This article discusses Bayesian and non-Bayesian estimations problems of the unknown parameters for the two-parameter bathtub-shaped lifetime distribution based on upper record values. The ML and the Bayes estimates based on record values are derived for the two unknown parameters as well as hazard function. When the Bayesian approach is considered, under the assumption that both parameters are unknown, the Bayes estimators cannot be obtained in explicit forms. An approximation form due to Soland (Soland (1969)) is used for obtaining the Bayes estimates based on a conjugate prior for the first shape parameter and a discrete prior for the second shape parameter of this model. This is done with respect to the squared error loss and LINEX loss functions. The estimation procedure is then applied to real data set and simulation data.

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This paper introduces a new three-parameter distribution called inverse generalized power Weibull distribution. This distribution can be regarded as a reciprocal of the generalized power Weibull distribution. The new distribution is characterized by being a general formula for some well-known distributions, namely inverse Weibull, inverse exponential, inverse Rayleigh and inverse Nadarajah-Haghighi distributions. Some of the mathematical properties of the new distribution including the quantile, density, cumulative distribution functions, moments, moments generating function and order statistics are derived. The model parameters are estimated using the maximum likelihood method. The Monte Carlo simulation study is used to assess the performance of the maximum likelihood estimators in terms of mean squared errors. Two real datasets are used to demonstrate the flexibility of the new distribution as well as to demonstrate its applicability.

Based on the Nadarajah Haghighi distribution and the Topp Leone-G family in view of the T-X family, we introduce a new generator of continuous distributions with three extra parameters called the Nadarajah Haghighi Topp Leone-G family. Three sub-models of the new class are discussed. Main mathematical properties of the new family are investigated such as; quantile function, raw and incomplete moments, Bonferroni and Lorenz curves, moment and probability generating functions, stress-strength model, Shanon and Rényi entropies, order statistics and probability weighted moments. The model parameters of the new family is estimated by using the method of maximum likelihood and the observed information matrix is also obtained. We introduce two real applications to show the importance of the new family.

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