Comparison of the Performance of the Gradient and Newton-Raphson Method to Estimate Parameters in Some Zero-Inflated Regression Models

This paper compares the performance of the gradient and Newton-Raphson (N-R) method to estimate parameters in some zero-in ated (ZI) regression models such as zero-in ated Poisson (ZIP) model, zero-in ated Bell (ZIBell) model, zero-in ated binomial (ZIB) model and zero-in ated negative binomial (ZINB) model. In the present work, rstly, we brie y present the approach of the gradient and N-R method. We then introduce the origin, formulas and applications of the ZI models. Finally, we compare the performance of two investigated approaches for these models through the simulation studies with numerous sample sizes and several missing rates. A real data set is investigated in this study. Speci cally, we compare the results and the execution time of the R code for two methods. Moreover, we provide some important notes on these two approaches and some scalable research directions for future work.


Introduction
In statistics and other sciences, estimating equations play a very essential and meaningful role in research because if there is an estimating equation, we can utilize estimating methods to nd its solutions. The gradient and N-R method are the most accustomed approaches to address this problem. Terlaky [30] indicate that the gradient method is one of the widespread approach to nd the optimal solution. Another regularly employed approach is the N-R method. This approach is named after two very renowned mathematicians in the world, Isaac Newton and Joseph Raphson. For the full details of these methods, see Pho et al. [24].
It will be known that, count data is a very conventional data in practice. It is often found in areas such as: transportation, education, economics, engineering, etc. In practice, the percentage of zero events in a count outcome variable is too excessive for the Poisson regression or negative binomial regression model to t (Cameron and Trivedi [2], Chapter 6). Thus a zero-inated (ZI) regression model was introduced and largely applied as a remedy to the zero-excess issue (Rideout et al. [27]). c 2020 Journal of Advanced Engineering and Computation (JAEC) 227 VOLUME: 4 | ISSUE: 4 | 2020 | December Some common ZI regression models, for example: Hall [5] who proposed the ZIB regression model to deal with excess zeros in the binomial regression. Asymptotic properties of the maximum-likelihood estimator (MLE) of the ZIB model are discussed in Diallo et al. [3]. The ZIP regression model is proposed by Lambert [11]. Lemonte et al. [12] introduce the ZIBell regression models for count data, the ZINB regression model is stated in Ridout et al. [28], etc.
As we know, among all ZI regression models, zero-inated Poisson (ZIP) models are the most widespread. This model has been developed and studied in a very diverse and plentiful manner including its theory and application. For instance, Lambert [11] presents the ZIP model with an application to defects in production. Li et al. [15] introduce the multivariate ZIP models and their applications. Xie et al. [39] provide the ZIP model in statistical process control. Jansakul et al. [10] propose the score tests for ZIP models. Li [13] develops a lack-of-t test for the ZIP models. Long et al. [16] oers a marginalized ZIP regression model with overall exposure effects. Huang et al. [8] illustrate the ZIP model relied likelihood ratio test for drug safety signal detection, and so on.
In addition, we often encounter the factual data sets that have missing values. The problems about missing data play an extremely vital and signicant role in the scientic research.
Missing data is a very widespread and prevalent problem in several research disciplines, e.g., nances, engineering and medical, etc. This is- where m k > 0 minimizes the following function The demonstration in (1) is the formula of the gradient approach in multi-dimension.
The formula root of the N-R approach in multi-dimension case is described as: where It should be borne in mind that: ∇g(t) and Hg(t) are called the gradient and Hessian matrix of g(t), respectively. 2) Recipe for ZIP model Let Y be a count random variable, X and Z be covariates, X = (1, X, Z) T . Lambert [11] proposes the parametric ZIP regression model in which the non-susceptible probability (mixing weight) p is linked to X via a logit-linear predictor, p = H(θ T X ) with H(u) = [1 + exp(−u)] −1 , and the Poisson mean λ is linked to X via a loglinear predictor, λ = exp(α T X ), where θ and α are unknown parameters. The ZIP model can be illustrated as: The likelihood function of the ZIP model is described as We present the main results of this paper in the next section.

Simulation study
We consider the simulation study to four ZI    c 2020 Journal of Advanced Engineering and Computation (JAEC) 231 sults will be slightly dierent from this article, this reason is quite simple because each time R code is run, the data generation function will not be the same. In addition, dierent computers will oer slightly dierent values.
In the current work, the methods are used to estimate parameters for ZIP models with It has been seen that the operation time of R code of the gradient method is shorter than the N-R method in most cases. This is also in line with the theory that is discussed in detail in the next section.

The other ZI models
For comparison purposes, we also considered some other ZI regression models, for instance: Parameterη FηCCηWηW sηFηCCηWηW s • On average 53% of X were missing in 1000 replications.
• The average rate of Y = 0 was 51% and 23% in 1000 full and validated simulated data sets.

Discussions
Firstly, we oer a general approach to apply the N-R method as well as the gradient method, it is required to execute the following ve-steps algorithm: .

The score function of the ZIB regression model
The log-likelihood of ψ = (β T , γ T ) T of the ZIB model, based on the observations (Z i , X i , W i ), i = 1, . . . , n, is given by (see, Diallo et al. [3]): Ji log e γ T W i + 1 + e β T X i −m i − log 1 + e γ T W i + (1 − Ji) Ziβ T Xi − mi log 1 + e β T X i c 2020 Journal of Advanced Engineering and Computation (JAEC) 245 VOLUME: 4 | ISSUE: 4 | 2020 | December T , (i = 1, . . . , n) be the score function of the ZIB model. Then S i1 (ψ) = ∂ i (ψ)/∂β T and S i2 (ψ) = ∂ i (ψ)/∂γ T are respectively provided by:  ∂ i (θ)/∂γ T and S i3 (ψ) = ∂ i (θ)/∂1/α are respectively provided by: 250 "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited (CC BY 4.0)."