Package 'spINAR'

(Semi)parametric estimation and bootstrapping of INAR models

Description

Semiparametric and parametric estimation of INAR models including a finite sample refinement for the semiparametric setting, different procedures to bootstrap INAR data and flexible simulation of INAR data.

Semiparametric INAR Model

The package provides a flexible simulation of INAR data by inserting a user-defined pmf argument in the spinar_sim function. Using spinar_est, it allows for semiparametric estimation of the INAR model along Drost et al. (2009) and additionally, it includes a small sample refinement spinar_penal (Faymonville et al., 2022) together with a validation of the upcoming penalization parameters (spinar_penal_val). Furthermore, it contains a semiparametric INAR bootstrap procedure implemented in spinar_boot (Jentsch and Weiß, 2017).

Parametric INAR Model

In addition to the semiparametric model, the package also allows for parametric simulation (spinar_sim), parametric estimation (spinar_est_param) and parametric bootstrapping (spinar_boot) of INAR data.

Author(s)

Maintainer: Maxime Faymonville [email protected] (ORCID)

Authors:

• Javiera Riffo [email protected] (ORCID)

• Jonas Rieger [email protected] (ORCID)

• Carsten Jentsch [email protected] (ORCID)

Other contributors:

• Christian H. Weiß [email protected] (ORCID) [contributor]

References

Faymonville, M., Riffo, J., Rieger, J. and Jentsch, C. (2024). "spINAR: An R Package for Semiparametric and Parametric Estimation and Bootstrapping of Integer-Valued Autoregressive (INAR) Models". Journal of Open Source Software 9(97), pp. 5386. doi:10.21105/joss.05386.

Faymonville, M., Jentsch, C., Weiß, C.H. and Aleksandrov, B. (2022). "Semiparametric Estimation of INAR Models using Roughness Penalization". Statistical Methods & Applications. doi:10.1007/s10260-022-00655-0.

Jentsch, C. and Weiß, C. H. (2017), "Bootstrapping INAR Models". Bernoulli 25(3), pp. 2359–2408. doi:10.3150/18-BEJ1057.

Drost, F., Van den Akker, R. and Werker, B. (2009), "Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models". Journal of the Royal Statistical Society. Series B 71(2), pp. 467–485. doi:10.1111/j.1467-9868.2008.00687.x.

See Also

Useful links:

• https://github.com/MFaymon/spINAR

• Report bugs at https://github.com/MFaymon/spINAR/issues

---

(Semi)parametric INAR bootstrap procedure

Description

INAR bootstrap procedures for the semiparametric and the parametric INAR setting, where the latter allows for moment- and maximum likelihood-based estimation and Poisson, geometrically and negative binomially distributed innovations.

Usage

spinar_boot(
  x,
  p,
  B,
  setting,
  type = "mom",
  distr = "poi",
  M = 100,
  level = 0.05,
  progress = TRUE
)

Arguments

x	[integer] vector with integer observations.
p	[integer(1)] order of the INAR model, where $\code{p} \in \{1,2\}$.
B	[integer(1)] number of bootstrap repetitions.
setting	[string(1)] estimation setting $\in \code{\{"sp", "p"\}}$, where "sp" defines a semiparametric setting and "p" a parametric setting.
type	[string(1)] type of estimation $\in \code{\{"mom", "ml"\}}$, where "mom" (default) performs moment-based estimation and "ml" maximum likelihood-based estimation.
distr	[string(1)] parametric family of innovation distribution $\in \code{\{"poi", "geo", "nb"\}}$, where "poi" (default) denotes Poi(lambda), "geo" Geo(prob) and "nb" NB(r, prob) distributions.
M	[integer(1)] upper limit for the innovations.
level	[numeric(1)] level for the bootstrap confidence intervals (percentile interval and Hall's percentile interval (bootstrap-t-interval without studentization)).
progress	[logical(1)] Should a nice progress bar be shown? Turning it off, could lead to significantly faster calculation. Default is TRUE.

Value

[named list] with entries

x_star

[matrix] of bootstrap observations with length(x) rows and B columns.

parameters_star

[matrix] of bootstrap estimated parameters with B rows. If setting = "sp", each row contains the estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the estimated entries of the pmf $\code{pmf}_0, \code{pmf}_1$, ... where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$. If setting = "p", each row contains the estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the estimated parameter(s) of the innovation distribution.

bs_ci_percentile

[named matrix] with the lower and upper bounds of the bootstrap percentile confidence intervals for each parameter in parameters_star.

bs_ci_hall

[named matrix] with the lower and upper bounds of Hall's bootstrap percentile confidence intervals for each parameter in parameters_star.

