Marshall-Olkin Exponential Pareto Distribution with Application on Cancer Stem Cells

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1 American Journal of Theoretical and Applied Statistics 2017; 6(5-1): doi: /jajtass ISSN: (Print); ISSN: (Online) Marshall-Olkin Exponential Pareto Distribution ith Application on Cancer Stem Cells Khairia El-Said El-Nadi L M Fatehy Nourhan Hamdy Ahmed Department of Mathematics and Computer Science Faculty of Science Alexandria University Alexandria Egypt address: Khairiaelsaid@hotmailcom (K El-Said El-Nadi) lmfatehy@yahoocom (L M Fatehy) nourhamdy2016@gmailcom (N H Ahmed) To cite this article: Khairia El-Said El-Nadi L M Fatehy Nourhan Hamdy Ahmed Marshall-Olkin Exponential Pareto Distribution ith Application on Cancer Stem Cells American Journal of Theoretical and Applied Statistics Special Issue: Statistical Distributions and Modeling in Applied Mathematics Vol 6 No pp 1-7 doi: /jajtass Received: December ; Accepted: December ; Published: January Abstract: A Marshall Olkin variant of exponential Pareto distribution is being introduced in this paper Some of its statistical functions and numerical characteristics among others characteristics function moment generalizing function central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed This density function is utilized to model a real data set of cancer stem cells patients The ne distribution provides a better fit than related distributions The proposed distribution could find applications for instance in the physical and biological sciences hydrology medicine meteorology and engineering Keyords: Pareto Distribution-Cancer Stem Cells-iological Sciences 1 Introduction Cancer stem cells (CSCs) are a subset of tumor cells that possess characteristics associated ith normal stem cells Specifically they have the ability to self-rene differentiate and generate the diverse cells that comprise the tumor CSCs have been identified and isolated in several human cancer types including breast brain colon head and neck leukemia liver ovarian pancreas and prostate These CSCs represent approximately 1% of the tumor as a distinct population and cause relapse and metastasis by giving rise to ne tumors While chemotherapy and other conventional cancer therapies may be more effective at killing bulk tumor cells CSCs may manage to escape and seed ne tumor groth due to the survival of quiescent CSCs Therefore traditional therapies often cannot completely eradicate tumors or prevent cancer recurrence and progression to metastasis With groing evidence supporting the role of CSCs in tumor genesis tumor heterogeneity resistance to chemotherapeutic and radiation therapies and the metastatic phenotype the development of specific therapies that target CSCs holds promise for improving survival and quality of life for cancer patients especially those ith metastatic disease Tumors consist of heterogeneous cell populations in hich only a small fraction less than 1% is able to seed ne tumors by transplantation functionally defined as cancer stem cells (CSCs) There is groing interest in identifying markers and therapeutically targeting the CSC population in tumors Recent studies have shon that CSCs have different drug sensitivities compared to the bulk population and represent an attractive therapeutic target Studying these cells hoever has been a challenge due to their lo abundance in vivo and the phenotypic plasticity they exhibit during expansion Using current methods isolated CSCs lose the expression of CSC markers and tumor initiating capacity hen cultured in vitro or in vivo in xenograft animal models The proportion of CSCs tends to an equilibrium level of less than 1% over time and the cell population derived from CSC cultures typically recapitulates the heterogeneous nature of the original population Thus the goal of this contract topic is to meet the critical need to develop cell culture systems that can specifically gro CSCs for basic and translational research Developments in stem cell engineering and tissue engineering have generated ne culture systems to accelerate the expansion of embryonic induced pluripotent and adult stem cell populations in vitro These systems include technologies such as three dimensional (3D) culture systems containing extracellular matrix components and topological features or bioreactors for large scale culture of cell spheroids Preliminary data suggest that these technologies or

