Abstract—Companies are viewing customers in terms of their lifetime duration. Customer lifetime duration is a powerful and straightforward measure that synthesizes churn (attrition) risk at individual customer level. For existing customers, customer lifetime duration can help companies develop customer loyalty and treatment strategies to maximize customer value. In this study, based on the Kumaraswamy distribution, the Kumaraswamy Lindley distribution is studied. Some mathematical properties of Kumaraswamy Lindley distribution such as moments, hazard function, quantile function, skewness, kurtosis are derived. The method of maximum likelihood is used to estimate the model parameters and the observed information matrix is derived. An application of our results is provided to show the applicability of this distribution, especially for customer lifetime duration. Therefore, the proposed distribution can be a useful tool to analyze customer lifetime duration in marketing research. Index Terms—Customer lifetime, Kumaraswamy distribution, Lindley distribution, moments, maximum likelihood estimation, marketing research. I. INTRODUCTION The Lindley distribution is introduced by Lindley in 1958 as a one-parameter distribution ߠ൐0 . Its probability density function(pdf) is given by ሺݔሻൌ ఏ మ ఏାଵ ሺ1 ൅ ݔሻ ఏ௫ . (1) Note that this distribution is a mixture of exponential and gamma (2, ߠ) distributions. The corresponding cumulative distribution function (cdf) is given by ܩሺݔሻൌ1െe θ୶ ቂ1 ൅ ఏ௫ ఏାଵ ቃ, ݔ൐0 , ߠ൐0 (2) Ghitany et al. [1] discussed various properties of this distribution and showed that the Lindley distribution provides a better model than the exponential distribution in many ways. A discrete version of the Lindley distribution suggested by Deniz and Ojeda [2] based on an application related to insurance. The Lindley mixture of Poisson distribution is obtained by Sankaran [3]. Ghitany et al. [4], [5] obtained size-biased and zero-truncated version of Poisson-Lindley distribution. Ghitany and Al-mutairi [6] discussed various estimation methods for the discrete Poisson-Lindley distribution. Bakouch et al. [7] proposed an extended Lindley distribution. Mazucheli and Achcar [8] Manuscript received April 10, 2014; revised June 13, 2014. The authors are with Department of Statistics, Hacettepe University, 06800, Turkey (e-mail: selencakmakyapan@ hacettepe.edu.tr,gamzeozl@ hacettepe.edu.tr). discussed the applications of Lindley distribution to competing risks lifetime data. Ghitany et al. [9] developed a two-parameter weighted Lindley distribution and discussed its applications to survival data. Recently, Zakerzadah and Dolati [10] and Elbatal et al. [11] have obtained the generalized Lindley distribution and the Kumaraswamy Quasi Lindley distribution, respectively. Although some studies have been conducted for the Lindley distribution, the Kumaraswamy distribution is not very common among statisticians and has been little explored in the literature. If G(x) is the baseline cdf of a random variable, the cdf of the Kumaraswamy-generalized (Kum-generalized) distribution, ܨሺݔሻ ൌ 1 െ ሺ1 െ ܩሺݔሻ ௔ ሻ ௕ ,0൏ ݔ൏1 (3) where  ൐ 0,  ൐ 0 are shape parameters. Then, the corresponding pdf is given by ሺݔሻ ൌ ሺݔሻܩሺݔሻ ௔ଵ ሺ1 െ ܩሺݔሻ ௔ ሻ ௕ଵ (4) Note that Eq. (4) can be unimodal, increasing, decreasing or constant, depending on the parameter values. As mentioned before, the Kumaraswamy distribution does not seem to be very familiar to statisticians or economists and has not been investigated systematically in much detail before, nor has its relative interchangeability with the beta distribution been widely appreciated. On the other hanf, in a very recent paper, Jones [12] explored the background of this distribution. Several advantages of the Kumaraswamy distribution over the beta distribution is listed: the normalizing constant is very simple; simple explicit formulae for the distribution and quantile functions which do not involve any special functions; explicit formulae for L-moments and simpler formulae for moments of order statistics. Besides, according to Jones [12], the beta distribution has the following advantages over the Kumaraswamy distribution: simpler formulae for moments and moment generating function (mgf); a one-parameter sub-family of symmetric distributions; simpler moment estimation and more ways of generating the distribution via physical processes. In this study we combine the Kumaraswamy distribution and the Lindley distribution and derive some statistical properties of this distribution to show the applicability in the customer lifetime duration in marketing research. II. KUMARASWAMY LINDLEY DISTRIBUTION In this section we introduce the Kumaraswamy Lindley (KL) distribution. Using (2) in (3), the cdf of the KL is A New Customer Lifetime Duration Distribution: The Kumaraswamy Lindley Distribution Selen Çakmakyapan and Gamze Özel Kadılar International Journal of Trade, Economics and Finance, Vol. 5, No. 5, October 2014 441 DOI: 10.7763/IJTEF.2014.V5.412