Proceedings of

MALAYSIAN

MATHEMATICAL SCIENCES SOCIETY

i

 

Table of Content i

Preface viii

Keynote Addresses

Numerical Analysis John C. Butcher

1

Applications of mathematical reliability theory to the problem of human aging, mortality and longevity

Leonid A. Gavrilov & Natalia S. Gavrilova

9

Pemodelan Berlaku Hujan Abdul Aziz Jemain & Sayang Mohd Deni

46

Practice and pricing in non-life insurance: the Malaysian experience Pan Wei Cheong, Abdul Aziz Jemain & Noriszura Ismail

65

Invited Papers

Sains dan teknologi dari perspektif Islam Mohd Yusof Hj Othman

79

Development of Mathematical Sciences in Our Own Mould (Perkembangan sains matematik dalam acuan sendiri)

Shaharir b Mohamad Zain

94

Some aspects of Islamic Cosmology and the current state of physics Shahidan Radiman

118

Mathematics and Operation Research

Explicit series solution for the Klein-Gordon equation by means of the homotopy analysis method

A.K. Al Omari, Mohd Salmi Md Noorani & Roslinda Mohd Nazar

125

Mendefinisikan matematik Abdul Latif Hj. Samian

138

Application of Double Laplace Transform and Green's function for solving non homogenous wave equation with double convolution properties

Adem Kilicman and Hassan Eltayeb

146

Constrained interpolant to geosciences chemical data Ahmad Abd. Majid, Azizan Saaban & Abd. Rahni Mt. Piah

155

Estimate on the second Hankel functional for functions whose derivative has a positive real part

Aini Janteng, Suzeini Abdul Halim & Maslina Darus

171

An integral equation method involving the Neumann kernel for conformal mapping of doubly connected regions onto a disc with a circular shift

Ali H.M. Murid, Laey-Nee Hu & Mohd Nor Mohamad

178

Some applicatons of Hermitian linear maps Ali Parsian

187

A mathematical model for surveying the destructive effects of some believes in human society

Ali Parsian

192

The Sf-function for s@-fibrations in the homotopy theory for topological semigroups Amin H. Saif & M. Alinor

196

ii

 

Similarity solutions for the laminiar boundary layer flow along a stretching cylinder Anuar Ishak, Roslinda Nazar & Ioan Pop

207

The mathematical model of the thermoregulation in human Ehsan Firouzfar & Maryam Attaran

215

A parallel overhead of the alternating direction implicit method (ADI) for the parabolic equation problem

Ewedafe Simon Uzezi & Rio Hirowati Shariffudin

229

2-D inverse dynamic model for three link kinematic chain of an arm via Kane’s method Fadiah Hirza Mohammad Ariff & Azmin Sham Rambely

237

New subclasses of analytic functions for operator on Hilbert space Faisal Al-Kasasbeh, and Maslina Darus

245

Model gerakan terbuka hayunan pada pukulan smesy badminton: Sebuah pendekatan rantai kinematik terbuka

Fazrolrozi & Azmin Sham Rambely

251

On a class of analytic functions with respect to symmetric points Firas Ghanim and Maslina Darus

262

Legendre wavelets and fractional differential equations H. Parsian

271

Red-black Qssor interative method for solving 2D Helmholtz equations Jumat Sulaiman, Mohamed Othman & Mohammad Khatim Hasan

278

Stability of a functional equation in quasi-Banach spaces K. Ravi and R. Kodandan

286

On a generalized n-dimensional additive functional equation with fixed point alternative K. Ravi & M. Arunkumar

314

Stability of a 3-variables quadratic functional equation K. Ravi & M. Arunkumar

331

A linear integral operator and its application Khalida Inayat Noor & Saqib Hussain

343

Applications of generalizations of Ruscheweyh derivatives and Hadamard products to harmonic functions

Khalifa Al-Shaqsi & Maslina Darus

352

Preconditioned nine-point formula in solving the Poisson’s Equation Lee Siaw Chong

362

Free convection boundary layer flow on a horizontal plate with variable wall temperature Leli Deswita, Roslinda Mohd Nazar, Rokiah @ Rozita Ahmad & Ioan Pop

372

Vehicle routing problem: A survey on models and algorithm Liong Choong Yeun, Wan Rosmanira Ismail, Kharuddin Omar & Mourad Zirour

379

On accuracy of Adomian decomposition method for Lü system M.Mossa Al-Sawalha & M.S.M. Noorani

392

Application of the homotopy-perturbation method to the nonlinear time-dependent singular initial value problems

Md. Sazzad Hossien Chowdhury & Ishak Hashim

397

403

iii

 

Approximate finite difference scheme for one dimensional Poisson’s equation Mohammad Khatim Hasan, Jumat Sulaiman & Mohamed Othman

Fluid motion due to a multi Fourier-component magnetic field, finite Rm Mohd Noor Saad

408

Pembelajaran matematik menggunakan teknik multimedia mesra pembelajaran Mohd Nor Hajar Hasrol Jono, Azlan Abdul Aziz , Prasanna Ramakrisnan, Nurul Hidayah Mat Zain & Suhaila Khalip

420

Natural convection boundary layer on a vertical surface with prescribed wall temperature and heat flux

Mohd Zuki Salleh & Roslinda Mohd Nazar

426

Conservation laws for nonholonomic systems via Poincaré-Lagrange-d’Alembert principle Naseer Ahmed and Kanwal Shahzadi

439

On the Fekete-Szegö problem Nik Nadhilah Nik Mohd Yusoff & Maslina Darus

446

New fifth-order Runge-Kutta methods for solving ordinary differential equation Noorhelyna Razali & Rokiah @ Rozita Ahmad

453

The symmetric squares on infinite non-Abelian 2-generator groups of nilpotency class 2 Nor Haniza Sarmin, Nor Muhainiah Mohd Ali & Luise-Charlotte Kappe

462

Mixed convection boundary layer flow of a viscoelastic fluid past a circular cylinder Nur Iliyana Anwar Apandi, Wan Mohd Khairy, Sharidan Sahfie & Norsarahaida Amin

469

Third order Runge-Kutta formulas based on harmonic mean Osama Yusuf Ababneh, Rokiah @ Rozita Ahmad & Eddie Shahril Ismail

479

Nonlinear response of concrete filled steel turbular composite columns under axial loading P. Vinayagam, K. Subramanian & R. Sundararajan

489

The cryptosystems on the decimal numbers Rand Al-Faris, Mohamed Rezal Kamel Ariffin & Mohamad Rushdan Md Said

502

Aplikasi pengaturcaraan gol 0-1 dalam penjadualan jururawat Ruzzakiah Jenal, Wan Rosmanira Ismail, Liong Choong Yuen, Masrio Ayob & Mohd Kairi Muda

511

Domination polynomial of paths and cycles Saeid Alikhani & Yee-hock Peng

522

Open problem for general class of superordination-preserving convex integral operator Saibah Siregar, Maslina Darus & Teodor Bulboacă

530

A simplified partial power model of a slightly unstable fused coupled fibers Saktioto, Jalil Ali, Rosly Abdul Rahman, Mohammed Fadhali & Jasman Zainal

539

Convexity-preserving interpolation by piecewise rational quintic generalized ball Samsul Ariffin Abdul Karim & Abd. Rahni Mt. Piah

554

Certain hybrid triple integral equations Sanjay Kumar Jain, Kuldeep Narain & Vijay Khare

564

Determining the preprocessing clustering algorithm in radial basis function neural network Sau Loong Ang, Hong Choon Ong & Heng Chin Low

572

Series solutions of boundary-layer flows induced by permeable stretching walls Seripah Awang Kechil & Ishak Hashim

580

iv

 

Optimization of crude palm oil and palm kernel transportation for northern Peninsular Malaysia

Shamsudin Ibrahim, Abbas F.M. Al-Karkhi & Omar A Kadir

589

Generating theorems for S-norms and T-norms Shawkat Mahmoud Rasheed Al-Khazale & Abdul Razak Salleh

598

Coefficient property for harmonic meromorphic functions which are convex of order β with respect to symmetric points

Suzeini Abdul Halim & Aini Janteng

611

A reliable algorithm for Blasius equation Syed Tauseef Mohyud-Din

616

The design of axisymmetric ducts for incompressible flow with Hagen-Poiseuille flow Vasos Pavlika

627

Fault diagnosis of linear time invariant systems – A Markov parameter approach Venugopal Maniknadan, N. Devarajan, S. Srikanth, R. Venkateswarabhupati & K. Ramakrishnan

642

Fuzzy clustering of microarray data Yahya Abu Hasan & Anita Talib

653

Determination of journal bearing stability using finite difference method Zaihar Yaakob, Mohammad Khatim Hasan & Jumat Sulaiman

