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
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Inf., Article in press.
Nitin Gupta
Department of Mathematics & Statistics,
Indian Institute of Technology, Kanpur
Kanpur - 208 016, INDIA
E-mail: [email protected]∗
Neeraj Misra
Department of Mathematics & Statistics,
Indian Institute of Technology, Kanpur
Kanpur - 208 016, INDIA
E-mail: [email protected]
Rameshwar D. Gupta
Department of Computer Science and Applied Statistics,
University of New brunswick,
St. John, New Brunswick, E2L 4L5, CANADA
E-mail: [email protected]
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