Introduction

Energy is an essential aspect of every single human action and supportable enhancements of the nations. The financial development and social success of any nation typically rely upon the non-sustainable power sources of that nation. Enhanced use of energy and its applications will be the key problem of the real world (Valasai et al. 2017). Among all renewable energy sources, the most dominant, rapidly creating, and extensively utilized wellspring of sustainable power source is wind energy, which is perfect and practical for the ecosystem (Alavi et al. 2016). Wind energy is considered an environment-friendly source all over the world and an increase in its use has been most important in late years (Kantar et al. 2018). In contrast with petroleum derivatives, the breeze vitality has not unfriendly impact on the environment. In recent years, in both public and private sectors, the investment and research on wind resources are increased (Arslan et al. 2017). Two basic segments to use the wind energy successfully given as (i) the decision of the zone for setting up a wind energy farm and (ii) For better modeling execution, the determination of the best statistical model. In determining districts, wind energy potential is checked by modeling the wind speed information. In this way, wind speed data is a fundamental parameter for the advancement of wind energy.

The usage of probabilistic models for the classification of wind speed is common in literature. Numerous authors had recent work on the wind speed analysis with different composite and mixture models such as Abbas et al. (2012), Arslan et al. (2017), Jung et al. (2017), Muhammandi et al. (2017), Kantar et al. (2018), Dey et al. (2019), Khan et al. (2019), Haq et al. (2020a), Kaseem et al. (2020), and Haq et al. (2020b). According to the previous studies, Weibull distribution (WD) is considered the best probabilistic distribution for wind speed modeling (Soukissian 2013). However, in some situations, WD does not provide sufficient information about the modeling of wind speed (Akgul et al. 2016). In other words, WD has inadequate proficiency to deal with bimodal, multimodal, and highly skewed wind speed data. Therefore, it is need of the hour to develop new extensions of probabilistic models for the betterment of wind speed analysis.

Lindley (1958) had suggested a model in the context of the Bayes theorem and named it Lindley distribution (LD). Various applications of Lindley distribution were described by Ghitny et al. (2008). Inverse Lindley (IL) distribution was developed by Sharma et al. (2015) for survival analysis and stress-strength reliability analysis of cancer patients. The density function (pdf) and distribution function (cdf) of IL distribution was given as \(f\left( x \right) = \frac{{\beta^{2} }}{1 + \beta }\left( {\frac{1 + x}{{x^{3} }}} \right)e^{{\left( { - \frac{\beta }{x}} \right)}}\) and \(F\left( x \right) = \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)x}}} \right)e^{{\left( { - \frac{\beta }{x}} \right)}}\) respectively, where \(x > 0\) and \(\beta > 0\) is the only scale parameter. The generalization and applications of IL distribution were discussed by numerous authors especially in hydrology such as Alkarni (2015), Sharma et al. (2016), and Eltehiwy (2020). Here, Barco et al. (2017) proposed an Inverse Lindley (IL) distribution and its generalized form called generalized inverse Lindley (GIL) by using the power transformation \(y = z^{{ - \frac{1}{\varphi }}}\) with the pdf and the cdf is given as, respectively.

$$g\left( y \right) = \frac{{\varphi \beta^{2} }}{1 + \beta }\left( {\frac{{1 + y^{\varphi } }}{{y^{2\varphi + 1} }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} ,\quad y,\beta , \varphi > 0 ,$$
(1)
$$G\left( y \right) = \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} .$$
(2)

It is found that the GIL distribution is more skewed to right than that of WD. Let Y follows the GIL distribution and X follows the Weibull distribution, then the upper tail probabilities at similar parameter combinations for both models is computed at \(\varphi = 3.5\) and \(\beta = 1\). As \(P\left( {Y > 2.5} \right) = 0.02023\) and \(P\left( {X > 2.5} \right) = 1.86 \times 10^{ - 11}\), while \(P\left( {Y > 2.11} \right) = 0.03661\) and \(P\left( {X > 2.11} \right) = 1.18 \times 10^{ - 6}\). Therefore, it is justified now from the above probabilities at considered percentiles that the GIL distribution has longer tails than that of the Weibull distribution. Therefore, GIL distribution and its generalizations are useful for the analysis of wind speed and these right-tailed probabilities can help the researchers who work in this area.

The main aim of this paper is to develop a new extension of GIL distribution as a wind speed distribution. Therefore, “Development of new model” represented the development of the new model and studied its failure and survival curves. “Mathematical properties of MOGIL distribution” investigated some mathematical properties of the new model. “Wind speed data” had explored the wind speed data and some descriptive measures of data sets. “Results” provided the results and discussion about wind speed analysis and the next section concluded the study.

