Abstract: Further to our earlier results, we derive exact explicit expressions for the triple and quadruple moments of order statistics from the generalized log-logistic distribution
INTRODUCTION
Recently Adeyemi[1], Adeyemi and Ojo[2] initiated the study into the recurrence relations for moments of order statistics from the generalized log logistic distribution. We have obtained recurrence relations for single and product moments of order statistics from a symmetric, Adeyemi[3] and the, generalized log logistic distribution Adeyemi and Ojo[2].
In this paper, we present further results on our earlier studies by presenting recurrence relations for triple and quadruple moments of order statistics from the generalized log logistic distribution.
The probability density function (pdf) of the GLL (m1, m2) distribution is given by
| (1.1) |
Letting
| (1.2) |
Note that if m1=m2=1, GLL(m1, m2)
becomes the log-logistic distribution. It is symmetric around
Let X1:n≤X2:n≤....≤Xn:n denote the order statistics obtained when the n Xi’s are arranged in increasing order of magnitude. We denote
| (1.3) |
and
| (1.4) |
Also
| (1.5) |
where
and
| (1.6) |
where
Adeyemi[3] and Adeyemi and Ojo[2] have obtained recurrence relations for and expressions for μr, s:n in both symmetric and general cases respectively.
In this paper, we obtain recurrence relations for and for positive integers m1 and m2.
Recurrence relations for triple moments: Theorem 2.1 for 1 ≤ r < s < t ≤ n - m1 - i and a, b, c ≥ 1
| (2.1) |
where
and
Proof
| (2.2) |
having used (1.1), (1.3) and (1.5) where
| (2.3) |
Integrating by parts, we have
(2.4) |
by putting (2.4) in (2.3) and after simplification, we have the relation (2.1)
Theorem 2.2 For 1 ≤ r < s ≤ n – 1 and a, b, c ≤ 1
where
and
| (2.5) |
| (2.6) |
having used (1.1), (1.3) and (1.5) where
| (2.7) |
Integrating by parts, we have
| (2.8) |
substracting (2.8) into (2.6) and simplyfying the resulting expression yields
the relation (2.5).
Theorem 2.3 For 1 ≤ r < s < t ≤ n and a, b, c ≥ 1
where
and
| (2.9) |
Proof
| (2.10) |
where
| (2.11) |
having used (1.1), (1.3) and (1.5). Upon writing F(x) = F(x) – F(w) + F(w) and 1-F(x) = F(y) – F(x) + 1 – F(y) and using binomial expansion, we have
| (2.12) |
Integrating (2.12) by parts, we have
By putting the above expression into (2.10) and after simplification, we have the relation (2.9).
Corollary 2.1 Setting s=r+1, t= r+2 we have
where
and
| (2.13) |
Corollary 2.2 For s–r ≥ 2 and t=s+1
where
and
| (2.14) |
Remark 2.1 In theorems 2.1, 2.2 and 2.3 if m1 = m2 = m we obtain relations for triple moments of order statistics from a symmetric generalized log-logistic distribution studied by Adeyemi[3].
Remark 2.2 In theorems 2.1, 2.2 and 2.3 if m1 = m2 =1 we obtain relations fro triple moments of order statistics from the ordinary log-logistics distribution studied by Ali and Khan[4].
Recurrence relations for quadruple moments
Theorem 3.1.For 1 ≤ r < s < t < u ≤ n and a, b, c, d ≥
1
where
| (3.1) |
Proof
| (3.2) |
where
| (3.3) |
having used (1.1), (1.4) and (1.6). Upon integrating (3.3) by parts writing F(z) = F(z) – F(y) + F(y), F(y) = F(y) –F(x) – F(x) and F(x) = F(x) – F(w) + F(w) and using binomial expansion, we have
| (3.4) |
Upon substituting (3.4) into (3.2) and simplifying, we have the relation (3.1).
Theorem 3.2. For 1 ≤ r < s < t < u ≤ n and a, b, c, d ≥ 1
where
| (3.5) |
Proof
| (3.6) |
where
having used (1.1), (1.4) and (1.6). Expressing 1–F(x) as 1 – F(y) + F(y) – F(x) and 1 – F(y) as F(z) – F(y) + 1 – F(z), we have
| (3.7) |
By integrating (3.7) by parts, we obtain
| (3.8) |
By substituting (3.8) into (3.6) and simplyfying the resulting expression, we obtain the relation (3.5)
Corollary 3.1. Setting s= r+1, t= r+2 and u= r+3, we have
Where
| (3.9) |
Corollary 3.2. For s≥r+2, t= s+1 and u= s+2, we have
where
| (3.10) |
Remark 3.1 In theorems 3.1 and 3.2 if we set m1=m2=m we obtain relations for quadruple moments of order statistics from a symmetric generalized log-logistic distribution studied by Adeyemi[3].
Remark 3.1 In theorems 3.1 and 3.2 if we set m1=m2=1 we obtain relations for quadruple moments of order statistics from the ordinary log-logistic distribution studied by Ali and Khan[4].