The fundamental assumption of variable control charts is that the data are normally distributed and spread randomly about the mean. Process data are not always normally distributed, hence there is need to set up appropriate control charts that gives accurate control limits to monitor processes that are skewed. In this study Shewhart-type control charts for monitoring positively skewed data that are assumed to be from Marshall-Olkin Inverse Loglogistic Distribution (MOILLD) was developed. Average Run Length (ARL) and Control Limits Interval (CLI) were adopted to assess the stability and performance of the MOILLD control chart. The results obtained were compared with Classical Shewhart (CS) and Skewness Correction (SC) control charts using the ARL and CLI. It was discovered that the control charts based on MOILLD performed better and are more stable compare to CS and SC control charts. It is therefore recommended that for positively skewed data, a Marshall-Olkin Inverse Loglogistic Distribution based control chart will be more appropriate.

Stochastic Volatility (SV) model usually assumes that the distribution of asset returns conditional on the latent volatility is normal. Previous approaches to estimation of SV model have mostly focused on Gaussian filters in practice. This paper analyzes SV model with the student-t distribution and compares the distribution with mixture-of-normal distributions of Kim and Stoffer [22]. A Sequential Monte Carlo with Expectation-Maximization (SMCEM) technique based on student-t distribution is developed to estimate the parameters for the extended volatility model. The SMC method, or particle filter based on student-t distribution, which is heavier tailed than Gaussians, provides an approximate solution to non-Gaussian estimation problem and hence more robust. Our empirical analysis indicates that extension of the SV model such as a specification of the error term with student-t distribution in the return equation dominates the normal mixture distribution. Additionally, the t-distribution based particle filter is applied to a multivariate stochastic volatility model. It is again shown that the student-t based algorithm performs quite well in explaining the joint dynamics in the volatility of a set of four exchange rates series. c ⃝ 2014 Production and Hosting by Elsevier B.V. on behalf of Nigerian Mathematical Society. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

This paper proposes a weighting of the exponentiated gamma distribution with a polynomial function called the poly-weighted exponentiated gamma distribution (PWEGD). It shows that the modified distribution harnesses the multi-dimensional effects of the distribution. We provided an extensive mathematical treatment of this proposed distribution: obtained its parameters, estimated its statistical properties with applicable, tests and compared the estimates with existing distribution. The study estimated the cumulative distribution function, hazard function, survival function, skewness, kurtosis, mode, median and quartiles of the distribution and evaluated the distribution with Monte Carlo simulated data and the data of the wind direction (degrees) in Lagos, Nigeria. Empirical analysis showed that with increased polynomial function, the estimates and the statistical properties like the expectation, variance, standard error, median, mode, hazard and survival functions, cumulative distribution function (CDF), moments, skewness and kurtosis were significantly better than the existing root distributions. The MSE of the parameters decreased with increased power and the parameter is significant (p <0.05). It is concluded that the proposed distribution does not only provide better fitting but also establishes an efficient structure for lifetime data modelling.

A stochastic volatility (SV) problem is formulated as a state space form of a Hidden Markov model (HMM). The SV model assumes that the distribution of asset returns conditional on the latent volatility is normal. This article analyzes the SV model with the student-t distribution and the generalized error distribution (GED) and compares these distributions with a mixture of normal distributions from Kim and Stoffer (2008). A Sequential Monte Carlo with Expectation Maximization (SMCEM) algorithm technique was used to estimate parameters for the extended volatility model; the Akaike Information Criteria (AIC) and forecast statistics were calculated to compare distribution fit. Distribution performance was assessed using simulation study and real data. Results show that, although comparable to the normal mixture SV model, the Student-t and GED were empirically more successful.

Models in behavioural relationship marketing suggest that relations between the customer and the company change over time as a result of the continuous encounter. Some theoretical models have been put forward concerning relationship marketing, both from the standpoints of consumer behaviour and empirical modelling. In addition to these, this study proposes the hidden Markov model (HMM) as a potential tool for assessing customer relationships. Specifically, the HMM is submitted via the framework of a Markov chain model to classify customers relationship dynamics of a telecommunication service company by using an experimental data set. We develop and estimate an HMM to relate the unobservable relationship states to the observed buying behaviour of the customers giving an appropriate classification of the customers into the relationship states. By merely accounting for the functional and unobserved heterogeneity with a two-state hidden Markov model and taking estimation into account via an optimal estimation method, the empirical results not only demonstrate the value of the proposed model in assessing the dynamics of a customer relationship over time but also gives the optimal marketing-mixed strategies in different customer state.