658

Feynman integral and the τ-function Zainal Abdul Aziz

671

A conflicting equation approach in multi-criteria selection Zamali Tarmudi, Mohd Lasim Abdullah & Abu Osman Md Tap

679

Forecasting as a tool in operation research Zuhaimy Ismail

686

The asphericity of relative presentation ( ) ( )1, 1 ,nH t ta t b a b H− = ∈

Abd Ghafur Ahmad

697

A problem of stochastic heat equation with weakly correlated coefficients Amr Mohamed Abdelrazak

700

A new subclass of P-valently analytic functions of Bazilevič type defined by a fractional calculus operator

Ajab Akbarally & Maslina Darus

707

Modelling of splicing system in DNA Fong Wan Heng, Nor Haniza Sarmin, Mohd Firdaus Abd Wahab & Nooraini Abdul Rashid

712

Chromatic uniqueness of certain complete t-particle graphs G.C. Lau & Y.H. Peng

719

Homotopy analysis method for solving linear fractional partial differential equations Omar Abdul Aziz & Ishak hashim

729

A Gregus type common fixed point theorem in normed spaces with application Rakesh Tiwari

741

v

 

On the bounds for radius of convexity of α-close-to-convex functions Shaharuddin Cik Soh & Daud Mohamad

754

Detection of lightning in space Shamsiah Suhaili, Mohd Suhaimi Sidek, Syed Idris Syed Hassan & Ngu Sze Song

762

Some third-order predictor-corrector method for solving nonlinear equation Muhammad Aslam Noor & Waseem Asghar Khan

772

Statistics and Actuarial Science

On the detection of ARCH effect in time series data A.H.M. Rahmatullah Imon, Md. Sirajud Doula & Nor Aishah Hamzah

783

Comparing the accuracy of density forecasts from competing GARCH models Abu Hassan Shaari Mohd Nor, A.Shamiri & Zaidi Isa

790

Evaluating the bi-Weibull distribution functions with a number of parameters Ahmad Mahir Razali & Ali A. Salih Al-Wakil

805

Measuring of exogamous marriage with weighted kappa through disagreement scaling Ahmad Mahir Razali & Chua Chee Ming

811

Comparison of M and MM-estimators in simple mediation model based on a single unusual observation

Anwar Fitrianto & Habshah Midi

818

Acceptance sampling based on truncated life tests in the Burr type X model Ayman Baklizi 

828

Comparison between rough set theory and logistic regression for classifying firm’s performance

Bahtiar Jamili Zaini, Siti Mariyam Shamsuddin & Saiful Hafizah Jaaman

835

Long-memory of foreign exchange rate by the fractional Brownian motion Chatchai Pesee

847

Designing 2n conjoint choice experiments using confounded factorial designs Chin Khian Yong & Kent M. Eskridge

854

Tests of random walk for Malaysian daily stock indices Chin Weng Cheong, Zaidi Isa & Abu Hassan Shaari Mohd Nor

864

A survey on principal-agent problem in financial market Eko Nugroho

873

Penyuaian taburan kebarangkalian ke atas data aliran maksimum sungai terpilih di Sabah G. Darmesah, A. Amran, S. Shamsiah & A. Noraini

887

The relationship between anxiety of suspicion and intelligence among sciences of students in high school

Habibollah Naderi

897

The relationship between intelligence and behavior of anxiety (tension) among sciences of students in high school

Habibollah Naderi

902

Transitory component of the multivariate economic time series: Some observations on four Asean countries

Hamizun Ismail

907

vi

 

Application of multivariate techniques in determining morphological variation in genus solen of Kuching bay area in Sarawak

Hung Tze Mau & Ruhana Hassan

917

On the ratio estimation using quartile ranked set sampling Kamarulzaman Ibrahim, Ameer Al-Omari & Abdul Aziz Jemain

929

Pembuatan keputusan pelbagai criteria dalam pemilihan pegawai akedamik di IPTA: Sati kajian kes di Universiti Malaya

Khairul Anuar Mohd Ali & Wu Ziou Hon

938

Momen ortogon Legendre dan teknik pengambangan setempat untuk pengecaman kedudukan penumpang

Liong Choong Yeun, Chris Thompson & Teo Yuan Chiu

949

Mengenalpasti tumpukan data (heaping) pada data tempoh penyusuan bayi dengan menggunakan ujian khi kuasa dua

Mahdiyah Mokhtar, Wan Norsiah Mohamed & Kamarulzaman Ibrahim

968

A study of nonlinear relationship between world oil price and Malaysian exchange rate Mohd Tahir Ismail & Zaidi Isa

975

Men-vriance and equilibrium pricing in incomplete model Munira Ismail

987

Stochastic comparisons and aging properties of multivariate reversed frailty models Nitin Gupta, Neeraj Misra & Rameshwar D. Gupta

993

Outcome-oriented cutpoint determination methods for competing risks Noor Akma Ibrahim, Abdul Kudus, Isa Daud & Mohd. Rizam Abu Bakar

1016

Anggaran kehilangan pendapatan akibat kematian pramatang pekerja Noriza Majid, Saiful Hafizah Jaaman & Noriszura Ismail

1033

Statistical analyses on ranking consistency between SVD and EM in AHP: A study of group decision making in faculty member selection

Nur Jumaadzan Zaleha Mamat & Afzan Adam

1040

Sikap dan tingkah laku pengguna terhadap keinginan memilih jenama telefon bimbit: Kajian kes di kawasan Bandar Baru Bangi Selangor

Nur Riza M Suradi, Petir Papilo & Zalina Mohd Ali

1050

Kriteria penentuan kepuasan pelancong hotel menggunakan analisis faktor Nur Riza M Suradi, Zainol Mustafa & Nur Diana Zamani

1060

A Markoc chain model for the occurrence of apnea Nur Zakiah Mohd Saat & Abdul Aziz Jemain

1069

Discriminating between Gamma and Weibull distributions Nur Zakiah Mohd Saat & Abdul Aziz Jemain

1076

Statistical approach for identifying defects in ceramics tiles using discrete wavelet transform P. Senthil Kumar, R. Arumuganathan & T. Veerakumar

1083

Penyuaian model polynomial bagi biodegrasi fenol oleh Candida Tropicalis Retr-Crl Piakong, M.T., Gabda, D., Sulaiman, J., Abdul Rashid, N.A. & M.Md. Salleh

1092

Kernel method in graduating Malaysian population data Rozita Ramli & Gan Shou Wan

1103

Understanding customer requirement in health care service quality Sal Hazreen Bugam, Wan Norsiah Mohamed & Faridatulazna A. Shahabudin

1109

vii

 

Analytical approximations for average run lengths in EWMA charts in case of light-tailed distributions

Saowanit Sukparungsee & Alexander Novikov

1117

Probability models for wet spells in Peninsular Malaysia Sayang Mohd Deni, Abdul Aziz Jemain & Suhaila Jamaludin

1125

Finding critical region for testing the presence of temporary change (TC) outliers in GARCH(1,1) processes

Siti Meriam Zahari & Mohamad Said Zainol

1136

Survey analysis on book buying behavior among School of Science and Technology student Suriani Hassan, Nortazi Sanusi, Sadaria Sahada, Rodeano Roslee & Farrah Anis Fazliatul Adnan

1143

Interpretation of river water quality data by chemometric techniques: Pahang River as a case study

Tan Kok Weng & Mazlin Mokhtar

1152

A modified SPCA for face recognition with one training image per person Umi Sabriah Haron@Saharon & Jacob K. Daniel

1164

Interrelationships between stock indices at Bursa Malaysia Veronica Tan Kah Min, Tan Kee Inn & Zainudin Arsad

1174

EWMA control charts for changes in exponential distribution Yupaporn Areepong & Alexander Novikov

1186

Identifying relevant economic factors affecting KLCI of KLSE using factor analysis Zainodin Hj. Jubok, Darmesah Gabda, Ho Chong Mun & Goh Cheng Hoe

1191

Penilaian tahap kepuasan pelanggan terhadap perkhidmatan Maxis di IPTA Zainol Mustafa, Nur Riza Mohd Suradi, Wan Norsiah Mohamed, Zalina Mohd Ali, Chai Chun Fatt & Juwairiah Mohd Ramli

1200

Fuzzy portfolio selection using semi-variance risk measures in Bursa Malaysia Zulkifli Mohamed, Daud Mohamad & Omar Samat

1211

Probality of correct selection for some discrete distributions with cubic variance function based on likelihood ratio statistic

Phang Y.N. & Ong S.H.

