Model specification and data interpretation of climate in Pakistan

Abstract

The issue of climate change has occurred very strongly during the last 2 decades on global scale in analysis of estimated inferences on the environment of vulnerable states. Pakistan is one of the states that has high vulnerability index to natural disaster. Currently, climate of Pakistan is growing warmer; and we are moving around different signals of change. Meanwhile, it is important to ascertain climate trend of temperature, rain, wind speed and humidity in Pakistan on sequential basis. Therefore, this study aims to ascertain the climate trend through model specification of Stretched distributions comprising Lindley Stretched Exponential distribution, Exponentiated Stretched Exponential distribution, Kumaraswamy Stretched Exponential distribution and Transmuted Stretched Exponential distribution. A sort of comparison among new proposed, classical and modern models is also put up to evaluate the relative quality of statistical models for Climate data through Maximized Log Likelihood, Akaike Information Criterion, Bayesian Information Criterion, Consistent Akaike Information Criterion, Hannan Quinn Information Criteria and Kolmogorov Smirnov test. Findings specify that Kumaraswamy Stretched Exponential distribution is the best fit for modeling mean minimum temperature (oC) data, Lindley Stretched Exponential distribution is the best fit for modeling total July’s rain (mm) data, Transmuted Stretched Exponential distribution is the best fit for modeling wind speed (knots) data, Exponentiated Stretched Exponential distribution is the best fit for modeling humidity (%) data according to model specification. Generally, the results demonstrate that and new proposed models are ascertained best models for specification of climate data in Pakistan. It is also observed that these models proved stretchy and flexible for modeling climate data in the state of Pakistan.

Introduction

On the whole, growing sea levels around the world, frequent and irregular precipitation distribution, glacial melting, flash flooding, stretched droughts, tropical cyclones, hurricanes, natural dust storms, and global warming are the results of changing climate directly or indirectly. Meanwhile, this article is an effort to ascertain climate trend of temperature, rain, wind speed and humidity in Pakistan on sequential basis.

According to Qamar (2017), the annual mean temperature in Pakistan has increased by roughly 0.5 °C in the last 50 years. The number of heat wave days per year has increased nearly fivefold in the last 30 years. Annual precipitation has historically shown high variability but has slightly increased in the last 50 years. Sea level along the Karachi coast has risen approximately 10 cm in the last century. By the end of this century, the annual mean temperature in Pakistan has expected to rise by 3–5 °C for a central global emissions scenario, while higher global emissions may yield a rise of 4–6 °C. Average annual rainfall has not expected to have a significant long-term trend, but expected to exhibit large inter-annual variability. Sea level has expected to rise by a further 60 cm by the end of the century and will most likely affect the low-lying coastal areas south of Karachi toward Keti Bander and the Indus River delta. Having kept in view the challenging state of affairs related to climate change, it is important to ascertain the climate trend in Pakistan.

In addition, innumerable studies based on various directions are shepherded for modeling climate data due to receiving attention from scientists on the basis of potential influence of climate variability and change in the recent years. Lau and Weng (1995) employed to report nonlinearity in climate records. IPCC (2001a, b) provided comprehensive review about potential impacts of climate change. Huang et al. (1998), Islam Molla et al. (2006) and Carmona and Poveda (2014) have analyzed temperature variability and underlying trend in surface by applying statistical approaches. Kunkel et al. (1999), performed analyses of historical station-level climate data by applying aggregation technique of area weighted averaging. Similarly Smith et al. (2013) implemented averaging of gridded cells and Caesar et al. (2006) executed weighting algorithms in the same context. IPCC (2007a, b) reported that he global average temperature has increased and continue to rise at rapid rate. Hartmann et al. (2013) globally discussed the changing climate. Deser et al. (2014) and Martel et al. (2018) indicated that climate is unique in each region and climate change is significantly varies in different locations. Shortle et al. (2015) considered national scale whereas Melillo et al. (2014) Wuebbles et al. (2017) studied states scales of regional climate change crucially. Kunkel et al. (2003) analyzed various climate variables in different studies. DeGaetano (2009) considered parameters of climate for modeling general extreme distribution. The World Meteorological Organization WMO (2011) recommended a period of 30 years to represent normal ad average values of measurements from climatology record. McKitrick and Vogelsang, (2014) presented comprehensive approaches to determine the magnitude of stochastic trends in climate record. Haleem and Abather (2016) suggested to assess the impact of climate change about air temperature as well as climate risk is assessed regionally in present and future basis.

Abhishek et al. (2019) used climate data to assess the performance of building develops under current and projected future climate are generated for Canadian cities. Hossein (2019) presented an outlook of big data in climate change. Lai and Dzombak (2019) used historical data to assess regional climate change across United States, etc. and continuous efforts of researchers are still going on the track to studying climate in different context.

Noticeably, there are numerous concerns and alarms in climate change faced by decision makers. In this regard, the motivation of this study is based on model specification and data interpretation of climate in Pakistan. We have considered Stretched exponential distribution as baseline distribution due to its potential in nature and economy especially earth temperature variations. In this connection, Stretched exponential distribution as baseline distribution has proved best choice for ascertainment of climate data.

Laherr`ere and Sornette (1998) proposed the stretched exponential family as a complement to frequently used power law distributions for interpretations of natural fat tail distributions. Stretched exponential distribution is special case of most famous Amoroso distribution. It is also known as complementary cumulative Weibull distribution in Statistics. In Mathematics, it is known as stretched exponential. That is why, in this study, we consider stretched exponential distribution as baseline distribution for purpose of converting into proposed Stretched distributions. A random variable \(X\) is said to have stretched exponential distribution, then its pdf with corresponding cdf are

$$f\left( {x;a,b} \right) = \frac{b}{\left| a \right|}\left( \frac{x}{a} \right)^{b - 1} e^{{ - \left( \frac{x}{a} \right)^{b} }} ; x > 0,a {\text{in }}{\mathbb{R}},b > 0$$
$$F\left( {x;a,b} \right) = 1 - e^{{ - \left( \frac{x}{a} \right)^{b} }} ,$$

where \(a\) and \(b\) are real-scale and shape parameters, respectively. If \(a\) is negative, then the distribution is inverse stretched exponential distribution, the parent of inverse Weibull distribution. If \(b\) inserts as zero and one in \(f\left( {x;a,b} \right)\), we recover power law distribution and exponential distribution, respectively. Indeed, Specification of baseline distribution towards Generalization is a classical and modern art of discrete probability distributions as well as continuous probability densities. Complementary, an extensive area to generalization of Statistical literature has been covered this requirement of time. Therefore, the major advantage of generalization of probability models is to provide new stretched as well as specified model for real-life datasets. Although a large number of probability models have generalized for modeling the real-life datasets. Furthermore, for adequate modeling and interpreting the real-life datasets, new families and class of distributions are presenting leading role to baseline distributions for developing better flexible and stretchy performance while modeling data.

