A New Extended Uniform Distribution

Transcription

1 International Journal of Statistical Distriutions and Applications 206; 2(3): doi: 0648/jijsd ISS: (Print); ISS: (Online) A ew Extended Uniform Distriution K K Sankaran K Jayakumar 2 Department of Statistics Sree arayana College Kerala India 2 Department of Statistics Uniersity of Calicut Kerala India address: snsankaran08@gmailcom (K K Sankaran) jkumar9@rediffmailcom (K Jayakumar) To cite this article: K K Sankaran K Jayakumar A ew Extended Uniform Distriution International Journal of Statistical Distriutions and Applications Vol 2 o pp 3-4doi: 0648/jijsd Receied: Octoer 4 206; Accepted: oemer 7 206; Pulished: Decemer 206 Astract: We introduce a new family of distriutions using truncated the Discrete Mittag- Leffler distriution It can e considered as a generalization of the Marshall-Olkin family of distriutions Some properties of this new family are deried As a particular case a three parameter generalization of Uniform distriution is gien special attention The shape properties moments distriutions of the order statistics entropies are deried and estimation of the unknown parameters is discussed An application in autoregressie time series modeling is also included Keywords: Discrete Mittag-Leffler Distriution Entropy Marshall-Olkin Family of Distriutions Maximum Likelihood Random Variate Generation Truncated egatie Binomial Distriution Uniform Distriution Introduction Many researchers are interested in search that introduces new families of distriutions or generalization of distriutions which can e used to descrie the lifetimes of some deices or to descrie sets of real data Exponential Rayleigh Weiull and linear failure rate are some of the important distriutions widely used in reliaility theory and surial analysis Howeer these distriutions hae a limited range of applicaility and cannot represent all situations found in application For example although the exponential distriution is often descried as flexile its hazard function is constant The limitations of standard distriutions often arouse the interest of researchers in finding new distriutions y extending ones The procedure of expanding a family of distriutions for added flexiility or constructing coariates models is a well known technique in the literature Uniform distriution is regarded as the simplest proaility model and is related to all distriutions y the fact that the cumulatie distriution function taken as a random ariale follows Uniform distriution oer (0) and this result is asic to the inerse method of random ariales generation This distriution is also applied to determine power functions of tests of randomness It is also applied in a power comparison of tests of non random clustering There are also numerous applications in non parametric inference such as Kolmogro-Smirno test for goodness of fit It is well known that Uniform distriution can e used as a representation distriution of round-off errors and it is also connected to proaility integral transformations Ristic and Popoic (2000a ) introduced and studied the properties of a first order autoregressie (AR()) time series model and discussed the parameter estimation of the uniform AR() process Jose and Krishna (20) introduced Marshall-Olkin extended uniform distriution as a generalization of uniform distriution and studied its properties Marshall and Olkin (997) introduced a new family of distriutions y adding a parameter to a gien family of distriutions They started with a parent surial function and considered a family of surial functions gien y = > 0 () They constructed their family of distriutions in the following way Let 2 e a sequence of independent and identically distriuted (iid) random ariales with surial function Let e a geometric random ariale with proaility mass function (pmf) Pr( =n) = α( α) n for n = 2 and 0<α< Then the random ariale U =min{ 2 } has the surial function gien y () If α > and is a geometric random ariale with pmf

