A Characterization and Recurrence Relations of Moments of the Size-Biased Power Function Distribution by Lower Record Values

A variety of research papers have been published on record values from various continuous distributions. This paper investigated lower record values from the size-biased power function distribution (LR-SPFD). Some basic properties including moments, skewness, kurtosis, Shannon entropy, cumulative distribution function, survival function and hazard function of the lower record values from SPFD have been discussed. The joint probability density function (pdf) of nth and mth lower record values from SPFD is developed. Recurrence relations of the single and product moments of the LR-SPFD have been derived. A characterization of the lower record values from SPFD is also developed.


Introduction
In the environmental and ecological work, observations usually fall in the non-experimental, non-replicated, and nonrandom categories.The problems like model specification and interpretation of data has special importance and great concern.The theory of weighted distributions provides a unique approach for these problems.Weighted distributions are applied in theory and applications of observational studies where biased data arise.Specifically, weighted distributions have applications in study of wildlife and fish populations that should motivate of those researchers who have an interest in environmental issues.Size-biased is one of the most commonly used type of various weight functions.
In this article we are introducing the lower record values from size-biased power function distribution (SPFD).Power function is a well known continuous distribution and a lot of work has been done on the record values from various continuous distributions.But in this article we weight the power function distribution named as size-biased power function distribution and derived record value from size-biased power function distribution.Some basic properties, recurrence relations for single and product moments, a characterization of the lower record values from SPFD are discussed.
Any observation is said to be a record value if its value is greater than or less than all the preceding observations.Chandler (1952) introduced the record value theory.Ahsanullah (1984) discussed the linear prediction of record values for the two parameter exponential distribution.Afterward various work on record values have been done including Ahsanullah (1986) rectangular distribution, Ahsanullah and Houchens (1989) Pareto distribution.Ahsanullah (1992) record values of independent and identically distributed continuous random variables, Balakrishnan and Chan (1993) Rayleigh and Weibull distribution, Balakrishnan et al. (1995) logistic record values, Ahsanullah and Bhoj (1996) Type-I extreme value distribution, Balakrishnan and Chan (1998) normal records, Sultan et al. (2008) gamma distribution based on record values, Teimouri and Gupta (2012) Weibull record values, Sultan (2010) inverse Weibull lifetime model, Shakil and Ahsanullah (2011) ratio of two independently distributed Rayleigh random variables, Bashir and Ahmad (2014) inverse Gaussian distribution, Bashir, Ahmed and Ahmad (2014) two sided power distribution.Raqab (2000) established the recurrence relations for the moment of record values.Khan and Zia (2009) gave some recurrence relations satisfied by single and product moments of upper record values from Gompertz distribution and also presented a characterization.Ahsanullah (2010) derived the r th concomitants and joint distribution of r th and s th concomitants of record values from the bivariate pseudo-Weibull distribution.He derived recurrence relation for the single moments.
Although lots of research had been done on all well known various continuous distributions but the work on record values from size-biased distributions started by the paper, Bashir and Ahmad (2014) developed record values from the size biased Pareto distributions and derived its various properties including a characterization.Bashir and Akhtar (2014) developed record values from size-biased student's t distribution.Bashir and Rasul (2015) developed record values from the weighted rectangular distribution.Bashir and Ahmad (2015) developed recurrence relations for single and product moments of record values from size-biased Pareto distribution.The pdf of the n th Lower record values is where, The joint pdf of n th and m th lower record values is The conditional pdf of lower record values Where β > 0 shape parameter and α > 0 scale parameter.The m th moments of the power function distribution is The size-biased power function distribution (SPFD) is The cumulative distribution function (cdf) of SPFD is

Record Values from the Size-Biased Power Function Distribution
Pdf of the lower record values from size-biased power function distribution (LR-SPFD) by using equation ( 7) and (8) in equation ( 1), Where β > 0 shape parameter and α > 0 scale parameter The cumulative distribution function LR-SPFD x ′ t n−1 e −t dt is incomplete gamma function and x ′ = −(β + 1) ln(x/α)

Moments
The r th moments of the LR-SPFD are First four raw moments of the LR-SPFD Mean, variance, 3 rd and 4 th mean moments of the LR-SPFD Coefficient of skewness and kurtosis of the LR-SPFD From equations ( 17) and ( 18) it can be seen that the coefficient of skewness and kurtosis are independent from scale parameter α  1, it can be seen that as shape parameter β increases the coefficient of skewness is decreases and approaches to zero.From table 1 and Figure 1, 2&3 it can be concluded that the shape of the LR-SPFD is approaching to symmetric for β ≥ 5, n ≥ 5.

Reliability Measures
The survival function of the LR-SPFD Where γ(n, x ′ ) = ∫ x ′ 0 t n−1 e −t dt is incomplete gamma function and x ′ = −(β + 1) ln(x/α) The hazard rate function of the LR-SPFD 3. Joint Record Values from the Size-Biased Power Function Distribution Using equations ( 7) and ( 8) in equation ( 2), we get the joint pdf of the lower record values from SPFD

Recurrence Relations for Moments
In this section some recurrence relations satisfied by the single and product moments of the lower record values from the SPFD are given.It is found that the size-biased power function distribution (SPFD) having the following relationship between pdf and cdf, Theorem 1: for n > 1 and r = 0, 1, 2, 3, . . .
after some simplifications we get The relation in equation ( 24) is developed by simply rewriting equation (25).
4)Ahsanullah and Lutful-Kabir (1974) developed a characterization from power function distribution.Saran and Pandey (2004) developed recurrence relations for moments, characterization and estimated parameters of a power function distribution by k th record values.Lim and Lee (2013) provided a characterization of the power function distribution by lower record values.Ahsanullah, et al (2013) provided a new characterization of power function distribution based on lower record values.The probability density function (pdf) of the power function distribution (PFD) is

Table 1 .
Values of coefficient of skewness for n = 5 & different values of shape parameter β