1219

Usage of computer applications among education students at Universiti Malaysia Sabah: A comparative analysis on genders

Fauziah Sulaiman & Suriani Hassan

1231

Analisis tahap kepuasan perkhidmatan internet di kalangan pelajar IPT Nur Riza Mohd Suradi, Rofizah Mohammad, Wan Rosmanira Ismail, Faridatulazna Ahmad Shahabuddin & Noor Azani Abdul Aziz

1238

On the number of families of branching processes with immigration with family sizes within random interval

Husna B. Hasan & George P Yanev

1246

 

 

viii  

Preface We would like to welcome each and every participant of the International Conference on Mathematical Sciences 2007 (ICMS 2007). ICMS 2007 is organized by School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia (National University of Malaysia) together with the Malaysian Mathematical Sciences Society (PERSAMA) to bridge as well as to nurture the understanding and collaboration amongst the regional and global mathematical scientists and practitioners. At a very least, ICMS 2007 intends to provide a platform for academicians and practitioners to share their work and exchange ideas related to the conference theme “Integrating Mathematical Sciences within Society”, which expresses the intention to highlight the crucial role played by mathematicians and statisticians in gaining a richer understanding of the fast-growing world. Over 120 papers from various universities and institutions all over Malaysia and abroad were accepted to be published in our proceeding. In this regard, on behalf of the conference organizing committee, the editors would like to express their deepest gratitude to all presenters, contributors/authors and participants of this conference for their overwhelming supports that turns this conference into a big success. While every single effort has been made to ensure consistency of format and layout of the proceedings, the Editors assume no responsibility for spelling, grammatical and factual errors. Besides, all opinion expressed in these papers are those of the authors and neither of the conference organizing committee nor the Editors. It is hoped that the conference papers included in this proceedings will not only benefit the conference participants, but also to all academicians, businesses, practitioners, policy-makers, researchers, graduate students and other interested readers who do not have the opportunity to attend the ICMS 2007 conference. Last but not least, thanks are due to all participants, members of various committees as well as supporting staffs, because without their supports, assistance and efforts, this conference will never be a success.

Editors: Abd Ghafur Ahmad Azmin Sham Rambely Fatimah Abdul Razak Hamizun Ismail Nur Riza Mohd Suradi Rokiah Rozita Ahmad Roslinda Mohd Nazar Syazwani Saadon Wan Norsiah Mohamed

November 2007 

STOCHASTIC COMPARISONS AND AGING PROPERTIES OF

MULTIVARIATE REVERSED FRAILTY MODELS

Nitin Gupta, (Neeraj Misra, and Rameshwar D. Gupta)

Abstract

Gupta and Wu (2001) and Gupta and Gupta (2007) have defined a model having

distribution function F (x) = (F1(x))V , x ≥ 0, where F1(x) is baseline distribution

function and V is a non-negative random variable, called frailty random variable. This

model is dual to frailty model introduced by Vapuel et al. (1979). The dual to frailty

model can be called as reversed frailty model. We have extended this model to multi-

variate set up, and discussed stochastic ordering and aging properties in multivariate

reversed frailty models.

Keywords: Failure rate order; Increasing concave order; Likelihood ratio order; Upper

orthant order; Usual stochastic order; Reversed failure rate order; Statistical aging;

Statistical dependence.

1 Introduction

Let X be a random variable (rv) having distribution function (df) F (x),−∞ < x < ∞,

survival function (sf) F (x) = 1 − F (x), −∞ < x < ∞ and Lebesgue probability density

function (pdf) f(x), −∞ < x < ∞. Gupta et al. (1988) proposed a proportional reversed

hazard rate model (PRHRM) in contrast to a proportional hazard model (PHM) which was

introduced by Vapuel et al. (1979). The df of PRHRM, corresponding to a rv X is given by

F (x) = (F1(x))v , −∞ < x < ∞, v ≥ 0, (1)

where F1(·) is a df (called baseline df), corresponding to a rv X∗ (called baseline rv). If v is

a positive integer, then PRHRM are called Lehman alternatives. Lehman (1953) has studied

such alternatives to define various nonparametric hypotheses. PRHRM are flexible enough to

accommodate both monotonic as well as non-monotonic failure rates even though the baseline

1

993

failure rates are monotonic. Gupta et al. (1988; 1997), Mudholkar et al. (1995), Mudholkar

and Srivastava (1993) and Mudholkar and Hustson (1996) has studied the monotonic and/or

non-monotonic failure rates of PRHRM, corresponding to various baseline distributions (e.g.,

gamma, lognormal, exponentiated expoential etc.)

Let V is a non-negative rv having Lebesgue pdf h(x), −∞ < x < ∞, df H(x), −∞ <

x < ∞ and sf H(x), −∞ < x < ∞. For model (1), if we replace non-negative constant v by

a non-negative random variable V, then PRHRM is given by

P (X ≤ x|V = v) = (F1(x))v , −∞ < x < ∞, v ≥ 0,

which is the conditional df of rv X given V = v. Then, the unconditional df of X is given by

F (x) = (F1(x))V , −∞ < x < ∞. (2)

Model (2) can be called reversed frailty model, since this model is in contrast to frailty model

of Vapuel et al. (1979). Gupta and Kirmani (2006) and Xu and Li (2007) has discussed

aging properties and stochastic ordering between between frailty models arising from taking

different baseline and/or baseline distributions.

The reversed frailty model (2) is an extension of PRHRM (1), where misspecified and

omitted factors are described by an unobservable rv V, called frailty rv. Univariate reversed

frailty model (2) have multivariate extensions. To define these multivariate extensions, sup-

pose that there is a population and related individuals in the population has a random

frailty V. Let X1, . . . , Xn be the failure times of n related individuals in the population.

Then PRHRM, corresponding to random vector X = (X1, . . . , Xn) is given by

P (X ≤ x|V = v) = P (X1 ≤ x1, . . . , Xn ≤ xn|V = v) =n∏

i=1

(Fi(xi))v, x ∈ Rn, v ≥ 0,

where, for i = 1, . . . , n, Fi(·) is the df of some rv X∗i , having support Rn

+. Here for a positive

integer n, Rn+ will denote the product space [0,∞)n = [0,∞)× · · · × [0,∞)︸ ︷︷ ︸

n times

and Rn will

denote the n-dimensional Euclidean space. Now the df of rvc X is given by

F (x) = P (X ≤ x) = P (X1 ≤ x1, . . . , Xn ≤ xn) = E

(n∏

i=1

Fi(xi)

)V , x ∈ Rn, (3)

2

994

The reversed frailty model (3) corresponding to random vector X is multivariate extensions

of univariate reversed frailty models (2).

In Section 2, we derive some results on the aging properties of multivariate reversed frailty

model. In general, there is no compelling reason, other than mathematical tractability, for

choosing the probability distribution of baseline distribution and/or frailty distribution in

practical situations. Thus, it is important to see how the reversed frailty model respond

to change in baseline distributions and/or frailty distributions. Therefore in Section 3,

using stochastic orders we make stochastic comparisons between frailty models arising from

different choices of baseline distributions and/or frailty distributions. We refer readers to

Shaked and Shanthikumar (2007) and Lai and Xi (2005) for details of various stochastic

ordering and aging properties.

Throughout the paper, the term increasing is used for non-decreasing and the term

decreasing is used for non-increasing. The minimum and the maximum operators are denoted

by ∧ and ∨, respectively, i.e., for real numbers x and y, x ∧ y (x ∨ y) denote the minimum

(maximum) of x and y. Also, for x, y ∈ Rn, x ∧ y = (x1 ∧ y1, x2 ∧ y2, . . . , xn ∧ yn) and

x ∨ y = (x1 ∨ y1, x2 ∨ y2, . . . , xn ∨ yn). For vectors x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ Rn,

the notation x ≤ y (or y ≥ x) means that xi ≤ yi, i = 1, 2, . . . , n. Similarly, for vectors

x, y ∈ Rn, the notation x < y (or y > x) means that xi < yi, i = 1, 2, . . . , n.

2 Aging Properties and Statistical Dependence

Since the overall population distribution of rv X is linked to baseline distribution X∗, through

frailty rv V, therefore for univariate reversed frailty models (2), Gupta and Wu (2001) proved

that E(V |X1 ≤ x) ≤ E(V |X1 ≤ x′), ∀x ≤ x′. In the following theorem, we extend this result

to multivariate reversed frailty models. We here recall the definition of likelihood ratio order.

Definition 2.1 Random variable X is said to be smaller than rv Y in the likelihood ratio

order (written as X ≤lr Y ) if f(y)g(x) ≤ f(x)g(y), whenever x, y ∈ R1 and x ≤ y.

Theorem 2.1 [V |X1 ≤ x1, . . . , Xn ≤ xn] ≤lr [V |X1 ≤ x′1, . . . , Xn ≤ x′n], ∀x = (x1, . . . , xn) ∈

3

995

Rn+, x′ = (x′1, . . . , x

′n) ∈ Rn

+, x ≤ x′. In particular E(V |X1 ≤ x1, . . . , Xn ≤ xn) ≤ E(V |X1 ≤x′1, . . . , Xn ≤ x′n), ∀x ≤ x′.

