Semiparametric and Parametric Estimation and Bootstrapping of Integer-Valued Autoregressive (INAR) Models.

The package provides flexible simulation of INAR data using a general pmf to define the innovations’ distribution. It allows for semiparametric and parametric estimation of INAR models and includes a small sample refinement for the semiparametric setting. Additionally, it provides different procedures to appropriately bootstrap INAR data.

- Faymonville, M., Jentsch, C., Weiß, C.H. and Aleksandrov, B. (2022). “Semiparametric Estimation of INAR Models using Roughness Penalization”. Statistical Methods & Applications. DOI
- Jentsch, C. and Weiß, C.H. (2017), “Bootstrapping INAR Models”. Bernoulli 25(3), pp. 2359-2408. DOI
- Drost, F., Van den Akker, R. and Werker, B. (2009), “Efficient estimation of auto-regression parameters and inovation distributions for semiparametric integer-valued AR(p) models”. Journal of the Royal Statistical Society. Series B 71(2), pp. 467-485. DOI

This R package is licensed under the GPLv3. For bug reports (lack of documentation, misleading or wrong documentation, unexpected behaviour, …) and feature requests please use the issue tracker. Pull requests are welcome and will be included at the discretion of the author.

For installation of the development version use devtools:

We simulate two datasets. The first consists of n = 100 observations resulting from an INAR(1) model with coefficient alpha = 0.5 and Poi(1) distributed innovations. The second consists of n = 100 observations from an INAR(2) model with coefficients alpha_1 = 0.3, alpha_2 = 0.2 and a pmf equal to (0.3, 0.3, 0.2, 0.1, 0.1).

```
set.seed(1234)
dat1 <- spinar_sim(100, 1, alpha = 0.5, pmf = dpois(0:20,1))
dat2 <- spinar_sim(100, 2, alpha = c(0.3, 0.2), pmf = c(0.3, 0.3, 0.2, 0.1, 0.1))
```

We estimate an INAR(1) model on the first dataset.

```
#semiparametrically
spinar_est(dat1, 1)
#parametrically (moment estimation, true Poisson assumption)
spinar_est_param(dat1, 1, "mom", "poi")
```

We estimate an INAR(2) model on the second dataset.

For small samples, it can be beneficial to apply a penalized version of the semiparametric estimation. For illustration, we restrict ourselves to the first 50 observations of the first dataset and apply semiparametric, parametric and penalized semiparametric estimation. We choose a small L2 penalization as this showed to be most beneficial in the simulation study in Faymonville et al. (2022) (see references). Alternatively, one could also use the spinar_penal_val function which validates the two penalization parameters.

```
dat1_50 <- dat1[1:50]
spinar_est(dat1_50, 1)
spinar_est_param(dat1_50, 1, "mom", "poi")
spinar_penal(dat1, 1, penal1 = 0, penal2 = 0.1)
```

Finally, we bootstrap INAR(1) data on the first data set. We perform a semiparametric and a parametric INAR bootstrap (moment estimation, true Poisson assumption).

```
spinar_boot(dat1, 1, 500, setting = "sp")
spinar_boot(dat1, 1, 500, setting = "p", type = "mom", distr = "poi")
```

The file vignette.md provides reproduced results from the literature for each provided functionality of the spINAR package.

A possible extension of the spINAR package is to not only cover INAR models but also the extension to GINAR (generalized INAR) models, see Latour (1997). This model class does not only cover the binomial thinning but also allows for other thinning operations, e.g. thinning using geometrically distributed random variables.