 1 \\ k \end{matrix}) θ (r!) e^{b^{- 1} (m - k - 1)}}{β^{k + 1} b^{r} (m - k - 1)} .

Figures 4 and 5 are the 3D graphs of the mean and variance and kurtosis and skewness, respectively, for various parametric values. These 3D plots of mean, variance, skewness, and kurtosis offer a visual representation for the exploration of patterns and trends, aiding in understanding the distributional characteristics and the impact of various factors on the data’s central tendency, spread, asymmetry, and tail behavior.

FIG. 4.

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3D plots of the mean and variance of the OLGo model.

FIG. 5.

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3D plots of skewness and kurtosis of the OLGo distribution.

IV. STOCHASTIC COMPARISONS

In reliability theory and other fields stochastic, ordering is an important tool to assess the comparative behavior. We assumed that X and Y are two random variables with distribution functions F and G and density functions f and g, respectively. Now let us recall some definitions: X is said to be smaller than Y in the usual stochastic order (denoted by X ≤ _stY) if $P \{X \leq x\} \geq P \{Y \leq x\}$ for all x. The random variable X is said to be smaller than the random variable Y in hazard rate order (denoted by X ≤ _hrY) if $P \{X > x\} / P \{Y > x\}$ is an increasing function of x. The random variable X is said to be smaller than the random variable Y in reversed hazard rate order (denoted by X ≤ _rhY) if $P \{X \leq x\} / P \{Y \leq x\}$ is an increasing function of x. The random variable X is said to be smaller than the random variable Y in the likelihood ratio order (denoted by X ≤ _lrY) if f/g is decreasing of x. The random variable X is said to be smaller than the random variable Y in the mean residual life order (denoted by X ≤ _mrlY) if m_X(x) ≤ m_Y(x) for all x where m_X(x) and m_Y(x) are the mean residual life functions of X and Y, respectively. The random variable X is said to be smaller than the random variable Y in the harmonic mean residual life order (denoted by X ≤ _hmrlY) if ${[\frac{1}{x} \int_{0}^{x} \frac{1}{m_{X} (u)} d u]}^{- 1} \leq {[\frac{1}{x} \int_{0}^{x} \frac{1}{m_{Y} (x)} d u]}^{- 1}$ for all x > 0. The random variable X is said to be smaller than the random variable Y in the increasing convex order (denoted by X ≤ _icxY) if $E [ζ (X)] \leq E [ζ (Y)]$ for all increasing convex functions $ζ : R \to R$ ⁠, provided the expectations exist.

For all the previous orders, we have the following chains of implications:

\begin{gathered} X \leq_{icx} Y \Leftarrow X \leq_{mrl} Y \Rightarrow X \leq_{hmrl} Y \Rightarrow X \leq_{icx} Y, \\ ⇑ \\ X \leq_{l r} Y \Rightarrow X \leq_{h r} Y \Rightarrow X \leq_{s t} Y \Rightarrow X \leq_{icx} Y, \\ ⇓ \\ X \leq_{r h} Y \Rightarrow X \leq_{s t} Y \Rightarrow X \leq_{icx} Y^{″} . \end{gathered}

It is clear that a stronger order is the likelihood ratio order.

Theorem.

Let X∼OLGo(θ₁, β₁, b) and Y∼OLGo(θ₂, β₂, b). If θ₁ > θ₂ and β₂ > β₁, then X ≤ _lrY.

Proof.

From Eq. , we have

f (x) = \frac{φ (x; θ_{1}, β_{1}, b)}{φ (x; θ_{2}, β_{2}, b)} = (\frac{θ_{1} β_{2}}{θ_{2} β_{1}}) \frac{{\{1 + \frac{1 - z}{β_{1} z}\}}^{- θ_{1} - 1}}{{\{1 + \frac{1 - z}{β_{2} z}\}}^{- θ_{2} - 1}},

where

z = e^{- \frac{1}{b} (e^{b x} - 1)}

⁠.

Hence,

\begin{array}{l} \ln f (x) & = \ln (\frac{θ_{1} β_{2}}{θ_{2} β_{1}}) - (θ_{1} + 1) \ln \{1 + \frac{1 - z}{β_{1} z}\} \\ + (θ_{2} + 1) \ln \{1 + \frac{1 - z}{β_{2} z}\} . \end{array}

By differentiating the last expression and after some algebra, we get

\frac{d}{d x} [\ln f (x)] = \frac{[(θ_{1} β_{2} - θ_{2} β_{1}) z + (β_{2} - β_{1}) z + (θ_{1} - θ_{2}) (1 - z)] (d z / d x)}{[β_{1} z + (1 - z)] [β_{2} z + (1 - z)] z},

where

d z / d x = - e^{b x} e^{- \frac{1}{b} (e^{b x} - 1)}

⁠. Now, if θ₁ > θ₂ and β₂ > β₁, then

\frac{d}{d x} [\ln f (x)] < 0

⁠, which implies that X ≤ _lrY.

Through the previous evidence, we find that the OLGo distribution is ordered with respect to the strongest likelihood ratio ordering.

V. MEAN RESIDUAL LIFE (MRL) AND MEAN INACTIVITY TIME FUNCTIONS

The mean residual life (MRL) function gives us the remaining life after age x. If the random variable X∼OLGo distribution, then MRL is given as

{\tilde{μ}}_{R L} (x) = E (X - x | X > x) = \frac{1}{\bar{Φ} (x)} \int_{x}^{\infty} \bar{Φ} (u) d u .