There are a lot of methods that have proved supportive for introducing more flexible and stretched probability distributions as well as probability densities. These methods are well known for innovators, for instance, transformation, Exponentiated Transmuted, and Kumaraswamy distributions. Thus, in the main, we consider Stretched exponential distribution as baseline distribution. Besides, this study was designed to throw light on the “Model Specification and Data Interpretation of Stretched distributions” due to the performance of different generators based on mixing criteria and induction of parameters. Consequently, we introduce four new stretched distributions named as Lindley Stretched Exponential distribution, Exponentiated Stretched Exponential distribution, Transmuted Stretched Exponential distribution and Kumaraswamy Exponentiated Stretched Exponential distribution.

The Lindley distribution was introduced by Lindley (1958). It is used to analyze failure time data of a process or device, especially in applications modeling stress–strength reliability. The Lindley distribution is generalization of an Exponential family. It is corresponded as a mixture of Exponential and Gamma distributions. Moreover, on account of less literature of Lindley distribution, Researchers are enhancing its worth, for instance, Adamidis and Loukas (1998) introduced two-parameter distribution based on Lindley distribution; Ghitany et al. (2011) introduced two-parameter Lindley distribution for modeling waiting and survival times data; Zakerzadeh and Dolati (2009) introduced three–parameter generalization of the Lindley distribution; Mahmoudi and Zakerzadeh (2010) obtained an extended version of the compound Poisson distribution by compounding the Poisson distribution with the generalized Lindley distribution; Merovci and Elbatal (2014) used the quadratic rank transmutation map to generate a flexible family of probability distributions taking Lindley-geometric distribution as the base value distribution by introducing a new parameter that would offer more distributional flexibility; Bhati and Malik (2014) introduced a new distribution generated by Lindley random variable which offered a more flexible model for modeling lifetime data; Min Wang, (2013) introduced a new three-parameter lifetime distribution named as Exponentiated Lindley distribution; Shanker et al. (2013) introduced a two-parameter Lindley distribution for modeling waiting and survival times data; Shanker et al. (2013) introduced a two-parameter Quasi Lindley distribution; lbrahim et al. (2013) presented a new class of distributions called New Generalized Lindley Distribution; Mavis et al. (2015) proposed a new class of distribution called the beta-exponentiated power Lindley distribution; Al-Babtain et al. (2015) proposed a five parameter Lindley Distribution as a new generalization of the basic Lindley distribution; Morad Alizadeh et al. (2017) introduced a new three-parameter Odd Log-Logistic Power Lindley distribution; Oluyede and Yang (2015) proposed a new four-parameter class of generalized Lindley distribution called the Beta-Generalized Lindley distribution; Joshi and Jose (2018) introduced a new circular distribution to be called as wrapped Lindley distribution; Ekhosuehi and Opone (2018) introduced a new class of lifetime distribution called a three-parameter generalized Lindley distribution; Shanker et al. (2019) proposed a generalization of two-parameter Lindley distribution.

There are many methods of adding one or more parameters to a distribution function. In this connection, the most simple method is exponentiation of a cumulative distribution function \(G\left( x \right)\) by a positive real number, say, \(\alpha\). According to Essam and Ahsanullah (2015), the applications of the Exponentiated model have been generally used for modeling of extreme value data, Actuarial science, Economics, Forestry, life testing and reliability. Further progressively, exponentiation is gaining popularity among innovators, such as, Gupta et al. (1998) first proposed the generalization of standard exponential distribution. They exposed a new family of distributions named as exponentiated exponential distribution. Pal et al. (2006) studied the family of distributions termed as Exponentiated Weibull distribution; Masoom et al. (2007) studied a number of new Exponentiated distributions; Shawky and Bakoban (2012) considered the Order statistics from an exponentiated gamma distribution; Huang and Oluyede (2014) proposed a new family of distributions called exponentiated Kumaraswamy–Dagum distribution; Youssef et al. (2015) introduced an extension of the Exponentiated Exponential distribution; Ahmed et al. (2016) defined and studied a new generalization of the Weibull–Pareto distribution called the exponentiated Weibull–Pareto distribution.

The Kumaraswamy distribution was originally introduced for modeling hydrology data. Accumulatively, Kumaraswamy distribution has acquired prominent place in generalization of probability models, for example, Jalmar et al. (2010) proposed and studied a new five-parameter continuous distribution based on generalization of the Kumaraswamy and the beta distributions along with some other familiar distributions; Shams (2013) introduced and studied the Kum-GEP distribution for the first time; Bourguignon et al. (2013) suggested the Kumaraswamy Pareto distribution; Artur et al. (2013) proposed a generalization of the Kumaraswamy distribution referred to as the exponentiated Kumaraswamy distribution; Elbatal (2013) presented Kumaraswamy Exponentiated Pareto distribution; Jailson and Silva (2015) proposed a new continuous distribution entitled exponentiated Kumaraswamy-exponential; Javanshiri et al. (2015) offered exp-kumaraswamy distribution; Ahsan (2016) hosted a four parameter Kumaraswamy Exponentiated Inverse Rayleigh distribution; Gauss et al. (2016) described the new Kumaraswamy exponential-Weibull distribution; Mahmoud and Abdullah (2016) planned a new family of distributions called Kumaraswamy-generalized power Weibull distribution; Ibrahim and Abdul-Moniem (2017) obtained Kumaraswamy Power function distribution; Fathy (2017) introduced a new five-parameter lifetime distribution called the exponentiated Kumaraswamy-Weibull distribution; Mahmoud et al. (2018) suggested a new five-parameter model entitled the Kumaraswamy exponentiated Fréchet distribution; Ibrahim et al. (2018) studied Kumaraswamy Extension Exponential Distribution; El-Sayed and Mahmoud (2019) presented a new family of distributions called the Kumaraswamy type I half logistic.