2 2 Khairia El-Said El-Nadi et al: Marshall-Olkin Exponential Pareto Distribution ith Application on Cancer Stem Cells similar culture systems may be applicable for quick and reproducible expansion of CSCs Thus commercial development of these culture systems specifically for CSC culture may have a significant impact in basic research and drug screening applications 2 Marshall-Olkin Method Marshall and Olkin (1997) [1] introduced a method of adding a ne parameter to an existing distribution The resulting ne distribution knon as the Marshall-Olkin extended distribution includes the original distributions a special case and gives more flexibility to model various types of data The Marshall-Olkin extended distribution has survival function 1 hich is the baseline survivor function Then the Marshall-Olkin extended distribution has survival function 1 given y or ritten in an equivalent form + The corresponding PDF takes the form: Finally central role is playing in the reliability theory by the quotient of the probability density function and the survival function h called hazard function (or also frequently called failure rate function) Here e introduce exponential Pareto distribution using the method proposed by (Marshall and Olkin 1997) inputting the exponential Pareto distribution Some statistical properties of the novel distribution are established and certain their special cases are discussed The associated density function is utilized to model a real data set The ne distribution provides a better fit than related distributions as measured by the Anderson Darling and Cram er von Mises statistics No e recall the exponential Pareto distribution by (Kareema Abed Al-Kadim Mohammad Abdalhussain oshi et al (2013) [2] The so called exponential Pareto function one defines " #! Where # is the Pareto distribution 1 % & and is the exponential distribution ( ) So that *;- -( )! % &/ Where is a constant of the Pareto distribution (the loer bound of the possible values that Pareto distributed rv can take on) is the shape parameter - is the number of events per unit time (rate parameter) Hence the CDF of the exponential Pareto distribution can be ritten in the form: *;- 1 ( )%0 1 & / Also the pdf of this distribution is given by: *;- 2" 31 4;) 2 ) % & ( )0 1 / This distribution is similar to Weibull distribution that given by: ( 89 here So no e can take the baseline CDF according to increase one more parameter to apply Marshall Olkin technique as follos: Hence the pdf is given by: 1 ( )%0 1 &/ (1) : ; ) % & +<( )%0 1 &/ (2) Applying Marshal-Olkin technique e ill get a ne distribution that called Marshall-Olkin exponential Pareto distribution (MOEP) hich have the CDF that denoted by: >% 1 &/?0 7 >%0 1 &/?0 According to (2) the related PDF reads 1 5( )% &/ - A@% +&( )% &/ )% ( &/ -1 5 A@ +1 5( )% &/ 1 5( )% &/ A

3 American Journal of Theoretical and Applied Statistics 2017; 6(5-1): After a fe steps e can get the final result as follo: 7>/ 1 /C >% 0 1 &/?0 D7 >%0 1 &/?0 E F (4) Here - >0 5 >0 α>0 >0 hile F H denotes the characteristic function of the set A that is * H 1 hen x A and vanishes elsehere Accordingly the four parameter distribution of the rv X having CDF in the form (2) e ill signify this correspondence X MOEP ( 5 - ) 3 Moments efore concentrating on the deviation of the I JK ra moment of the MOEP () distribution e introduce the foxright function Ψ M [ 3] [4] hich is a generalization of the familiar generalized hyper geometric function F M ith N numerator parameters Q Q C and T N denominator parameters U U C Z defined by Ψ Q X Q X U Y U M Y M [A\ ΓQ X ^ Q X ^ [ ΓU Y ^ U M Y M^ Where the empty products are conventionally taken to be equal 1 hile X b > 0 e ; 1 Y f > 0 g ; 1T M 1+\Y b \X b 0 bj bj We no derive closed form representations of the real order moments of a rv * MOEP () l* m m! p - m o 5+ ( )% &/ s r o r! o@1 1 5( )% &/ A r n q It is easy to expand the denominator of the PDF (4) into a poer series as follos: y 1 1 5( )% &/ ()% &/ v( )% &/ y v( )% &/ ! 2! 3! Or e can rite it on another form: 1 1 5( )% &/ \^+11 5 v( )% &/ j \ v( )% &/ j No interchanging the integral and sum e have: l* m 5\ m j + - ( )% &/ v( )% &/! 5 \ m ( )% &/! + 5- \2 1 5 m ( )% &/! The I JK moment is a linear combination of integrals I (considered already for a similar purpose by Nadarajah and kotz in [5] [6] here I f ( }~! g Q > 0 The folloing representation of this integral for general parameter values as obtained by Pogany and Saxena in [5]: see [6 7] I f Ψ ; g Q < 0 < < 1 Γ g +Q f 1 1 Ψ D Šg 1 Œ Q E > 1 Q f Thus for all 01 e have

