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Recently, Kharazmi and Saadatinik (2016) introduced a new family of lifetime distributions called hyperbolic cosine – F (HCF) distribution. In the present paper, it is focused on a special case of HCF family with exponentiated exponential distribution as a baseline distribution (HCEE). Various properties of the proposed distribution including explicit expressions for the moments, quantiles, mode, moment generating function, failure rate function, mean residual lifetime, order statistics and expression of the entropy are derived. Estimating parameters of HCEE distribution are obtained by eight estimation methods: maximum likelihood, Bayesian, maximum product of spacings, parametric bootstrap, non-parametric bootstrap, percentile, least-squares and weighted least-squares. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators. Finally, one real data set has been analyzed for illustrative purposes and it is observed that the proposed model fits better than Weibull, gamma and generalized exponential distributions.