Transmuted distribution is gained by adding a new parameter, say, λ, a real number;\(\left| {\uplambda } \right| \le 1,\) to existing distribution.The concept of the Transmuted distribution was first suggested by Shaw and Buckley (2007). As well, induction of transmuted parameter in baseline distribution is advance skill, because quadratic behavior of distributions is keenly observed by researchers belonging to different fields of life. For this purpose, Aryal and Tsokos (2009) suggested Transmuted extreme value distribution; Merovci (2013a) developed a Transmuted generalized Rayleigh distribution; Merovci (2013b) studied Transmuted Lindley distribution; Merovci (2013c) suggested Transmuted Rayleigh distribution; Aryal and Tsokos (2011) embedded Transmuted Weibull distribution; Ashour and Eltehiwy (2013) proposed Transmuted Lomax distribution; Khan and King (2013) introduced a Transmuted modified Weibull distribution; Aryal (2013) proposed Transmuted log-logistic distribution; Elbatal (2013) introduced Transmuted modified inverse Weibull distribution; Elbatal and Aryal (2013) proposed Transmuted additive Weibull distribution; Mahmoud and Mandouh (2013) introduced Transmuted Fréchet distribution; Merovci et al. (2014) proposed Transmuted generalized inverse Weibull distribution; Merovci and Puka (2014) suggested Transmuted Pareto distribution; Abdul-Moniem and Seham (2015) introduced Transmuted Gompertz distribution; Fatima and Roohi (2015) introduced Transmuted Exponentiated Pareto-I distribution.

Hence, having kept in view, the climate data of state of Pakistan have been ascertained by model specification of Lindley Stretched Exponential distribution (LSED), Exponentiated Stretched Exponential distribution (ESED), Kumaraswamy Stretched Exponential distribution (KwESED) and Transmuted Stretched Exponential distribution (TSED). Therefore, the particulars of these four innovative models have been as followed next.

Lindley stretched exponential distribution

The first innovative model is referred as Lindley Stretched Exponential Distribution (LSED). It is semi-infinite range probability density function and based on continuous, three parameter, univariate and unimodal.

Definition

If a random variable \(Y\) follows Lindley distribution, then the random variable \(X = aY^{1/b}\) follows LSED with a scale parameter \({\text{a }}\) and two shape parameters \({\text{b}}\) and \({\uptheta }\), i.e., X ~ LSED(\(x;{\text{a}},{\text{b}}\), \({\uptheta }\)).Then, we have the cdf and \({\text{pdf}}\), respectively, as:

$$G_{L} \left( {x;a,b,\theta } \right) = 1 - \frac{{e^{{ - \theta \left( \frac{x}{a} \right)^{b} }} }}{1 + \theta }\left[ {1 + \theta + \theta \left( \frac{x}{a} \right)^{b} } \right],$$
(1)
$$g_{L} \left( {x;a,b,\theta } \right) = \frac{{b\theta^{2} }}{{\left| a \right|\left( {1 + \theta } \right)}}\left( \frac{x}{a} \right)^{b - 1} \left[ {1 + \left( \frac{x}{a} \right)^{b} } \right]e^{{ - \theta \left( \frac{x}{a} \right)^{b} }} ;x > 0,a in {\mathbb{R}} , b > 0, \theta > 0.$$
(2)

Hence, the \({\text{LSED}}\left( {x;a,b,\theta } \right)\) contains a real-scale parameter \(a\) and two shape parameters \(b\) and \(\theta\). If \(a\) is negative, then the distribution is inverse Lindley Stretched Exponential distribution, the parent of inverse Weibull. The possible shapes of the pdf and corresponding cdf of LSED for different values of parameters \(a{\text{ in}}{\mathbb{R}}\), \(b > 0\) and \(\theta > 0\) have been illustrated in Fig. 1. Figure 1 shows the fluctuate performance on different values of \(a,b\) and \(\theta\), respectively. We have observed the following results on the basis of pdf plots. For increasing values of \(a,b {\text{and}} \theta\), the curve indicates its behavior exponentially decreases as well as the peakedness of the curve decreases representing shift towards right tail in \({\text{pdf}}\) plot of figure. Likewise, \({\text{pdf }}\) plot also indicates its behavior similar to positive skewed. The \({\text{cdf}}\) behavior of LSED has also displayed in \({\text{cdf}}\) plot of Fig. 1 with selected values of \(a,b\) and\(\theta\).

Fig. 1
figure1

pdf and cdf Plots of LSED

Exponentiated stretched exponential distribution

The second innovative model is known as Exponentiated Stretched Exponential Distribution (ESED). Like LSED, this model has semi-infinite range probability density function and based on continuous, three parameter, univariate and unimodal.

Definition

A random variable \(X\) is said to have Exponentiated distribution with probability density function \(\left( {{\text{pdf}}} \right)\), \(f\left( x \right)\), and distribution function \(\left( {{\text{cdf}}} \right)\), \(F\left( x \right)\), defined by.

$$F\left( x \right) = \left[ {G\left( x \right)} \right]^{\alpha } .$$
(3)

The pdf of X is given by

$$f\left( x \right) = \alpha \left[ {G\left( x \right)} \right]^{\alpha - 1} g\left( x \right) ; x \in {\mathbb{R}},\alpha > 0,$$
(4)

where \(g\left( x \right)\) is corresponding pdf to \(G\left( x \right)\) of any arbitrary baseline distribution, where \(\alpha\) is shape parameter. We consider Stretched Exponential Distribution as baseline distribution. Then, the resultant distribution is known as Exponentiated Stretched Exponential Distribution (ESED). Hence, \(g_{{\text{E}}} \left( {x;a,b,\alpha } \right)\) is the corresponding pdf to \(G_{{\text{E}}} \left( {x;a,b,\alpha } \right),\) cdf of ESED, which is given by

$$G_{{\text{E}}} \left( {x;a,b,\alpha } \right) = \left[ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right]^{\alpha } ,$$
(5)
$$g_{{\text{E}}} \left( {x;a,b,\alpha } \right) = \frac{b\alpha }{{\left| a \right|}}\left( \frac{x}{a} \right)^{b - 1} e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} \left[ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right]^{\alpha - 1} ;x > 0, a{\text{ in}}{\mathbb{R}},b > 0,\alpha > 0.$$
(6)

Thus, the three parameters of Exponentiated Stretched Exponential distribution comprehend a real-scale parameter \(a\) and two shape parameters \(b\) and \(\alpha\). If \(a\) is negative, then the distribution is inverse Exponentiated Stretched Exponential distribution, the parent of various inverse distributions. The possible shapes of the corresponding pdf to cdf plots of ESED for different values of parameters \(a in{\mathbb{R}}\), \(b > 0\) and \(\alpha > 0\) have been revealed in Fig. 2. These figures have been shown the oscillate situations with different values of \(a,b\) and \(\alpha\), respectively. We have perceived the following consequences on the basis of pdf plots; for increasing values of \(a\) and \(b\), the peak of the curve flatters displaying a positively skewed behavior as well as the peakedness of the curve increases in \(\mathrm{pdf}\) plot of Fig. 2.