4 4 Khairia El-Said El-Nadi et al: Marshall-Olkin Exponential Pareto Distribution ith Application on Cancer Stem Cells l* m 5\ IŠI+1^+1^ \2 1 5 IŠI+^+1^+1-5 m \2 1 5 ^+1 m hen 1 e have I A ^+1 m \2 1 5 ^+1 m l* m 5Γ r \ ^+1 m m I+ - A ^+1 The remaining values of the parameter g > 1 lead to the expected value: l* m 5 Š - m \ ^+1 m Ψ D ŠI+1 1 Œ ^+1 - E + 5 Š - m\ ^+1 m p I o Š o +11 Œ o Š - n Ψ s ^+1 q rrr Thus e get the folloing result: Theorem Let the rv * l g g % ) /5& and all parameters > 0 then for I > 1 e have: 5 m \2 1 5 ^+1 m I A ^+1 m \2 1 5 ^+1 m I+ - <1 < 0 A ^+1 5Γr l* m Š+ - m\ ^+1 m 1 5 Š - m\ ^+1 m Ψ D ŠI+1 1 Œ ^+1 5 E+ - Š - m\ 2 p I 1 5 ^+1 m Ψ o Š o +11 s Œ o Š - >1 n ^+1 q rrr estimated by the maximum likelihood in conjunction ith the 4 Parameter Estimation N Maximize command in the symbolic computational In this section e ill make use of the MOEP extended package Mathematica Additionally to goodness-of-fit Weibull (Ex W) (Peng and Yan 2014) exponential-weibull measures are proposed to compare the density estimates (EW) (Cordeiro et al 2013c) [8-13] to parameter Weibull In order to estimate the parameters of the proposed MOEP (Weibull) distribution to model to ell knon real data density function as defined in Equation (4) the Log sets namely the Carbon fibers (Nichols and Padgett 2006) Likelihood of the sample is maximized ith respect to the and the Cancer patients (Lee and Wang 2003) data parameters iven sets[14] The parameters of the MOEP distribution can be the data xi i 1 n the Log Likelihood function is ^ :5 +\log + - \-Š j j \log@1 1 5( )% &/ A >% 0 1 /?0 & 7 7 j 0 (5) j + j 7 >%0 1 & /?0 œ % >/ 1 &%0 1 &/ >%0 1 & /?0 j 0 (6) D7 >%0 /?0 1 & E / 1 &%0 % 1 &/C % ) & + 1 &/ j j j 0 (7) 7% >/ 1 &%0 1 &/C >% 0 1 & /?0 % 0 1 &/ % 0 D7 >%0 /?0 1 & E

5 American Journal of Theoretical and Applied Statistics 2017; 6(5-1): >% 0 1 /?0 & )% 0 1 &/ >/ žÿ 1 &%0 1 &/C žÿ % > 1 &%0 1 &/C j 0 (8) D7 >%0 1 /?0 & >/ 1 &%0 1 &/C A - % & : 7 j + % j Solving these equations simultaneously yields the maximum likelihood estimates (MLEs) of the four parameters Numerical iterative techniques are then necessary to estimate the model parameters It is possible to determine the global maxima of the log-likelihood by taking different initial values for the parameters Hoever e observed that the MLEs for this model are not very sensitive to the initial estimates For interval estimation on the model parameters e require the Fisher information matrix; hoever in this article e leave this routine calculation to the interested reader 5 oodness-of-fit Statistics The Anderson-Darling and the Cram er-von Mises statistics are idely utilized to determine ho closely a specific distribution hose associated cumulative distribution function is denoted by cdf (000) fits the empirical distribution associated ith a given data set These statistics are: X % + y +1& ^+ 2e 1log%[ bj b 1 [ b & Š 2^+1 \Š[ 1 b 2e 1 2^ ^ bj respectively here [ b ª b the ª b values being the ordered observations The smaller these statistics are the better the fit 6 Application No e ill make use of the MOEP to parameter gamma (amma) to parameter Weibull [23] generalized gamma () [13] provost type gammaweibull [10][11] extended Weibull (ExtW)[12] distributions to model to ell knon real data set namely the Cancer patients [14] data set The parameters of the MOEP distribution can be estimated from the loglikelihood of the samples in conjunction ith the N Maximize command in the symbolic computational package Mathematica More specifically the models being considered are: The classical gamma distribution ith density function: The classical Weibull distribution ith density function: g f( - % - & ) > 0g > 0- > 0 The generalized gamma () distribution [13-stacy 1962] ith density function: g- ( ) > 0 > 0g > 0- > 0 Γ g The Provost type gamma Weibull distribution (Provost et al 2011)[11] ith density function: g-f f ( ) > 0 +g > 0- > 0 Γ1+ g The extended Weibull (ExtW) distribution [Peng and Yan 2014] [12] ith density function: Comp [15-20] Q +U ( ± ~² ³ 0 > 0Q > 0U > The Cancer Stem Cells Patients Data Set The second data set represents the remission times (in months) of a random sample of 128 bladder cancer patients as reported in Lee and Wang (2003) The data are If e take the special case of MOEP hen p1 then the PDF and CDF estimates of the MOEP distribution for Cancer patients data are plotted in Figure (1) ( «> 0 > 0 > 0 Γ