Hence,

\int_{x}^{\infty} \bar{Φ} (u) d u = \sum_{i = 0}^{\infty} \sum_{j = 0}^{i} {(- 1)}^{j} β^{- i} (\begin{matrix} - θ \\ i \end{matrix}) (\begin{matrix} i \\ j \end{matrix}) e^{\frac{1}{b} (j - i)} \int_{x}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u \cdot

(17)

Now, we suppose that

= \frac{1}{b} (j - i) e^{b u}

⁠, and thus,

\int_{x}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u = \frac{1}{b} \int_{\frac{1}{b} (j - i) e^{b x}}^{\infty} \frac{e^{- v} d v}{v} = b^{- 1} E_{1} [b^{- 1} (j - i) e^{b x}],

where

E_{1} (s) = \int_{1}^{\infty} \frac{e^{- st} d t}{t} = \int_{s}^{\infty} \frac{e^{- t} d t}{t}

is the exponential integral. Therefore, the MRL function of the random variable X is

{\tilde{μ}}_{R L} (x) = \frac{1}{\bar{Φ} (x)} \sum_{i = 0}^{\infty} \sum_{j = 0}^{i} ρ_{i, j} (θ, β, b) E_{1} [b^{- 1} (j - i) e^{b x}],

where

ρ_{i, j} (θ, β, b) = {(- 1)}^{j} {(b β^{i})}^{- 1} \exp (b^{- 1} (j - i)) (\begin{matrix} i \\ j \end{matrix}) (\begin{matrix} - θ \\ i \end{matrix})

⁠.

The Mean Inactivity Time (MIT) function serves as a valuable tool for assessing and forecasting the precise moment when failure occurs. Consequently, the MIT function holds a prominent position as a reliability metric, finding wide-ranging applications across various fields, including reliability theory, survival analysis, bio-medicine, epidemiology, medical research, and actuarial studies. The MIT function at a specific point “x,” for a lifetime random variable “X” following an OLGo distribution, is formally defined as follows:

{\tilde{μ}}_{I T} (x) = E (x - X | X \leq x) = \frac{1}{Φ (x)} \int_{0}^{x} Φ (u) d u .

(18)

Thus, based on Eq. and changing the limits of integration, we can get

\begin{align} \int_{0}^{x} Φ (u) d u & = x - \sum_{i = 0}^{\infty} \sum_{j = 0}^{i} {(- 1)}^{j} β^{- i} (\begin{matrix} - θ \\ i \end{matrix}) (\begin{matrix} i \\ j \end{matrix}) \\ \times e^{\frac{1}{b} (j - i)} \int_{0}^{x} e^{- \frac{1}{b} (j - i) e^{b u}} d u . \end{align}

(19)

Since,

\int_{0}^{x} e^{- \frac{1}{b} (j - i) e^{b u}} d u = \int_{0}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u - \int_{x}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u .

Using the same substitution mentioned above, we find that

\int_{0}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u = \frac{1}{b} \int_{\frac{1}{b} (j - i)}^{\infty} \frac{e^{- v} d v}{v} = b^{- 1} E_{1} (b^{- 1} (j - i))

and

\int_{x}^{\infty} e^{- \frac{1}{b} (j - i) e^{b u}} d u = \frac{1}{b} \int_{\frac{1}{b} (j - i) e^{b x}}^{\infty} \frac{e^{- v} d v}{v} = b^{- 1} E_{1} (b^{- 1} (j - i) e^{b x}) .

Therefore,

\int_{0}^{x} e^{- \frac{1}{b} (j - i) e^{b u}} d u = b^{- 1} [E_{1} (b^{- 1} (j - i)) - E_{1} (b^{- 1} (j - i) e^{b x})] .

From Eqs. and , the MIT function of the OLGo random variable X is

\begin{array}{l} {\tilde{μ}}_{I T} (x) & = \frac{1}{Φ (x)} \{x - \sum_{I = 0}^{\infty} \sum_{j = 0}^{i} ρ_{i, j} (θ, β, b) [E_{1} (b^{- 1} (j - i)) \\ - E_{1} (b^{- 1} (j - i) e^{b x})]\}, \end{array}

where

ρ_{i, j} (θ, β, b) = {(- 1)}^{j} {(b β^{i})}^{- 1} \exp (b^{- 1} (j - i)) (\begin{matrix} i \\ j \end{matrix}) (\begin{matrix} - θ \\ i \end{matrix})

⁠.

A. Entropy

Entropy is the metric that quantifies the information needed to characterize a random variable. Shannon entropy stands as the prevalent measure for assessing information content. Nevertheless, Rényi entropy serves as a broader framework, encompassing Shannon entropy and several other entropy measures as specific instances. Rényi entropy finds utility across diverse fields, including physics, mathematics, astronomy, computer science, engineering, medicine, material science, and more. When applied to a non-negative continuous random variable X with probability density function f(x), Rényi entropy is formally defined as

E n_{R} (δ) = \frac{1}{1 - δ} \log \{\int_{0}^{\infty} f {(x)}^{δ} d x\},

where δ > 0 and δ ≠ 1. If a random variable X has the OLGo distribution, then from Eq. ,

\begin{array}{l} I = \int_{0}^{\infty} f {(x)}^{δ} d x & = {(\frac{θ}{β})}^{δ} \int_{0}^{\infty} e^{δ b x + \frac{δ}{b} (e^{b x} - 1)} \\ \times {\{1 + \frac{1 - e^{- \frac{1}{b} (e^{b x} - 1)}}{β e^{- \frac{1}{b} (e^{b x} - 1)}}\}}^{- (θ + 1) δ} . d x \end{array}

Using power series expansion and replacing z = e^bx, we get

\begin{array}{l} I & = \sum_{k = 0}^{\infty} \sum_{v = 0}^{k} {(- 1)}^{v} θ^{δ} β^{- δ - k} b^{- 1} e^{\frac{1}{b} (v - δ - k)} (\begin{matrix} k \\ v \end{matrix}) (\begin{matrix} - (θ + 1) δ \\ k \end{matrix}) \int_{1}^{\infty} z^{δ - 1} \\ \times e^{- \frac{1}{b} (v - δ - k) z} d z . \end{array}

