Article
The Combined-Unified Hybrid Censored Samples from Pareto
Distribution: Estimation and Properties
Khalaf S. Sultan *
,†
and Walid Emam
†
Citation: Sultan, K.S.; Emam, W. The
Combined-Unified Hybrid Censored
Samples from Pareto Distribution:
Estimation and Properties. Appl. Sci.
2021, 11, 6000. https://doi.org/
10.3390/app11136000
Academic Editor: Antonio
López-Quílez
Received: 30 April 2021
Accepted: 24 June 2021
Published: 28 June 2021
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Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia; wemam.c@ksu.edu.sa
* Correspondence: ksultan@ksu.edu.sa
† The authors contributed equally in this work.
Abstract:
In this paper, we use the combined-unified hybrid censoring samples to obtain the
maximum likelihood estimates of the unknown parameters, survival, and hazard functions of
Pareto distribution. Next, we discuss some efficiency criteria of the maximum likelihood estimators,
including; the unbiasedness, consistency, and sufficiency. Additionally, we use MCMC to obtain the
Bayesian estimates of the unknown parameters. In addition, we calculate the intervals estimation of
the unknown parameters. Finally, we analyze a set of real data in view of the theoretical findings of
the paper.
Keywords:
maximum likelihood estimates; unbiased estimator; interval estimation; minimum vari-
ance bound and relative efficiency; combined hybrid censored and unified hybrid
censored samples
1. Introduction
Pareto distribution was introduced by Pareto [
1
] for the distribution of income. The
importance of Pareto distribution lies in its applications in economics and reliability studies.
Arnold [
2
] has given a wide historical aacount of Pareto distribution and its applications.
Estimation and characteristics of Pareto distribution were investigated by many authors,
among researchers, see for examples, Malik [
3
], Arnold and Press [
4
], Tiwari, Yang and
Zalkikar [
5
], Abdel-Ghaly, Attia and Aly [
6
], Hossain and Zimmer [
7
], and Soliman [
8
].
Saldaña-Zepeda et al. [
9
] have proposed a goodness-of-fit test for Pareto distribution when
the observations are Type-II right censoring. Wu [
10
] has constructed an interval estimation
for Pareto distribution using a doubly Type-II censored sample. Recently, Han [
11
] has
investigated the expected Bayesian estimation and its expected mean square error of Pareto
distribution parameter under different loss functions and Poudyal [
12
] has investigated the
truncated, censored, and actuarial payment-type moments of the robust fitting of a single
parameter Pareto distribution.
A random variable
X
follows Pareto distribution
P(k
,
α)
if its probability density
function (pdf) is given by
f (x) = αk
α
x
−α−1
, α > 0, k > 0, x ≥ k, (1)
with the corresponding cumulative distribution function (cdf) is given by
F(x) = 1 −
k
x
α
, x ≥ k. (2)
The reliability function R(t) and hazard function H(t) are given, respectively, as
R(t) =
k
t
α
and H(t) =
α
t
. (3)
Appl. Sci. 2021, 11, 6000. https://doi.org/10.3390/app11136000 https://www.mdpi.com/journal/applsci
