Joe.Markov.MLE {Copula.Markov}

Maximum Likelihood Estimation and Statistical Process Control Under the Joe Copula

Description

The maximum likelihood estimates are produced and the Shewhart control chart is drawn with k-sigma control limits (e.g., 3-sigma). The dependence model follows the Joe copula and the marginal (stationary) distribution follows the normal distribution.

Usage

Joe.Markov.MLE(Y, k = 3, D = 1, plot = TRUE,GOF=FALSE,method = "nlm")

Arguments

Y

vector of datasets

k

constant determining the length between LCL and UCL (k=3 corresponds to 3-sigma limit)

D

diameter for U(-D, D) used in randomized Newton-Raphson

plot

show the control chart if TRUE

GOF

show the model diagnostic plot if TRUE

method

apply "nlm" or "Newton" method

Value

mu

estimate, SE, and 95 percent CI

sigma

estimate, SE, and 95 percent CI

alpha

estimate, SE, and 95 percent CI

Control_Limit

Center = mu, LCL = mu - k*sigma, UCL = mu + k*sigma

out_of_control

IDs for out-of-control points

Gradient

gradients (must be zero)

Hessian

Hessian matrix

Eigenvalue_Hessian

Eigenvalues for the Hessian matrix

KS.test

KS statistics

CM.test

CM statistics

log.likelihood

Log-likelihood value for the estimation

Author(s)

Long TH, Huang XW and Takeshi Emura

References

Emura T, Long TH, Sun LH (2017), R routines for performing estimation and statistical process control under copula-based time series models, Communications in Statistics - Simulation and Computation, 46 (4): 3067-87

Long TH and Emura T (2014), A control chart using copula-based Markov chain models, Journal of the Chinese Statistical Association 52 (No.4): 466-96

Examples

n=1000
alpha=2.856 ### Kendall's tau =0.5 ###
mu=2
sigma=1
Y=Joe.Markov.DATA(n,mu,sigma,alpha)
mean(Y)
sd(Y)
cor(Y[-1],Y[-n],method="kendall")

Joe.Markov.MLE(Y,k=2)