Proof Fix x, x′ ∈ Rn+. The pdf of [V |X1 ≤ x1, . . . , Xn ≤ xn] is given by

px(x) =(∏n

i=1 Fi(xi))xh(x)

F (x), x > 0.

Now, for x > 0,

px′(x)

px(x)=

(n∏

i=1

Fi(x′i)

F i(xi)

)xF (x)

F (x′).

Since F i(x′i)/F i(xi) ≥ 1, i = 1, 2, . . . , n, it follows that px′(x)/px(x) is increasing in x on

(0,∞). Hence [V |X1 ≤ x1, . . . , Xn ≤ xn] ≤lr [V |X1 ≤ x′1, . . . , Xn ≤ x′n].¤

Note that, for t ∈ Rn+,

E(V n|X1 ≤ t1, . . . , Xn ≤ tn) =rX(t)∏n

i=1 r0,X∗i(ti)

,

where rX(t) = f(t)/F (t) is the overall population reversed failure rate function of random

vector X and r0,X∗i(·) is reversed failure rate function of baseline rv X∗

i , i = 1, 2, . . . , n. The

above corollary has an obvious interpretation in terms of reversed failure rate taking frailty

into account relative to the reversed failure rate when frailty is ignored.

If frailty rv V is considered as some index of risk and X is the total life time of the

system, then the following theorem claims that higher the risk, shorter is the life of system.

Theorem 2.2 [V |X1 ≤ x1, . . . , Xn ≤ xn] ≤lr [V |X1 = x1, . . . , Xn = xn], ∀x = (x1, . . . , xn) ∈Rn

+. In particular E(V |X1 ≤ t1, . . . , Xn ≤ tn) ≤ E(V |X1 = t1, . . . , Xn = tn), ∀x =

(x1, . . . , xn) ∈ Rn+.

Proof Fix x ∈ Rn+. It is easy to see that pdf of rvs [V |X1 ≤ x1, . . . , Xn ≤ xn] and [V |X1 =

x1, . . . , Xn = xn] are given by

px(x) =(∏n

i=1 Fi(xi))xh(x)

F (x), x > 0, and, qx(x) =

xn (∏n

i=1 Fi(xi))xh(x)

E(V n (

∏ni=1 Fi(xi))

V) , x > 0,

4

996

respectively. Now, for x > 0,

qx(x)

px(x)=

xnF (x)

E(V n (

∏ni=1 Fi(xi))

V) .

Clearly, qx(x)/px(x) is increasing in x on (0,∞). Hence [V |X1 ≤ x1, . . . , Xn ≤ xn] ≤lr

[V |X1 = x1, . . . , Xn = xn] ¤

If we consider frailty as an index of risk and if we consider individual whose frailty

exceeds a certain threshold ξ as constituting the high risk group, then it is important to

have information about the distribution function of this group. Thus, P (X ≤ t|V > v) is a

quantity of interest. In the next theorem we provide stochastic bounds for the distribution

function of a high risk group. Here we define following definition.

Definition 2.2 Random variable X is said to be new better than used of second stochastic

order (NBU(2)) if∫ x

0F (u)du ≥ ∫ x

0F (t + u)/F (t)du, for all x, t ≥ 0.

Theorem 2.3 (i) P (X ≤ t|V > ξ) ≤ ∏ni=1 F ξ

i (ti), ∀t ∈ Rn+, ∀ξ ∈ R1

+.

(ii) If V is NBU(2) and E(V ) ≤ 1, then P (X ≤ t|V > ξ) ≥ ∏ni=1 F ξ+1

i (ti), ∀t ∈ Rn+, ∀ξ ∈

R1+.

Proof (i) For fixed t ∈ Rn+ and ξ ∈ R1

+, we have

P (X ≤ t|V > ξ) =

∫∞ξ

P (X ≤ t|V = v)h(v)dv

P (V > ξ)

=

∫∞ξ{∏n

i=1 F vi (ti)}h(v)dv

P (V > ξ)

≤n∏

i=1

F ξi (ti),

since the function φt(x) =∏n

i=1 F xi (ti) is decreasing in x on (0,∞). Hence the assertion

holds.

(ii) Since V is NBU(2), we have∫ x

0P (V > v + ξ)/P (V > ξ)dv ≤ ∫ x

0P (V > v)dv for

all x, ξ ∈ R1+. Also the function φt(x) =

∏ni=1 F x

i (ti) is non-negative and decreasing in x

5

997

on (0,∞), then using Lemma 7.1 (b) given on page 120 of Barlow and Proschan (1981), it

follows that

∫ ∞

0

φt(v)P (V > v + ξ)

P (V > ξ)dv ≤

∫ ∞

0

φt(v)P (V > v)dv, ∀ξ ∈ R1+.

Now using the above inequality, we have

P (X ≤ t|V > ξ) =

∫∞0

φt(v)h(v)dv

P (V > ξ)

= φt(ξ)

[1 + ln (φt(1))

∫ ∞

0

φt(v)P (V > v + ξ)

P (V > ξ)dv

]

≥ φt(ξ)

[1 + ln (φt(1))

∫ ∞

0

φt(v)P (V > v)dv

]

= φt(ξ)F (t). (4)

Since φt(x) is a convex function of x in [0,∞), using the Jensen inequality and the fact that

E(V ) ≤ 1, it follows that

F (t) = E (φt(V )) ≥ φt(E(V )) ≥ φt(1) =n∏

i=1

Fi(ti), ∀t ∈ Rn+.

Using this in (4), we have

P (X ≤ t|V1 > ξ) ≥n∏

i=1

F ξ+1i (ti).

Hence the assertion holds. ¤

Random vector X has pdf

f(x) = f(x1, . . . , xn) =

∫ ∞

0

vn

{n∏

i=1

(F v−1

i (xi)fi(xi))}

h(v)dv, x ∈ Rn. (5)

Definition 2.3 Random vector X is said to be TP2 in pairs if, for every i, j ∈ {1, . . . , n}, i 6=j, for fixed 0 < x < y < ∞ and 0 < xl < ∞, l 6= i, j,

f(x1, . . . , xi−1, y, xi+1, . . . , xn)/f(x1, . . . , xi−1, x, xi+1, . . . , xn) is increasing in xj ∈ (0,∞).

The following theorem states that X is TP2 in pairs.

Theorem 2.4 Random vector X is TP2 in pairs.

6

998

Proof For fixed k, r ∈ {1, . . . , n}, k 6= r, 0 < xk < x′k < ∞ and 0 < xr < x′r < ∞, define

∆ = f(x1, . . . , xk, . . . , xr, . . . , xn)f(x1, . . . , x′k, . . . , x

′r, . . . , xn)

−f(x1, . . . , x′k, . . . , xr, . . . , xn)f(x1, . . . , xk, . . . , x

′r, . . . , xn).

We will show that ∆ ≥ 0. Using (5), we have

∆ =∫ ∞

0

∫ ∞

0

xnyn

n∏i=1

i6=k,r

{fi(xi)F x−1

i (xi)}

n∏i=1

i6=k,r

{fi(xi)F

y−1i (xi)

} fk(xk)fk(x′k)fr(xr)fr(x′r)

F x−1r (xr)F y−1

r (x′r)(F x−1

k (xk)F y−1k (x′k)− F x−1

k (x′k)F y−1k (xk)

)h(x)h(y)dxdy

=∫ ∫

0<x<y<∞xnyn

n∏i=1

i 6=k,r

{fi(xi)F x−1

i (xi)}

n∏i=1

i 6=k,r

{fi(xi)F

y−1i (xi)

} fk(xk)fk(x′k)fr(xr)

fr(x′r)Fx−1k (xk)F x−1

k (x′k)F x−1r (xr)F x−1

r (x′r)(F y−x

k (x′k)− F y−xk (xk)

) (F y−x

r (x′r)− F y−xr (xr)

)

h(x)h(y)dxdy

≥ 0,

since 0 < Fk(xk) ≤ Fk(x′k), 0 < Fr(xr) ≤ Fr(x

′r) and y − x > 0. Hence X is TP2 in pairs.¤

Definition 2.4 Random vector X = (X1, . . . , Xn) is said to be a conditionally increasing

sequence (CIS) if, for i = 2, 3, . . . , n, FXi|X1=x1,...,Xi−1=xi−1(t) ≤ FXi|X1=y1,...,Xi−1=yi−1

(t),

whenever 0 < xj ≤ yj, j = 1, 2, . . . , i− 1, and t ∈ R1, where FXi|X1=x1,...,Xi−1=xi−1(t) denotes

the conditional sf of Xi given that X1 = x1, . . . , Xi−1 = xi−1.