Hence, the Rényi entropy of the OLGo distribution after some algebra operations is

E_{R} (δ) = \frac{1}{1 - δ} \log \{\sum_{k = 0}^{\infty} \sum_{v = 0}^{k} \frac{{(- 1)}^{v} θ^{δ} b^{δ - 1} e^{\frac{1}{b} (v - δ - k)} (\begin{matrix} k \\ v \end{matrix}) (\begin{matrix} - (θ + 1) δ \\ k \end{matrix}) Γ (δ, b^{- 1} (v - δ - k))}{β^{δ + k} {[b^{- 1} (v - δ - k)]}^{δ}}\},

where

Γ (δ, b^{- 1} (v - δ k))

is the upper incomplete gamma function. Figure 6 is the 3D display of the Renyi entropy for various parametric values.

FIG. 6.

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3D plot of Renyi entropy of the OLGo model.

VI. ORDER STATISTICS, MOMENTS OF ORDER STATISTICS, AND L-MOMENTS

Suppose X₁, …, X_n is a random sample of size n from

O L G o (θ, β, b)

⁠. Then, the pdf of the ith order statistic is given by

f_{i : n} (x) = \frac{1}{B (i, n - i + 1)} \sum_{j = 0}^{n - i} {(- 1)}^{j} (\begin{matrix} n - i \\ j \end{matrix}) φ (x) Φ^{i + j - 1} (x) .

(20)

From Eqs. and , we can get

\begin{align} φ (x) Φ^{i + j - 1} (x) & = \sum_{s = 0}^{i + j - 1} \frac{{(- 1)}^{s} (\begin{matrix} i + j - 1 \\ s \end{matrix})}{s + 1} (\frac{(s + 1) θ}{β}) e^{b x + \frac{1}{b} (e^{b x} - 1)} \\ \times {\{1 + \frac{1 - e^{- \frac{1}{b} (e^{b x} - 1)}}{β e^{- \frac{1}{b} (e^{b x} - 1)}}\}}^{- (s + 1) θ - 1} . \end{align}

(21)

Combining Eqs. and yields

f_{i : n} (x) = \sum_{j = 0}^{n - i} \sum_{s = 0}^{i + j - 1} ϖ_{j, s} (n, i) φ (x; (s + 1) θ, β, b),

where

ϖ_{j, s} (n, i) = \frac{{(- 1)}^{j + s} (\begin{matrix} n - i \\ j \end{matrix}) (\begin{matrix} i + j - 1 \\ s \end{matrix})}{B (i, n - i + 1) (s + 1)},

and

φ (x; (s + 1) θ, β, b)

is the pdf of OLGo distribution with parameters

(s + 1) θ, β

⁠, and b. Hence, it is easy to find the rth moment of the ith order statistic as follows:

E (X_{i : n}^{r}) = \sum_{j = 0}^{n - i} \sum_{s = 0}^{i + j - 1} \sum_{k = 0}^{\infty} \sum_{m = 0}^{k} ϖ_{j, s} (n, i) ξ_{k, m, s, j}^{(r)} E_{1}^{r - 1} (b^{- 1} (m - k - 1)),

(22)

where

ξ_{k, m, s, j}^{(r)} = \frac{{(- 1)}^{m} (\begin{matrix} - (s + 1) θ - 1 \\ k \end{matrix}) (\begin{matrix} k \\ m \end{matrix}) r! (s + 1) θ \exp (b^{- 1} (m - k - 1))}{β^{k + 1} b^{r} (m - k - 1)}

⁠.

L-moments exist whenever the mean of the distribution exists, even though some higher moments may not exist. They are analogous to the ordinary moments but can be obtained by linear combinations of order statistics. The kth L-moment is defined by

ϒ_{k} = \sum_{u = 0}^{k - 1} {(- 1)}^{k - u - 1} (\begin{matrix} k - 1 \\ u \end{matrix}) (\begin{matrix} k + u - 1 \\ u \end{matrix}) λ_{u},

where

λ_{u} = E (X F^{u} (X))

⁠, so it can be computed using Eq. . In particular, ϒ₁ = λ₀, ϒ₂ = 2λ₁ − λ₀, ϒ₃ = 6λ₂ − 6λ₁ + λ₀, ϒ₄ = 20λ₃ − 30λ₂ + 12λ₁ − λ₀.

In general, $λ_{u} = {(u + 1)}^{- 1} E (X_{u + 1 : u + 1})$ ⁠; thus, it can be calculated using Eq. (22) Figure 7 illustrates the minimum and maximum values of order statistics. These plots are helpful for understanding the behavior and variability of extreme values.

FIG. 7.

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Min-Max plots of the OLGo model for various parametric values.

VII. STATISTICAL INFERENCE

A. Maximum likelihood estimators

In this section, the maximum likelihood estimation (MLEs) of the parameters of the OLGo distribution are determined. Let x₁, x₂, …, x_n be a random sample of size n from the

O L G o (θ, β, b)

distribution. The log-likelihood function for the vector parameter

δ = {(θ, β, b)}^{T}

is

\begin{array}{l} l (δ) & = n (\frac{θ}{β}) - \frac{n}{b} + b \sum_{i = 1}^{n} x_{i} + \frac{1}{b} \sum_{i = 1}^{n} e^{b x_{i}} - (θ + 1) \\ \times \sum_{i = 1}^{n} \ln [β e^{- z_{i}} - e^{- z_{i}} + 1] + (θ + 1) \sum_{i = 1}^{n} \ln (β e^{- z_{i}}), \end{array}

where

z_{i} = \frac{1}{b} (e^{b x_{i}} - 1)

