Best Linear Unbiased Estimation and Prediction of Record Values Based on Kumaraswamy Distributed Data

To predict a future upper record value based on Kumaraswamy distributed data, an explicit expression for single and product moments has been established along with some enhanced expressions that makes the applying process on mathematical softwares easier. The best linear unbiased estimator approach for estimating the parameters and the prediction of future record values have been considered and some important tables have been created to help in the calculation processes. Two illustrative examples based on a simulation study and a real-life data are provided to assess the performance of the introduced results


Introduction
Let X 1 , X 2 , X 3 , ..... be a sequence of independent and identically distributed (iid) random variables. If X j is greater than X 1 , X 2 , ...., X j−1 , then X j is an upper record value. Thus X j > X i for all i < j. The record times U (n), n ≥ 1 at which the records appear, is define as U (n) = min{j : j > U (n − 1), X j > X U (n−1) }, n > 1.
U (1) = 1 with probability 1 and X U (1) is the first upper (lower) record. The sequence X U (n) , n ≥ 1 defines a sequence of upper record values.
The theory of record values was first introduced by Chandler [8]. Feller [10] gave some examples of record values in gambling. For more reading about the details of the record value and its applications from the point view of the statisticians, the reader can refer to Arnold et al. [3], [1], Khan et al. [13], and Nevzorov [21]. Record values appears in many real-life events, such as sports, economics, weather, pollution levels, industry and so on. Like in earthquakes, it might be interesting to know if there is an upcoming harder than the last greater earthquake, while the amount of rainfall that is greater than or smaller than the previous ones is important to people studying hydrology and climatology. In some industrial experiment, it's important to observe the greatest number of defective products in an operation run and expect if there is a greater number in the next runs and compare both 1252 RECORD VALUES BASED ON KUMARASWAMY DISTRIBUTED DATA quantity-based estimation for the parameters then found the exact confidence intervals and confidence regions for them based on k-records.
Kumaraswamy distribution has two parameters a and b and according to their values the shape of the distribution will change between 9 different shapes which can help in studding some life events. For example, the daily rainfall follows a pattern when a = 1 and b > 1. And on many occasions, it happens for values of a > 1 and a < 1. While for a = 1 and b < 1, a water levels in a cistern from which outflow occurs only through a crest spillway will follow that pattern.
A random variable x is considered to be Kumaraswamy distributed if: where z is a random variable of process [z], z min is the lower bound of the random variable and z max is the upper bound.
The pdf and the cdf of the Kumaraswamy distribution are given by where 0 < x < 1, a > 0 and b > 0, and In this paper, record values based on Kumaraswamy distribution are considered. In section2, the single moment from which we can find the mean and variance of record values based on Kumaraswamy distribution also the product moment to find covariance between two records has introduced. In section 3, the BLUE of the parameters along with some important tables that will make the calculations easier has established along with the BLUP method for future record values. In section 4, to check the efficiency of the study a simulation study and a real data example has performed.

Moments of the Record Values
Let X U (1) , X U (2) , ..., X U (n) be the first n upper record values that comes from a sequence of iid Kumaraswamy distributed random variables. We will denote E(X j U (n) ) by α j n , V ar(X U (n) ) by σ 2 n , E(X U (m) , X U (n) ) by α m,n and Cov(X U (m) , X U (n) ) by σ m,n , where j ≥ 1. Theorem 1. The jth moment of the nth upper record value for n ≥ 1 can be calculated from the following formula For n = 1 .
While the product moments of the mth and nth upper record values for m < n will be given by Stat., Optim. Inf. Comput. Vol. 10, September 2022 Now upon using (1), (3) and (4) in (7) we get the following We will take the transformation x a = u, we get Then we will use the logarithmic expansion introduced by Balakrishnan and Cohen [4] By integrating the integal part we reach (5). Now for the product moments, we have By substituting (2), (3) and (4) in (9) we get Using the binomial expansion, we get After using the logarithm expansion on the term [−ln(1 − y a )] k and binomial expansion on (1 − y a ) b−1 , we reach the following When substituting I(x) in α m,n and by taking the same two expansions on the relative terms we get to final expression (6).

