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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
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Basic continuous-time signals include the unit step function, unit impulse function, and unit ramp function, collectively referred to as singularity functions. Singularity functions are characterized by discontinuities or discontinuous derivatives.
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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
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Drugs administered through various routes can lead to nonlinear elimination, resulting in complex pharmacokinetic behaviors crucial to understanding efficacious drug dosing.
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Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
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A New First-Order Integer-Valued Autoregressive Model with Bell Innovations.

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  • 1School of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, China.

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|July 2, 2021
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Summary

This study introduces a new integer-valued autoregressive model using Bell innovations, offering greater flexibility than Poisson models for count time series data exhibiting overdispersion.

Keywords:
Bell distributionINARcount time seriesestimationoverdispersion

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Area of Science:

  • Statistics
  • Time Series Analysis
  • Count Data Modeling

Background:

  • Integer-valued autoregressive (INAR) models commonly use Poisson innovations, which have restrictive mean-variance equality.
  • This limitation hinders flexibility in modeling count data, especially when overdispersion is present.

Purpose of the Study:

  • Introduce a novel first-order, non-negative, integer-valued autoregressive model (Bell-INAR(1)) utilizing Bell innovations.
  • Provide a more flexible alternative to Poisson-based INAR models, particularly for overdispersed count time series.

Main Methods:

  • The proposed Bell-INAR(1) model is constructed using the binomial thinning operator.
  • Key model properties, including mean, variance, joint distributions, and multi-step-ahead conditional measures, are derived.
  • Parameter estimation is performed using conditional least squares, Yule-Walker, and conditional maximum likelihood methods.

Main Results:

  • The Bell-INAR(1) model demonstrates superior flexibility compared to traditional Poisson-based models.
  • The model effectively captures overdispersion in count time series data.
  • Simulation studies assess the performance of the parameter estimation techniques.

Conclusions:

  • The developed Bell-INAR(1) model offers a simple yet powerful tool for analyzing overdispersed count time series.
  • The Bell distribution serves as a viable and flexible alternative to the Poisson distribution in INAR modeling.
  • The study validates the model's applicability through real data examples.