2 36 K K Sankaran and K Jayakumar: A ew Extended Uniform Distriution = = n = 2 then the random ariale V = max{ 2 } also has the surial function as in () Many authors hae studied arious uniariate distriutions elonging to the Marshall-Olkin family of distriutions; see Ristic et al (2007) Jose et al (200) and Cordeioro and Lemente (203) Jayakumar and Thomas (2008) proposed a generalization of the family of Marshall-Olkin distriution as ;! = " #$ for > 0! > 0 () + (2) adarajah et al (203) introduced a new family of life time models as follows: Let 2 e a sequence of independent and identically distriuted random ariales with surial function Let e a truncated negatie inomial random ariale with parameters α (0) and θ > 0 That is = = = 2 Consider U = min{ 2 } Then That is 3 = 2 > = 3 = "7 + 8 # (3) Similarly if α > and is a truncated negatie inomial random ariale with parameters /α and θ > 0 then V = max{ 2 } also has the surial function (3) This implies that we can consider a new family of distriutions gien y the surial function 3; - = "7 + 8 # > 0 - > 0 and + ote that 3; - as α This family of distriutions is a generalization of the Marshall-Olkin family in the sense that the family is reduced to the Marshall-Olkin family of distriutions when θ= The aim of this paper is to introduce a new family of uniariate distriutions y using discrete Mittag-Leffler truncated distriution In section 2 we introduce a new family of uniariate distriutions for a gien parent distriution function F This family contains the well-known Marshall- Olkin family of distriutions We study some properties of this family including random ariate generation In section3 we introduce a new family of uniariate distriution which contains Uniform distriution and Marshall-Olkin extended Uniform distriution We derie its shape properties moments median mode quantiles distriution of order statistics entropies and estimation procedure In section 4 we discuss the estimation of parameters of DMLU y the method of maximum likelihood An application in autoregressie time series modeling is presented in Section Conclusions are presented in Section 6 2 Truncated Discrete Mittag-Leffler Family of Distriutions Pillai and Jayakumar (99) introduced the discrete Mittag-Leffler distriution and studied its properties The mathematical origin of the discrete Mittag-Leffler distriution can e descried as follows: Consider a sequence of independent Bernoulli trails in which the k th trail has proaility of success α/k with 0 <α< and k = 2 3 Let e the trail numer in which the first success occurs Then the proaility that {=r} is gien y p > = α α 2 α 3 α r α r = A B C C! (4) Proaility generating function (pgf) of is gien y G(z)= ( z) α Let 2 n e independent and identically distriuted random ariales as and let 0 =0 Let M e geometric distriuted random ariale with parameter p ie Pr(M =k)=q k p k = 02 ; 0 < p < q = p Then M has generating function E = F GH I = JH I () with p = /(+c) The distriution with pgf () is known as Discrete Mittag-Leffler distriution with parameters α and c Define now a new random ariale Y such that Then Therefore K = = O78 = L = M = 2 3 OP = QP R = 4 PS ML = T M S6 = U4 P S ML = T ML = 0V M S6 = WP M + Y + = W + P Y + Z + [ Hence we otain a new family of distriutions with

3 International Journal of Statistical Distriutions and Applications 206; 2(3): parameters α and c haing surial function = I J I (6) The corresponding distriution function is gien y JI J I (7) Since discrete Mittag-Leffler distriution can e considered as a generalization of geometric distriution the family of distriutions gien y (7) can e considered appropriate for life time modeling From (7) the pdf is gien y \; JI]^_ `J I a 0 0 c+ (8) where f(x) is the pdf of F(x) while the hazard rate function is gien y d e f JI]^_ ` I a`j I The distriution elonging to (7) can easily e simulated For a gien parent distriution function F random ariale with distriution function (7) can e simulated as L " #^ R I JJR (9) (0) where Y is uniformly distriuted random ariale on (0) The newly constructed truncated Discrete Mittag-Leffler distriution can e considered as a generalization of Marshall-Olkin family of distriutions since it reduces to Marshall-Olkin family when α = In (7) when F(x) is exponential G(x) ecomes the Marshall-Olkin generalized exponential distriution studied in Ristic and Kundu (20) When F(x) in (7) is Weiull G(x) reduces to Marshall-Olkin exponentiated Weiull distriution studied in Bidram et al (20) Hence (7) is a rich class in the sense that it leads to arious generalizations of existing distriutions that hae the capaility of modeling real data sets Figure Distriution function of truncated Discrete Mittag-Leffler Uniform Distriution(i) α = c=233 (ii) α = c=3 (iii) α = c= (i) α= c=00 () α= c=0 (i) α=2 c=233 (ii) α= c= Proaility Density Function The proaility density function is gien y \; - I J I]^ ` I J I a (3) for 0 < x < θ α > 0 c > 0 and θ > 0 The graph of g(x) for different alues of α for c=233 and θ=0 are gien in Figure 2 3 A ew Family of Uniform Distriution 3 Distriution Function Let Uniform (0 θ) distriution where θ > 0 Then 0 h h - () Using (7) we get the distriution function G(x) for F(x) in () as JI I J I (2) We refer to this distriution as truncated Discrete Mittag- Leffler Uniform Distriution (DMLU) with parameters α c and θ; and write it as DMLU(αcθ) The graph of G(x) for different alues of α and c for θ = 0 is gien in Figure Figure 2 Proaility density function of DMLU(α 233 0) (i) α =0 (ii) α = (iii) α=20 Some special cases of the DMLU(α c θ) are: Case I: When α = c = q/ p \ i; j M -k M- `M- ja B 0 h h - 0 h M h j M This is Marshall-Olkin Extended Uniform (MOEU) distriution studied in Jose and Krishna (20) Case II: When α = and c 0

