Order statistics and record values moments from the Topp-Leone Lomax distribution with applications to entropy

In this paper, we derive exact expressions, as well as recurrence relations, for single and product moments of order statistics (OSs) and record values of the Topp-Leone Lomax (TLLo) distribution proposed by Oguntunde et al. [29] (Wireless Personal Comm., 109, 349-360, 2019). Moreover, we study the L-moments of this distribution. Also, we estimate the parameters of the TLLo distribution through the L-moments and compare to other models. We ﬁt ﬂood level data to the TLLo distribution because Okorie and Nadarajah [30] (Wireless Personal Comm., 115, 589-596, 2020) have shown that the TLLo distribution may be used to model datasets other than airborne communication transceiver data used by Oguntunde et al. [29]. Finally, we reveal a relationship between the entropy and the upper and lower record values, as well as OSs .


Introduction
Recently, the data sets resulted from the different applications of the recent sciences are not simple to fit the classical distributions.In order to overcome this problem many authors obtained more flexible and capable distributions by adding further shape parameters to these classical distributions or mixing them.The TLLo distribution is one of these flexible and capable generalized distributions, it is a sub model of Topp-Leone family distribution.The three parameter TLLo distribution was introduced by [29].A random variable X is said to have a TLLo distribution, if its probability density function (pdf ) is of the form and the corresponding distribution function (df ) is The relation between the pdf and df can be seen as The Lomax, exponentiated Pareto, and standard Pareto distributions are special cases of the TLLo distribution at the different values of the parameters.
Oguntunde et al. [29] explored some particular properties of the TLLo distribution with considering the estimation issue of the model parameters, while Okorie and Nadarajah [30] have demonstrated that the TLLo distribution may be used to simulate datasets other than one utilised by Oguntunde et al. [29].The earlier limitations of [29] motivate us to explore further properties for the TLLo distribution with fitting it to flood data.Due to several applications of the OSs and the record values, as stated below, we derive the exact expressions for the moments of OSs and record values that which show the charateristic of the TLLo distribution and find the information theory as entropy measure of OSs and record values.
Records (or simply upper record values) which are greater than all previous values.Suppose the X 1 , X 2 , • • • , X n is a sequence of independent and identically distributed with a df F(x) and pdf f(x).
, n ≥ 2} with U (1) = 1 denote the times of lower record values.Then X L(n) and X U (n) are the nth lower and upper record values, respectively.The concept of the record values was coined by [16].For a survey on important results in this area one may refer to [1], [2], [7], [11], [12], [24].
The marginal pdf of the nth lower record value is given as By the same analogous, we defined the marginal pdf of upper record value replacing F (.) by 1 − F (.) in the equation (5).
Sometimes data are more specified by the order random variable, such as extreme OSs and record values, e.g.weather (hottest day, coolest day, flood levels) in a decade, economic condition in yearly, sports (highest jump and minimum time to complete swimming etc.), industrial stress testing (minimum and maximum time to failure), seismology, mining surveys, shock models, reliability testing and meteorological data analysis.Prediction about future record values are very important such as intensity of the next earthquake, highest (record) level of water dam hold or discharge, extreme point of market index.For more applications, see [4], [5], [9], [26] etc.
The rest of the paper is organized as follow: In Section 2, the exact expressions as well as the recurrence relations for single and product moments are obtained for OSs.In Section 3, the exact expressions and a recurrence relation for single moments are obtained from record values.Section 4 contains two applications, the first is the kth moments of OSs for the flood level data set.The second application reveals a relationship of entropy to the record (upper and lower) values.Finally, the study is concluded in Section 5.

Moments of the TLLo distribution and OSs based on it
In this section, we derive the exact expression as well as recurrence relations for the single (Subsection 2.1) and product moment (Subsection 2.2) from the TLLo distribution and the OSs based on it.