Methodology

Development of new model

Marshall and Olkin (1997) proposed a new family named as Marshall–Olkin G (MO–G) family of distribution with an additional shape parameter \(\psi > 0\). Let \(\overline{G\left( z \right)} = 1 - G\left( z \right)\) be an arbitrary baseline survival function for a continuous random variable \(X,\) then the MO-G family has survival function given as

$$\overline{{F\left( {z,\gamma } \right)}} = \frac{{\psi \overline{G\left( z \right)} }}{{1 - \overline{\left( \psi \right)L\left( z \right)} }},\quad - \infty < z < \infty ,$$
(3)

where \(\overline{\psi } = 1 - \psi\) is a tilt parameter. The cdf and the pdf of the MO-G family are given as

$$F\left( z \right) = \frac{G\left( z \right)}{{1 - \left( {1 - \psi } \right)\left( {1 - G\left( z \right)} \right)}} ,$$
(4)
$$f\left( z \right) = \frac{\psi g\left( z \right)}{{\left[ {1 - \left( {1 - \psi } \right)\left( {1 - G\left( z \right)} \right)} \right]^{2} }},$$
(5)

respectively. By substituting (2) in (4), the cdf of Marshall–Olkin generalized inverse Lindley (MOGIL) distribution is obtained as

$$F\left( y \right) = \frac{{\left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} }}{{1 - \left( {1 - \psi } \right)\left( {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right) e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)}}\quad {\text{for}} \,y \ge 0,\; \beta ,\psi ,\varphi > 0.$$
(6)

The corresponding pdf of Eq. (6) is defined as

$$f\left( y \right) = \frac{{\psi \left( {\frac{{\varphi \beta^{2} }}{1 + \beta }\left( {\frac{{1 + y^{\varphi } }}{{y^{2\varphi + 1} }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)}}{{\left[ {1 - \left( {1 - \psi } \right)\left\{ {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right\}} \right]^{2} }}.$$
(7)

The reliability of any system is checked by the survival function. For MOGIL, the survival rate function is

$$S\left( y \right) = \frac{{\psi \left( {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)}}{{1 - \left( {1 - \psi } \right)\left( {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right) e^{{ - \frac{\theta }{{y^{\varphi } }}}} } \right)}}.$$
(8)

Hazard rate or failure rate function has many demographics uses in many research areas such as actuarial sciences and reliability analysis. The hazard rate function of MOGIL distribution from Eq. (7) is defined as

$$H\left( y \right) = \frac{{\left( {\frac{{\varphi \beta^{2} }}{1 + \beta }\left( {\frac{{1 + x^{\varphi } }}{{y^{2\varphi + 1} }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)}}{{\left( {1 - \left( {1 - \psi } \right)\left( {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)*\left( {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right)} \right)}} .$$
(9)

The shapes of MOGIL distribution

The density curves of MOGIL distribution have unimodal behavior and the tails of the distribution curves become heavier by an increase in shape parameters as represented in Fig. 1.

Fig. 1
figure 1

pdf curves for MOGIL distribution

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Mathematical properties of MOGIL distribution

Some mathematical properties are derived in this section those are essential to explain a distribution and useful to explain real-life applications. The rth moment of the MOGIL distribution is formulated as

$$\begin{aligned} \mu_{r}^{\prime } & = \frac{\psi }{\varphi }\mathop \sum \limits_{{k,m,{\mathcal{T}} = 0}}^{\infty } \tau_{{k,m,{\mathcal{T}}}} \left( {\beta \left( {m + 1} \right)} \right)^{{\frac{{r - \varphi - {\mathcal{T}}}}{\varphi }}} \\ & \times \left[ {\left( {\beta \left( {m + 1} \right)} \right)^{\varphi } \Gamma \left( {\frac{{ - r + 2\varphi + {\mathcal{T}}}}{\varphi }} \right) + \Gamma \left( {\frac{{ - r + \varphi + {\mathcal{T}}}}{\varphi }} \right)} \right]\quad \varphi > r. \\ \end{aligned}$$
(10)

The moment generating function of the MOGIL distribution is given as

$$\begin{aligned} M_{y} \left( t \right) & = \frac{\psi }{\varphi }\mathop \sum \limits_{{k,m,{\mathcal{T}},q = 0}}^{\infty } \frac{{\left( t \right)^{q} }}{q!}\tau_{{k,m,{\mathcal{T}}}} \left( {\beta \left( {m + 1} \right)} \right)^{{\frac{{q - \varphi - {\mathcal{T}}}}{\varphi }}} \\ & \times \left[ {\left( {\beta \left( {m + 1} \right)} \right)^{\varphi } \Gamma \left( {\frac{{ - q + 2\varphi + {\mathcal{T}}}}{\varphi }} \right) + \Gamma \left( {\frac{{ - q + \varphi + {\mathcal{T}}}}{\varphi }} \right)} \right]. \\ \end{aligned}$$
(11)