⁠. The components of the score vector

U (δ) = {(U_{θ}, U_{β}, U_{b})}^{T}

are given by

U_{θ} = \frac{n}{β} - \sum_{i = 1}^{n} \ln [1 + \frac{1 - e^{- z_{i}}}{β e^{- z_{i}}}],

U_{β} = \frac{- n θ}{β^{2}} + \frac{n (θ + 1)}{β} - (θ + 1) \sum_{i = 1}^{n} \frac{e^{z_{i}}}{β e^{- z_{i}} - e^{- z_{i}} + 1},

and

\begin{array}{l} U_{b} & = \frac{n}{b^{2}} + \sum_{i = 1}^{n} x_{i} + \frac{1}{b^{2}} \sum_{i = 1}^{n} \{(b x_{i} - 1) e^{b x_{i}}\} - \frac{(θ + 1)}{b} \\ \times \sum_{i = 1}^{n} \frac{e^{- z_{i}} (x_{i} e^{b x_{i}} - z_{i}) (1 - β)}{β e^{- z_{i}} - e^{- z_{i}} + 1} - \frac{(θ + 1)}{b} \sum_{i = 1}^{n} (x_{i} e^{b x_{i}} - z_{i}) . \end{array}

The MLEs of the parameters can be obtained by solving the nonlinear equations U_θ = 0, U_β = 0, and U_b = 0 numerically by statistical software.

B. Simulation analysis

This section offers a simulation analysis to show how the parameter estimates of the OLGo model perform over a specified replication with r = 1000. In order to mix the parameters θ, β, and b in different methods, we create N = 1000 samples of various sizes (n = 20, 50, 100, and 200). As an illustration, let us look at the values for sets VS-I [θ = 0.75, β = 2.75, b = 1.5], VS-II [θ = 0.15, β = 1.5, b = 0.5], and VS-III [θ = 0.1, β = 1.5, b = 0.1], respectively. The unknown parameters are approximated utilizing the MLEs, Maximum Product of Spacing estimate (MPSE), least square estimate (LSE), weighted least square estimate (WLSE), and Cramer–Von-Mises estimate (CME), methods as specified in the previous section. The results of the simulation study showed that biases and mean square errors (MSEs) decrease with the sample size. Table II displays the outcomes of the simulation study of MSE and average estimates (AEs), as well as estimate biases. All estimation methods are successful, but MLE appears to be stable as MSEs reduces with the increase in sample sizes. Figures 7–9 are the graphical evidences of our claim that MLE is the best estimation approach for the OLGo model. All of the computations for the different estimating techniques are carried out using the R computation programming language.

TABLE II.

Summary of the AEs and MSEs from simulations using different estimation techniques.

Parameter	$\hat{θ}$				$\hat{β}$				$\hat{b}$
VS-I
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.409	0.414	0.428	0.430	0.990	1.039	1.040	1.045	0.845	0.847	0.845	0.845
MPSE	0.463	0.478	0.483	0.486	0.993	1.042	1.047	1.048	0.849	0.850	0.849	0.849
OLSE	0.426	0.434	0.436	0.437	0.981	1.038	1.046	1.049	0.848	0.849	0.849	0.848
WLSE	0.443	0.458	0.466	0.471	0.980	1.032	1.042	1.047	0.859	0.849	0.847	0.846
CME	0.431	0.435	0.437	0.438	0.991	1.038	1.047	1.049	0.849	0.859	0.850	0.847
MSE
MLE	0.056	0.035	0.056	0.043	3.100	2.898	2.876	2.865	0.404	0.420	0.421	0.323
MPSE	0.083	0.074	0.071	0.069	3.105	2.919	2.899	2.897	0.424	0.423	0.423	0.393
OLSE	0.108	0.102	0.100	0.099	3.149	2.935	2.902	2.895	0.425	0.423	0.413	0.413
WLSE	0.097	0.087	0.081	0.078	3.154	2.955	2.918	2.900	0.424	0.423	0.423	0.403
CME	0.105	0.101	0.099	0.098	3.111	2.932	2.901	2.895	0.424	0.423	0.423	0.333
VS-II
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.310	0.245	0.256	0.254	0.967	1.015	1.025	1.040	0.654	0.698	0.702	0.714
MPSE	0.313	0.303	0.296	0.291	0.976	1.025	1.044	1.049	0.677	0.706	0.722	0.734
OLSE	0.293	0.278	0.273	0.267	0.977	1.018	1.039	1.046	0.737	0.788	0.809	0.828
WLSE	0.291	0.274	0.268	0.262	0.993	1.025	1.042	1.048	0.743	0.793	0.812	0.828
CME	0.289	0.275	0.271	0.266	1.004	1.025	1.040	1.047	0.772	0.804	0.817	0.832
MSE
MLE	0.020	0.015	0.013	0.010	0.256	0.230	0.209	0.200	0.058	0.058	0.048	0.041
MPSE	0.033	0.028	0.024	0.021	0.302	0.235	0.210	0.203	0.060	0.061	0.060	0.061
OLSE	0.026	0.019	0.016	0.014	0.300	0.242	0.215	0.206	0.077	0.092	0.100	0.109
WLSE	0.026	0.018	0.015	0.013	0.278	0.233	0.211	0.205	0.079	0.094	0.101	0.109
CME	0.024	0.018	0.016	0.014	0.262	0.232	0.213	0.206	0.088	0.099	0.104	0.111
VS-III
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.256	0.265	0.271	0.238	1.035	1.018	1.026	1.020	0.710	0.698	0.710	0.618
MPSE	0.296	0.302	0.299	0.290	1.044	1.023	1.044	1.029	0.722	0.701	0.719	0.735
OLSE	0.273	0.276	0.273	0.256	1.039	1.016	1.038	1.026	0.809	0.787	0.811	0.628
WLSE	0.268	0.272	0.270	0.252	1.042	1.022	1.044	1.027	0.812	0.792	0.812	0.528
CME	0.271	0.273	0.272	0.245	1.040	1.024	1.040	1.046	0.817	0.801	0.818	0.731
MSE
MLE	0.010	0.014	0.012	0.010	0.201	0.228	0.202	0.102	0.054	0.045	0.040	0.008
MPSE	0.024	0.027	0.024	0.021	0.210	0.237	0.209	0.104	0.060	0.050	0.043	0.062
OLSE	0.016	0.018	0.016	0.014	0.215	0.245	0.216	0.107	0.100	0.093	0.051	0.019
WLSE	0.015	0.017	0.016	0.013	0.211	0.237	0.209	0.105	0.101	0.094	0.036	0.016
CME	0.016	0.018	0.016	0.012	0.213	0.234	0.213	0.106	0.104	0.099	0.065	0.010