Definition 2.5 Random vector X is said to be associated if Cov(φ1(X), φ2(X)) ≥ 0 for all

increasing functions φ1 and φ2 for which E(φ1(X)), E(φ2(X)) and E(φ1(X)φ2(X)) exist.

Since X is TP2 in pairs ⇒ X is CIS ⇒ X is associated, we have the following corollary.

Corollary 2.1 (i) X is a conditionally increasing sequence (CIS).

(ii) X is associated.

7

999

3 Stochastic Ordering Between Reversed Frailty Mod-

els

Let V1 (V2) is a non-negative rv having Lebesgue pdf h1(x) (h2(x)), −∞ < x < ∞, df

H1(x) (H2(x)), −∞ < x < ∞ and sf H1(x) (H2(x)), −∞ < x < ∞. Suppose that there

are two populations and related individuals in the first (second) population have random

frailty V1 (V2). Let X1, . . . , Xn (Y1, . . . , Yn) be the survival times of n related individuals in

the first (second) population. Then reversed frailty model corresponding to random vector

X = (X1, . . . , Xn) has df given by

F (x) = E

(n∏

i=1

Fi(xi)

)V1 , x ∈ Rn, (6)

where, for i = 1, . . . , n, Fi(·) is the baseline df of some rv X∗i , having support Rn

+. Similarly,

reversed frailty model corresponding to random vector Y = (Y1, . . . , Yn) has df given by

G(x) = E

(n∏

i=1

Gi(xi)

)V2 , x ∈ Rn, (7)

where, for i = 1, . . . , n, Gi(·) is the baseline df of some rv Y ∗i , having support Rn

+.

Stochastic comparisons that translate the order between random variables V1 and V2 into

that between random vectors X and Y helps in understanding the effect of mis-specification

of the frailty distributions. Therefore, our interest is to see if there is a multivariate stochastic

ordering between V1 and V2 and baseline distribution are the same (i.e., Fi(·) = Gi(·), i =

1, . . . , n), then how these translate into a multivariate stochastic ordering between random

vectors X and Y.

Random vector X and Y have pdfs

f(x) =

∫ ∞

0

vn

{n∏

i=1

(F v−1

i (xi)fi(xi))}

h1(v)dv, x ∈ Rn, (8)

and

g(x) =

∫ ∞

0

vn

{n∏

i=1

(Gv−1

i (xi)gi(xi))}

h2(v)dv, x ∈ Rn, (9)

respectively.

8

1000

Definition 3.1 Random vector X is said to be smaller than rvc Y in the likelihood ratio

order (written as X ≤lr Y) if f(x)g(y) ≤ f(x ∧ y)g(x ∨ y), for all x,y ∈ Rn.

The following theorem states that if there is likelihood ratio ordering between random vari-

ables V1 and V2, with the same baseline distributions, then random vectors X and Y preserve

this ordering.

Theorem 3.1 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤lr V2. Then X ≤lr Y.

Proof For fixed x,y ∈ Rn, define ∆1 = f(x ∧ y)g(x ∨ y) − f(x)g(y). We will show that

∆1 ≥ 0. Since the sign of ∆1 depends on the joint distribution of (V1, V2) only through

marginal distributions, without loss of generality, we assume that V1 and V2 are statistically

independent. Then, using (8) and (9), we have

∆1 = E (φ2(V1, V2))− E (φ1(V1, V2)) ,

where, for (x, y) ∈ R2+,

φ1(x, y) = xnyn

n∏i=1

{fi(xi)fi(yi)F

x−1i (xi)F

y−1i (yi)

},

and,

φ2(x, y) = xnyn

n∏i=1

{fi(xi)fi(yi)F

x−1i (xi ∧ yi)F

y−1i (xi ∨ yi)

}.

Define ∆φ21(x, y) = φ2(x, y) − φ1(x, y), (x, y) ∈ R2+. If we can show that ∆φ21(x, y) ≥ 0

and ∆φ21(x, y) ≥ −∆φ21(y, x), whenever 0 ≤ x < y < ∞, then using Theorem 1.C.22 given

on page 51 of Shaked and Shanthikumar (2007), it would follow that ∆1 ≥ 0 and hence

X ≤lr Y. Let x and y be real numbers such that 0 ≤ x < y < ∞. Then,

∆φ21(x, y) = xnyn

[n∏

i=1

(fi(xi)fi(yi))

][n∏

i=1

(F x−1

i (xi ∧ yi)Fy−1i (xi ∨ yi)

)

−n∏

i=1

(F x−1

i (yi)Fy−1i (xi)

)]

.

9

1001

The following three cases arise:

Case I: −∞ < xi ≤ yi < ∞, i = 1, . . . , n.

In this case, ∆φ21(x, y) = ∆φ21(y, x) = 0.

Case II: −∞ < yi ≤ xi < ∞, i = 1, . . . , n.

In this case,

∆φ21(x, y) = xnyn

[n∏

i=1

{fi(xi)fi(yi)F

x−1i (xi)F

x−1i (yi)

}][

n∏i=1

F y−xi (xi)

−n∏

i=1

F y−xi (yi)

]≥ 0,

since 0 < Fi(yi) ≤ Fi(xi), i = 1, . . . , n and y − x > 0.

Also, ∆φ21(x, y) + ∆φ21(y, x) = 0.

Case III: For some l ∈ {1, 2, . . . , n− 1} and a permutation (i1, . . . , in) of (1, . . . , n), −∞ <

xij ≤ yij < ∞, j = 1, . . . , l, and −∞ < yij ≤ xij < ∞, j = l + 1, . . . , n.

In this case,

∆φ21(x, y) = xnyn

[n∏

i=1

(fi(xi)fi(yi))

][l∏

j=1

(F x−1

ij(xij)F

y−1ij

(yij))]

[n∏

j=l+1

(F x−1

ij(yij)F

x−1ij

(xij))][

n∏

j=l+1

F y−xij

(xij)−n∏

j=l+1

F y−xij

(yij)

]≥ 0,

since 0 < Fij(yij) ≤ Fij(xij), j = l + 1, . . . , n, and y − x > 0.

Also,

∆φ21(x, y) + ∆φ21(y, x)

= xnyn

[n∏

i=1

(fi(xi)fi(yi))

][n∏

i=1

(F x−1

i (xi)Fx−1i (yi)

)]

[l∏

j=1

F y−xij

(yij)−l∏

j=1

F y−xij

(xij)

][n∏

j=l+1

F y−xij

(xij)−n∏

j=l+1

F y−xij

(yij)

]≥ 0,

since 0 < Fij(xij) ≤ Fij(yij) j = 1, . . . , l, 0 < Fij(yij) ≤ Fij(xij), j = l + 1, . . . , n and

y − x > 0.

On combining the above three cases we conclude that ∆φ21(x, y) ≥ 0 and ∆φ21(x, y) ≥

10

1002

−∆φ21(y, x), whenever 0 ≤ x < y < ∞. Now the result follows on using Theorem 1.C.22

given on page 51 of Shaked and Shanthikumar (2007). ¤

Definition 3.2 Random vector X is said to be smaller than rvc Y in the weak likelihood

ratio order (written as X ≤wlr Y) if f(y)g(x) ≤ f(x)g(y), whenever x,y ∈ Rn and x ≤ y.

Since X ≤lr Y ⇒ X ≤wlr Y, we have the following corollary.

Corollary 3.1 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤lr V2. Then X ≤wlr Y. ¤

Definition 3.3 Random vector X is said to be smaller than rvc Y in the reversed failure

rate or strong lower orthant increasing ratio order (written as X ≤rfr Y or X ≤sloir Y) if

F (x)G(y) ≤ F (x ∧ y)G(x ∨ y), for all x,y ∈ Rn.

The following theorem states that if there is reversed failure rate ordering between random

variables V1 and V2, with the same baseline distributions, then random vectors X and Y

preserve this ordering.

Theorem 3.2 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤rfr V2. Then X ≤rfr Y.

Proof For fixed x,y ∈ Rn, define ∆2 = F (x ∧ y)G(x ∨ y) − F (x)G(y). We will show that

∆2 ≥ 0. Since the sign of ∆2 depends on the joint distribution of (V1, V2) only through

marginal distributions, without loss of generality, we assume that V1 and V2 are statistically

independent. Then

∆2 = E (φ2(V1, V2))− E (φ1(V1, V2)) ,

where, for (x, y) ∈ R2+,

φ1(x, y) =n∏

i=1

(F xi (xi)F

yi (yi)) , and φ2(x, y) =

n∏i=1

(F xi (xi ∧ yi)F

yi (xi ∨ yi)) .