Fig. 2
figure2

Pdf and cdf Plots of ESED

Kumaraswamy exponentiated stretched exponential distribution

The third innovative model is recognised as Kumaraswamy Exponentiated Stretched Exponential Distribution (KwESED). It is semi-infinite range probability density function. It is based on continuous, four parameter, univariate and unimodal.

Definition

A random variable \(X\) is said to have Kumaraswamy Exponentiated distribution with probability density function \(\left( {{\text{pdf}}} \right)\), \(f\left( x \right)\), and distribution function \(\left( {{\text{cdf}}} \right)\), \(F\left( x \right)\), defined by.

$$F\left( x \right) = 1 - \left[ {1 - G\left( x \right)^{c} } \right]^{d} ,$$
(7)

where \(c > 0\) and \(d > 0\) are additional shape parameters. Then, the resultant pdf of \(X\) is given by

$$f\left( x \right) = cdg\left( x \right)G\left( x \right)^{c - 1} \left\{ {1 - G\left( x \right)^{c} } \right\}^{d - 1} ;x > 0, c > 0,d > 0.$$
(8)

\(G\left( x \right)\) is corresponding cdf to pdf of any arbitrary baseline distribution \(g\left( x \right).\) Let \(X\) be random variable of Kumaraswamy Exponentiated Stretched Exponential Distribution (KwESED) with parameters \(a,b,c {\text{and}} d\), then pdf \(g_{{\text{k}}} \left( {x;a,b,c,d} \right)\) is the corresponding probability density function to \(G_{{\text{k}}} \left( {x;a,b,c,d} \right),\) cumulative distribution function of KwESED:

$$G_{{\text{k}}} \left( {x;a,b,c,d} \right) = 1 - \left[ {1 - \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}^{c} } \right]^{d} ,$$
(9)
$$g_{k} \left( {x;a,b,c,d} \right) = \frac{bcd}{{\left| a \right|}}\left( \frac{x}{a} \right)^{b - 1} e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}^{c - 1} [1 - \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}^{c} ]^{d - 1}$$
(10)
$$x > 0, a {\text{in}}{\mathbb{R}}, b > 0, c > 0,d > 0.$$

Hence, the four parameter Kumaraswamy Exponentiated Stretched Exponential distribution contains a real-scale parameter \(a\) and three shape parameters \(b, c\) and \(d\). If \(a\) is negative, then the distribution is inverse Kumaraswamy Exponentiated Stretched Exponential distribution, the parent of various inverse distributions. The possible shapes of the corresponding pdf to cdf of KwESED for different values of parameters \(a {\text{in}}{\mathbb{R}}\), \(b > 0, c > 0\) and \(d > 0\) have been established in Fig. 3. Figure has explained the possible shapes of the KwESED that showed the vary performance on different values of \(a,b,c\) and \(d\), respectively. We have observed the following remarks on the basis of pdf plots; for fixed values of \(c\) and \(d\). with increasing values of \(a\) and \(b,\) the curve presented exponential and positively ewed behavior in peakedness in pdf plot of Figure. Figure 3 has also presented possible shapes for cdf of KwESED with certain values of \(a,b,c\) and \(d\).

Fig. 3
figure3

pdf and cdf plots of KwESED

Transmuted stretched exponential distribution

The fourth innovative model is referred as Transmuted Stretched Exponential Distribution (TSED). It is three parameter, continuous, univariate and unimodal with semi-infinite range probability density function like LSED and ESED.

Definition

A random variable \(X\) is said to have Transmuted distribution with cumulative distribution function \(\left( {{\text{cdf}}} \right)\), \(F\left( x \right)\), corresponding to probability density function \(\left( {pdf} \right)\), \(f\left( x \right)\), respectively, defined by

$$F\left( x \right) = \left( {1 + {\uplambda }} \right)G\left( x \right) - {\lambda }\left[ {{\text{G}}\left( x \right)} \right]^{2} ,$$
(11)
$$f\left( x \right) = \left( {1 + \lambda } \right)g\left( x \right) - 2\lambda G\left( x \right)g\left( x \right) ;\;\; x > 0,\left| {\uplambda } \right| \le 1$$
(12)

where \(g\left( x \right)\) is the corresponding pdf to arbitrary cdf \(G\left( x \right).\)

Let \(X\) be random variable of Transmuted Stretched Exponential Distribution (TSED) with parameters \(a,b {{{\rm and} \lambda }}\), then \(G_{{\text{T}}} \left( {x;a,b,{\uplambda }} \right)\) is the corresponding cumulative distribution function to probability density function \(g_{{\text{T}}} \left( {x;a,b,{\uplambda }} \right)\) of TSED:

$$G_{{\text{T}}} \left( {x;a,b,\lambda } \right) = \left( {1 + \lambda } \right)\left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\} - \lambda \left[ {\left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}} \right]^{2} ,$$
(13)
$$g_{{\text{T}}} \left( {x;a,b,\lambda } \right) = \frac{b}{\left| a \right|}\left( \frac{x}{a} \right)^{b - 1} \left[ {\left( {1 + \lambda } \right) - 2\lambda \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}} \right]e^{{ - \left( \frac{x}{a} \right)^{b} }}$$
(14)
$$;x > 0, a {\text{in}}{\mathbb{R}},b > 0,\left| {\uplambda } \right| \le 1.$$

Hence, three parameter Transmuted Stretched Exponential distribution contains a real-scale parameter \(a\), a shape parameter \(b\) and a transmuted parameter \(\lambda\). If \(a\) is negative then the distribution is inverse Transmuted Stretched Exponential distribution, the parent of various inverse distributions. The probable shapes of the corresponding pdf to cdf of TSED for different values of parameters \(a in{\mathbb{R}}\), \(b >0\) and \(\lambda >0\) have been established in Fig. 4. Figure has explained the possible shapes of pdf and cdf plots for TSED that showed the vary performance on different values of \(a,b\) and \(\lambda\), respectively. We have observed the following results on the basis of pdf plots; for fixed \(b \mathrm{and}\lambda\) with increasing values of \(a\), the peak of the curve flatters presenting exponential behavior, whereas, for fixed \(b \mathrm{and}\lambda\) with increasing values of \(a,\) the curve has its showed gradual decreasing performance in pdf plot of figure. The cdf plot in Fig. 4 has been exhibited for TSED with selected values of \(a,b\) and \(\lambda\).

Fig. 4
figure4

pdf and cdf plots of TSED

Thus, according to the study requirement, a comparison among four models along with classical and modern models is also highlighted via evaluation criterion.

