Clayton.Markov2.MLE: Maximum Likelihood Estimation and Statistical Process Control...

In Copula.Markov: Copula-Based Estimation and Statistical Process Control for Serially Correlated Time Series

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)

Example output

$estimates
         mu       sigma       alpha         UCL         LCL 
0.261049287 0.005741491 1.368887433 0.266790777 0.255307796 

$out_of_control
 [1]  9 10 15 16 17 18 20 21 27 28 32 33 34 35 36 37

$gradient
[1] -1.137437e-06  2.660385e-08 -1.023182e-08

$hessian
            [,1]       [,2]       [,3]
[1,] 547688.6475 -886.90112 2212.90224
[2,]   -886.9011   91.08253  -28.79977
[3,]   2212.9022  -28.79977   18.62747

$CM.test
[1] 0.1054486

$KS.test
[1] 0.1110295

$log_likelihood
[1] 152.4118

Warning messages:
1: In nlm(logL, initial, hessian = TRUE) :
  NA/Inf replaced by maximum positive value
2: In nlm(logL, initial, hessian = TRUE) :
  NA/Inf replaced by maximum positive value
3: In nlm(logL, initial, hessian = TRUE) :
  NA/Inf replaced by maximum positive value
4: In nlm(logL, initial, hessian = TRUE) :
  NA/Inf replaced by maximum positive value
$estimates
         mu       sigma       alpha         UCL         LCL 
 0.04778644  0.98485893  8.18568800  3.00236323 -2.90679034 

$out_of_control
[1] "NONE"

$gradient
[1] -2.773959e-05 -4.320100e-05 -8.652019e-07

$hessian
          [,1]       [,2]       [,3]
[1,]  697.8434  -989.1547   902.4891
[2,] -989.1547  2245.1051 -1566.8685
[3,]  902.4891 -1566.8685  1274.1062

$CM.test
[1] 5.321814

$KS.test
[1] 0.1276037

$log_likelihood
[1] 185.9784

Copula.Markov documentation built on Nov. 29, 2021, 9:07 a.m.