4 38 K K Sankaran and K Jayakumar: A ew Extended Uniform Distriution \; 0-0 h h - which is Uniform distriution in ( 0 θ ) In order to derie the shape properties of the pdf (3) we consider the function log \ I J I BJ I I J I (4) The following shapes are possile: Let α (0) Then g(x) is a decreasing function with g(0) = and g- J 2 Let α > Then g(x) is a unimodal function with mode at x 0 Furthermore g(0)=0 and \- 33 Hazard Rate Function The hazard rate function is gien y d I J I]^ I I I J I J () The shapes of hazard rate function aries with respect to α: If 0 < α < then h(x) initially decreasing and then moing constantly and then increasing steeply 2 If α > h(x) is moing constantly at the initial alues and then increasing steeply The graph of hazard function for different alues for α and c when θ = 0 are gien in Figure 3 From Prudniko et al (986) equation (222) is q t s ( ) ) u ( ( )u ) 4 w u ) x ( ) x x6 where a c and d are real numers with (ac+d)(c+d) > 0; Real part of α > 0 and B(a ) = ytys yts Hence if r/α is a positie integer we hae Therefore In particular QL QL B I r A I q - r B )r ApI J^pA I QL C ^ JI JI A A JI x6 B x6 C x6 C x6 2 x A I xa I J } x A I xa I J } (7) yx^ I yx^ I yb^ I J} (8) yx I yx I yb I J} (9) Var () = E( 2 ) E() 2 The q-th quantile of a random ariale following DMLU(α c θ) is gien through the quantile function as G j- ^ j W j Y I 0 ~ j ~ where G - () denote the inerse distriution function of G() In particular the median of is gien y Figure 3 Hazard rate function of DMLU(αc0) (i) α=0c = (ii) α =c = (iii) α=0c =2 34 Moments Suppose that has the DMLU(α c θ) The r th moment can e written as QL C - o Let x α = u aoe equation reduces to ApI]^ I J I ) (6) Median Finally the mode of is gien y ^ BJI #^ Mode - " I J The mean and ariance of DMLU for different alues of α and c when θ =0 are calculated (ia MATHCAD) and gien in Tale QL C - q I r A I - r B )r

5 International Journal of Statistical Distriutions and Applications 206; 2(3): Tale Mean and Variance of DMLU for different alues of α and c when θ = 0 c α Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance Order Statistics Assume 2 n are independent random ariales haing the DMLU(α c θ) distriution Let i:n denote the i th order statistic The pdf of i:n is \ : ; - = 36 Renyi and Shannon Entropies!!! \; - ; - ; - =!I J I ]^ I I ]!! I J I p (20) Entropy is in principle a measure of ariation or uncertainty The Renyi entropy of a random ariale with pdf g() is defined as ˆ! =! q \$ )! > 0! The Shannon entropy of a random ariale is defined y E[ log g()] It is the particular (limiting) case of the Renyi entropy for γ Let us first derie the Renyi entropy We hae Let u = x α Then o \ $ ) = `- + a $ o " q I]^ I J I #$ ) = `- + a $ o " $ Z - + B$[ ) = r^ I`$a q - + r B$ )r ŠI]^ I J I Š# Using eq(222) from Prudniko et al (986) and if!! is a positie integer the aoe integral ecomes - + ^ Therefore the Renyi entropy is The Shannon entropy is $^ I $ 4 w `! + a +!! I`$Ba + x - $$ ˆ! = $^ I $ x6! log`- + + a - + ^ I`$Ba 4 w `! + a +!! + x - $$ x6 Q` log \La = log `- + a Q`logLa + 2Q`log- + a )