Evaluation criterion

A sort of comparison among new proposed, classical and modern models is also put up to evaluate the relative quality of statistical models for Climate data through measures of evaluation criterion including Maximized Log Likelihood \(\left( { - 2lnL} \right),\) Akaike Information Criterion \(\left( {{\text{AIC}}} \right)\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \(\left( {{\text{CAIC}}} \right)\), Hannan Quinn Information Criteria \(\left( {{\text{HQIC}}} \right)\) and Kolmogorov Smirnov (KS) test. The evaluation criterion is based on goodness of fit.

The measures to evaluate the relative quality of statistical models for different sets of data are the following:

  1. (a)

    Maximized Log likelihood \(\left( { - 2lnL} \right)\)

  2. (b)
    $${\text{AIC}} = 2p - 2lnL$$
  3. (c)
    $${\text{BIC}} = pln\left( n \right) - 2lnL$$
  4. (d)
    $${\text{CAIC}} = \frac{2pn}{{n - p - 1}} - 2lnL$$
  5. (e)
    $${\text{HQIC}} = 2pln\left( {ln\left( n \right)} \right) - 2lnL$$

where \(L\) is the maximized likelihood function. \(p\) is the number of parameters in the model. \(n\) is the total number of observations.

The smallest value of these measures indicates the best fit to the data. R software has been used for the analysis. The list of classical, modern and new proposed models is given by

  • Stretched Exponential distribution (SED)

    $$f_{{{\text{SE}}}} \left( {x;a,b} \right) = \frac{b}{\left| a \right|}\left( \frac{x}{a} \right)^{b - 1} e^{{ - \left( \frac{x}{a} \right)^{b} }} ; \;\;\;x > 0,a{\text{ in}}{\mathbb{R}},b > 0$$
  • Exponential distribution (ED)

    $$f_{{\text{E}}} \left( {x;\lambda } \right) = \lambda e^{ - \lambda x} ; \;\;x \ge 0,\lambda > 0$$
  • Lindley distribution (LD)

    $$f_{{\text{L}}} \left( {x;\theta } \right) = \frac{{\theta^{2} }}{1 + \theta }\left( {1 + x} \right)e^{ - \theta x} ; \;\;x > 0,\theta > 0$$
  • Weibull distribution (WD)

    $$g_{{\text{W}}} \left( {x;k,\theta } \right) = \frac{k}{\theta }\left( \frac{x}{k} \right)^{\theta - 1} e^{{ - \left( \frac{x}{k} \right)^{\theta } }} ;\;\;x \ge 0,k > 0,\theta > 0$$
  • A Three Parameter Lindley Distribution (ATPLD)

    $$g_{{{\text{ATPL}}}} \left( {x;\theta ,\alpha ,\beta } \right) = \frac{{\theta^{2} }}{\alpha \theta + \beta }\left( {\alpha + \beta x} \right)e^{ - \theta x}$$
    $$;x > \theta , \theta > 0 ,\alpha > 0,\beta > 0,\alpha \theta + \beta > 0$$
  • Transmuted Exponentiated Pareto-I Distribution (TEPID)

    $$g_{{{\text{TEPI}}}} \left( {x;a,k,\lambda } \right) = ak^{a} e^{ - ax} \left[ {1 - \lambda \left( {1 - 2k^{a} e^{ - ax} } \right)} \right]$$
    $$;x > lnk, a > 0,k > 0,\left| {\uplambda } \right| \le 1$$
  • Lindley Weibull Distribution (LWD)

    $$g_{{{\text{LW}}}} \left( {x;a,b,\theta } \right) = \frac{{a\theta^{2} }}{{b\left( {\theta + 1} \right)}}\left( \frac{x}{b} \right)^{a - 1} \left[ {1 + \left( \frac{x}{b} \right)^{a} } \right]e^{{ - \theta \left( \frac{x}{b} \right)^{a} }}$$
    $$;x \ge \theta , a > 0 ,b > 0,\theta > 0$$
  • Kumaraswamy Exponentiated Inverse Rayleigh Distribution (KEIRD)

    $$g_{{{\text{KEIR}}}} \left( {x;a,b,\alpha ,\theta } \right) = \frac{2ab\alpha \theta }{{x^{3} }}e^{{ - {\raise0.7ex\hbox{$\theta $} \!\mathord{\left/ {\vphantom {\theta {x^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${x^{2} }$}}}} \left[ {1 - \left( {e^{{ - {\raise0.7ex\hbox{$\theta $} \!\mathord{\left/ {\vphantom {\theta {x^{2} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${x^{2} }$}}}} } \right)^{a\alpha } } \right]^{b - 1}$$
    $$;x \ge 0, a > 0, b > 0, \alpha > 0,\theta > 0$$
  • Exponentiated Weibull Distribution (EWD)

    $$g_{{{\text{EW}}}} \left( {x;a,b,\theta } \right) = a\alpha \theta^{\alpha } x^{\alpha - 1} e^{{ - \left( {\theta x} \right)^{\alpha } }} \left[ {1 - e^{{ - \left( {\theta x} \right)^{\alpha } }} } \right]^{a - 1}$$
    $$;x > 0, a > 0,b > 0,\theta > 0$$
  • Lindley Stretched Exponential distribution (LSED)

    $$g_{{\text{L}}} \left( {x;a,b,\theta } \right) = \frac{{b\theta^{2} }}{{\left| a \right|\left( {1 + \theta } \right)}}\left( \frac{x}{a} \right)^{b - 1} \left[ {1 + \left( \frac{x}{a} \right)^{b} } \right]e^{{ - \theta \left( \frac{x}{a} \right)^{b} }}$$
    $$;x \ge \theta , a {\text{in}} {\mathbb{R}},b > 0,\theta > 0$$
  • Exponentiated Stretched Exponential distribution (ESED)

    $$g_{{\text{E}}} \left( {x;a,b,\alpha } \right) = \frac{b\alpha }{{\left| a \right|}}\left( \frac{x}{a} \right)^{b - 1} e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} \left[ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right]^{\alpha - 1}$$
    $$;x > 0, a {\text{in}}{\mathbb{R}},b > 0,\alpha > 0$$
  • Transmuted Stretched Exponential distribution (TSED)

    $$g_{{\text{T}}} \left( {x;a,b,\lambda } \right) = \frac{b}{\left| a \right|}\left( \frac{x}{a} \right)^{b - 1} \left[ {\left( {1 + \lambda } \right) - 2\lambda \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}} \right]e^{{ - \left( \frac{x}{a} \right)^{b} }}$$
    $$;x > 0, a {\text{in }}{\mathbb{R}},b > 0,\left| {\uplambda } \right| \le 1$$
  • Kumaraswamy Exponentiated Stretched Exponential distribution (KwESED)