Examples

# generate data
dat1 <- spinar_sim(n = 200, p = 1, alpha = 0.5,
                   pmf = c(0.3, 0.3, 0.2, 0.1, 0.1))
dat2 <- spinar_sim(n = 200, p = 2, alpha = c(0.2, 0.3),
                   pmf = dgeom(0:60, 0.5))


# semiparametric INAR(1) bootstrap
spinar_boot(x = dat1, p = 1, B = 50, setting = "sp")
# parametric Geo-INAR(2) bootstrap using moment-based estimation
spinar_boot(x = dat2, p = 2, B = 50, setting = "p", type = "mom", distr = "geo")

---

Semiparametric estimation of INAR models

Description

Semiparametric estimation of the autoregressive parameters and the innovation distribution of INAR(p) models, $\code{p} \in \{1,2\}$. The estimation is conducted by maximizing the conditional likelihood of the model.

Usage

spinar_est(x, p)

Arguments

x	[integer] vector with integer observations.
p	[integer(1)] order of the INAR model, where $\code{p} \in \{1,2\}$.

Value

Vector containing the estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the estimated entries of the pmf $\code{pmf}_0, \code{pmf}_1$,... where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$.

Examples

# generate data
dat1 <- spinar_sim(n = 200, p = 1, alpha = 0.5,
                   pmf = c(0.3, 0.3, 0.2, 0.1, 0.1))
dat2 <- spinar_sim(n = 200, p = 2, alpha = c(0.2, 0.3),
                   pmf = c(0.25, 0.2, 0.15, 0.1, 0.1, 0.1, 0.1))


# semiparametric estimation of INAR(1) model
spinar_est(x = dat1, p = 1)
# semiparametric estimation of INAR(2) model
spinar_est(x = dat2, p = 2)

---

Parametric estimation of INAR models

Description

Parametric estimation of the autoregressive parameters and the innovation distribution of INAR(p) models, $\code{p} \in \{1,2\}$, with Poisson, geometrically or negative binomially distributed innovations. The estimation can either be moment- or maximum likelihood-based.

Usage

spinar_est_param(x, p, type, distr)

Arguments

x	[integer] vector with integer observations.
p	[integer(1)] order of the INAR model, where $\code{p} \in \{1,2\}$.
type	[string(1)] type of estimation $\in \code{\{"mom", "ml"\}}$, where "mom" performs moment-based estimation and "ml" maximum likelihood-based estimation.
distr	[string(1)] parametric family of innovation distribution $\in \code{\{'poi', 'geo', 'nb'\}}$, where "poi" denotes Poi(lambda), "geo" Geo(prob) and "nb" NB(r, prob) distributions.

Value

Named vector containing the estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the estimated parameter(s) of the innovation distribution.

Examples

# generate data
# Poi-INAR(1) data
dat1 <- spinar_sim(n = 200, p = 1, alpha = 0.5, pmf = dpois(0:20, 1))
# Geo-INAR(2) data
dat2 <- spinar_sim(n = 200, p = 2, alpha = c(0.2, 0.3),
                   pmf = dgeom(0:60, 0.5))
# NB-INAR(1) data
dat3 <- spinar_sim(n = 200, p = 1, alpha = 0.5, pmf = dnbinom(0:40, 2, 2/3))

# moment-based parametric estimation of Poi-INAR(1) model
spinar_est_param(x = dat1, p = 1, type = "mom", distr = "poi")
# moment-based parametric estimation of Geo-INAR(2) model
spinar_est_param(x = dat2, p = 2, type = "mom", distr = "geo")
# maximum likelihood-based parametric estimation of NB-INAR(1) model
spinar_est_param(x = dat3, p = 1, type = "ml", distr = "nb")

---

Penalized semiparametric estimation of INAR models

Description

Semiparametric penalized estimation of the autoregressive parameters and the innovation distribution of INAR(p) models, $\code{p} \in \{1,2\}$. The estimation is conducted by maximizing the penalized conditional likelihood of the model. If both penalization parameters are set to zero, the function coincides to the spinar_est function of this package.

Usage

spinar_penal(x, p, penal1 = 0, penal2 = 0)

Arguments

x	[integer] vector with integer observations.
p	[integer(1)] order of the INAR model, where $\code{p} \in \{1,2\}$.
penal1	$L_1$ penalization parameter (default value zero results in no $L_1$ penalization)
penal2	$L_2$ penalization parameter (default value zero results in no $L_2$ penalization)

Value

Vector containing the penalized estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the penalized estimated entries of the pmf $\code{pmf}_0, \code{pmf}_1$,... where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$.

