This paper aims to model the returns on a financial asset with volatilities expressed through a Gaussian process. This approach avoids a rigid structure for the functional form (over time) of volatility. Inference about the unknown quantities of the models is made under a Bayesian approach with the application of numerical methods such as Gibbs sampling and particle filter. Although the model is defined with a Markovian structure, with the introduction of the Gaussian process, there is a dependence between the states in the unconditional distributions. The state prior is complicated, so to approximate it and speed up the estimation process, a data window is proposed. In the applications, a small Monte Carlo study is presented to evaluate the estimation process and compare performance with other models, as well as with real data.

KEYWORDS: Gibbs sampling, Sequential Monte Carlo methods, State space model.

The ultimate objective of constructing time series models is to forecast future data accurately. However, the presence of outliers within the series can impact all phases of model development, inevitably affecting the accuracy of forecasts. This study seeks to examine how disregarding Seasonal Level Shift (SLS) outlier influences point forecasts generated by SARIMA models. Through analytical methods, we obtain the expression for the rise in the mean square error of h-step ahead forecasts attributable to the existence of an SLS outlier. To gain deeper insights into the findings, we conduct a simulation study. A key revelation is that SLS outlier notably levate the mean square forecast error. Nevertheless, the extent of this escalation hinges not only on when the outlier appears relative to the forecast origin but also on its magnitude, number of years considered in the data, sample size, variance of errors and the parameter of the specific SARIMA model under consideration.

KEYWORDS: Time series, Forecasts, Seasonal level shift, Outlier, Mean square forecast error

In modern statistics, the big data issue is increasingly widespread. Long since principal component analysis (PCA) is a technique for reducing the dimensionality of such data sets. Principal component regression further reduces this large number of explanatory variables to a more handy model. This article explains the relationship of no of variables(p) and no of observations(n) in principal component-based statistical classification techniques both in the parametric and non-parametric ways. It discusses on the amount of misclassification error through the adaptive data analysis technique. In reality, we established that reducing a large number of candidate explanatory variables does not make principal component-based classification more worthy. In fact for non-Gaussian populations, variable-based non-parametric classification comes out more convincing.

KEYWORDS: Statistical classification, linear discriminant analysis, nonparametric classification, kernel, big data analytics

We generalize Ornstein-Uhlenbeck process to include non-normal innovations. This model captures the stylized facts of financial markets as it preserves jumps in the volatility process. We study the asymptotic behavior of some estima-tors of the drift parameter in the Gamma-Ornstein-Uhlenbeck, Inverse-Gaussian-Ornstein-Uhlenbeck, Modified-Tempered-Stable-Ornstein-Uhlenbeck volatility pro-cesses based on discrete equally spaced observations of the price process. The esti-mators are explicit. We study robustness and eciency of the estimators.

KEYWORDS: Stochastic differential equation, Stochastic volatility, Levy process, Inverse Gaussian-Ornstein-Uhlenbeck process, Gamma-Ornstein-Uhlenbeck process, Modified Tempered Stable-Ornstein-Uhlenbeck process, Jumps, Infinite Divisibility, Heavy Tails, Skewness, Kurtosis, Discrete Observations, Robust Estimator, Moment Estimator, Mixing, Factor Model.

maximum likelihood estimator of the (scale) parameter in the Akash distribution. The latter distribution has flexible features that make it attractive for modelling lifetime data. Based on a simulation experiment, all three bias-reduction methods are found to be highly effective, and have the added merit of also reducing the mean squared error of the maximum likelihood estimator. The analytical results are also illustrated with six real-life data-sets.

KEYWORDS: Akash distribution, lifetime data, maximum likelihood estimation, bias reduction.

Markov Chain Monte Carlo methods are widely used in Bayesian statistical inference to sample from the posterior distribution from a target distribution. However, for non-Gaussian and non-linear state space models, one can find difficulties in calculating the exact likelihood. To overcome problems in calculating the likelihood function, it is possible to use approximations made by particle filter methods. Furthermore, an adaptive Metropolis-Hastings algorithm may be applied since its proposal distribution is updated with previous draws from the posterior distribution. In this way, this paper discusses the applicability of adaptive Metropolis-Hastings (AMH) algorithms with random walk or independent proposals combined with estimated likelihoods through particle filters. We also propose a few model comparison criteria that can be easily integrated to the AMH. Moreover, we estimate non-linear and non-Gaussian volatility models for three time series of real index returns.

KEYWORDS: Diminishing adaptation, sequential Monte Carlo methods, state space model

Autoregressive Fractionally Integrated Moving Average (ARFIMA) has been successfully applied in modelling and forecasting with linear economic time series with long memory components. In order to capture additional complex nonlinear economic relationships with many unknown patterns, another popular approach known as the Artificial Neural Network (ANN) can be used. It has been recognised that a combination of both ARFIMA and ANN can be used to capture both the linear and nonlinear components of a time series. This paper proposes an alternative hybrid model, which is distinctive in integrating both the linear and nonlinear components
of applied time series with long memory - and in considering both additive and multiplicative models. A simulation study has been carried out to investigate the properties of this ARFIMA-ANN hybrid modelling and forecasting. We justify the usefulness of this proposed hybrid model in practice using empirical data sets from various domains - financial, environmental (pollution), climate (El Ni˜no) and energy (electricity load) - and compare the accuracy of forecasting with existing models. We have shown that, in general, these hybrid models will often produce more accurate forecast values than other - individual - models. We also discuss explainability and interpretability.

KEYWORDS: Long memory, Fractional difference, Heteroscedastic, ARFIMA, Forecasting, neural net, ANN, Hybrid, Explainable AI, XAI, Interpretable.