Relations for single moments
Let X 1 , X 2 , • • • , X n be a random sample of size n from the TLLo distribution.The pdf f r:n (x) of the rth OS X r:n , 1 ≤ r ≤ n, is given as where Theorem 1 Let 1 ≤ r ≤ n and k be any integer for which k < 2β.Then, for the TLLo distribution (1), we get and for 1 ≤ r < n, where B(a, b) is the complete beta function.
In the next theorem, we find the exact moments and their recurrence relation of TLLo distribution directly by putting the values of n and r.
Theorem 2 Let k < 2β.Then, for the TLLo distribution (1), we get and Proof.The proof is directly followed by putting r = n = 1 in ( 7) and ( 8).This completes the proof of the theorem.
The following interesting theorem treats with µ r:n := E(X −1 r:n ) as inverse moment of OSs.In the following theorem, we find the recurrence relations for the inverse moment of TLLo distribution.
Theorem 3 For the distribution given in (1), suppose that 2β + 1 is a positive integer, then the moment µ (−1) r:n exists, for any n > r ≥ 1.Moreover, Proof.The relation between the pdf and df can be seen as in (3).Thus, under the assumption that 2β + 1 is an integer, we have Therefore, by using (15), we get The existence of the right hand side (RHS) of ( 16) implies that the left hand side (LHS) of ( 16) exists, thus upon integrating ( 16) by part with respect to x, we get which after arranging yields (14).On the other side, the existence of the moments on the RHS of ( 14) implies that the moments on the LHS of ( 14) exist.This completes the proof of the theorem.
The numerical computations of the moments from the TLLo distribution are given in Table 1, with the different values of parameters which show that the moments are increasing when we increase the value of α, shape parameter at the fixed value of β, scale parameter.Moreover, in Tables 2 and 3, we calculate the first and second order moments of OSs from the TLLo distribution, the value of moments increases with increasing the values of r at the fixed value α and β.At the fixed value of r, the moments increase when shape parameter α increases.

Relations for product moments of OSs
The next result is about some relationships between the product moments of OSs.The joint pdf f r,s:n (x, y), 1 ≤ r < s ≤ n, of X r:n and X s:n , is given as (0 where Theorem 4 Let 1 ≤ r < s ≤ n, and k, l ∈ N, k, l < 2β.Then, for the TLLo distribution (1), we get where for p = q + 1 and q j=1 b j − p j=1 a j > 0, see, [27].
The following interesting theorem treats with µ r,s:n := E( X l s:n Xr:n ) as ratio moment of OSs.In the following theorem we find the recurrence relations for the ratio moment of TLLo distribution.
Theorem 5 For the distribution given in (1), suppose that 2β + 1 is a positive integer, then the moment µ (−1,l) r,s:n := E( X l s:n Xr:n ) exists, for any n > r ≥ 1.Moreover, Proof.The proof of this theorem follows as the proof of Theorem 3.
Corollary 1 When l = 0, in Theorem 5 we get the recurrence relations for single moment as given in Theorem 3 from the TLLo distribution.

Moments of record values of the TLLo distribution
In this section, we derive the exact expression and a recurrence relation for the single moments of lower record values from the TLLo distribution, noting that the marginal pdf of the nth lower record value is defined in (5).
Tables 4 and 5 explain the moments of the first and second order lower record values from the TLLo distribution.The variances are computed as in Table 6, we see that the variance decreases when we increase the values of n, which is universally expected.4 Applications with a practical data example In this section, we study properties of L-moments of the TLLo distribution as L-coefficient of variation, measure of variability, L-skewness, L-kurtosis etc. Furthermore, we fit the distribution of the flood level data, which was already studied by several authors (cf.[9], [26]) and are commonly used in OSs area.Because Okorie and Nadarajah [30] HAVE shown that the TLLo distribution does not the best fit model amongst a range of comparable models, flood data was used instead of airborne communication transceiver data.Also, we estimate the parameters of TLLo distribution through the L-moments.Finally, we reveal the relationship between the entropy and upper and lower record values.

L-Moments
L-moments are more robust in case of outliers in comparison to ordinary moments and have minimum sample variance.Apart from summarization of observed data, L-moments can also be used to characterize probability distributions in the model specification to the parameter estimation and hypothesis testing.The L-moments are expectations of certain linear combinations of OSs [23].The mth L-moment of a distribution can be defined as where µ i:m := µ i:m .The first four L-moments are , respectively.L-moments are direct analogous to the conventional moments, such as mean, variance, skewness, kurtosis, and so on.The properties and applications of L-moments were much explored by [23], which contain L-coefficient of variation (L-CV) that is a dimensionless measure of variability and defined by L 2 /L 1 , and L-skewness and L-kurtosis, which are dimensionless measures of asymmetry and kurtosis, respectively, defined by τ 3 = L 3 /L 2 and τ 4 = L 4 /L 2 .
In Equation (27) setting n = 1, 2, 3, and 4 and using the results given in [18], the first four L-moments easily follow.The L-moments for the TLLo distribution can be written as The L-moments for the TLLo distribution are computed for selected parameter values, and the results are reported in Table 7.
Table 7: L-moments for the TLLo distribution at different parameters Equating the population L-moments with sample l -moments and after simplification, we get the following results that can be used for the parameter estimation of TLLo distribution. and The sample l 1 and l 2 are defined as Equating L m and l m , m = 1, 2, . . ., gives reasonable estimators for the population parameters.This method is called L-moment estimation.By this approach, estimators of the parameters α and β of the TLLo distribution in (1) are simply obtained as , where Similarly for the l 2 , we have , where