Parameter	$\hat{θ}$				$\hat{β}$				$\hat{b}$
VS-I
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.409	0.414	0.428	0.430	0.990	1.039	1.040	1.045	0.845	0.847	0.845	0.845
MPSE	0.463	0.478	0.483	0.486	0.993	1.042	1.047	1.048	0.849	0.850	0.849	0.849
OLSE	0.426	0.434	0.436	0.437	0.981	1.038	1.046	1.049	0.848	0.849	0.849	0.848
WLSE	0.443	0.458	0.466	0.471	0.980	1.032	1.042	1.047	0.859	0.849	0.847	0.846
CME	0.431	0.435	0.437	0.438	0.991	1.038	1.047	1.049	0.849	0.859	0.850	0.847
MSE
MLE	0.056	0.035	0.056	0.043	3.100	2.898	2.876	2.865	0.404	0.420	0.421	0.323
MPSE	0.083	0.074	0.071	0.069	3.105	2.919	2.899	2.897	0.424	0.423	0.423	0.393
OLSE	0.108	0.102	0.100	0.099	3.149	2.935	2.902	2.895	0.425	0.423	0.413	0.413
WLSE	0.097	0.087	0.081	0.078	3.154	2.955	2.918	2.900	0.424	0.423	0.423	0.403
CME	0.105	0.101	0.099	0.098	3.111	2.932	2.901	2.895	0.424	0.423	0.423	0.333
VS-II
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.310	0.245	0.256	0.254	0.967	1.015	1.025	1.040	0.654	0.698	0.702	0.714
MPSE	0.313	0.303	0.296	0.291	0.976	1.025	1.044	1.049	0.677	0.706	0.722	0.734
OLSE	0.293	0.278	0.273	0.267	0.977	1.018	1.039	1.046	0.737	0.788	0.809	0.828
WLSE	0.291	0.274	0.268	0.262	0.993	1.025	1.042	1.048	0.743	0.793	0.812	0.828
CME	0.289	0.275	0.271	0.266	1.004	1.025	1.040	1.047	0.772	0.804	0.817	0.832
MSE
MLE	0.020	0.015	0.013	0.010	0.256	0.230	0.209	0.200	0.058	0.058	0.048	0.041
MPSE	0.033	0.028	0.024	0.021	0.302	0.235	0.210	0.203	0.060	0.061	0.060	0.061
OLSE	0.026	0.019	0.016	0.014	0.300	0.242	0.215	0.206	0.077	0.092	0.100	0.109
WLSE	0.026	0.018	0.015	0.013	0.278	0.233	0.211	0.205	0.079	0.094	0.101	0.109
CME	0.024	0.018	0.016	0.014	0.262	0.232	0.213	0.206	0.088	0.099	0.104	0.111
VS-III
n	20	50	100	200	20	50	100	200	20	50	100	200
AEs
MLE	0.256	0.265	0.271	0.238	1.035	1.018	1.026	1.020	0.710	0.698	0.710	0.618
MPSE	0.296	0.302	0.299	0.290	1.044	1.023	1.044	1.029	0.722	0.701	0.719	0.735
OLSE	0.273	0.276	0.273	0.256	1.039	1.016	1.038	1.026	0.809	0.787	0.811	0.628
WLSE	0.268	0.272	0.270	0.252	1.042	1.022	1.044	1.027	0.812	0.792	0.812	0.528
CME	0.271	0.273	0.272	0.245	1.040	1.024	1.040	1.046	0.817	0.801	0.818	0.731
MSE
MLE	0.010	0.014	0.012	0.010	0.201	0.228	0.202	0.102	0.054	0.045	0.040	0.008
MPSE	0.024	0.027	0.024	0.021	0.210	0.237	0.209	0.104	0.060	0.050	0.043	0.062
OLSE	0.016	0.018	0.016	0.014	0.215	0.245	0.216	0.107	0.100	0.093	0.051	0.019
WLSE	0.015	0.017	0.016	0.013	0.211	0.237	0.209	0.105	0.101	0.094	0.036	0.016
CME	0.016	0.018	0.016	0.012	0.213	0.234	0.213	0.106	0.104	0.099	0.065	0.010

TABLE III.

Descriptive statistics for the dataset I and dataset II.

Data sets	N	Min	Mean	Median	S.D.	Skewness	Kurtosis	First Q	Third Q	Max
I	63	0.55	1.51	1.59	0.32	−0.88	0.29	1.375	1.685	2.24
II	100	0.39	2.62	2.7	1.01	0.36	0.04	1.840	3.220	5.56

Data sets	N	Min	Mean	Median	S.D.	Skewness	Kurtosis	First Q	Third Q	Max
I	63	0.55	1.51	1.59	0.32	−0.88	0.29	1.375	1.685	2.24
II	100	0.39	2.62	2.7	1.01	0.36	0.04	1.840	3.220	5.56

FIG. 8.

