Biometrics & Biostatistics International Journal Zero- Truncated Discrete Shanker Distribution and Its Applications Submit Manuscript | http://medcraveonline.com Volume 5 Issue 6 - 2017 Department of Mathematical Sciences, Tezpur University, India *Corresponding author: Krishna Ram Saikia, Department of Mathematical Sciences, Tezpur University, Napaam, Tezpur, Assam, India, Email: Received: March 10, 2017 | Published: May 16, 2017 Research Article Biom Biostat Int J 2017, 5(6): 00152 Abstract Discrete analogue of the continuous Shanker distribution, which may be called a discrete Shanker distribution, has been introduced. The probability mass function and probability generating function of the distribution have been obtained. Zero truncated form of the distribution has been investigated. Certain recurrence relations for probabilities and moments have been also derived. The parameters of Zero- truncated discrete Shanker distribution have been estimated by using Newton- Raphson method. The distributions have been fitted to eight numbers of well- known data sets, which are used by other authors. A comparative study has been made among ZTP, ZTPL and ZTDS distributions, using the same data set based on the goodness of fit test. It has been observed that in most cases ZTPL gives much closer fit than ZTP distribution. While ZTDS gives very closer fit to ZTPL and in some cases ZTDS gives better fit than ZTPL distribution. Keywords: Discrete Shanker distribution; Zero-truncated discrete Shanker distribution; Zero- truncated Poisson- Lindley distribution; Recurrence Relations; Survival function Abbreviations: DS: Discrete Shanker; ZTP: Zero–Truncated Poisson; ZTPL: Zero–truncated Poisson Lindley; ZTDS: Zero– Truncated Discrete Shanker; PDF: Probability Density Function; pmf: probability mass function; () S x : survival function; () r x : failure hazard rate, () * r x : reversed failure rate; ( ) ; D f x θ : pmf of DS distribution, ( ) ; z f x θ : pmf of ZTDS distribution; [] ' r η : [] ' r ì factorial moment of ZTDS distribution; [] ' r µ : th r raw moment of DS distribution; r P : th r Probability of DS distribution; z r P : th r Probability of ZTDS distribution Introduction It is sometimes inconvenient to measure the life length of a device, on a continuous scale. In practice, we come across situation, where lifetime of a device is considered to be a discrete random variable. For example, in the case of an on off switching device, the lifetime of the switch is a discrete random variable. If the lifetimes of individuals in some populations are grouped or when lifetime refers to an integral numbers of cycles of some sort, it may be desirable to treat it as a discrete random variable. When a discrete model is used with lifetime data, it is usually a multinomial distribution. This arises because effectively the continuous data have been grouped. Such situations may demand another discrete distribution, usually over the non negative integers. Such situations are best treated individually, but generally one tries to adopt one of the standard discrete distribution. Some of those works are by Nakagawa and Osaki [1], where the discrete Weibull distribution is obtained; Roy [2] studied discrete Rayleigh distribution; Kemp [3] derived discrete Half normal distribution. Krishna and Pundir [4] investigated the discrete Burr and the discrete Pareto distribution. Gomez-Deniz [5] derived a new generalization of the geometric distribution obtained from the generalized exponential distribution of Marshall and Olkin [6]. Borah et al. [7,8] studied on two parameter discrete quasi- Lindley and discrete Janardan distributions respectively. Borah and Saikia [9] introduced discrete Sushila distribution. Dutta and Borah [10] studied zero- modified Poisson- Lindley distribution. Derivation of the Proposed Distribution One parameter continuous Shanker distribution introduced by Shanker [11] with parameter θ is defined by its probability density function (pdf) ( ) ( ) 2 2 : . 0. 0. 1 x f x xe x θ θ θ θ θ θ − = + > > + (2.1) Discretization of continuous distribution can be done using different methodologies. In this paper we deal with the derivation of a new discrete distribution which may be called discrete Shanker (DS) distribution. It takes values in {0, 1, 2, . . .,}. This distribution is generated by discretizing the survival function of the continuous Shanker distribution () ( ) : x S x f x dx θ ∞ =∫ 2 2 1 , 0. 0. 1 x x e x θ θ θ θ θ + + − = > > + (2.2)