    $$g_{{\text{K}}} \left( {x;a,b,c,d} \right) = ~\frac{{bcd}}{{\left| a \right|}}\left( {\frac{x}{a}} \right)^{{b - 1}} e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}^{{c - 1}} \left[ {1 - \left\{ {1 - e^{{ - \left( {{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right)^{b} }} } \right\}^{c} } \right]^{{d - 1}}$$
    $$;x > 0, a {\text{in}}{\mathbb{R}}, b > 0, c > 0,d > 0$$
  • Lomax Distribution (LxD)

    $$g_{{{\text{Lx}}}} \left( {x; \alpha ,\beta } \right) = \alpha \beta^{\alpha } \left( {\beta + x} \right)^{\alpha + 1}$$
    $$;x > 0, \alpha > 0, \beta > 0$$
  • Power Lomax Distribution (PLxD)

    $$g_{{{\text{PLx}}}} \left( x \right) = \frac{{\alpha \beta \lambda^{\alpha } x^{\beta - 1} }}{{\left( {\lambda + x^{\beta } } \right)^{\alpha + 1} }}$$
    $$;x > 0, \alpha > 0, \beta > 0, \lambda > 0$$

Now, the applicability of all considered models has been analyzed through climate data.

Analysis of climate data

The crucial causes of climate change in Pakistan due to global warming, emission of greenhouse gases, burning of fossil fuels, deforestation, increasing population, increasing livestock farming, excessive use of fertilizers, and usage of aerosol sprays have been observed on the behalf of model specification of Stretched distributions in this study. The climate data of observed cities in Pakistan containing four parameters were selected. These parameters are Temperature (Minimum/Maximum), Rainfall, Wind speed and Humidity. Consequently, the information of these four climate parameters in brief with analysis are as follows:

Temperature

Temperature quantitatively expresses hot and cold physical characteristic of matter or surface. It is the indicator of heat present in all matter. Most common scales of temperature are used, e.g., kelvin (K), Celsius (C) “OR” Centigrade and Fahrenheit (F). Besides, Fourth scale of temperature is Rankine (R) that is less used. Hence, the data of temperature are analyzed in the zone of monthly mean minimum temperature (oC) of 26 cities of Pakistan from 2007 to 2014. Data Source is MET Office. The sample of monthly mean minimum Temperature (oC) data of Cherat, Pakistan is provided in Table 1.

Table 1 Data of monthly mean minimum temperature (oC) of Cherat, Pakistan (2007–2014)

The descriptive statistics with Box plots for monthly mean minimum temperature (oC) data of 26 observed cities of Pakistan are reported in Table 2.

Table 2 Descriptive statistics for monthly mean minimum temperature (oC) data of Pakistan’s cities

Box plots in Fig. 5 display that monthly mean minimum temperature (oC) data of all cities except Balakot and Pattan which is skewed right. Descriptive statistics monthly mean minimum temperature (oC) data of all cities also confirm the right skewness because median is almost greater than mean. Dot plots represent lengthy diagonals as well as mostly continuous that ascertain the sequences are connected along their entire length.

The ML estimates for all classical, modern and proposed models are reported in Table 3. According to Tables 17, 18, 19, 20 in “Appendix A1”, LSED, ESED, TSED and KwESED fit best to the monthly mean minimum temperature (oC) data for Cherat with minimum measures of evaluation criteria among other Pakistan’s cities. Therefore, we consider data of monthly mean minimum temperature (oC) for Cherat, Pakistan only for purpose of comparison among models.

Fig. 5
figure5figure5

Box and Dot plots for monthly mean minimum temperature (oC) data of Pakistan’s cities

Table 3 specifies the ML estimates of considered distributions for monthly mean minimum temperature (oC) data of Cherat, Pakistan. Table 4 indicates that KwESED provides best competitor to other distributions used for fitting monthly mean minimum temperature (oC) data of Cherat, Pakistan. Hence, KwESED is the best fit according to model specification due to its minimum measures of Maximized Log Likelihood \(\left(-2lnL\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), and Hannan Quinn Information Criteria \(\left(\mathrm{HQIC}\right).\)

Table 3 ML estimates for temperature (oC) data of Cherat, Pakistan
Table 4 − 2lnL, AIC, CAIC, BIC and HQIC and KS statistic for temperature (oC) data of Cherat, Pakistan

Figure 6 indicates that the estimated densities plots, empirical cumulative distribution function (ECDF) plots and TTT plots for LSED, ESED, TSED and KwESED on temperature (oC) data, respectively. Since TTT plots are concave shaped, thus the shape of hazard rate is increasing for LSED, ESED, TSED and KwESED on temperature (oC) data, respectively.

Fig. 6
figure6

Estimated density plots of LSED, ESED, TSED and KwESED for monthly mean minimum temperature (oC) data of Cherat, Pakistan

Figure 7 shows Probability–Probability plots for all considered distributions. Figure indicates that LSED, ESED, TSED and KwESED are best fit for monthly mean minimum temperature (oC) data of Cherat, Pakistan.

Fig. 7
figure7

PP plots on temperature (oC) data of Cherat, Pakistan

Rain

Rain is a kind of precipitation that reduces from atmospheric water vapor and then results heavy to fall under gravity. Rain is a major component of the water cycle and is responsible for depositing most of the fresh water on the Earth. It provides suitable conditions for many types of ecosystem, as well as water for hydroelectric power plants and crop irrigation. It is measured in millimeters (mm).Therefore, the data of rain are analyzed for July’s total rain (mm) of 19 cities of Pakistan from 1985 to 2014. Data Source is MET Office. The sample of July’s total rain (mm) data of Gilgit, Pakistan is provided in Table 5.

Table 5 Data of July’s total rain (mm) of Gilgit, Pakistan (1985–2014)

The descriptive statistics with Box plots for July’s total rain (mm) data of 19 cities of Pakistan are summarized in Table 6.

Table 6 Descriptive statistics for July’s total rain (mm) data of Pakistan’s cities

Box plots in Fig. 8 display that July’s total rain (mm) data of observed cities skewed left for majority of cities except Lahore and Muzaffarabad. Descriptive statistics July’s total rain (mm) data of cities also confirm the left skewness because values of median are lower than values of mean. Dot plots represent lengthy diagonals with decreasing pattern as well as mostly continuous that ascertain the sequences are connected along their entire length.