6 6 Khairia El-Said El-Nadi et al: Marshall-Olkin Exponential Pareto Distribution ith Application on Cancer Stem Cells Figure 1 The Cancer Patients data fitted using the maximum likelihood approach; Left panel: The MOEP PDF estimates superimposed on the histogram for Cancer patients data Right panel: The MOEP CDF estimates and empirical CDF The estimates of the parameters and the values of the Anderson-Darling and Cram er-von Mises goodness of fit statistics are given in Table (1) It is seen that the proposed MOEP model provides the best fit for the both data sets Table 1 Estimates of the Parameters and oodness-of-fit Statistics for the Cancer Patients Data Distribution Estimates µ µ amma ( ) Weibull (g-) (g- ) g ExtW (a b c) * MOEP (-5g) *10 (For different applications see [21-30] Acknoledgments We ould like to thank the referees and Professor Dr Judy arland for their careful reading of the paper and their valuable comments References [1] Marshall A M and Olkin I A ne method for adding a parameter to a family of distributions ith application to the exponential and Weibull families iometrika [2] Kareema Abed Al-Kadim and Mohammad Abdalhussain oshi Exponential Pareto Distribution Mathematical theory and modeling no 1 volume3 no [3] H M Srivastava SOME FOX-WRIHT ENERALIZED HYPEREOMETRIC FUNCTIONS AND ASSOCIATED FAMILIES OF CONVOLUTION OPERATORS Applicable Analysis and Discrete Mathematics 1 (2007) [4] Miller A R and Moskoitz I S Reduction of a class of Fox-Wright Psi functions for certain rational parameters Comput Math Appl [5] Nadarajah S Cordeiro M and Ortega Edin M M eneral results for the beta modified Weibull distribution J Statist Comput Simulation [6] Nadarajah S and Kotz S On a distribution of Leipnik and Pearce ANZIAM J [7] Cordeiro M Ortega Edin M M and Cunha D C C The exponentiated generalized class of distributions J Data Sci [8] Cordeiro M Ortega Edin M M and Lemonte A J The exponential Weibull lifetime distribution J Statist Comput Simulation [9] Pogany T K and Saxena R K The gamma-weibull distribution revisited Anais Acad rasil Ci encias [10] Provost S Saboor A and Ahmad M The gamma Weibull distribution Pak J Statist [11] Peng X and Yan Z Estimation and application for a ne extended Weibull distribution Reliab Eng Syst Safety [12] Stacy E W A generalization of the gamma distribution Ann Math Stat [13] Lee E and Wang J Statistical Methods for Survival Data Analysis (Ne York: Wiley &Sons 2003) [14] Kilbas A A Srivastava H M and Trujillo JJ Theory and Applications of Fractional Differential Equations North- Holland Mathematical Studies Vol 204 (Amsterdam: Elsevier (North-Holland) Science Publishers 2006) [15] Leipnik R and Pearce C E M Independent non identical five parameter gamma Weibull variates and their sums ANZIAM J [16] Luke Y L The Special Functions and Their Approximations Vol I (San Diego: Academic Press 1969 [17] Nichols M D and Padgett W J A bootstrap control chart for Weibull percentiles Qual Reliab Eng Int [18] Pal M and Tiensuan M The beta transmuted Weibull distribution Austrian Journal of Statistics 43 (2)

7 American Journal of Theoretical and Applied Statistics 2017; 6(5-1): [19] P olya y and Szeg o Problems and Theorems in Analysis I: Series Integral Calculus Theory of Functions Third edition (Mosco: Nauka 1978) (in Russian) [20] Risti c M M and alakrishnan N The gamma exponentiated exponential distribution J Statist Comput Simulation 82 (8) [21] Saboor A and Pog any T K Marshall Olkin gamma Weibull distribution ith applications Commun Stat Theor Methods 2014 [22] Khairia El-Said El-Nadi Asymptotic formulas for some cylindrical functions and generalized functions Uspekhi Math Nauk [23] Khairia El-Said El-Nadi Asymptotic methods for non-central Chi square distributions The Proceeding of the Eight Conference of Statistics and Computer Sciences April [24] Mahmoud M El-orai Khairia El-Said El-Nadi Osama L and Hamdy M Numerical methods for some nonlinear stochastic differential equations Applied mathematics and computations [25] Mahmoud M El-orai Khairia El-Said El-Nadi Osama L Mostafa and Hamdy M Ahmed Semigroup and some fractional stochastic integral equations International Journal Pure & Applied Mathematical Sciences Vol 3 No 1 pp [26] Khairia El-Said El-Nadi On some stochastic parabolic differential equations in a Hilbert space Journal of Applied mathematics and Stochastic Analysis [27] Mahmoud M El-orai Khairia El-Said El-Nadi and Hoda A Foad On some fractional stochastic delay differential equations Computers and Mathematics ith Applications [28] Mahmoud M El-orai Khairia El-Said El-Nadi amd Eman El-Akabay On some fractional evolution equations Computers and Mathematics ith Applications [29] Khairia El-Said El-Nadi Wagdy El-Sayed and Ahmed Khdher Qassem Mathematical model of brain tumor International Research Journal of Engineering and Technology (IRJET) Vol 2 Aug 2015 [30] Khairia El-Said El-Nadi Wagdy El-Sayed and Ahmed Khdher Qassem On some dynamical systems of controlling tumor groth International Journal of Applied Science and Mathematics Vol 2 Issue