The Exponentiated Burr XII Distribution: Moments and Estimation Based on Lower Record Values

In this paper, we have established several explicit expressions and recurrence relations for single and product moments of k -th lower record values from exponentiated Burr XII distribution. Two characterizing results of exponentiated Burr XII distribution has been obtained by using the recurrence relation for single moments and conditional expectation based on lower record values. The method of maximum likelihood is adopted for estimating the model parameters based on k -th lower record values. We carried out Monte Carlo simulations to compare the performances of the proposed methods and providing one real data case study for illustration of the results obtained.


Introduction
Record values find extensive applications in many real life situations involving data relating to weather, sport, economics, life testing studies and so on. There are several situations like Guinness World Records where only record values are observed. News items like fastest time taken to recite the periodic table of the elements, shortest ever tennis matches both in terms of number of games and duration in terms of time, fastest indoor marathon, longest time to hop on one foot, etc are of immense interest to people . Several attempts are made to make a record and records are made only when attempts are successful. Usually, we do not get the data on all of the attempts made to break the records around the world. The data that we have are the records. Another example is the situation in the Devendra Kumar, Jagdish Saran and Neetu Jain 2 ISSN-2424-6271 IASSL assessment of glucose level among diabetic patients, the researchers may be interested to study the behaviour of the ordered records of glocodine. Also, there are several situations where the lower record values are of special interest. For example, if various voltages of equipment are considered, only the voltages less than the previous one can be recorded. These recorded voltages are the lower record value sequence. Many scientists specially the statisticians have become interested in record values over the past 60 years or so since 1952 when Chandler (1952) first studied the distributions of lower records, record times and interrecord times for iid sequences of random variables. There are hundreds of papers and several books published on record-breaking data and its distributional properties see, for instance, Chandler (1952), Feller (1966), Resnick (1973), Shorrock (1973), Glick (1978), Nevzorov (1987), Balakrishnan and Ahsanullah (1995), Ahsanullah (1995), Kumar (2012Kumar ( , 2015Kumar ( , 2016, Kumar and Kulshrestha (2013) and Kumar and Saran (2014), Kumar et al. (2015) and so on.
be a sequence of independent and identically distributed .
. For convenience, we shall also take 0 Ahsanullah, 1995 (1) are given respectively by Further, for lower record values, the conditional pdf of A random variable X is said to have exponentiated Burr XII (EB XII) distribution if its pdf is of the form (5) and the corresponding cdf is given, by Further, the survival function and hazard rate function of EB XII distribution are given, respectively by and ] } ) The paper is organized as follows. Section 2 gives explicit expressions and recurrence relations for single moments of k -th lower record values from exponentiated Burr XII distribution. The obtained relations are used to compute first four moments, variance, skewness, kurtosis and coefficient of variation of lower record values. In Section 3, explicit expressions and recurrence relations for product moments of k -th lower record values from Devendra Kumar, Jagdish Saran and Neetu Jain 4 ISSN-2424-6271 IASSL exponentiated Burr XII distribution are derived. Further, in Section 4, two characterization theorems of this distribution are also obtained on using a recurrence relation for single moments and conditional expectation of record values. Also, maximum likelihood estimates of the parameters of exponentiated Burr XII distribution based on k -th lower record valuesare derived and the confidence intervals using Fisher information matrix are obtained in Section 5. Section 6 consists of simulation study based on the maximum likelihood estimates of the parameters of exponentiated Burr XII distribution based on lower record values. In Section 7, a case study is provided to illustrate the performance of maximum likelihood estimates of exponentiated Burr XII distribution.

Relations for single moments
In the section, we obtain the explicit expressions and recurrence relations for the single moments of the k -th lower record values from the EB XII distribution.
Note that, the marginal distribution of lower record values is Again by putting Remark 1: For 1  k in (11), we deduce the explicit expression for single moments of lower record values from the EB XII distribution.
Proof: Clearly,We have from (5) and (6), we see that Therefore, for Integrating by parts taking for integration and the rest of the integrand for differentiation, we get the constant of integration vanishes since the integral considered in (14) is a definite integral. On using (13), we obtain and hence the result given in (12). (12), we deduce the recurrence relation for single moments of lower record values from the EB XII distribution.

Relations for product moments
In this section, we obtain the explicit expressions and recurrence relations for product moments of the k-th lower record values from the EB XII distribution.
n m  is given by The explicit expression for the product moments of k-th lower record values can be obtained as where By setting On substituting the above expression of ) (x G in (15), we obtain By setting the constant of integration vanishes since the integral in ) (x G is a definite integral. On using the relation (13), we obtain (   1  1 and hence the result given in (19).

Remark 4:
in (19), we deduce the recurrence relation for product moments of lower record values from the EB XII distribution. One can also note that Theorem 1 can be deduced from Theorem 2 by setting 0  r and replacing s by r .

Characterization
This section contains characterizations of EB XII distribution by using the recurrence relation for the moments of a.e. on ) , ( b a . We start with the following result of Lin (1986).

Proposition 1: Let 0
n be any fixed non-negative integer, , an absolutely continuous function, with 0 ) Using the above Proposition we get a stronger version of Theorem 1.

Theorem 3:
be a fixed positive integer, r be a non-negative integer and X be an absolutely continuous random variable with cdf if and only if Proof: The necessary part follows immediately from equation (12). On the other hand if the recurrence relation in equation (21) is satisfied, then on using equation (1), we have Integrating the first integral on the right hand side of equation (22) (23) It now follows from Proposition 5.1 that Theorem 4: Let X be an absolutely continuous random variable with if and only if (25) By setting ) ( Simplifying the above expression, we derive the relation given in (24). To prove the sufficiency part, we have from (24) and (25)

Maximum likelihood estimation
In this section, we obtain the maximum likelihood estimators of the parameters , is a lower record value of this sequence, if it is greater than all preceding observations that is Suppose we observe n lower record values denoted by, say, following EB XII distribution with pdf (5). The likelihood function based on the random sample of size n is obtained from By using (5), equation (27) can be rewritten as The log-likelihood function The Exponentiated Burr XII Distribution IASSL ISSN-2424- 6271 11 We assume that the parameters  ,  and are unknown. To obtain the normal equations for the unknown parameters, we differentiate (28) partially with respect to  ,  and and equate to zero. The resulting equations are The solutions of the above equations are the maximum likelihood estimators of the EB XII distribution parameters ,  and , denoted MLE  , MLE ˆ and MLE ˆ, respectively. As the equations expressed in (30), (31) and (32) cannot be solved analytically, one must use a numerical procedure to solve them.

Approximate confidence intervals
In this section, we present the asymptotic confidence intervals for the parameters of the EB XII distribution. For interval estimation and hypothesis testing on the model parameters, we require the 3  3 Fisher information matrix and  decreases as the sample size n increases which quantifies the consistency of the estimation procedures.

Real data analysis
To illustrate the result of this paper, we analyze a real data set.  Form this data set, we extract the 4  n lower record values 1460, 1300, 318 and 170.Using the methods described in Section 5 we compute the maximum likelihood estimates as well as 95% confidence interval for  ,  and  as given in Table 4.