According to Tables 21, 22, 23 and 24 in “Appendix A2”, LSED, ESED TSED and KwESED fit best to July’s total rain (mm) data for Gilgit with minimum measures of evaluation criteria among other Pakistan’s cities. Therefore, we consider data of July’s total Rain (mm) for Gilgit, Pakistan only for purpose of comparison with other classical and modern models.

Fig. 8
figure8figure8

Box and Dot plots for July’s total rain (mm) data of Pakistan’s cities

Table 7 specifies the ML estimates of considered distributions for July’s total rain (mm) data of Gilgit, Pakistan. Table 8 indicates that LSED gives best competitor among all considered models used for fitting July’s total rain (mm) data of Gilgit, Pakistan. Hence, LSED is the best fit according to model specification due to its minimum measures of Maximized Log Likelihood \(\left(-2lnL\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), and Hannan Quinn Information Criteria \(\left(\mathrm{HQIC}\right).\)

Table 7 ML estimates for July’s total rain (mm) data
Table 8 − 2lnL, AIC, CAIC, BIC and HQIC and KS statistic for July’s total rain (mm) data

Figure 9 shows that the estimated densities plots, empirical cumulative distribution function (ECDF) plots and TTT plots for LSED, ESED, TSED and KwESED on rain (mm) data, respectively. Since TTT plots are concave shaped, thus the shape of hazard rate is increasing for LSED, ESED, TSED and KwESED on Rain (mm) data, respectively.

Fig. 9
figure9

Estimated density, ECDF and TTT plots of LSED for July’s total rain (mm) data of Gilgit, Pakistan

Figure 10 displays Probability–Probability plots for all considered distributions. Figure specifies that LSED, ESED, TSED and KwESED are good fit for July’s total rain (mm) data of Gilgit, Pakistan.

Fig. 10
figure10

PP plots on  July’s total rain (mm) data

Wind speed

Wind speed is a fundamental atmospheric quantity caused by air moving from maximum to low pressure, usually due to changes in temperature. It is noticeable that wind direction is usually almost parallel to isobars due to rotation of Earth. Wind speed affects weather forecasting, aviation and maritime operations, construction projects, growth and metabolism rate of many plant species, and has countless other implications. Wind speed is now commonly measured with an anemometer. The most common unit of wind speed is meters per second (m/s), the SI unit for velocity, kilometers per hours (km/h). Some units are used for historical reasons such as miles per hour (mph), knots (kn) or feet per seconds (ft/s). Therefore, the data of wind speed are analyzed for wind speed (knots) of 13 cities of Pakistan from 2010 to 2014. Data Source is MET Office. The sample of wind speed (knots) data of Gilgit, Pakistan is provided in Table 9.

Table 9 Data of wind speed (knots) of Gilgit, Pakistan (2010–2014)

The descriptive statistics with Box plots for wind speed (knots) data of 13 cities of Pakistan are summarized in Table 10.

Table 10 Descriptive statistics for wind speed (knots) data of Pakistan’s cities

Box plots in Fig. 11 display that wind speed (knots) data of observed cities is skewed left for Astore, Balakot, Gupis, Kotli, Lahore, Mandi Bahu Din and Sialkot. Descriptive statistics of wind speed (knots) data of Astore, Balakot, Gupis, Kotli, Lahore, Mandi Bahu Din and Sialkot also confirm the left skewness because values of median are lower than values of mean. Dot plots represent lengthy diagonals as well as mostly continuous that ascertain the sequences are connected along their entire length. On the other hand, box plots show that wind speed (knots) data of observed cities is skewed right for Bahawalpur, Chilas, Chitral, Drosh, Faisalabad and Gilgit. Descriptive statistics of wind speed (knots) data of Bahawalpur, Chilas, Chitral, Drosh, Faisalabad and Gilgit also confirm the right skewness because values of median are greater than values of mean. Dot plots denote long diagonals as well as frequently continuous that find out the sequences are linked along with full length.

According to Tables 25, 26, 27 and 28 in “Appendix A3”, LSED, ESED, TSED and KwESED fit best to Wind speed (knots) data for Gilgit with minimum measures of goodness of fit among other Pakistan’s cities. Therefore, we consider data of wind speed (knots) for Gilgit, Pakistan only for purpose of comparison with other existing models.

Fig. 11
figure11

Box and Dot plots for wind speed (knots) data of Pakistan’s cities

Table 11 specifies the ML estimates of considered distributions for wind speed (knots) data of Gilgit, Pakistan. Table 12 indicates that TSED provides best competitor to other models used for fitting wind speed (knots) data of Gilgit, Pakistan. Hence, TSED is the best fit according to model specification due to its minimum measures of Maximized Log Likelihood \(\left(-2\mathrm{lnL}\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), and Hannan Quinn Information Criteria \(\left(\mathrm{HQIC}\right).\)

Table 11 ML estimates for wind speed (knots) data
Table 12 − 2lnL, AIC, CAIC, BIC and HQIC and KS statistic for wind speed (knots) data

Figure 12 displays that the estimated densities plots, empirical cumulative distribution function (ECDF) plots and TTT plots for LSED, ESED, TSED and KwESED on wind speed (knots) data, respectively. Since TTT plots are concave shaped, thus the shape of hazard rate is increasing for LSED, ESED, TSED and KwESED on wind speed (knots) data, respectively.

Fig. 12
figure12

Estimated density, ECDF and TTT plots of LSED, ESED, TSED and KwESED  for wind speed (knots) data of Gilgit, Pakistan

Figure 13 shows Probability–Probability plots for all considered distributions. Figure indicates that LSED, ESED, TSED and KwESED are best fit for average wind speed (knots) data of Gilgit, Pakistan.

Fig. 13
figure13

PP plots on wind speed (knots) data

Humidity

Humidity refers to the ratio of the amount of moisture in the air at a certain temperature to the maximum amount of moisture that the air can retain at the same temperature. In other words, humidity measures how much of the moisture capacity of the air is used. Humidity is recorded as a percentage and the highest during rain, usually reaching 100%. Accordingly, the data of humidity are analyzed for humidity (%) of 18 cities of Pakistan from 2004 to 2014. Data Source is MET Office. The sample of humidity (%) data of Muzaffarabad, Pakistan is provided in Table 13.

Table 13 Data of humidity (%) of Muzaffarabad, Pakistan (2004–2014)

The descriptive statistics with Box plots for humidity (%) data of 18 cities of Pakistan are summarized in Table 14.

