第 50 卷 第 2 期
2014 年 4 月
兰 州 大 学 学 报（自然科学版）
Journal of Lanzhou University (Natural Sciences)
Vol. 50 No. 2
Apr. 2014
Articlcal ID: 0455-2059(2014)02-0255-07
Statistical inference on generalized Pareto distribution
with progressive type -Ⅰ censoring scheme
CHENG Cong-hua
1
, CHEN Jin-yuan
2
1. School of Mathematics and Computation Science, Zhanjiang Normal University, Zhanjiang 524048, China
2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Abstract: The statistical inference problems on generalized Pareto distribution with progressive type-Ⅰ cen-
soring scheme were studied. The maximum likelihood estimates (MLEs) of parameter were discussed and the EM
algorithm was applied to deal with this incomplete data case for computing MLEs. Using the missing informa-
tion principle, the observed Fisher information matrix was obtained for constructing the asymptotic confidence
interval for parameters. The method was discussed via simulation studies.
Key words: maximum likelihood estimation; EM algorithm; progressive censoring; Fisher information matrix
CLC number: O212.1; O213.2 Document code: A
AMS Subject Classifications(2000): 62F10
基于循序 -Ⅰ 型删失数据的广义 Pareto 分布统计推断
程从华
1
, 陈进源
2
1. 湛江师范学院 数学与计算科学学院, 广东 湛江 524048
2. 兰州大学 数学与统计学院 , 兰州 730000
摘 要: 研究了在循序 -Ⅰ 型删失数据情形下广义 Pareto 分布的统计推断问题, 讨论了广义 Pareto 分布未知参
数的最大似然估计 (MLE). 由于数据的缺失, 采用 EM 方法来获得参数的 MLE. 基于遗失信息原则, 给出了观
测 Fisher 信息矩阵, 进而给出未知参数的区间估计. 通过数值模拟, 讨论本文 提出的方法.
关键词: 最大似然估计; EM 算法; 循序型删失; Fisher 信息矩阵
中图分类号: O212.1; O213.2 文献标识码: A
The generalized Pareto distribution was intro-
duced by Pickands
[1]
. A random variable X is said
to have generalized a Pareto distribution if its prob-
ability density function (PDF) is given by
f(x; ξ, σ, µ) =
1
σ
(1 + ξ
x − µ
σ
)
−(
1
ξ
+1)
, (1)
where µ, ξ ∈ R and σ ∈ (0, +∞). For convenience ,
we reparametrize this distribution by defining
ξ
σ
= λ,
1
ξ
= α and µ = 0. Therefore,
f(x; α, λ) = αλ(1 + λx)
−(α+1)
. (2)
The cumulative distribution function is
F (x; α, λ) = 1 − (1 + λx)
−α
, x, α, λ > 0. (3)
Here α and λ are the s hape and scale parameters,
respective ly. It is well known as Pareto type Ⅱ dis tri-
bution or Lomax distribution with decreasing failure
rate property. There is an interpretation of Pareto
type Ⅱ distribution. In reliability study, the life-
time of a particular component has an exponential
distribution with failure ra te ν; let the ν follow a
Gamma dis tribution with scale parameter
1
λ
and
shape pa rameter α. Then the failure time Y of a
component selected at random from such a mixed
population has a Pareto type Ⅱ distribution.
The applica tion of generalized Pareto distribution
Received date: 2013-05-17
Found term: Supported by the Guangdong Natural Science Foundation (S2012040007369); the Foundation for Distinguished
Young Talents in Higher Education of Guangdong(2012LYM
−
0089)
Biography: CHENG Cong-hua (1981−), male, born in Neijiang, Sichuan Province, lecturer, doctor, e-mail: cch0971@126.com,
majoring in survival analysis, applied statistics.
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