11

1003

Define, ∆φ21(x, y) = φ2(x, y) − φ1(x, y), (x, y) ∈ R2+. If we can show that, for each fixed

y ∈ R1+, ∆φ21(x, y) decreases in x on [0, y) and ∆φ21(x, y) ≥ −∆φ21(y, x), whenever 0 ≤ x <

y < ∞, then using Theorem 1.B.48 given on page 38 of Shaked and Shanthikumar (2007) it

would follow that ∆2 ≥ 0 and hence X ≤rfr Y. For 0 ≤ x < y < ∞, we have

∂

∂x∆φ21(x, y) =

{n∏

i=1

(F xi (xi ∧ yi)F

yi (xi ∨ yi))

}ln

(n∏

i=1

Fi(xi ∧ yi)

)

−{

n∏i=1

(F xi (xi)F

yi (yi))

}ln

(n∏

i=1

Fi(xi)

),

and

∆φ21(x, y) + ∆φ21(y, x) =n∏

i=1

(F xi (xi ∧ yi)F

yi (xi ∨ yi)) +

n∏i=1

(F xi (xi ∨ yi)F

yi (xi ∧ yi))

−n∏

i=1

(F xi (xi)F

yi (yi))−

n∏i=1

(F

x

i (yi)Fy

i (xi)).

Let x and y be real numbers such that 0 ≤ x < y < ∞. The following three cases arise:

Case I: −∞ < xi ≤ yi < ∞, i = 1, . . . , n.

In this case,

∂

∂x∆φ21(x, y) = 0, and, ∆φ21(x, y) + ∆φ21(y, x) = 0.

Case II: −∞ < yi ≤ xi < ∞, i = 1, . . . , n.

In this case,

∂

∂x∆φ21(x, y) =

{n∏

i=1

F xi (xi)F

xi (yi)

}[{n∏

i=1

F y−xi (xi)

}ln

(n∏

j=1

Fj(yj)

)

−{

n∏i=1

F y−xi (yi)

}ln

(n∏

j=1

Fj(xj)

)]≤ 0,

since 0 < y−x < ∞, 0 <∏n

i=1 Fi(yi) ≤∏n

i=1 Fi(xi) ≤ 1 and ln (∏n

i=1 Fi(yi)) ≤ ln (∏n

i=1 Fi(xi)) ≤0.

Also, ∆φ21(x, y) + ∆φ21(y, x) = 0.

Case III: For some l ∈ {1, 2, . . . , n− 1} and a permutation (i1, . . . , in) of (1, . . . , n), −∞ <

12

1004

xij ≤ yij < ∞, j = 1, . . . , l, and −∞ < yij ≤ xij < ∞, j = l + 1, . . . , n.

In this case,

∂

∂x∆φ21(x, y) =

l∏

j=1

F xij

(xij )Fyij

(yij )

n∏

j=l+1

F xij

(xij )Fxij

(yij )

n∏

j=l+1

F y−xij

(xij )

l∑

j=1

ln(Fij (xij ))

+n∑

j=l+1

ln(Fij (yij ))

−

n∏

j=l+1

F y−xij

(yij )

l∑

j=1

ln(Fij (xij )) +n∑

j=l+1

ln(Fij (xij ))

≤ 0,

since 0 < y − x < ∞, 0 ≤ ∏nj=l+1 F y−x

ij(yij) ≤

∏nj=l+1 F y−x

ij(xij) and

l∑j=1

ln(Fij(xij)) +n∑

j=l+1

ln(Fij(yij)) ≤l∑

j=1

ln(Fij(xij)) +n∑

j=l+1

ln(Fij(xij)) ≤ 0.

Also,

∆φ21(x, y) + ∆φ21(y, x) =

[n∏

i=1

{F xi (xi)F

xi (yi)}

] [l∏

j=1

F y−xij

(yij)−l∏

j=1

F y−xij

(xij)

]

[n∏

j=l+1

F y−xij

(xij)−n∏

j=l+1

F y−xij

(yij)

]≥ 0,

since 0 < y − x < ∞, 0 ≤ ∏lj=1 Fij(xij) ≤

∏lj=1 Fij(yij) and 0 ≤ ∏n

j=l+1 Fij(yij) ≤∏nj=l+1 Fij(xij).

On combining the above three cases it follows that, for each fixed y ∈ R1+, ∆φ21(x, y) de-

creases in x on [0, y) and ∆φ21(x, y) ≥ −∆φ21(y, x), whenever 0 ≤ x < y < ∞. Now the

result follows on using Theorem 1.B.48 given on page 38 of Shaked and Shanthikumar (2007).

¤

Definition 3.4 Random vector X is said to be smaller than rvc Y in the weak reversed

failure rate or lower orthant increasing ratio order (written as X ≤wrfr Y or X ≤loir Y) if

F (y)G(x) ≤ F (x)G(y), whenever x,y ∈ Rn and x ≤ y.

Since X ≤rfr Y ⇒ X ≤wrfr Y, the following corollary to the above theorem is immediate.

Corollary 3.2 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤rfr V2. Then X ≤wrfr Y. ¤

13

1005

The following lemma will be useful in proving the next result of this section.

Lemma 3.1 (i) Let 0 < α < β < ∞ and let ψ1 : (0, 1) → R1 be defined by

ψ1(x) =1− xβ

1− xα, 0 < x < 1.

Then ψ1(x) is an increasing function of x on (0, 1).

(ii) Let 0 < α < ∞ and let 0 < z1 ≤ z2 < 1. Then

1− zα1

1− zα2

≥ zα1 (ln z1)

zα2 (ln z2)

.

Proof (i) We have, for 0 < x < 1,

ψ′1(x) =d

dxψ1(x) =

xα−1ψ2(x)

(1− xα)2, 0 < x < 1,

where ψ2(x) = (β − α)xβ − βxβ−α + α, 0 < x < 1. Clearly ψ′2(x) = β(β − α)xβ−α−1(xα −1) < 0, ∀0 < x < 1. It follows that ψ2(x) ≥ limx→1− ψ2(x) = 0, ∀0 < x < 1. Therefore

ψ′1(x) ≥ 0, ∀0 < x < 1.

(ii) Consider the functions k1(x) = 1 − zx1 , x ∈ R1

+, and k2(x) = 1 − zx2 , x ∈ R1

+. Using

the generalized mean value theorem, it follows that

1− zα1

1− zα2

=k1(α)− k1(0)

k2(α)− k2(0)=

k′1(ξ)k′2(ξ)

=zξ1(ln z1)

zξ2(ln z2)

,

for some ξ ∈ (0, α). Since 0 < z1/z2 ≤ 1, ξ ∈ (0, α) and ln z1/ ln z2 > 0, it follows that

1− zα1

1− zα2

=zξ1(ln z1)

zξ2(ln z2)

≥ zα1 (ln z1)

zα2 (ln z2)

. ¤

Using (8)− (9), and the Fubini theorem, it follows that random vectors X and Y have sfs

F (x) =

∫ ∞

0

{n∏

i=1

(1− F vi (xi))

}h1(v)dv, x ∈ Rn, (10)

and

G(x) =

∫ ∞

0

{n∏

i=1

(1−Gvi (xi))

}h2(v)dv, x ∈ Rn, (11)

respectively.

14

1006

Definition 3.5 Random vector X is said to be smaller than rvc Y in the failure rate

or strong upper orthant increasing ratio order (written as X ≤fr Y or X ≤suoir Y) if

F (x)G(y) ≤ F (x ∧ y)G(x ∨ y), for all x,y ∈ Rn.

The following theorem states that if there is failure rate ordering between random vari-

ables V1 and V2, with the same baseline distributions, then random vectors X and Y preserve

this ordering.

Theorem 3.3 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤fr V2. Then X ≤fr Y.

Proof For fixed x, y ∈ Rn, define ∆3 = F (x ∧ y)G(x ∨ y)− F (x)G(y). We will show that

∆3 ≥ 0. If Fi(xi) = 1 for some i ∈ {1, . . . , n} or Fj(yj) = 1 for some j ∈ {1, . . . , n} then,

using (10) and (11), F (x) = F (y) = G(x) = G(y) = 0 and so ∆3 = 0. Now suppose that,

for every i, j ∈ {1, . . . , n}, Fi(xi) < 1 and Fj(yj) < 1. Since the sign of ∆3 depends on the

joint distribution of (V1, V2) only through marginal distributions, without loss of generality,

we assume that V1 and V2 are statistically independent. Then, using (10) and (11), we have

∆3 = E (φ2(V1, V2))− E (φ1(V1, V2)) ,

where, for (x, y) ∈ R2+,

φ1(x, y) =n∏

i=1

{(1− F xi (xi)) (1− F y

i (yi))} , and

φ2(x, y) =n∏

i=1

{(1− F xi (xi ∧ yi)) (1− F y

i (xi ∨ yi))} .

