Properties and Dynamics of the Beta Gompertz Generalized Makeham Distribution
Beta distribution has been widely used in many scientific areas. It is applied very often and is known as a probability distribution with wide application in the management of various kinds of problems, such as in survival analysis. In recent years, devel
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Properties and Dynamics of the Beta Gompertz Generalized Makeham Distribution Panagiotis Andreopoulos, Alexandra Tragaki, George Antonopoulos, and Fragkiskos G. Bersimis
18.1 Introduction 18.1.1 Beta Distribution Beta distribution has been widely used in a variety of scientific fields (Abramowitz and Stegun 2006; Gupta and Nadarajah 2004) due to the fact that it is able to describe a wide range of different data with bounded support. More specifically, beta distribution with two parameters—i.e., left parameter a (shape of beta) and right parameter b (scale of beta)—is used for modeling data, taking values within the interval (0,1). Beta density is expressed as follows: f (x; α, β) =
1 x α−1 (1 − x)β−1 , 0 < x < 1, B (α, β)
α, β > 0
where B(α, β) is given by the integral
P. Andreopoulos () · A. Tragaki Department of Geography, Harokopio University, Athens, Greece e-mail: [email protected]; [email protected] G. Antonopoulos MSc: Statistics and Operational Research, Athens, Greece Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece e-mail: [email protected] F. G. Bersimis Department of Informatics and Telematics, Harokopio University, Tavros, Greece e-mail: [email protected] © Springer Nature Switzerland AG 2020 C. H. Skiadas, C. Skiadas (eds.), Demography of Population Health, Aging and Health Expenditures, The Springer Series on Demographic Methods and Population Analysis 50, https://doi.org/10.1007/978-3-030-44695-6_18
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1 B (α, β) =
uα−1 (1 − u)β−1 du 0
Beta density function may have different shapes (Venter 1983) including left- and right-skewed or the flat shape of uniform density, depending on the combination of its parameters. In the case that all scores are equally likely, then the parameters could be set as α = 1 and β = 1, as illustrated in Panel B; this gives a “flat” probability density function (pdf). Some other special cases are: • If parameters α = β < 1, the curve of the beta pdf in Panel B takes the shape of a symmetrical bathtub. If α = β, then the shape of a non-symmetrical bathtub α−1 appears with a minimum value at x0 = α+β−2 . • If α = β = 1, the beta distribution B(1, 1) reduces to the uniform distribution U(0, 1) (the pdf is constant and equal to one; Panel B). • If α = 1 and β > 1, beta pdf is decreasing as x tends to 1, and if α > 1 and β = 1, beta pdf is increasing as x tends to 1. • If α < 1 and β ≥ 1, beta pdf is decreasing as x tends to 1. • If α ≥ 1 and β < 1, beta pdf is increasing as x tends to 1. • If α > 1 and β > 1, beta pdf (Panel C) is increasing in the domain (0, x0 ) and decreasing in the domain (x0 , 1). The peak of the density is in the interior of [0,1] and the mode of the beta distribution is x0 (Fig. 18.1).
18.1.2 Generated Beta Distribution In mathematical statistics, the study of the beta distribution is useful in many ways: one can relate the beta distribution to other well-known distributions (uniform, gamma, exponential, normal, etc.) or just present it as an important application of the ga
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