Selçuk J. Appl. Math. Selçuk Journal of Vol. 13. No. 1. pp. 75-82, 2012 Applied Mathematics
Recurrence Relations for Moments of k Record Values from Generalized Beta II Distribution and a Characterization
Devendra Kumar, M. I. Khan
Department of Statistics and Operations Research Aligarh Muslim University, Aligarh-202 002, India
e-mail: devendrastats@ gm ail.com, izhar.stats@ gm ail.com
Received Date: September 30, 2011 Accepted Date: March 19, 2012
Abstract. In this study we give some explicit expressions and recurrence rela-tions satisfied by single and product moments of k record values from generalized Beta II distribution. Further, using a recurrence relation for single moments we obtain characterization of generalized Beta II distribution.
Key words: Record values; Single moments; Product moments; Recurrence relations; Generalized Beta II distribution; Characterization.
2000 Mathematics Subject Classification: 62G30, 62G99, 62E10. 1. Introduction
Record values are found in many situations of daily life as well as in many sta-tistical applications. Often we are interested in observing new records and in recording them: for example, Olympic records or world records in sport. Record values are used in reliability theory. Moreover, these statistics are closely con-nected with the occurrences times of some corresponding non homogeneous Poisson process used in shock models. The statistical study of record values started with Chandler [6], he formulated the theory of record values as a model for successive extremes in a sequence of independently and identically random variables. Feller [8] gave some examples of record values with respect to gam-bling problems. Resnick [16] discussed the asymptotic theory of records. Theory of record values and its distributional properties have been extensively studied in the literature, for example, see, Ahsanullah [1], Arnold et al. [2, 3], Nevzorov [12] and Kamps [10] for reviews on various developments in the area of records. We shall now consider the situations in which the record values (e.g. successive largest insurance claims in non-life insurance, highest water-levels or highest temperatures) themselves are viewed as outliers and hence the second or third
largest values are of special interest. Insurance claims in some non life insurance can be used as one of the examples. Observing successive k largest values in a sequence, Dziubdziela and Kopocinski [7] proposed the following model of k record values, where k is some positive integer.
Let {Xn, n ≥ 1} be a sequence of identically independently distributed (i.i.d)
random variables with probability density function (pdf ) f (x) and cumulative density function (cdf ) F (x). The j− th order statistics of a sample (X1, X2, · · · , Xn)
is denoted by Xj:n. For a fix k ≥ 1 we define the sequence {Unk, n ≥ 1} of k
upper record times of {Xn, n ≥ 1} as follows
U1(k) = 1,
Un+1(k) = min{j > Un(k): Xj : j + k + 1 > XU(k)
n :Un(k)+k−1}.
The sequence {Yn(k), n ≥ 1} with Yn(k) = XU(k)
n :Un(k)+k−1, n = 1, 2, . . . are called
the sequences of k upper record values of {Xn, n ≥ 1}.
For k = 1 and n = 1, 2, . . . we write U1(1)= Un. Then {Un, n ≥ 1} is the sequence
of record times of {Xn, n ≥ 1}. The sequence {Yn(k), n ≥ 1}, where Yn(k)= XU(k) n
is called the sequence of k upper record values of {Xn, n ≥ 1}. For convenience,
we shall also take Y0(k)= 0. Note that k = 1 we have Yn(1)= XUn, n ≥ 1, which
are record value of {Xn, n ≥ 1}. Moreover Y1(k)= min{X1, X2, . . . , Xk = X1:k}.
Let Xn(k), n ≥ 1 be the sequence of k upper record values from (1.3). Then the
pdf of Xn(k), n ≥ 1, is given by (1.1) fX(k) n (x) = kn (n − 1)![−ln( ¯F (x))] n−1[ ¯F (x)]k−1f (x)
and the joint pdf of Xm(k)and Xn(k), 1 ≤ m < n, n > 2 is given by
fX(k) m ,Xn(k)(x, y) = kn (m − 1)!(n − m − 1)![−ln( ¯F (x))] m−1 (1.2) ×[−ln ¯F (y) + ln ¯F (x)]n−m−1[ ¯F (y)]k−1f (x)¯ F (x)f (y), x < y. We Shall denote μ(r)n:k = E((Xn(k))r), r, n = 1, 2, . . . , μ(r,s)m,n:k = E((Xm(k))r(Xn(k))s), 1 ≤ m ≤ n − 1 and r, s = 1, 2, . . . , μ(r,0)m,n:k = E((Xm(k))r) = μ(r)m:k, 1 ≤ m ≤ n − 1 and r = 1, 2, . . . , μ(0,s)m,n:k = E((Xn(k))s) = μ(s)n:k, 1 ≤ m ≤ n − 1 and s = 1, 2, . . . , Recurrence relations for single and product moments of k record values from Weibull, Pareto, generalized Pareto, Burr, exponential and Gumble distribution
are derived by Pawalas and Szynal [13, 14, 15]. Sultan [18], Saran and Singh [17] are established recurrence relations for moments of k record values from modified Weibull and linear exponential distribution respectively. Balakrishnan and Ahsanullah [4, 5] have proved recurrence relations for single and product moments of record values from generalized Pareto, Lomax and exponential dis-tributions respectively.
