Research Article
Lifetime Distribution Based on Generators for Discrete Mixtures with Application to Lomax Distribution
Peter Nyandiri Mecha,
Isaac Chumba Kipchirchir,
George Odweso Muhua,
Joseph Antony Makoteku Ottieno
Issue:
Volume 14, Issue 2, April 2025
Pages:
51-60
Received:
15 July 2024
Accepted:
24 December 2024
Published:
13 March 2025
DOI:
10.11648/j.ajtas.20251402.11
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Abstract: In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible distributions that can address these complexities. Such distributions can be generated through techniques like reparameterization, generalization, compounding, and mixing. This paper focuses on deriving generators for survival functions of discrete mixtures using minimum and maximum order statistic distributions. The approach leverages the probability generating function (PGF) techniques of mixing distributions, including zero-truncated Poisson, shifted geometric, zero-truncated binomial, zero-truncated negative binomial, and logarithmic series distributions. Specifically, the derived generator was applied to Lomax distributions to construct survival functions. Consequently, the probability density function (PDF) and failure rate of the resulting discrete mixtures were also obtained. Furthermore, the paper examines the shapes of the PDF and failure rate for discrete mixtures derived from the zero-truncated Poisson distribution. Notably, the failure rates of discrete mixtures generated using minimum and maximum order statistics from the Lomax distribution exhibited distinct behaviors. The failure rate for the minimum order statistic was observed to decrease, while the failure rate for the maximum order statistic showed a combination of non-decreasing and bathtub-shaped patterns.
Abstract: In various application areas, data is frequently collected and analyzed using basic statistical distributions such as exponential, Poisson, and gamma distributions. However, these traditional distributions often fail to adequately capture the inherent heterogeneity present in real-world data. This limitation highlights the need for more flexible di...
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Research Article
Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model
Issue:
Volume 14, Issue 2, April 2025
Pages:
61-76
Received:
12 February 2025
Accepted:
3 March 2025
Published:
24 March 2025
DOI:
10.11648/j.ajtas.20251402.12
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Abstract: Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W1,..., Wq) is a Lipschitz continuous function defined on ℝq→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.
Abstract: Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of inter...
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