Table 14 Descriptive statistics for humidity (%) data of Pakistan’s cities

Box plots in Fig. 14 display that humidity (%) data of observed cities is skewed right. Descriptive statistics of humidity (%) data also confirm the right skewness because values of median are lower than values of mean. Dot plots denote long diagonals with increasing pattern as well as frequently continuous that find out the sequences are associated along with full length.

Fig. 14
figure14figure14

Box and Dot plots for humidity (%) data of observed Pakistan’s cities

According to Tables 29, 30, 31 and 32 in “Appendix A4”, LSED, ESED, TSED and KwESED fit best to humidity (%) data for Muzaffarabad with minimum measures of goodness of fit among other Pakistan’s cities. Therefore, we consider data of humidity (%) for Muzaffarabad, Pakistan only for purpose of comparison among models.

Table 15 ML estimates of models for humidity (%) data
Table 16 − 2lnL, AIC, CAIC, BIC and HQIC and KS statistic for humidity (%) data

Table 15 specifies the ML estimates of considered distributions for humidity (%) data of Muzaffarabad, Pakistan. Table 16 indicates that ESED provide best competitor to other distributions used for fitting humidity (%) data of Muzaffarabad, Pakistan. Hence, ESED is the best fit according to model specification due to its minimum measures of Maximized Log Likelihood \(\left(-2lnL\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), and Hannan Quinn Information Criteria \(\left(\mathrm{HQIC}\right).\)

Figure 15 displays that the estimated densities plots, empirical cumulative distribution function (ECDF) plots and TTT plots for LSED, ESED, TSED and KwESED on Humidity (%) data, respectively. Since TTT plots are concave shaped, thus the shape of hazard rate is increasing for LSED, ESED, TSED and KwESED on humidity (%) data, respectively.

Fig. 15
figure15

Estimated density, ECDF and TTT plots for humidity (%) data of Muzaffarabad, Pakistan

Figure 16 shows Probability–Probability plots for all considered distributions. Figure indicates that LSED, ESED, KwESED and TSED are best fit for humidity (%) data of Muzaffarabad, Pakistan.

Fig. 16
figure16

PP plots on humidity (%) data

Conclusion

Climate trend in Pakistan has based on model specification of new proposed models through evaluation criteria. In this study, we proposed four new models named as Lindley Stretched Exponential Distribution, Exponentiated Stretched Exponential Distribution, Kumaraswamy Exponentiated Stretched Exponential Distribution and Transmuted Stretched Exponential Distribution. The estimation of parameters of new proposed models is estimated using climate data of observed cities of Pakistan based on mean minimum temperature (oC), total July’s rain (mm), wind speed (knots) and humidity (%) data. R is the best software to gain the estimates of parameters with Maximized Log Likelihood \(\left(-2lnL\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), Hannan Quinn Information Criteria \((\mathrm{HQIC})\) and Kolmogorov Smirnov (KS) test. It is observed that new models LSED, ESED, KwESED and TSED fit best for city of Cherat related to temperature data. Similarly, LSED, ESED, KwESED and TSED fit best for city of Gilgit related to rain and wind speed data. In addition, LSED, ESED, KwESED and TSED fit best for city of Muzaffarabad related to humidity data. Hence, we used climate data of Cherat, Gilgit and Muzaffarabad only for the purpose of comparison as well as evaluation of new models. Finally, it resulted that LSED, ESED, KwESED and TSED are ascertained best models for specification of climate data in Pakistan. For this reason, KwESED is the best fit for modeling mean minimum temperature (oC) data, LSED is the best fit for modeling total July’s rain (mm)data, TSED is the best fit for modeling wind speed (knots) data, ESED is the best fit for modeling humidity (%) data according to model specification due to its minimum measures of evaluation criteria of goodness of fit, i.e., Maximized Log Likelihood \(\left(-2\mathrm{lnL}\right),\) Akaike Information Criterion \((\mathrm{AIC})\), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion \((\mathrm{CAIC})\), and Hannan Quinn Information Criteria \(\left(\mathrm{HQIC}\right)\) among classical and modern models.

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Appendices

Appendix

Appendix A1

Temperature (oC) data.

See Table 17, 18, 19, 20

Table 17 ML estimates of LSED for monthly mean minimum temperature (oC) data of Pakistan’s cities with measures of goodness of fit
Table 18 ML estimates of ESED for monthly mean minimum temperature (oC) data of Pakistan’s cities with measures of goodness of fit
Table 19 ML estimates of TSED for monthly mean minimum temperature (oC) data of Pakistan’s cities with measures of goodness of fit
Table 20 ML estimates of KwESED for monthly mean minimum temperature (oC) data of Pakistan’s cities with measures of goodness of fit

Appendix A2

Rain (mm) data.

See Tables 21, 22, 23, 24

Table 21 ML estimates of LSED for July’s total rain (mm) data of Pakistan’s cities with measures of goodness of fit
Table 22 MLE estimates of ESED for July’s total rain (mm) data of Pakistan’s cities with measures of goodness of fit
Table 23 ML estimates of TSED for July’s total rain (mm) data of Pakistan’s cities with measures of goodness of fit
Table 24 ML estimates of KwESED for July’s total rain (mm) data of Pakistan’s cities with measures of goodness of fit

Appendix A3

Wind speed (knots) data.

See Tables 25, 26, 27, 28

Table 25 ML estimates of LSED for average wind speed (knots) data of Pakistan’s cities with measures of goodness of fit
Table 26 ML estimates of ESED for wind speed (knots) data of Pakistan’s cities with measures of goodness of fit
Table 27 ML estimates of TSED for wind speed (knots) data of Pakistan’s cities with measures of goodness of fit
Table 28 ML estimates of KwESED for wind speed (knots) data of Pakistan’s cities with measures of goodness of fit

Appendix A4

Humidity (%) data.

See Table 29, 30, 31, 32

Table 29 ML estimates of LSED for humidity (%) data of Pakistan’s cities with measures of goodness of fit
Table 30 ML estimates of ESED for humidity (%) data of Pakistan’s cities with measures of goodness of fit
Table 31 ML estimates of TSED for humidity (%) data of Pakistan’s cities with measures of goodness of fit
Table 32 ML estimates of KwESED for humidity (%) data of Pakistan’s cities with measures of goodness of fit

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Majid, G.A., Akhter, A.S. Model specification and data interpretation of climate in Pakistan. Model. Earth Syst. Environ. (2021). https://doi.org/10.1007/s40808-020-01072-6

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Keywords

  • Climate data
  • Lindley distribution
  • Exponentiated distribution
  • Kumaraswamy distribution
  • Transmuted distribution
  • Maximum likelihood estimation
  • Evaluation measures criteria