Research Article A Flexible Reduced Logarithmic-X Family of Distributions with Biomedical Analysis Yinglin Liu, 1 Muhammad Ilyas, 2 Saima K. Khosa, 3 Eisa Muhmoudi , 4 Zubair Ahmad , 4 Dost Muhammad Khan , 5 and G. G Hamedani 6 1 College of Pharmacy and Chemistry, Dali University, Dali City, Yunnan Province, China 2 Department of Statistics, University of Malakand, Dir (L), Chakdara, Khyber Pakhtunkhwa, Pakistan 3 Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan 4 Department of Statistics, Yazd University, P.O. Box 89175-741, Yazd, Iran 5 Department of Statistics, Abdul Wali University Mardan, Mardan, Khyber Pakhtunkhwa, Pakistan 6 Department of Mathematical and Statistical Sciences, Marquette University, Milwaukee, WI 53201-1881, USA Correspondence should be addressed to Zubair Ahmad; z.ferry21@gmail.com Received 15 September 2019; Revised 19 January 2020; Accepted 20 January 2020; Published 20 February 2020 Academic Editor: Chuangyin Dang Copyright © 2020 Yinglin Liu et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Statistical distributions play a prominent role in applied sciences, particularly in biomedical sciences. e medical data sets are generally skewed to the right, and skewed distributions can be used quite effectively to model such data sets. In the present study, therefore, we propose a new family of distributions to model right skewed medical data sets. e proposed family may be named as a flexible reduced logarithmic-X family. e proposed family can be obtained via reparameterizing the exponentiated Kumaraswamy G-logarithmic family and the alpha logarithmic family of distributions. A special submodel of the proposed family called, a flexible reduced logarithmic-Weibull distribution, is discussed in detail. Some mathematical properties of the proposed family and certain related characterization results are presented. e maximum likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is done to evaluate the performance of these estimators. Finally, for the illustrative purposes, three applications from biomedical sciences are analyzed and the goodness of fit of the proposed distribution is compared to some well-known competitors. 1. Introduction e statistical analysis and modeling of lifetime phenomena are essential in almost all areas of applied sciences, partic- ularly, in biomedical sciences. A number of parametric continuous distributions for modeling lifetime data sets have been proposed in literature including exponential, Rayleigh, gamma, lognormal, and Weibull, among others. e ex- ponential, Rayleigh, and Weibull distributions are more popular than the gamma and lognormal distributions since the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and hence both require numerical integration to arrive at the mathe- matical properties. e exponential and Rayleigh distribu- tions are commonly used in lifetime analysis. ese distributions, however, are not flexible enough to counter complex forms of the data. For example, the exponential distribution is capable of modeling data with constant failure rate function, whereas the Rayleigh distribution offers data modeling with only increasing failure rate function. e Weibull distribution, also known as the super exponential distribution, is more flexible than the aforementioned dis- tributions. e Weibull distribution offers the characteristics of both the exponential and Rayleigh distributions and is capable of modeling data with monotonic (increasing, de- creasing, and constant) hazard rate function. Unfortunately, the Weibull distribution is not capable of modeling data with nonmonotonic (unimodal, modified unimodal, and bath- tub-shaped) failure rate function. In some medical situa- tions, for example, neck cancer, bladder cancer, and breast Hindawi Computational and Mathematical Methods in Medicine Volume 2020, Article ID 4373595, 15 pages https://doi.org/10.1155/2020/4373595