Define, ∆φ21(x, y) = φ2(x, y) − φ1(x, y), (x, y) ∈ R2+. If we show that, for each fixed x ∈

R1+, ∆φ21(x, y) is an increasing function of y on (x,∞) and ∆φ21(x, y) ≥ −∆φ21(y, x),

whenever 0 ≤ x < y < ∞, then using Theorem 1.B.10 given on page 22 of Shaked and

Shanthikumar (2007) it would follow that ∆3 ≥ 0 and hence X ≤fr Y.

For 0 ≤ x < y < ∞, we have

∂

∂y∆φ21(x, y) =

[n∏

i=1

(1− F xi (xi ∧ yi))

][n∏

i=1

(1− F yi (xi ∨ yi))

]n∑

k=1

F yk (xk ∨ yk) (− ln(Fk(xk ∨ yk)))

1− F yk (xk ∨ yk)

−[

n∏

i=1

(1− F xi (xi))

][n∏

i=1

(1− F yi (yi))

]n∑

k=1

F yk (yk) (− ln(Fk(yk)))

1− F yk (yk)

,

15

1007

and

∆φ21(x, y) + ∆φ21(y, x) =n∏

i=1

{(1− F xi (xi ∧ yi)) (1− F y

i (xi ∨ yi))} −n∏

i=1

{(1− F xi (xi)) (1− F y

i (yi))}

+n∏

i=1

{(1− F xi (xi ∨ yi)) (1− F y

i (xi ∧ yi))} −n∏

i=1

{(1− F xi (yi)) (1− F y

i (xi))} .

Let x and y be real numbers such that 0 ≤ x ≤ y < ∞. The following three cases arise:

Case I: −∞ < xi ≤ yi < ∞, i = 1, . . . , n.

In this case,

∂

∂y∆φ21(x, y) = 0, and, ∆φ21(x, y) + ∆φ21(y, x) = 0.

Case II: −∞ < yi ≤ xi < ∞, i = 1, . . . , n.

In this case,

∂

∂y∆φ21(x, y) =

[n∏

i=1

(1− F xi (yi))(1− F y

i (xi))

]n∑

k=1

F yk (xk) (− ln(Fk(xk)))

1− F yk (xk)

−[

n∏i=1

(1− F xi (xi))(1− F y

i (yi))

]n∑

k=1

F yk (yk) (− ln(Fk(yk)))

1− F yk (yk)

. (12)

Using Lemma 3.1, along with the facts that 0 < Fi(yi) ≤ Fi(xi) < 1, i = 1, . . . , n and

0 ≤ x < y < ∞, we have

(1− F xi (xi))(1− F y

i (yi)) ≤ (1− F xi (yi))(1− F y

i (xi)), ∀i ∈ {1, 2, . . . , n}, (13)

andF y

k (xk) (− ln Fk(xk))

1− F yk (xk)

≤ F yk (yk) (− ln Fk(yk))

1− F yk (yk)

, ∀k ∈ {1, 2, . . . , n}. (14)

Now, on using (13) and (14) in (12), it follows that

∂

∂y∆φ21(x, y) ≥ 0.

Also, in this case,

∆φ21(x, y) + ∆φ21(y, x) = 0.

16

1008

Case III: For some l ∈ {1, 2, . . . , n − 1} and a permutation (i1, . . . , in) of (1, . . . , n), 0 ≤xij ≤ yij < ∞, j = 1, . . . , l, and 0 ≤ yij ≤ xij < ∞, j = l + 1, . . . , n.

In this case,

∂

∂y∆φ21(x, y) =

l∏

j=1

{(1− F x

ij(xij ))(1− F y

ij(yij ))

}

n∏

j=l+1

{(1− F x

ij(yij ))(1− F y

ij(xij ))

}

(l∑

k=1

F yk (yk) (− ln(Fk(yk)))

1− F yk (yk)

+n∑

k=l+1

F yk (xk) (− ln(Fk(xk)))

1− F yk (xk)

)−

n∏

j=l+1

{(1− F x

ij(xij ))(1− F y

ij(yij ))

}

(n∑

k=1

F yk (yk) (− ln(Fk(yk)))

1− F yk (yk)

) . (15)

Using Lemma 3.1, along with the facts that 0 < Fij(yij) ≤ Fij(xij) < 1, j = l + 1, . . . , n,

and 0 ≤ x < y < ∞, it follows that

(1− F x

ij(xij)

)(1− F y

ij(yij)

)≤

(1− F x

ij(yij)

)(1− F y

ij(xij)

), j = l + 1, . . . , n (16)

andF y

ij(xij)

(− ln(Fij(xij))

1− F yij(xij)

≤ F yij(yij)

(− ln(Fij(yij))

1− F yij(yij)

, j = l + 1, . . . , n. (17)

Now, on using (16) and (17) in (15), it follows that

∂

∂y∆φ21(x, y) ≥ 0.

Also, in this case,

∆φ21(x, y) + ∆φ21(y, x)

=

[l∏

j=1

{(1− F x

ij(yij)

)(1− F y

ij(xij)

)}−

l∏j=1

{(1− F x

ij(xij)

)(1− F y

ij(yij)

)}]

[n∏

j=l+1

{(1− F x

ij(xij)

)(1− F y

ij(yij)

)}−

n∏

j=l+1

{(1− F x

ij(yij)

)(1− F y

ij(xij)

)}].

(18)

On using Lemma 3.1 (i), along with the facts that 0 < Fij(xij) ≤ Fij(yij), j = 1, . . . , l and

0 ≤ x < y < ∞, it follows that

(1− F x

ij(xij)

)(1− F y

ij(yij)

)≤

(1− F x

ij(yij)

)(1− F y

ij(xij)

), j = 1, . . . , l. (19)

17

1009

Now, on using (18) and (19) in (17), it follows that ∆φ21(x, y) + ∆φ21(y, x) ≥ 0.

On combining the above three cases we conclude that, for each fixed x ∈ R1+, ∆φ21(x, y)

is an increasing function of y on (x,∞) and ∆φ21(x, y) ≥ −∆φ21(y, x), whenever 0 ≤ x <

y < ∞. Now the result follows on using Theorem 1.B.10 given on page 22 of Shaked and

Shanthikumar (2007). ¤

Definition 3.6 Random vector X is said to be smaller than rvc Y in the weak failure rate

or upper orthant increasing ratio order (written as X ≤wfr Y or X ≤uoir Y) if F (y)G(x) ≤F (x)G(y), whenever x,y ∈ Rn and x ≤ y.

Since X ≤fr Y ⇒ X ≤wfr Y, the following corollary to the above theorem is immediate.

Corollary 3.3 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤fr V2. Then X ≤wfr Y. ¤

Definition 3.7 Random vector X is said to be smaller than rvc Y in the usual stochastic

order (written as X ≤st Y) if E(φ(X)) ≤ E(φ(Y)), for all increasing functions φ : Rn → R1

for which the expectations exist.

The following theorem states that if there is usual stochastic ordering between random

variables V1 and V2, with the same baseline distributions, then random vectors X and Y

preserve this ordering.

Theorem 3.4 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤st V2. Then X ≤st Y.

Proof Let φ : Rn → R be an increasing function, such that E(φ(X)) and E(φ(Y)) exist.

18

1010

Then, using (8), (9) and the Fubini theorem, we have

∆4 = E(φ(Y))− E(φ(X))

=

∫ ∞

0

vnh2(v)

(∫

Rn+

φ(t)

{n∏

i=1

Fv−1

i (ti)fi(ti)

}dt

)dv

−∫ ∞

0

vnh1(v)

(∫

Rn+

φ(t)

{n∏

i=1

F v−1i (ti)fi(ti)

}dt

)dv

=

∫ ∞

0

h2(v)

(∫

[0,1]nφ(F−1

1 (u1v1 ), . . . , F−1

n (u1vn ))du

)dv

−∫ ∞

0

h1(v)

(∫

[0,1]nφ(F−1

1 (u1v1 ), . . . , F−1

n (u1vn ))du

)dv

= E(k(V2))− E(k(V1)),

where dt = dt1 . . . , dtn, du = u1, . . . , un and

k(v) =

∫

[0,1]nφ(F−1

1 (u1v1 ), . . . , F−1

n (u1vn ))du, v ≥ 0.

Clearly k(v) is an increasing function of v on Rn+. Since V1 ≤st V2, we conclude that ∆4 =

E(k(V2))− E(k(V1)) ≥ 0, and therefore X ≤st Y. ¤

Definition 3.8 Random vector X is said to be smaller than rvc Y in the upper orthant

order (written as X ≤uo Y) if F (x) ≤ G(x) , for every x ∈ Rn.

