DERIVATION OF A GENERALIZED DISTRIBUTION FOR MATERNAL MORTALITY RATIO
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Date
2018-05
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UNIVERSITY OF ILORIN
Abstract
Maternal mortality is an outcome of adverse health and remains a major challenge in some societies. Study on distribution fitting for global Maternal Mortality Ratio (MMR) is therefore needful. Analysis of data without a pre-knowledge of the distribution that describes data may lead to misleading or irrelevant results. Distribution fitting to data provides the best fitting distribution for data analysis. Limitations in characteristics of existing distributions motivate generalization of distributions in order to improve goodness of fit and introduce more flexibility. The aim of this study was to derive a generalized distribution for MMR while the specific objectives were to: (i) generate families of generalized distributions; (ii) illustrate the flexibility of generalized distributions; (iii) fit some existing distributions to MMR to determine the best fitting distribution; (iv) derive generalized distributions for the best fitted distribution; and (v) fit and assess the derived generalized distributions to MMR.
Five existing parameter induction methods were applied to generate families of generalized distributions with an additional parameter. Methods in permutations of these five existing parameter induction methods, namely: Lehmann Alternative 1; Lehmann Alternative 2, Marshall and Olkin Method, Power Transformation Method and α-Power Transformation Method taking two methods at a time were applied sequentially to obtain generalized families with two additional shape parameters. Same methods were also applied twice. Exponentiated families of Generalized Pareto Distribution (GPD) were generated and flexibility of generalized families was illustrated by showing effects of introduced parameters on the Probability Density and Hazard Function shapes. Histogram plot, Kolmogorov-Smirnov (K-S) distances and Akaike Information Criterion (AIC) were employed in determining the best fitting distribution for MMR. Using best fitted distribution as base distribution, its generalized distributions were derived from families generated and subsequently fitted to MMR using AIC and K-S distances for selection.
Findings of the study were that:
(i) seventeen distinct generalized families of distributions having two additional parameters were generated for commutativity and idempotency of some methods. Five families with one additional parameter were also generated;
(ii) Lehmann type I GPD improved flexibility by introducing the unimodal and bathtub shapes in both Probability Density and Hazard Functions shapes;
(iii) Frechet distribution provided the best fit for MMR amongst plausible distributions studied;
(iv) four generalized Frechet distributions with an additional shape parameter and two generalized Frechet distributions with two additional shape parameters were generated; and
(v) generalized Frechet distributions improved goodness of fit based on K-S distances with Lehman type II generalized Frechet giving the best fit while AIC selected Marshall-Olkin generalized Frechet distribution as a good fitting but parsimonious distribution for MMR.
In conclusion, families of generalized distribution introducing n parameters may be generated by sequentially applying methods in permutations of s (s≥n) distinct parameter induction methods taking n methods at a time. Lehmann type I GPD introduced flexibility thereby illustrating flexibility of generalized distributions. Generalized Frechet distributions improved goodness of fit but for parsimony, Marshal-Olkin generalized Frechet distribution is recommended as an alternative reference distribution to the Frechet distribution for modelling MMR.
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Keywords
DERIVATION, GENERALIZED DISTRIBUTION, MATERNAL MORTALITY, Generalized Pareto Distribution (GPD), Maternal Mortality Ratio (MMR), Akaike Information Criterion (AIC), Kolmogorov-Smirnov (K-S) distances