estCopC {CopCTS}

Pseudo maximum likelihood estimator of the copula parameter

Description

Obtains the pseudo maximum likelihood estimator of the copula parameter based on censored time series.

Usage

estCopC(cop="Gaussian",Yc,d,delta,nIS=500,jumps=NULL,MARGIN=NULL,...,interval=NULL)

Arguments

cop

the choice of copula function. There are currently five available copula funcitons, including Clayton copula, Gaussian copula, Gumbel copula, Joe copula and Frank copula. Specify one from "Clayton","Gaussian","Gumbel","Joe" and "Frank". The default is "Gaussian".

Yc

the Nx1 vector of observed response variable that is subject to lower detection limit.

d

the lower detection limit.

delta

the Nx1 vector of censoring indicator with 1 indicating uncensored and 0 indicating left censored.

nIS

the size for sequential importance sampling. The default is 500.

jumps

the Nx1 vector indicating whether each time t is a start of a new time series, which is deemed to be independent from the previous series. By default, jumps = c(1,rep(0,n-1)) indicating the data is one Markov sequence.

MARGIN

the marginal distribution function of the latent time series. The default is the empirical cdf:

\frac{1}{n+1}\sum_{t=1}^n I_{Y_t<=y}

. MARGIN can also be specified as other existing distribution functions such as pnorm.

...

additional parameters for the marginal distribution of the latent time series.

interval

the lower and upper bound for the copula paraameter. By default, interval= c(-1,1) for Gaussian copula, c(-1,Inf) for Clayton copula, c(1,Inf) for Gumbel and Joe copula and c(-Inf,Inf) for Frank copula.

Value

estCopC returns a list of components including.

para

the pseudo maximum likelihood estimator of the copula parameter.

likelihood

the negative log-likelihood value corresponding to the estimated copula parameter.

copula

the estimated copula object, with estimated copula parameter plugged in.

References

Li, F., Tang, Y. and Wang, H. (2018). Copula-based Semiparametric Analysis for Time Series Data with Detection Limits, technical report.

Examples

### Using a simulated data for demonstration:
set.seed(20)
Y = genLatentY(cop="Clayton",1,30,MARGIN.inv = qt,df=3)
d = -1
Yc = pmax(d,Y)
delta = (Y>d)
## CopC estimator
estCopC(cop = "Clayton",Yc,d,delta,nIS = 50,interval = c(1,10))
## Omniscient estimator
estCopC(cop = "Clayton",Y,d,delta=rep(TRUE,length(Y)),interval = c(1,10))
## CopC estimator under true marginal
estCopC(cop = "Clayton",Yc,d,delta,nIS = 50,MARGIN=pt,df=3,interval = c(1,10))
### Analyze the water quality data:
attach(water)
Yc = TNH3[1:30]
delta = Delta[1:30]
jumps = Indep[1:30]
set.seed(1)
estCopC(cop="Clayton",Yc=Yc,d=0.02,delta=delta,jumps=jumps,interval = c(1,10),nIS=50)