Clayton.Markov2.MLE {Copula.Markov}

Maximum Likelihood Estimation and Statistical Process Control Under the Clayton Copula with a 2nd order Markov chain.

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 Clayton copula and the marginal (stationary) distribution follows the normal distribution. The model diagnostic plot is also given (by the option "GOF=TRUE"). See Huang and Emura (2019) for the methodological details.

Usage

Clayton.Markov2.MLE(Y, k = 3, D = 1, plot = TRUE, GOF=FALSE)

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

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)

Xinwei Huang and Takeshi Emura

References

Huang XW, Emura T (2021), Model diagnostic procedures for copula-based Markov chain models for statistical process control, Communications in Statistics - Simulation and Computation, doi: 50(8):2345-67

Examples

Y = c(0.265, 0.256, 0.261, 0.261, 0.260, 0.257, 0.258, 0.263, 0.254, 0.254,
      0.258, 0.256, 0.256, 0.265, 0.270, 0.267, 0.270, 0.267, 0.266, 0.271,
      0.270, 0.264, 0.261, 0.264, 0.266, 0.264, 0.269, 0.268, 0.264, 0.262,
      0.257, 0.255, 0.255, 0.253, 0.251, 0.254, 0.255)
Clayton.Markov2.MLE(Y, k = 1, D = 1, plot = TRUE)

Y=Clayton.Markov2.DATA(n=1000,mu=0,sigma=1,alpha=8)
Clayton.Markov2.MLE(Y, plot=TRUE)