The explicit expression for the incomplete moment of the MOGIL distribution is given as

$$\begin{aligned} M_{r} \left( {\mathcal{G}} \right) & = \frac{\psi }{\varphi }\mathop \sum \limits_{{k,m,{\mathcal{T}} = 0}}^{\infty } \tau_{{k,m,{\mathcal{T}}}} \left( {\beta \left( {m + 1} \right)} \right)^{{\frac{{r - \varphi - {\mathcal{T}}}}{\varphi }}} \\ & \times \left[ {\left( {{\upbeta }\left( {m + 1} \right)} \right)^{\varphi } {\Gamma }\left( {\frac{{ - {\text{r}} + 2\varphi + {\mathcal{T}}}}{\varphi }} \right),\frac{\beta }{{{\mathcal{G}}^{\varphi } }}\left( {m + 1} \right) + {\Gamma }\left( {\frac{{ - {\text{r}} + \varphi + {\mathcal{T}}}}{\varphi }} \right),\frac{\beta }{{{\mathcal{G}}^{\varphi } }}\left( {m + 1} \right)} \right]. \\ \end{aligned}$$
(12)

The stress-strength reliability of the MOGIL distribution is given as

$$\begin{aligned} R & = \frac{\psi }{\varphi }\mathop \sum \limits_{{k,m,{\mathcal{T}} = 0}}^{\infty } \tau_{{k,m,{\mathcal{T}}}} \left( {\beta \left( {m + 2} \right)} \right)^{{\frac{{ - \varphi - {\mathcal{T}} - 1}}{\varphi }}} \\ & \times \left[ {\left( {\beta \left( {m + 2} \right)} \right)^{\varphi } \Gamma \left( {\frac{{3\varphi + {\mathcal{T}} + 1}}{\varphi }} \right),\frac{\beta }{{z^{\varphi } }}\left( {m + 2} \right) + \Gamma \left( {\frac{{2\varphi + {\mathcal{T}} + 1}}{\varphi }} \right),\frac{\beta }{{z^{\varphi } }}\left( {m + 2} \right)} \right]. \\ \end{aligned}$$
(13)

The density of rth order statistics for the MOGIL distribution is:

$$\begin{aligned} \mu_{r}^{\prime } & = \frac{\rho \psi }{\varphi }\mathop \sum \limits_{{z,k,m,{\mathcal{T}} = 0}}^{\infty } \tau_{{k,m,{\mathcal{T}}}} \left( {\beta \left( {z + {\mathcal{T}} + 1} \right)} \right)^{{\frac{{r - 2\varphi - z\varphi - {\mathcal{T}}\varphi }}{\varphi }}} \\ & \times \left[ {\left( {\beta \left( {z + {\mathcal{T}} + 1} \right)} \right)^{\varphi } \Gamma \left( {\frac{{ - r + 3\varphi + z\varphi + {\mathcal{T}}\varphi }}{\varphi }} \right) + \Gamma \left( {\frac{ - r + 2\varphi + z\varphi + j\varphi }{\varphi }} \right)} \right]. \\ \end{aligned}$$
(14)

Let \(X_{1} ,X_{2} ,X_{3} , \ldots ,X_{n}\) is a random sample that follows MOGIL distribution, and then the log-likelihood function of the distribution is

$$\begin{aligned} n\left( L \right) & = n\ln \left( \psi \right) + n\ln \left( {\varphi \beta^{2} } \right) - n\ln \left( {1 + \beta } \right) + \,\mathop \sum \limits_{i = 1}^{n} \ln \left( {1 + y^{\varphi } } \right) - \left( {2\varphi + 1} \right)\mathop \sum \limits_{i = 1}^{n} \ln \left( y \right) \\ & - \mathop \sum \limits_{i = 1}^{n} \frac{\beta }{{y^{\varphi } }} - 2\mathop \sum \limits_{i = 1}^{n} \ln \left[ {1 - \left( {1 - \psi } \right)\left\{ {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right\}} \right] . \\ \end{aligned}$$
(15)

