New Control Charts for a Multivariate Gamma Distribution

In this study, a multivariate gamma distribution is first introduced. Then, by defining a new statistic, three control charts called the MG charts, are proposed for this distribution. The first control chart is based on the exact distribution of this statistic, the second control chart is based on the Satterthwaite approximation, and the last is based on the normal approximation. Efficiency of the proposed control charts is evaluated by average run length (ARL) criterion.

On a Multiplicative Multivariate Gamma Distribution with Applications in Insurance

One way to formulate a multivariate probability distribution with dependent univariate margins distributed gamma is by using the closure under convolutions property. This direction yields an additive background risk model, and it has been very well-studied. An alternative way to accomplish the same task is via an application of the Bernstein–Widder theorem with respect to a shifted inverse Beta probability density function. This way, which leads to an arguably equally popular multiplicative background risk model (MBRM), has been by far less investigated. In this paper, we reintroduce the multiplicative multivariate gamma (MMG) distribution in the most general form, and we explore its various properties thoroughly. Specifically, we study the links to the MBRM, employ the machinery of divided differences to derive the distribution of the aggregate risk random variable explicitly, look into the corresponding copula function and the measures of nonlinear correlation associated with it, and, last but not least, determine the measures of maximal tail dependence. Our main message is that the MMG distribution is (1) very intuitive and easy to communicate, (2) remarkably tractable, and (3) possesses rich dependence and tail dependence characteristics. Hence, the MMG distribution should be given serious considerations when modelling dependent risks.

A Hybrid Statistical Method for Accurate Prediction of Supplier Delivery Times of Aircraft Engine Parts

Aircraft engine assembly operations require thousands of parts provided by several geographically distributed suppliers. A majority of the operation steps are sequential, necessitating the availability of all the parts at appropriate times for these steps to be completed successfully. Thus, being able to accurately predict the availabilities of parts based on supplier deliveries is critical to minimizing the delays in meeting the customer demands. However, such accurate prediction is challenging due to the large lead times of these parts, limited knowledge of supplier capacities and capabilities, macroeconomic trends affecting material procurement and transportation times, and unreliable delivery date estimates provided by the suppliers themselves. We address these challenges by developing a statistical method that learns a hybrid stepwise regression — generalized multivariate gamma distribution model from historical transactional data on closed part purchase orders and is able to infer part delivery dates sufficiently before the supplier-promised delivery dates for open purchase orders. The hybrid form of the model makes it robust to data quality and short-term temporal effects as well as biased toward overestimating rather than underestimating the part delivery dates. Test results on real-world purchase orders demonstrate effective performance with low prediction errors and constantly high ratios of true positive to false positive predictions.

Finite-Dimensional Distributions of a Square-Root Diffusion

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

Finite-Dimensional Distributions of a Square-Root Diffusion

We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.