Remark 1.
Another form of α j n could be conclude if after we took the logarithmic expansion, we used the binomial expansion on the term (1 − u) b−1 which will lead to the following expression Remark 2. To calculate the variance of the nth upper record for n ≥ 1, we will use the will known rule By substituting in (5) once for j = 1 and another for j = 2 we reached the following variance expression Corollary 1. The followings are another easier applicable forms on mathematical softwares for formula (6) 1. When m = 1 and n = 2 ] .

And for
.
Stat., Optim. Inf. Comput. Vol. 10, September 2022 Note: I applied all previous formulas for different values of n and m to find values of α j n or α m,n with summations to ∞ on MATHEMATICA. And found that after 1000 iterations, the values do not change that much.

BLUE of the Parameters
, ..., T U (n) be the first n upper record values that comes from a sequence of iid Kumaraswamy distributed random variables on the form .., n be the column vector of n upper records from a population with standard Kumaraswamy distribution which its pdf is given by (1) and its joint pdf is given by (2). Then, the BLUEs of the two parameters µ and σ can be given by the following (see, Balakrishnan and Cohen [4] where X U : represents the column vector of the existed upper records. α: represents the column vector of the expected values for these upper record values from our distribution. Σ: represents the variance-covariance matrix of the upper record values from our distribution. 1: a column vector of dimension n with all entries of the number 1.  Also, the variances and covariance of these BLUEs was given by (see, Balakrishnan and Cohen [4]) Other forms have been introduced by Arnold et al. [3], which will make calculating the formulas from (11) to (15) easier. The values of c i and d i that we introduced in formulas (11) and (12) to find their values, will be the outcomes of the following matrix While, the values of the formulas from (13) to (15) are simply the outcomes of the following matrix Now to get the values of c i and d i for calculatingμ andσ we can use Tables 1 and 2. In Table 1, the mean values of upper records are calculated for a = 1 and for values of b from 1 to 10 (since some important Kumaraswamy patterns occurs around these value) and this table can help into finding the values of the column vector α. Table  2 contains the variances (for value of m = n) and covariances of upper records at the same values of a and b for different values of m and n to specify the values of the matrix Σ. And Table 3 will introduce the values of c i and d i at the previous values of a and b for different values of n to the 10th record which has been calculated using the help of the values from Tables 1 and 2. While in Table 4, the variances ofμ andσ along with their covariances has been calculated in terms of σ 2 where the fisrt and second values will represent the variance ofμ andσ respectively, while the third value will be for the covariance between them.     and Cov(μ,σ) when a = 1.

BLUP of a Future Record
Suppose we have the following sequence or record values X U (1) , X U (2) , ..., X U (m) and we want to predict X U (n) , where 1 ≤ m < n. Since our distribution function belongs to a location -scale family, the BLUP of X U (n) can be Clearly whenâ = 3.52629 andb = 5, the values of all goodness of fit tests are better. After that, the records among the values are 0.35970, 0.36977, 0.68141, 0.68902 and 0.75595. When using the same steps in Example 4.1 to predict the 5th upper record from the first four records, we will getX U (5) = 0.74018 which has a small difference from the original value (0.75595).

Conclusion
The work that has been done in this paper focuses on predicting a future upper record based on Kumaraswamy distributed data with a very small part of error in the prediction by using the method of BLUE for estimating the parameters and the method of BLUP for the prediction process. The results are useful when people are interesting into knowing the next biggest number for some natural phenomena. Section 2 introduces new formulas for the jth moment of the nth upper record and the product moment of two not necessarily sequential upper records along with two various versions for the expression introduced to make the programming process (on a mathematical software such as MATHEMATICA) of the expression much easier. And of course, using these mathematical softwares gives credibility for the work that has been done.In section 3, the method of finding the BLUE for the parameters along with the rule for predicting the PLUP was founded along with some tables that were built for specific most repeated values of the parameters to make it easier for the practitioners to do their work. The simulated data in Example 4.1 was generated to clarify the steps that is needed to be done to find the predictive value of the future upper record value and to show that it will bring a good result for the predicted record. While in Example 4.2, a real data was tested for their distribution and whether if follows Kumaraswamy distribution or not by using different testing methods and found that it does. Then the same steps used in Example 4.1 applied here and we found an even better results.