The components of the score vectors can be obtained by taking the partial derivatives of \(n\left(L\right)\) for \(\uppsi\), \(\varphi\) and \(\upbeta\), and equate them to zero. The observed likelihood equations are

$$\frac{d\ln \left( L \right)}{{d\psi }} = \frac{n}{\psi } - \frac{{2\mathop \sum \nolimits_{i = 1}^{n} \left( {1 - e^{{ - \frac{\beta }{{y^{\varphi } }}}} \left( {1 + \frac{{\beta y^{ - c} }}{1 + \beta }} \right)} \right) }}{{1 - \left( {1 - \psi } \right)\left\{ {1 - \left( {1 + \frac{\beta }{{\left( {1 + \beta } \right)y^{\varphi } }}} \right)e^{{ - \frac{\beta }{{y^{\varphi } }}}} } \right\}}},$$
(16)
$$\begin{aligned} \frac{d\ln \left( L \right)}{{d\varphi }} & = \frac{n}{\varphi } + \beta \mathop \sum \limits_{i = 1}^{n} x^{ - \varphi } \ln \left( y \right) + \mathop \sum \limits_{i = 1}^{n} \left( { - 2\ln \left( y \right) + \frac{{y^{\varphi } \ln \left( y \right)}}{{1 + y^{\varphi } }}} \right) \\ & + \frac{{2\left( {1 - \psi } \right)\mathop \sum \nolimits_{i = 1}^{n} \left( {\frac{{\beta e^{{ - \frac{\beta }{{x^{\varphi } }}}} x^{ - \varphi } \ln \left( x \right)}}{1 + \beta } - \beta e^{{ - \frac{\beta }{{y^{\varphi } }}}} y^{ - \varphi } \left( {1 + \frac{{\beta y^{ - \varphi } }}{1 + \beta }} \right)\ln \left( y \right)} \right)}}{{1 - \left( {1 - \psi } \right)\left( {1 - e^{{ - \frac{\beta }{{y^{\varphi } }}}} \left( {1 + \frac{{\beta y^{ - \varphi } }}{1 + \beta }} \right)} \right)}}, \\ \end{aligned}$$
(17)
$$\begin{aligned} \frac{d\ln \left( L \right)}{{d\beta }} & = \left( {\frac{2}{\beta } - \frac{1}{1 + \beta }} \right)n - \mathop \sum \limits_{i = 1}^{n} y^{ - \varphi } \\ & + \frac{{2\left( {1 - \psi } \right)\mathop \sum \nolimits_{i = 1}^{n} \left( { - e^{{ - \frac{\beta }{{y^{\varphi } }}}} \left( { - \frac{{\beta y^{ - \varphi } }}{{\left( {1 + \beta } \right)^{2} }} + \frac{{y^{ - \varphi } }}{1 + \beta }} \right) + e^{{ - \frac{\beta }{{y^{\varphi } }}}} y^{ - \varphi } \left( {1 + \frac{{\beta y^{ - \varphi } }}{1 + \beta }} \right)} \right)}}{{1 - \left( {1 - \psi } \right)\left( {1 - e^{{ - \frac{\beta }{{y^{\varphi } }}}} \left( {1 + \frac{{\beta y^{ - \varphi } }}{1 + \beta }} \right)} \right)}} . \\ \end{aligned}$$
(18)

Since the derived equations have complex structures, therefore, it is hard to get a closed-form of the estimated parameters. Therefore, we use these equations by iterative procedures such as the Newton–Raphson algorithm for the estimation of parameters during wind speed analysis.

Wind speed data

Pakistan is the second biggest nation in South Asia and it is the 33rd biggest nation on the planet. Geographically, Pakistan is separated into three unique regions, for example, the Northern Mountains, the Indus River plain, and Baluchistan level.

For this study, data of wind speed measured at 20 m height with the 10-min interval from four distinct stations (Bahawalpur. Gwadar, Peshawar and Quetta) from different geographical locations in Pakistan. The data is taken online from the world bank data catalog. The map of Pakistan shows the locations of selected distinct stations in Fig. 2. Station A (Bahawalpur) is situated between the Indus River plains and south of the Sutlej River. It has 214 m altitude and is situated at 29.3544° N and 71.6911° E. Station B (Gwadar) has a hot desert climate and it also has a high variation between summer and winter temperatures. It has 15 m altitude and situated at 25.2460° N and 62.2861° E. Station C (Peshawar) lies in the Northern Highlands region.

Fig. 2
figure 2

Considered wind stations of Pakistan

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Peshawar has the same behavior as that of the previous two stations and it has 331 m altitude and situated at 34.0151° N and 71.5249° E. In the last, Station D (Quetta) has diversity in its climate with respect to summer and winter temperatures. It has 1679 m altitude and situated at 30.1798° N and 66.9750° E.

