scholarly journals A Noncentral Lindley Construction Illustrated in an INAR(1) Environment

Stats ◽  
2022 ◽  
Vol 5 (1) ◽  
pp. 70-88
Author(s):  
Johannes Ferreira ◽  
Ané van der Merwe
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Count Data ◽  
Structural Characteristics ◽  
Poisson Model ◽  
Real Data ◽  
Error Distribution ◽  
Noncentrality Parameter ◽  
Lindley Distribution ◽  
Count Model

This paper proposes a previously unconsidered generalization of the Lindley distribution by allowing for a measure of noncentrality. Essential structural characteristics are investigated and derived in explicit and tractable forms, and the estimability of the model is illustrated via the fit of this developed model to real data. Subsequently, this model is used as a candidate for the parameter of a Poisson model, which allows for departure from the usual equidispersion restriction that the Poisson offers when modelling count data. This Poisson-noncentral Lindley is also systematically investigated and characteristics are derived. The value of this count model is illustrated and implemented as the count error distribution in an integer autoregressive environment, and juxtaposed against other popular models. The effect of the systematically-induced noncentrality parameter is illustrated and paves the way for future flexible modelling not only as a standalone contender in continuous Lindley-type scenarios but also in discrete and discrete time series scenarios when the often-encountered equidispersed assumption is not adhered to in practical data environments.

Continuous Autoregressive Moving Average Models

2021 ◽  
pp. 118-141
Author(s):  
Yakup Ari
Keyword(s):  
Time Series ◽  
Parameter Estimation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Real Data ◽  
Autoregressive Moving Average ◽  
The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

scholarly journals Second-Order Least Squares Estimation in Nonlinear Time Series Models with ARCH Errors

Econometrics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 41
Author(s):  
Mustafa Salamh ◽  
Liqun Wang
Keyword(s):  
Time Series ◽  
Real Data ◽  
Error Distribution ◽  
Likelihood Method ◽  
Strong Mixing ◽  
Efficiency Gain ◽  
Economic Time Series ◽  
Mixing Condition ◽  
Conditional Moments ◽  
Dynamic Regression Model

Many financial and economic time series exhibit nonlinear patterns or relationships. However, most statistical methods for time series analysis are developed for mean-stationary processes that require transformation, such as differencing of the data. In this paper, we study a dynamic regression model with nonlinear, time-varying mean function, and autoregressive conditionally heteroscedastic errors. We propose an estimation approach based on the first two conditional moments of the response variable, which does not require specification of error distribution. Strong consistency and asymptotic normality of the proposed estimator is established under strong-mixing condition, so that the results apply to both stationary and mean-nonstationary processes. Moreover, the proposed approach is shown to be superior to the commonly used quasi-likelihood approach and the efficiency gain is significant when the (conditional) error distribution is asymmetric. We demonstrate through a real data example that the proposed method can identify a more accurate model than the quasi-likelihood method.

A new one-parameter discrete distribution with associated regression and integer-valued autoregressive models

2020 ◽  
Vol 70 (4) ◽  
pp. 979-994
Author(s):  
Emrah Altun
Keyword(s):  
Time Series ◽  
Regression Model ◽  
Count Data ◽  
Discrete Distribution ◽  
Real Data ◽  
Autoregressive Process ◽  
Important Application ◽  
Statistical Properties ◽  
Data Sets ◽  
Application Fields

AbstractThis study introduces the Poisson-Bilal distribution and its associated two models for modeling the over-dispersed count data sets. The Poisson-Bilal distribution has tractable properties and explicit forms for its statistical properties. A new over-dispersed count regression model and integer-valued autoregressive process with flexible innovation distribution are defined and studied comprehensively. Two real data sets are analyzed to prove empirically the importance of proposed models. Empirical findings show that the Poisson-Bilal distribution has important application fields in time series and regression modeling.

scholarly journals Statistical analysis of no observed effect concentrations or levels in eco-toxicological assays with overdispersed count endpoints

2020 ◽  
Author(s):  
Ludwig A. Hothorn ◽  
Felix M. Kluxen
Keyword(s):  
Count Data ◽  
Poisson Model ◽  
Real Data ◽  
Test Methods ◽  
Statistical Testing ◽  
Data Sets ◽  
Mixing Distribution ◽  
Flexible Modeling ◽  
Hazard Characterization ◽  
Over Dispersion

AbstractIn (eco-)toxicological hazard characterization, the No Observed Adverse Effect Concentration or Level (NOAEC or NOAEL) approach is used and often required despite of its known limitations. For count data, statistical testing can be challenging, due to several confounding factors, such as zero inflation, low observation numbers, variance heterogeneity, over- or under-dispersion when applying the Poisson model or hierarchical experimental designs. As several tests are available for count data, we selected sixteen tests suitable for overdispersed counts and compared them in a simulation study. We assessed their performance considering data sets containing mixing distribution and over-dispersion with different observation numbers. It shows that there is no uniformly best approach because the assumed data conditions and assumptions are very different. However, the Dunnett-type procedure based on most likely transformation can be recommended, because of its size and power behavior, which is relatively better over most data conditions as compared to the available alternative test methods, and because it allows flexible modeling and effect sizes can be estimated by confidence intervals. Related R-code is provided for real data examples.

scholarly journals Comparing Bayesian Spatial Conditional Overdispersion and the Besag–York–Mollié Models: Application to Infant Mortality Rates

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 282
Author(s):  
Mabel Morales-Otero ◽  
Vicente Núñez-Antón
Keyword(s):  
Infant Mortality ◽  
Count Data ◽  
Spatial Dependence ◽  
Poisson Model ◽  
Mortality Rates ◽  
Estimation Methods ◽  
Neighborhood Structure ◽  
Specific Context ◽  
Count Data Models ◽  
Conditional Mean

In this paper, we review overdispersed Bayesian generalized spatial conditional count data models. Their usefulness is illustrated with their application to infant mortality rates from Colombian regions and by comparing them with the widely used Besag–York–Mollié (BYM) models. These overdispersed models assume that excess of dispersion in the data may be partially caused from the possible spatial dependence existing among the different spatial units. Thus, specific regression structures are then proposed both for the conditional mean and for the dispersion parameter in the models, including covariates, as well as an assumed spatial neighborhood structure. We focus on the case of response variables following a Poisson distribution, specifically concentrating on the spatial generalized conditional normal overdispersion Poisson model. Models were fitted by making use of the Markov Chain Monte Carlo (MCMC) and Integrated Nested Laplace Approximation (INLA) algorithms in the specific context of Bayesian estimation methods.

scholarly journals Calibrating the CreditRisk+ Model at Different Time Scales and in Presence of Temporal Autocorrelation †

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1679
Author(s):  
Jacopo Giacomelli ◽  
Luca Passalacqua
Keyword(s):  
Time Series ◽  
Default Risk ◽  
Real Life ◽  
Statistical Error ◽  
Real Data ◽  
Sampling Period ◽  
Calibration Procedure ◽  
Industry Standards ◽  
Default Rate

The CreditRisk+ model is one of the industry standards for the valuation of default risk in credit loans portfolios. The calibration of CreditRisk+ requires, inter alia, the specification of the parameters describing the structure of dependence among default events. This work addresses the calibration of these parameters. In particular, we study the dependence of the calibration procedure on the sampling period of the default rate time series, that might be different from the time horizon onto which the model is used for forecasting, as it is often the case in real life applications. The case of autocorrelated time series and the role of the statistical error as a function of the time series period are also discussed. The findings of the proposed calibration technique are illustrated with the support of an application to real data.

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