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Plots of AEs and MSEs for some parameter values (VS–I).

FIG. 9.

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Plots of AEs and MSEs for some parameter values (VS-II).

C. Value at risk (VaR) and expected shortfall (ES) simulation analysis

Here, we use simulation analysis at various parametric values to evaluate the performance of both risk measures, namely, VaR and ES. Both VaR and ES are important tools for risk management and decision-making in financial institutions. They help investors and portfolio managers to set risk limits, evaluate the risk reward trade off, and assess the potential impact of adverse market conditions. Using N = 1000 and n = 500 from the OLGo model, a simulation analysis is created. At x = seq (0.01, 0.99, 0.01), the value of VR and ES are calculated together with 95% lower and upper confidence bounds. The scale and form characteristics are combined in various ways. I = [1.5, 2.5, 0.2], II = [0.5, 0.5, 0.5], and III = [1.5, 1, 1.2]. VR and ES are represented graphically, respectively, in Figs. 10–12 with lower and upper confidence limits of 95% for each.

FIG. 10.

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Plots of AEs and MSEs for some parameter values (VS-III).

FIG. 11.

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Plots of VaR and ES for some parameter values (I).

FIG. 12.

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Plots of VaR and ES for some parameter values (II).

VIII. APPLICATIONS ON REAL LIFE DATASETS

In this section, modeling of two datasets with different shapes is considered to illustrate the suitability of the proposed OLGo distribution. The first dataset is the strength of glass fibers of length 1.5 cm. These data have been analyzed by Smith and Naylor (1987).²⁵

0.55, 0.74, 0.77, 0.81, 0.84, 0.93,1.04, 1.11, 1.13, 1.24, 1.25, 1.27, 1.28, 1.29, 1.30, 1.36, 1.39, 1.42, 1.48, 1.48, 1.49, 1.49, 1.50,1.50, 1.51, 1.52, 1.53, 1.54, 1.55, 1.55, 1.58, 1.59, 1.60, 1.61, 1.61, 1.61, 1.61, 1.62, 1.62, 1.63,1.64, 1.66, 1.66, 1.66, 1.67, 1.68, 1.68, 1.69, 1.70, 1.70, 1.73, 1.76, 1.76, 1.77, 1.78, 1.81, 1.82,1.84, 1.84, 1.89, 2.00, 2.01, 2.24.

The breaking stress of carbon fibers (measured in Gba) is the second dataset. These data have also been analyzed by and Cordeiro and Lemonte (2011).

3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 2.05, 3.65.

To ascertain the unknown parameters of the OLGo distribution, we employed the Maximum Likelihood Estimation (MLE) approach. The selection of the optimal model hinges on various criteria, including the minimization of the following metrics: AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), CAIC (Consistent Akaike Information Criterion), HQIC (Hannan-Quinn Information Criterion), A* (Anderson–Darling criterion), W* (Cramér–von Mises criterion), and the Kolmogorov–Smirnov (KS) test, along with higher p-values. The model exhibiting the lowest values across these statistical measures is deemed the most suitable for fitting the data. For both the OLGo distribution and competing distributions, such as the Exponentiated Lomax distribution (ExpLx), generalized Gompertz distribution (GG), Poisson Lomax distribution (PLx),²⁶ and Lomax distribution (Lx), we have tabulated the MLE estimates and standard errors in Tables IV and VI. Additionally, Tables V and VII present the AIC, CAIC, BIC, HQIC, and other goodness-of-fit measures for the OLGo distribution and competing distributions (G, ExpLx, GG, PLx, and Lx) for datasets I and II, respectively.

TABLE IV.

MLEs, standard error (in parentheses) values for dataset I.

Models	$\hat{θ}$	$\hat{β}$	$\hat{b}$	$\hat{λ}$
OLGo(θ, β, b)	1.0738(0.5202)	86.6440(36.7292)	1.1553(0.1393)	⋯
G(b)	⋯	⋯	0.2163(0.0968)	⋯
ExpLx(α, θ, λ)	69.3808(42.3568)	33.1261(10.3749)	⋯	0.0392(0.0250)
GG(λ, θ, b)	1.7921(0.9040)	⋯	2.7819(0.7151)	0.1170(0.1443)
PLx(λ, b)	⋯	⋯	0.0160(0.0146)	68.6311(62.3911)
Lx(β)	⋯	1.0991(0.365)	⋯	⋯

Models	$\hat{θ}$	$\hat{β}$	$\hat{b}$	$\hat{λ}$
OLGo(θ, β, b)	1.0738(0.5202)	86.6440(36.7292)	1.1553(0.1393)	⋯
G(b)	⋯	⋯	0.2163(0.0968)	⋯
ExpLx(α, θ, λ)	69.3808(42.3568)	33.1261(10.3749)	⋯	0.0392(0.0250)
GG(λ, θ, b)	1.7921(0.9040)	⋯	2.7819(0.7151)	0.1170(0.1443)
PLx(λ, b)	⋯	⋯	0.0160(0.0146)	68.6311(62.3911)
Lx(β)	⋯	1.0991(0.365)	⋯	⋯

TABLE V.

Summary of goodness-of-fit criteria for comparative distributions of real dataset I.