Kamps [11] investigated the importance of recurrence relations of order statis-tics in characterization.
In this paper, we established some explicit expressions and recurrence relations satisfied by the single and product moments of k upper record values from the generalized Beta II distribution. A characterization of this distribution has also been obtained on using a recurrence relation for single moments.
A random variable X is said to have generalized Beta II distribution if its pdf is of the form (1.3) f (x) = αλβ−αxα−1 ∙ 1 + µ x β ¶α¸−(λ+1) , x > 0, α, β > 0, and λ > 0 and the corresponding df is
(1.4) F (x) =¯ ∙ 1 + µ x β ¶α¸−λ , x > 0, α, β > 0, and λ > 0 where ¯ F (x) = 1 − F (x). 2. Relations for single moments
Note that for generalized Beta II distribution defined in (1.3)
(2.1) F (x) =¯ β αx−(α−1) αλ ∙ 1 + µ x β ¶α¸ f (x).
The relation in (2.1) will be exploited in this paper to derive recurrence relations for the moments of record values from the generalized Beta II distribution. We shall first establish the explicit expression for single moment k record values E((Xn(k))r). Using (1.1), we have
(2.2) μ(r)n:k= k n (n − 1)! Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx. By setting t = [ ¯F (x)]1/λ in (2.2), we get μ(r)n:k =β r(λk)n (n − 1)! [r/α]X a=0 (−1)a µ [r/α] a ¶ Z 1 0 tλk+a−(r/α)−1[−lnt]n−1dt.
Further by setting, w = −lnt, we get (2.3) μ(r)n:k= βr(αλk)n [r/α] X a=0 (−1)a µ [r/α] a ¶ 1 [α(λk + a) − r]n.
Remark 2.1. For k = 1 in (2.3) we deduce the explicit expression for single moments of upper record values from the generalized Beta II distribution. Recurrence relations for single moments of k upper record values from df (1.4) can be derived in the following theorem.
Theorem 2.1. For a positive integer k ≥ 1 and for n ≥ 1 and r = 0, 1, 2, . . . ,
(2.4) ³1 − r αλk ´ μ(r)n:k = μ(r)n−1:k+ rβ α αλkμ (r−α) n:k .
Proof. We have from (1.1) (2.5) μ(r)n:k= k n (n − 1)! Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx.
Integrating by parts treating [ ¯F (x)]k−1f (x) for integration and the rest of the
integrand for differentiation, we get μ(r)n:k= μ(r)n−1:k+ rk n k(n − 1)! Z ∞ 0 xr−1[ ¯F (x)]k[−ln( ¯F (x))]n−1dx
the constant of integration vanishes since the integral considered in (2.5) is a definite integral. On using (2.1), we obtain
μ(r)n:k = μ(r)n−1:k+ rk n αλk(n − 1)! ½Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx +βα Z ∞ 0 xr−α[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx ¾
and hence the result given in (2.4).
Remark 2.2. Setting k = 1 in (2.4) we deduce the recurrence relation for single moments of upper record values from the generalized Beta II distribution. 3. Relations for product moments
On using (1.2), the explicit expression for the product moments of k record values μ(r,s)m,n:k can be obtained
(3.1) μ(r,s)m,n:k = k n (m − 1)!(n − m − 1)! Z ∞ 0 xr[−ln( ¯F (x))]m−1 f (x) [ ¯F (x)]I(x)dx
where,
(3.2) I(x) = Z ∞
0
ys[ln( ¯F (x)) − ln( ¯F (y))]n−m−1[ ¯F (x)]k−1f (y)dy. By setting w = ln( ¯F (x)) − ln( ¯F (y)) in (3.2), we obtain
I(x) = βs(αλ)n−m [s/α] X a=0 (−1)a µ [s/α] a ¶ [ ¯F (x)]k+(a/λ)−(s/αλ)Γ(n − m) [α(λk + a) − s]n−m .
On substituting the above expression of I(x) in (3.1) and simplifying the result-ing equation, we obtain
μ(r,s)m,n:k= β(r+s)(αλk)n [s/α] X a=0 [r/α] X b=0 (−1)a+b µ [s/α] a ¶ µ [r/α] b ¶ (3.3) × 1 [α(λk + a) − s]n−m[α(λk + a + b) − (r + s)]m.