Examples

# generate data
dat1 <- spinar_sim(n = 50, p = 1, alpha = 0.5,
                   pmf = c(0.3, 0.25, 0.2, 0.15, 0.1))

# penalized semiparametric estimation
spinar_penal(x = dat1, p = 1, penal1 = 0, penal2 = 0.1)

---

Validated penalized semiparametric estimation of INAR models

Description

Semiparametric penalized estimation of the autoregressive parameters and the innovation distribution of INAR(p) models, $\code{p} \in \{1,2\}$. The estimation is conducted by maximizing the penalized conditional likelihood of the model. Included is a possible validation of one or both penalization parameters. If no validation is wanted, the function coincides to the spinar_penal function of this package.

Usage

spinar_penal_val(
  x,
  p,
  validation,
  penal1 = NA,
  penal2 = NA,
  over = NA,
  folds = 10,
  init1 = 1,
  init2 = 1,
  progress = TRUE
)

Arguments

x	[integer] vector with integer observations.
p	[integer(1)] order of the INAR model, where $\code{p} \in \{1,2\}$.
validation	[logical(1)] indicates whether validation is wanted.
penal1	[numeric(1)] $L_1$ penalization parameter. It will be ignored if validation = TRUE and over $\in \{"both", "L_1"\}$. It is mandatory if validation = FALSE.
penal2	[numeric(1)] $L_2$ penalization parameter. It will be ignored if validation = TRUE and over $\in \{"both", "L_2"\}$. It is mandatory if validation = FALSE.
over	[string(1)] validation over "both" penalization parameters or only over "L_1" or "L_2". It is mandatory if validation = TRUE, otherwise it will be ignored.
folds	[integer(1)] number of folds for (cross) validation.
init1	[numeric(1)] initial value for penal1 in validation. Default value is init1 = 1.
init2	[numeric(1)] initial value for penal2 in validation. Default value is init2 = 1
progress	[logical(1)] Should a nice progress bar be shown? Turning it off, could lead to significantly faster calculation. Default is TRUE.

Value

If validation = FALSE, the function returns a vector containing the penalized estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the penalized estimated entries of the pmf $\code{pmf}_0, \code{pmf}_1$... where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$.

If validation = TRUE, the function returns a named list, where the first entry contains the penalized estimated coefficients $\code{alpha}_1,...,\code{alpha}_p$ and the penalized estimated entries of the pmf $\code{pmf}_0, \code{pmf}_1$,... where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$. The second (and if over = both also the third entry) contain(s) the validated penalization parameter(s).

Examples

# generate data
dat1 <- spinar_sim(n = 50, p = 1, alpha = 0.5,
                   pmf = c(0.3, 0.3, 0.2, 0.1, 0.1))


# penalized semiparametric estimation with validation over L1
spinar_penal_val(x = dat1, p = 1, validation = TRUE, penal2 = 0.1,
                 over = "L1")
# penalized semiparametric estimation with validation over both L1 and L2
spinar_penal_val(x = dat1, p = 1, validation = TRUE, over = "both")

---

Simulation of (semi)parametric integer autoregressive (INAR) models

Description

Generating INAR(p) observations, where p $\in \{1,2\}$. It allows for general pmfs which can be generated parametrically or "manually" (semiparametrically).

Usage

spinar_sim(n, p, alpha, pmf, prerun = 500)

Arguments

n	[integer(1)] number of observations.
p	[integer(1)] lag of the INAR(p) model, where p $\in \{1,2\}$.
alpha	[integer(p)] vector of INAR coefficients $\code{alpha}_1,...,\code{alpha}_p$.
pmf	[numeric] vector of probability mass function $\code{pmf}_0,..., \code{pmf}_k$ where $\code{pmf}_i$ represents the probability of an innovation being equal to $i$.
prerun	[integer(1)] number of observations which are generated additionally and then omitted (to ensure stationarity).

Value

Vector with $n$ INAR(p) observations.

Examples

# generate (semiparametrically) 100 INAR(1) observations with
# alpha_1 = 0.5 and a manually set pmf
spinar_sim(n = 100, p = 1, alpha = 0.5, pmf = c(0.3, 0.3, 0.2, 0.1, 0.1))

# generate 100 obervations of an INAR(2) model with
# alpha_1 = 0.2, alpha_2 = 0.3 and Poi(1)-innovations
spinar_sim(n = 100, p = 2, alpha = c(0.2, 0.3), pmf = dpois(0:20,1))

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