Bayesian Analysis of Cancer Data Using a 4-Component Exponential Mixture Model

Cancer is among the major public health problems as well as a burden for Pakistan. About 148,000 new patients are diagnosed with cancer each year, and almost 100,000 patients die due to this fatal disease. Lung, breast, liver, cervical, blood/bone marrow, and oral cancers are the most common cancers in Pakistan. Perhaps smoking, physical inactivity, infections, exposure to toxins, and unhealthy diet are the main factors responsible for the spread of cancer. We preferred a novel four-component mixture model under Bayesian estimation to estimate the average number of incidences and death of both genders in different age groups. For this purpose, we considered 28 different kinds of cancers diagnosed in recent years. Data of registered patients all over Pakistan in the year 2012 were taken from GLOBOCAN. All the patients were divided into 4 age groups and also split based on genders to be applied to the proposed mixture model. Bayesian analysis is performed on the data using a four-component exponential mixture model. Estimators for mixture model parameters are derived under Bayesian procedures using three different priors and two loss functions. Simulation study and graphical representation for the estimates are also presented. It is noted from analysis of real data that the Bayes estimates under LINEX loss assuming Jeffreys’ prior is more efficient for the no. of incidences in male and female. As far as no. of deaths are concerned again, LINEX loss assuming Jeffreys’ prior gives better results for the male population, but for the female population, the best loss function is SELF assuming Jeffreys’ prior.

Sequential Test for a Mixture of Finite Exponential Distribution

Testing the number of components in a finite mixture is considered one of the challenging problems. In this paper, exponential finite mixtures are used to determine the number of components in a finite mixture. A sequential testing procedure is adopted based on the likelihood ratio test (LRT) statistic. The distribution of the test statistic under the null hypothesis is obtained using a resampling technique based on B bootstrap samples. The quantiles of the distribution of the test statistic are evaluated from the B bootstrap samples. The performance of the test is examined through the empirical power and application on two real datasets. The proposed procedure is not only used for testing the number of components but also for estimating the optimal number of components in a finite exponential mixture distribution. The innovation of this paper is the sequential test, which tests the more general hypothesis of a finite exponential mixture of k components versus a mixture of k + 1 components. The special case of testing an exponential mixture of one component versus two components is the one commonly used in the literature.

Explicit results on conditional distributions of generalized exponential mixtures

For independent exponentially distributed random variables $X_i$ , $i\in {\mathcal{N}}$ , with distinct rates ${\lambda}_i$ we consider sums $\sum_{i\in\mathcal{A}} X_i$ for $\mathcal{A}\subseteq {\mathcal{N}}$ which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum $\sum_{i\in {\mathcal{N}}}X_i$ given that a subset sum $\sum_{j\in \mathcal{A}}X_j$ exceeds a certain threshold value $t>0$ , and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for $t\to\infty$ . Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.

Calculation of the age of the first infection for skin sores and scabies in five remote communities in northern Australia

Prevalence of skin sores and scabies in remote Australian Aboriginal communities remains unacceptably high, with Group AStreptococcus(GAS) the dominant pathogen. We aim to better understand the drivers of GAS transmission using mathematical models. To estimate the force of infection, we quantified the age of first skin sores and scabies infection by pooling historical data from three studies conducted across five remote Aboriginal communities for children born between 2001 and 2005. We estimated the age of the first infection using the Kaplan–Meier estimator; parametric exponential mixture model; and Cox proportional hazards. For skin sores, the mean age of the first infection was approximately 10 months and the median was 7 months, with some heterogeneity in median observed by the community. For scabies, the mean age of the first infection was approximately 9 months and the median was 8 months, with significant heterogeneity by the community and an enhanced risk for children born between October and December. The young age of the first infection with skin sores and scabies reflects the high disease burden in these communities.