Remark 3.1. Setting k = 1 in (3.3) we deduce the explicit expression for product moments of record values from the generalized Beta II distribution. Making use of (2.1), we can derive recurrence relations for product moments of k upper record values
Theorem 3.1: For 1 ≤ m ≤ n − 2 and r, s = 1, 2, . . . ,
(3.4) ³1 − s αλk ´ μ(r,s)m,n:k = μ(r,s)m,n−1:k+sβ α αλkμ (s−α) m,n:k.
Proof: From (1.2)for 1 ≤ m ≤ n − 1 and r, s = 0, 1, 2, . . . (3.5) μ(r,s)m,n:k = k n (m − 1)!(n − m − 1)! Z ∞ 0 xr[−ln( ¯F (x))]m−1 f (x) [ ¯F (x)]G(x)dx, where, G(x) = Z ∞ x ys[ln( ¯F (x)) − ln( ¯F (y))]n−m−1[ ¯F (x)]k−1f (y)dy.
Integrating G(x) by parts treating [ ¯F (x)]k−1f (y) for integration and the rest
of the integrand for differentiation, and substituting the resulting expression in (3.5), we get μ(r,s)m,n:k= μ(r,s)m,n−1:k+ sk n k(m − 1)!(n − m − 1)! Z ∞ 0 Z ∞ x xrys−1[−ln( ¯F (x))]m−1
×[ln( ¯F (x)) − ln( ¯F (y))]n−m−1[ ¯F (y)]k f (x) [ ¯F (x)]dydx
the constant of integration vanishes since the integral in G(x) is a definite inte-gral. On using the relation (2.1), we obtain
μ(r,s)m,n:k= μ(r,s)m,n−1:k+ sk n αλk(m − 1)!(n − m − 1)! ½Z ∞ 0 Z ∞ x xrys[−ln( ¯F (x))]m−1 ×[ln( ¯F (x)) − ln( ¯F (y))]n−m−1[ ¯F (y)]k−1 f (x) [ ¯F (x)]f (y) dydx + β αZ ∞ 0 Z ∞ x xrys−α ×[−ln( ¯F (x))]m−1[ln( ¯F (x)) − ln( ¯F (y))]n−m−1[ ¯F (y)]k−1 f (x) [ ¯F (x)]f (y) dydx ¾
and hence the result given in (3.4).
Remark 3.2. Setting k = 1 in (3.4) we deduce the recurrence relation for prod-uct moments of upper record values from the generalized Beta II distribution. 4. Characterization
Theorem 4.1. Let k ≥ 1 is a fix positive integer, r be a non- negative integer and X be an absolutely continuous random variable with pdf f (x) and cdf on the support (o, ∞), then
(4.1) ³1 − r αλk ´ μ(r)n:k= μ(r)n−1:k+ rβ α αλkμ (r−α) n:k if and only if ¯ F (x) = ∙ 1 + µ x β ¶α¸−λ , x > 0, α, β > 0, and λ > 0.
Proof. The necessary part follows immediately from equation (2.4). On the other hand if the recurrence relation in equation (4.1) is satisfied, then on using equation (1.4), we have kn (n − 1)! Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx = k n (n − 2)! Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−2f (x)dx + rβ α kn αλk(n − 1)! Z ∞ 0 xr−α[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx (4.2) + rk n αλk(n − 1)! Z ∞ 0 xr[ ¯F (x)]k−1[−ln( ¯F (x))]n−1f (x)dx.
Integrating the first integral on the right hand side of equation (4.2) by parts and simplifying the resulting expression, we find that
rkn k(n − 1)! Z ∞ 0 xr−1[ ¯F (x)]k−1[−ln( ¯F (x))]n−1 (4.3) ½ −[ ¯F (x)] + β α αλx −(α−1)f (x) + x αλf (x) ¾ dx = 0.
Now applying a generalization of the Müntz-Szász Theorem (Hwang and Lin, [9]) to equation (4.3), we get f (x) ¯ F (x) = αλxα−1 βαh1 +³xβ´ αi
which proves that ¯ F (x) = ∙ 1 + µ x β ¶α¸−λ , x > 0, α, β > 0, and λ > 0. 5. Conclusion
In this study some explicit expression and recurrence relations for single and product moments of k upper record values from the generalized Beta II distri-bution have been established. Further, characterization of this distridistri-bution has also been obtained on using a recurrence relation for single moments.
6. Acknowledgement
The authors are grateful to Dr. R.U. Khan, Aligarh Muslim University, Aligarh for his help and suggestions thought the preparation of this paper. The authors also acknowledge with thanks to referees and Editor Selcuk J. Appl. Math for carefully reading the paper and for helpful suggestions which greatly improved the paper.
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