The results of Table 1 showed the maximum and minimum wind speed for all stations in different seasons. Moreover, variance, skewness and kurtosis are also calculated for all seasons of all selected stations. As a matter of fact, that all the stations at all seasons have positively skewed behavior, therefore, a positively skewed distribution can significantly explore the aspects.

Table 1 Descriptive statistics for the wind speed data at four stations in Pakistan
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Table 2 enlightened the pdf and cdf of the existing wind speed models such as Weibull, inverse exponential, inverted Kumaraswamy, inverse Weibull, inverse Lindley, and generalized inverse Lindley distribution.

Table 2 The pdf and cdf of the corresponding distributions
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We have estimated the parameters by using MOGIL distribution and considered wind speed distributions using the maximum likelihood estimation method. We have fitted the observed data sets of wind speed season wise for four stations of Pakistan. The performance and enactment of the models are evaluated based on certain criteria such as Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and Kolmogorov–Smirnov test (KS) given in Table 3.

Table 3 The formulas of criteria for model evolution
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Results

Table 4 showed the estimated parameters, \(- 2\ln L\), AIC, BIC and KS statistic for the MOGIL distribution and the other well-known wind speed distributions for the different seasons at station A.

Table 4 ML estimates with the goodness of fit measures for Station A
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Since the MOGIL model has minimum values of \(- 2\ln L\), AIC, BIC, and KS values for all four seasons of Bahawalpur while IL and IW distributions have maximum values for these goodness-of-fit criteria. Therefore, the MOGIL is considered as the best-fitted model for station A. Figure 3 represented the fitted curves of the pdf and it is also could easily be seen that the MOGIL distribution has the best fit for the heavy-tailed data sets.

Fig. 3
figure 3

Graphs for fitted pdf for all seasons at Station A

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See Fig. 4, it could be seen from the histogram of the wind speed data at Gwadar station that the data is more skewed and have a long tail towards the right. Therefore, graphically it is feasible to fit MOGIL distribution.

Fig. 4
figure 4

Graphs for fitted pdf for all seasons at Station B

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Table 5 justified the betterment of the MOGIL distribution as compared to existing wind speed models for Gwadar station. Based on all goodness of fit criteria such as \(- 2\ln L\), AIC, BIC, and KS we could see that MOGIL distribution has minimum values. Moreover, it seems a good competitor of Weibull distribution to deal with heavy-tailed data. The goodness of fit measures shows that the new proposed model has the best fit for all seasons of Gwadar station.

Table 5 ML estimates with the goodness of fit measures for the Station B
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On the same lines, Table 6 showed the estimated parameters as well as the goodness of fit measures for MOGIL and other wind speed distributions for all seasons of Peshawar station. The minimum values for all goodness of fit measures demonstrated that the MOGIL distribution has the least values for all goodness of fit measures. Therefore, MOGIL distribution seems the best competitor of all existing wind speed distributions for heavy-tailed data sets at station C.

Table 6 ML estimates with the goodness of fit measures for the Station C
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We justified these measures graphically in Fig. 5 and provided evidence about the superiority of MOGIL distribution.

Fig. 5
figure 5

Graphs for fitted pdf for all seasons at Station C

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Table 7 shows that the developed model has a better fit as compared to considered inverse models for all seasons at Quetta station. Based on all goodness of fit criteria, MOGIL distribution seems a good competitor of Weibull and other wind speed distributions to deal with heavy-tailed data. It is justified from Fig. 6 that the new proposed model is best fitted than other considered inverted distributions for Station D.

Table 7 ML estimates with the goodness of fit measures for the Station D
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Fig. 6
figure 6

Graphs for fitted pdf for all seasons at Station D

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Conclusion

It is justified in the study that the well-known Weibull distribution is not always be used as a wind speed distribution to deal with heavy-tailed data sets. In this research, we propose MOGIL distribution as a new heavy-tailed distribution for the modeling of wind speed for all seasons at selected stations of Pakistan. However, some mathematical properties of new models are also evaluated theoretically for a better understanding of the model. We had estimated the parameters by using the ML approach and evaluated some goodness of fit criteria such as \(- 2\ln L\), AIC, BIC, and KS test. The MLE method is used for the estimation of parameters. We compare the performance of the given model with well-known wind speed models such as Weibull, inverse exponential, inverted Kumaraswamy, inverse Weibull, inverse Lindley, and generalized inverse Lindley distribution. It is justified that the MOGIL distribution has favorable performance for all four regions of Pakistan as compared to other heavy-tailed distributions. It could also be noted that the addition of one shape parameter can increase the flexibility of existing models and improve the potentiality.