The kth moments of OSs of the flood levels data
Flood levels data: Consider the data given by [20], which represents the maximum flood levels (in millions of cubic feet per second) of the Susquehenna River at Harrisburg, Pennsylvenia over 20 four-year periods (1890-1969) as: 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.324, 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265.From Tables 8 and 9, the flood data is leptokurtic and fits to the TLLo distribution.In the next section, we give some computations related to the moments of OSs of such data.The kth moments of OSs: In case of prediction for future emergencies to flood events, it is of interest to have the largest observations of the OSs.For this purpose, we will use the kth moments of some OSs for the data sets using Equation ( 7) from the TLLo distribution, where the parameters of the TLLo distribution are replaced by their corresponding L-moments estimates for the flood data set.Those OSs are given in Table 10, where the moments increase with increasing k.

Entropies
Entropy provides a measure of the average amount of information needed to represent an event drawn from a probability distribution for a random variable.First time or the classical entropy as the information is mathematically calculated by C. E. Shannon and called the Shannon entropy.If X is a non-negative absolutely continuous random variable associated with the information then the Shannon entropy is defined by [34] as There are several different versions of entropy, each one suitable for a specific situation.
Recently, various authors have discussed different versions of entropy and their applications.
For more details the reader may see [13], [14], [15], [25], and [28].The cumulative residual entropy (CRE) of X given by [31] and the cumulative entropy (CE) of X given by [19] are ones of the most important versions of the entropy.They are defined respectively as In the next two subsections we reveal relationships between these measures of entropy and OSs and record values, from the TLLo distribution.

Relationships between entropies and OSs from the TLLo distribution
Theorem 7 Let X have the TLLo distribution with finite mean, i.e., β > 0.5.Then, the CRE of X is given by Moreover, the CE of X is given by Proof.In view of [8], we have Thus, the proof follows by applying Theorems 1 and 2 for k = 1 and bearing in mind that n(n+1) = 1.This completes the proof of the theorem.

Relationships between entropies and record values from TLLo distribution
Theorem 8 Let X have the TLLo distribution with finite mean, i.e., β > 0.5.Then, the CRE of X can also be written in terms upper record values as follows: Upon simplifying (30), we get the second result.This completes the proof of the theorem.

Concluding remarks
The current study demonstrated the moments of OSs and lower record values from the TLLo distribution.The exact expressions for single and product moments were obtained.The recursive relations for single and product moments were also developed to make ease of computation of moments.The recurrence relations for ratio and inverse momemts of OSs were also obtained.Furthermore, we revealed several properties of the L-moments and entropy based on moments of OSs and record values.Finally, we successfully fitted a flood level data set using the TLLo distribution.Our studies may be extended for the generalized OSs which contain OSs, record and progressive censoring. of the TLLo distribution.

Conflicts of Interest:
The authors are declare no competing interests.
. Therefore, by using the obvious relation n−r j
Authors' Contributions: Mahfooz Alam proposed the concept the paper and calculate the existing results.Haroon M. Barakat contributed in writing original draft preparation and calculate the existing results.Haroon M. Barakat and Christophe Chesneau performed revision and improve the quality of the draft.All authors have read and agreed to the published version of the manuscript.Funding Statement: This research received no external funding.

Table 1 :
Behaviours of ordinary moments for various values of α from the TLLo distribution

Table 2 :
The value of µ

Table 3 :
The value of µ

Table 4 :
The values of E[X 1 L(n) ] from the TLLo distribution at different parameters

Table 5 :
The values of E[X 2 L(n) ] from the TLLo distribution at different parameters

Table 8 :
The flood levels data

Table 9 :
Estimates, KS statistic and its p-values for the flood levels data

Table 10 :
The k-th moments of order statistics of the flood level data