Since X ≤st Y ⇒ X ≤uo Y we have the following corollary.

Corollary 3.4 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤st V2. Then X ≤uo Y. ¤

Definition 3.9 Random vector X is said to be smaller than rvc Y in the lower orthant

order (written as X ≤lo Y) if F (x) ≥ G(x), for every x ∈ Rn.

In the following theorem, we provide conditions under which X ≤lo Y holds.

Theorem 3.5 For reversed frailty models (6) and (7), suppose that baseline distributions

are identical (i.e., Fi(·) = Gi(·), i = 1, . . . , n), and V1 ≤icv V2. Then X ≤lo Y.

19

1011

Proof For fixed x ∈ Rn, consider

∆5 = F (x)−G(x) = E(ψ(V2))− E(ψ(V1)),

where ψ(v) = − (∏n

i=1 Fi(xi))v, v ∈ R1

+. For fixed x ∈ Rn, since ψ(v) is an increasing and

concave function of v on R1+, and V1 ≤icv V2, it follows that ∆5 ≥ 0. Therefore F (x) ≥

G(x), ∀x ∈ Rn, and the assertion is proved. ¤

Now we will compare reversed frailty models (6) and (7), when baseline distributions

differ but frailty rvs V1 and V2 are identically distributed. For such models, the following

theorem provides conditions under which X ≤st Y holds.

Theorem 3.6 For reversed frailty models (6) and (7), suppose that frailty rvs V1 and V2 are

identically distributed (i.e., H1(·) ≡ H2(·)) and X∗i ≤st Y ∗

i , i = 1, . . . , n. Then X ≤st Y.

Proof Let φ : Rn → R be an increasing function such that E(φ(X)) and E(φ(Y)) exist.

Then, using (8), (9) and the Fubini theorem, we have

∆6 = E(φ(Y))− E(φ(X))

=

∫ ∞

0

vnh1(v)

(∫

Rn+

φ(t)n∏

i=1

(gi(ti)G

v−1i (ti)

)dt

)dv

−∫ ∞

0

vnh1(v)

(∫

Rn+

φ(t)n∏

i=1

(fi(ti)F

v−1i (ti)

)dt

)dv

=

∫ ∞

0

h1(v)

(∫

[0,1]nφ(G−1

1 (u1v1 ), G−1

2 (u1v2 ), . . . , G−1

n (u1vn ))du

)dv

−∫ ∞

0

h1(v)

(∫

[0,1]nφ(F−1

1 (u1v1 ), F−1

2 (u1v2 ), . . . , F−1

n (u1vn ))du

)dv.

Since X∗i ≤st Y ∗

i , i = 1, . . . , n, we have F−1i (u

1/vi ) ≤ G−1

i (u1/vi ), ∀ui ∈ [0, 1] and v >

0, i = 1, . . . , n. Also since φ(·) is an increasing function, we conclude that ∆6 ≥ 0 and hence

X ≤st Y. ¤

Theorem 3.7 For reversed frailty models (6) and (7), suppose that X∗i ≤st Y ∗

i , i = 1, . . . , n,

and V1 =st V2. If t = (t1, . . . , tn) ∈ Rn+, then [V1|Y1 ≤ t1, . . . , Yn ≤ tn] ≤lr [V1|X1 ≤

t1, . . . , Xn ≤ tn].

20

1012

Proof Fix t ∈ Rn+. It is easy to see that the pdfs of rvs [V1|X1 ≤ t1, . . . , Xn ≤ tn] and

[V1|Y1 ≤ t1, . . . , Yn ≤ tn] are given by

pt(x) =(∏n

i=1 Fi(ti))xh1(x)

F (t), x > 0, and, qt(x) =

(∏n

i=1 Gi(ti))xh1(x)

G(t), x > 0,

respectively. Now, for x > 0,

pt(x)

qt(x)=

G(t)

F (t)

(n∏

i=1

Fi(ti)

Gi(ti)

)x

.

Since, X∗i ≤st Y ∗

i , we have Fi(ti)/Gi(ti) ≥ 1, i = 1, 2, . . . , n. It follows that pt(x)/qt(x) is

increasing in x on (0,∞) and therefore [V1|Y1 ≤ t1, . . . , Yn ≤ tn] ≤lr [V1|X1 ≤ t1, . . . , Xn ≤tn]. ¤

Since, for t ∈ Rn+, [V1|Y1 ≤ t1, . . . , Yn ≤ tn] ≤lr [V1|X1 ≤ t1, . . . , Xn ≤ tn] ⇒ E(ψ(V1|Y1 ≤

t1, . . . , Yn ≤ tn)) ≤ E(ψ(V1|X1 ≤ t1, . . . , Xn ≤ tn)), for any increasing function ψ : R1 → R1

for which expectations exist, we have the following corollary.

Corollary 3.5 For reversed frailty models (6) and (7), suppose that X∗i ≤st Y ∗

i , i = 1, . . . , n,

and V1 =st V2. If t ∈ Rn+ and α > 0, then E(V α

1 |Y1 ≤ t1, . . . , Yn ≤ tn) ≤ E(V α1 |X1 ≤

t1, . . . , Xn ≤ tn). ¤

Note that, for t ∈ Rn+,

E(V n1 |X1 ≤ t1, . . . , Xn ≤ tn) =

rX(t)∏ni=1 r0,X∗

i(ti)

, and, E(V n1 |Y1 ≤ t1, . . . , Yn ≤ tn) =

rY(t)∏ni=1 r0,Y ∗i (ti)

,

where rX(t) = f(t)/F (t) and rY(t) = g(t)/G(t) are the overall population reversed failure

rate functions of random vectors X and Y, respectively, and r0,X∗i(·) and r0,Y ∗i (·) are reversed

failure rate functions of baseline rvs X∗i and Y ∗

i , respectively, i = 1, 2, . . . , n. The above

corollary has an obvious interpretation in terms of reversed failure rate taking frailty into

account relative to the reversed failure rate when frailty is ignored.

Acknowledgements

The first author would like to acknowledge the financial assistance received from the C.S.I.R.,

India for carrying out this research work.

21

1013

References

Gupta, R.C., Gupta, P.L. and Gupta, R.D. (1998). Modeling failure time data

by Lehman alternatives Comm. Stat.- Theory Methods , 27 (4), 887–904.

Gupta, R.C. and Gupta, R.D. (2007). Proportional reversed hazard rate model

and its applications. J. Stat. Plann. Inf., Article in press.

Gupta, R.C., Kannan, N. and Raychaudhari, A. (1997). Analysis of lognormal

survival data. Mathematical Biosciences, 139, 103–115.

Gupta, R.C. and Kirmani, S.N.U.A. (2006). Stochastic comparisons in frailty

model. J. Stat. Plann. Inf., 136 (10), 3647–3658.

Gupta, R.C. and Wu, H. (2001). Analyzing survival data by proportional re-

versed hazard model. Int. J. Reliab. Appl., 2 (1), 1–26.

Lai, C.D. and Xie, M. (2005). Stochastic Ageing and Dependence for Reliability.

New York: Springer.

Lehman, E.L. (1953). The power of rank test. Ann. of Math. Statist., 24, 28–43.

Mudholkar, G.S. and Hustson, A.D. (1996). The exponentiated weibull family:

some properties and a flood data application. Comm. Stat.- Theory Methods,

25 (12), 3059–3083.

Mudholkar, G.S. and Srivastva, D.K. (1993). Exponentiated weibull family for

analyzing bathtub failure-rate data. IEEE Trans. Reliability, 42 (2), 299–

302.

Mudholkar, G.S., Srivastva, D.K. and Freimer, M. (1995). The exponentiated

weibull family: a reanalysis of the bus-motar-failure data. Technometrics, 37

(4), 436–445.

Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. Springer.

Vaupel, J.W., Manton, K.G. and Stallard, E. (1979). The impact of heterogeneity

in individual frailty on the dynamics of mortality. Demography, 16 (3), 439–

454.

Xu, M. and Li, X. (2007). Negative dependence in frailty models. J. Stat. Plann.

22

1014

Inf., Article in press.

Nitin Gupta

Department of Mathematics & Statistics,

Indian Institute of Technology, Kanpur

Kanpur - 208 016, INDIA

E-mail: guptan@iitk.ac.in∗

Neeraj Misra

Department of Mathematics & Statistics,

Indian Institute of Technology, Kanpur

Kanpur - 208 016, INDIA

E-mail: neeraj@iitk.ac.in

Rameshwar D. Gupta

Department of Computer Science and Applied Statistics,

University of New brunswick,

St. John, New Brunswick, E2L 4L5, CANADA

E-mail: gupta@unbsj.ca

23

1015