6 40 K K Sankaran and K Jayakumar: A ew Extended Uniform Distriution 4 Estimation Since the moments of a DMLU random ariale cannot e otained in closed from we consider estimation of the unknown parameters y the method of maximum likelihood For a gien sample := (x x 2 x n ) the log-likelihood function is gien y log Œ; - = log `- + a + log 2 log - + a = log + log - + log + + log 2 log - + a The partial deriaties of the log-likelihood function with respect to the parameters are log Œ = + Ž \- + 4 log 2 4 log log Œ = log Œ - = The maximum likelihood estimates can e otained numerically soling the equation e e = 0 e J = 0 = 0 We can use for example the function nlm from the programming language R The second deriaties of the log-likelihood function of DMLU with respect to α c and θ are gien y B = B log log- + B = + B B - + B - B = - B 2 4 ` - B - B - + B - = ` log- + a B = 2 4 ` log- + a B - = B If we denote the MLE of β:= (α c θ) y = - then the osered information matrix is gien y š B = Q - B - - œ œ œ - B œ log Œœ - B and hence the ariance coariance matrix of the estimate ector would e I ( ) The approximate ( ž% confidence interals for the parameters α c θ are ± ± - ± - respectiely where V( V andv- ) are the ariances of and - which are gien y the diagonal elements of I - (β) and is the upper ( ) percentile of B standard normal distriution Autoregressie Time Series Modeling Time series models with uniform marginal distriution are studied y arious researchers (see Ristic and Popoic (2000a) and Jose and Krishna (20)) Here we deelop an AR() model with DMLU as marginal distriution which is considered as a generalization of the existing time series models with uniform marginal Consider the AR() process L = c ª «h M u(u Ž «T L c ª «h M u(u Ž «T (2) where 0 and {c }is a sequence of iid random ariales with power function distriution in (0 θ) Theorem : Consider the AR() structure gien in (2) with 0 distriuted as a DMLU(αcθ) distriution Let = Then { n n } is a stationary Markoian autoregressie model with DMLU(αcθ) marginals iff {c }is distriuted as power distriution in (0 θ) with surial function Proof: From the aoe expression (2) it follows that = + ]^ Using the fact that 0 has DMLU(α θ) distriution and c has a power distriution in (0 θ) with distriution function we otain that for n = ^ = ± + ² ³ ^ - = U V W - Y = = - - +

7 International Journal of Statistical Distriutions and Applications 206; 2(3): which means that has DMLU(αcθ) distriution where = Assume that n- is distriuted as DMLU(αcθ) Then y induction method we can estalish that { n } is distriuted as DMLU(αcθ) Hence the process {n} is stationary Conersely if { n n } is stationary with DMLU(α θ) marginal s it can e easily shown that {c }has power distriution F(x) = 0 < < - Hence the theorem Een if 0 is aritrary it is easy to estalish that {n} is asymptotically stationary 6 Conclusion Marshall and Olkin (997) introduced a way of expanding a gien family of distriution y adding a parameter In this paper we introduced the Discrete Mittag-Leffler truncated distriution as a generalization of Marshall-Olkin family of distriutions and studied its properties This class is a rich class in the sense that some of the recently inestigated distriutions are memers of this family; see Ristic and Kundu (20) and Bidram et al (20) As a particular case a three parameter generalization of Uniform distriution was gien special attention The shape properties of the distriution were studied The expression for moments distriutions of the order statistics and entropies were also deried Moreoer we discussed the maximum likelihood method of estimation of the distriution s parameters An application on the autoregressie time series modeling was also presented References [] H Bidram M H Alamastsaz and V ekoukhou (20) On an extension of the exponentiated Weiull distriution Communications in Statistics - Simulation and Computation [2] G M Cordeiro and A J Lemonte (203) On the Marshall- Olkin extended Weiull distriution Statistical Papers [3] K Jayakumar and M Thomas (2008) On a generalization to Marshall-Olkin scheme and its application to Burr type II distriution Statistical Papers [4] K K Jose S R aik and M M Ristic (200) Marshall- Olkin q Weiull distriution and maximin processes Statistical Papers [] K K Jose and E Krishna (20) Marshall-Olkin extended uniform distriution ProStat Forum [6] A W Marshall and I Olkin (997) A new method for adding a parameter to a family of distriutions with application to the exponential and Weiull families Biometrika [7] S adarajah K Jayakumar and M M Ristic (203) A new family of life- time models Journal of Statistical Computation and Simulation [8] R Pillai and K Jayakumar (99) Discrete Mittag-Leffler distriutions Statistics and Proaility Letters [9] A P Prudniko Y A Brychko and OI Mariche (986) Integrals and series Vol I Gordeon and Breach Sciences Amsterdam etherlands [0] M M Ristic and B C Popoic (2000 a) A new Uniform AR() time series Model (UAR()) Pulications De Linstitut Mathematique ouelle series [] M M Ristic and B C Popoic (2000 ) Parameter estimation for Uniform autoregressie processes oi Sad Journal of Mathematics [2] M M Ristic K K Jose and A Joseph (2007) A Marshall- Olkin gamma distriution and minification process STARS International Journal (Science) 07-7 [3] M M Ristic and D Kundu (20) Marshall-Olkin generalized exponential distriution Metron