Models	AIC	BIC	CAIC	HQIC	A*	W*	KS (p-value)
OLGo $(\hat{θ}, \hat{β}, \hat{b})$	30.4907	36.920 18	30.8975	33.0195	0.4868	0.0832	0.0981(0.5786)
G $(\hat{b})$	187.3017	189.4448	187.3673	188.1446	2.6893	0.4902	0.6159(10^–16)
ExpLx (⁠ $\hat{θ}, \hat{β}, \hat{λ}$ ⁠)	69.9505	76.3799	70.3573	72.4792	0.8019	4.3702	0.2289(0.0027)
GG (⁠ $\hat{λ}, \hat{θ}, \hat{b}$ ⁠)	91.6520	85.2220	91.2450	89.1230	0.1584	0.8937	0.1308(0.2308)
PLx (⁠ $\hat{λ}, \hat{b}$ ⁠)	233.6310	237.9180	233.8310	235.3170	3.8933	0.7122	0.4447(3.015 × 10⁻¹¹)
Lx (⁠ $\hat{β}$ ⁠)	230.7300	232.8730	230.7960	231.5730	3.8781	0.7093	0.4487(1.909 × 10⁻¹¹)

Models	AIC	BIC	CAIC	HQIC	A*	W*	KS (p-value)
OLGo $(\hat{θ}, \hat{β}, \hat{b})$	30.4907	36.920 18	30.8975	33.0195	0.4868	0.0832	0.0981(0.5786)
G $(\hat{b})$	187.3017	189.4448	187.3673	188.1446	2.6893	0.4902	0.6159(10^–16)
ExpLx (⁠ $\hat{θ}, \hat{β}, \hat{λ}$ ⁠)	69.9505	76.3799	70.3573	72.4792	0.8019	4.3702	0.2289(0.0027)
GG (⁠ $\hat{λ}, \hat{θ}, \hat{b}$ ⁠)	91.6520	85.2220	91.2450	89.1230	0.1584	0.8937	0.1308(0.2308)
PLx (⁠ $\hat{λ}, \hat{b}$ ⁠)	233.6310	237.9180	233.8310	235.3170	3.8933	0.7122	0.4447(3.015 × 10⁻¹¹)
Lx (⁠ $\hat{β}$ ⁠)	230.7300	232.8730	230.7960	231.5730	3.8781	0.7093	0.4487(1.909 × 10⁻¹¹)

TABLE VI.

MLEs, standard error (in parentheses) values for dataset II.

Models	$\hat{θ}$	$\hat{β}$	$\hat{b}$	$\hat{λ}$
OLGo (θ, β, b)	0.4535(0.3867)	15.0440(9.5683)	0.3140(0.1949)	⋯
G (b)	⋯	⋯	0.1923(0.0434)	⋯
ExpLx (α, θ, λ)	69.9260(26.0504)	8.1348(1.5964)	⋯	0.0150(0.0057)
GG (λ, θ, b)	162.3873(40.9572)	⋯	0.2114(0.0975)	1.8288(0.4732)
PLx (λ, b)	⋯	⋯	0.0122(0.0078)	65.7771(42.3444)
Lx (β)	⋯	0.8024(0.0802)	⋯	⋯

Models	$\hat{θ}$	$\hat{β}$	$\hat{b}$	$\hat{λ}$
OLGo (θ, β, b)	0.4535(0.3867)	15.0440(9.5683)	0.3140(0.1949)	⋯
G (b)	⋯	⋯	0.1923(0.0434)	⋯
ExpLx (α, θ, λ)	69.9260(26.0504)	8.1348(1.5964)	⋯	0.0150(0.0057)
GG (λ, θ, b)	162.3873(40.9572)	⋯	0.2114(0.0975)	1.8288(0.4732)
PLx (λ, b)	⋯	⋯	0.0122(0.0078)	65.7771(42.3444)
Lx (β)	⋯	0.8024(0.0802)	⋯	⋯

TABLE VII.

Summary of goodness-of-fit criteria for comparative distributions of real dataset II.

Models	AIC	BIC	CAIC	HQIC	A*	W*	KS (p-value)
OLGo $(\hat{θ}, \hat{β}, \hat{b})$	291.125	298.941	291.375	294.288	0.4473	0.0619	0.0541(0.9314)
G $(\hat{b})$	502.970	505.576	503.011	504.025	1.3292	0.2518	0.6120(2.2 × 10⁻¹⁶
ExpLx (⁠ $\hat{θ}, \hat{β}, \hat{λ}$ ⁠)	299.227	307.043	299.477	302.391	1.2486	0.2377	0.1088(0.1872)
GG (⁠ $\hat{λ}, \hat{θ}, \hat{b}$ ⁠)	389.199	397.0146	389.449	392.362	0.4520	0.0773	0.0700(0.711)
PLx (⁠ $\hat{λ}, \hat{b}$ ⁠)	498.711	503.921	498.835	500.82	1.7368	0.3214	0.4038(1.377 × 10¹⁴)
Lx (⁠ $\hat{β}$ ⁠)	495.264	497.87	495.305	496.319	1.7368	0.3214	0.4038(1.377 × 10¹⁴)

Models	AIC	BIC	CAIC	HQIC	A*	W*	KS (p-value)
OLGo $(\hat{θ}, \hat{β}, \hat{b})$	291.125	298.941	291.375	294.288	0.4473	0.0619	0.0541(0.9314)
G $(\hat{b})$	502.970	505.576	503.011	504.025	1.3292	0.2518	0.6120(2.2 × 10⁻¹⁶
ExpLx (⁠ $\hat{θ}, \hat{β}, \hat{λ}$ ⁠)	299.227	307.043	299.477	302.391	1.2486	0.2377	0.1088(0.1872)
GG (⁠ $\hat{λ}, \hat{θ}, \hat{b}$ ⁠)	389.199	397.0146	389.449	392.362	0.4520	0.0773	0.0700(0.711)
PLx (⁠ $\hat{λ}, \hat{b}$ ⁠)	498.711	503.921	498.835	500.82	1.7368	0.3214	0.4038(1.377 × 10¹⁴)
Lx (⁠ $\hat{β}$ ⁠)	495.264	497.87	495.305	496.319	1.7368	0.3214	0.4038(1.377 × 10¹⁴)

Figures 13–19 have been constructed to facilitate visual comparisons of all the fitted probability density functions (PDFs). These figures also depict the fitted cumulative distribution functions (CDFs) alongside the corresponding observed histograms. We have included various graphical representations, such as TTT, probability–probability (P–P) plots, Quantile–Quantile (Q–Q) plots, boxplots, ogives, and profile log-likelihood plots for both datasets I and II. To provide a comprehensive overview of the data, Table III presents descriptive statistics for both datasets I and II.

FIG. 13.

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Plots of VaR and ES for some parameter values (III).

FIG. 14.

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Plots: (a) TTT, (b) boxplot, (c) density, and (d) ogives of dataset I.

FIG. 15.

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Plots: (a) P–P, (b) Q–Q, (c) estimated density, and (d) estimated CDF of dataset I.

FIG. 16.

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Plots: (a) TTT, (b) boxplot, (c) density, and (d) ogives of dataset II.

FIG. 17.

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Plots: (a) P–P, (b) Q–Q, (c) estimated density, and (d) estimated CDF of dataset II.

FIG. 18.

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Profile log-likelihood plots of the OLGo model for dataset I.

We have chosen two datasets, both displaying concave-shaped TTT plots (refer to Figs. 13 and 15), indicating an increasing HRF for both sets. In Table III, the first dataset exhibits negative skewness and leptokurtosis, while the second dataset displays positive skewness and leptokurtosis.

FIG. 19.

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Profile log-likelihood plots of the OLGo model for dataset II.

Furthermore, we have presented model selection and goodness-of-fit statistics in Tables V and VII Our findings reveal that the OLGo distribution outperforms all competing models, as evidenced by its minimum values for AIC, BIC, CAIC, and HQIC, along with the lowest goodness-of-fit statistics (A*, W*, and KS with highest). These results collectively confirm that the suggested model is superior to the other models under consideration.

We have plotted the PDF fit and CDF fit (see Figs. 20, 21, and 23) plots of all models under study and found that the suggested model fits the data better than competing models. We have also displayed the PP-plots of all models under study in Figs. 22 and 24.

FIG. 20.

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PDF fit of all competing models for datasets I and II.

FIG. 21.

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Fitted CDF plot of all competing models for dataset I.

FIG. 22.

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PP-plot of all competing models for dataset I.

FIG. 23.

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Fitted CDF plot of all competing models for dataset II.

FIG. 24.

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PP-plot of all competing models for dataset II.

IX. CONCLUDING REMARKS

This study introduces a new three-parameter model called the odd Lomax–Gompertz distribution. The proposed model offers increased flexibility, allowing decreasing, increasing, and bathtub shaped failure rates. Various properties of the proposed model are explored and discussed, particularly in the context of reliability analysis. Notably, the new model exhibits strong ordering properties under certain parameter restrictions, specifically in terms of likelihood ratio order. The estimation of model parameters is conducted using the maximum likelihood of model parameters after verifying the asymptotic behavior of the maximum likelihood estimators through a simulation exercise. The simulation analysis of the two-risk metrics, namely, value at risk and expected shortfall, revealed the ability of the distribution to capture diverse failure rate patterns, which makes it particularly relevant for assessing financial risks. Additionally, the efficiency of the new distribution is demonstrated through two real-world applications and found that the proposed model can fit datasets under study very nicely. The empirical applications further highlighted the practical relevance and usefulness of the proposed model in real world scenarios.

ACKNOWLEDGMENTS

This study was funded by the Researchers Supporting Project number (RSPD2023R969), King Saud University, Riyadh, Saudi Arabia.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Authors have worked equally to write and review the manuscript.

Waleed Marzouk: Conceptualization (equal); Formal analysis (equal); Methodology (equal). Shakaiba Shafiq: Data curation (equal); Investigation (equal); Resources (equal). Sidra Naz: Software (equal); Supervision (equal); Validation (equal). Farrukh Jamal: Resources (equal); Validation (equal); Writing – review & editing (equal). Laxmi Prasad Sapkota: Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Nagy: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal). A. H. Mansi: Resources (equal); Software (equal). Eslam Hussam: Conceptualization (equal); Investigation (equal); Resources (equal). Ahmed M. Gemeay: Conceptualization (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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A new univariate continuous distribution with applications in reliability

I. INTRODUCTION

II. THE OLGo DISTRIBUTION

III. MAIN PROPERTIES

A. Density function expansion

B. Measures of position and random number generation

C. Ordinary moments

D. Probability weighted moments

IV. STOCHASTIC COMPARISONS

V. MEAN RESIDUAL LIFE (MRL) AND MEAN INACTIVITY TIME FUNCTIONS

A. Entropy

VI. ORDER STATISTICS, MOMENTS OF ORDER STATISTICS, AND L-MOMENTS

VII. STATISTICAL INFERENCE

A. Maximum likelihood estimators

B. Simulation analysis

C. Value at risk (VaR) and expected shortfall (ES) simulation analysis

VIII. APPLICATIONS ON REAL LIFE DATASETS

IX. CONCLUDING REMARKS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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A new univariate continuous distribution with applications in reliability

I. INTRODUCTION

II. THE OLGo DISTRIBUTION

III. MAIN PROPERTIES

A. Density function expansion

B. Measures of position and random number generation

C. Ordinary moments

D. Probability weighted moments

IV. STOCHASTIC COMPARISONS

V. MEAN RESIDUAL LIFE (MRL) AND MEAN INACTIVITY TIME FUNCTIONS

A. Entropy

VI. ORDER STATISTICS, MOMENTS OF ORDER STATISTICS, AND L-MOMENTS

VII. STATISTICAL INFERENCE

A. Maximum likelihood estimators

B. Simulation analysis

C. Value at risk (VaR) and expected shortfall (ES) simulation analysis

VIII. APPLICATIONS ON REAL LIFE DATASETS

IX. CONCLUDING REMARKS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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