Nothing
R/hCGF_plot.R
In PlotNormTest: Graphical Univariate/Multivariate Assessments for Normality Assumption
Defines functions
d4hCGF_plot
d3hCGF_plot
Documented in
d3hCGF_plot
d4hCGF_plot
## -- Filename: hCGF_plot.R
#' @title Graphical plots to assess multivariate normality assumption.
#' @name Multivariate_CGF_PLot
#' @description Cumulant generating functions of normally distributed
#' random variables has derivatives of order higher than 3 are all 0.
#' Hence, plots of empirical third/fourth order derivatives with large value
#' or high slope gives indication of non-normality.
#' \code{Multivariate_CGF_PLot} estimates and provides confidence region for
#' average (or any linear combination) of third/fourth derivatives of empirical
#' cumulant function at the points \eqn{t = t^*1_p}. Plots for
#' \eqn{p = 2, 3, \dots, 10} will be faster to obtain, as confidence regions
#' and other necessary parameters are available in \code{mt3_lst_param.rda} and
#' \code{mt4_lst_param.rda}.
#' Higher dimension requires expensive computational cost.
#' @param x Data matrix of size \eqn{n \times p}
# @param l Vector of linear combination, having length is the number of
# unique third/fourth derivatives. Default is \code{NULL} where the algorithm
# will use the average of distinct derivatives.
#' @param alpha Significant level (default is \eqn{.05})
#' @return \code{d3hCGF_plot} returns plot relying in third derivatives.
#' @seealso \code{\link[=dhCGF_plot1D]{dhCGF_plot1D()}}
#' @export
#' @examples
#' set.seed(1234)
#' p <- 3
#' x <- MASS::mvrnorm(500, rep(0, p), diag(p))
#' d3hCGF_plot(x)
#' d4hCGF_plot(x)
d3hCGF_plot <- function(x,
# l = NULL,
alpha = 0.05){
bigt <- seq(-1, 1, by = 0.05)
p <- ncol(x); n <- nrow(x)
lT <- p + choose(p, 2)*2 + choose(p, 3)
numt <- length(bigt)
l <- rep(1/sqrt(lT), lT)
# This part allows other values of l
# if (is.null(l)){
# l <- rep(1/sqrt(lT), lT)
# }
if (!(p %in% 1:10)){
lst <- mt3_get_param(p, l = l)
warning("Heavy computation")
} else {
Hvar <-
matrix(
nrow = 10, ncol = 21, byrow = T,
data = c(
92.42158217, 74.73877805, 60.7302487, 49.59500711, 40.7146832,
33.61041073, 27.91023887, 23.3244267, 19.62665994, 16.63973104,
14.22459407, 12.2719798, 10.69596014, 9.429002869, 8.418171,
7.622206539, 7.00930269, 6.555417571, 6.243020088, 6.060187813,
6,
51.64735474, 42.06049707, 34.43383781, 28.34512273, 23.46757024,
19.54747827, 16.3872706, 13.83262291, 11.76265797, 10.08245759,
8.717328805, 7.608403151, 6.70925268, 5.983284647, 5.401735469,
4.942128632, 4.58709455, 4.323475808, 4.141660713, 4.035103292,
4,
38.88149786, 31.69987493, 25.98318627, 21.41640716, 17.75565754,
14.81153012, 12.43646553, 10.51516324, 8.957276844, 7.691833972,
6.662962446, 5.826609195, 5.148016599, 4.599779034, 4.160346054,
3.812871459, 3.544332294, 3.344860732, 3.207246338, 3.126577521,
3.1,
32.68733899, 26.64565406, 21.8369797, 17.99606145, 14.91756688,
12.44204029, 10.4452675, 8.830198359, 7.520794105, 6.457327483,
5.592783363, 4.890095636, 4.320022099, 3.859508021, 3.490425881,
3.198606386, 2.973096793, 2.805598483, 2.690047987, 2.622315183,
2.6,
29.04678297, 23.66659881, 19.38571068, 15.96742076, 13.22853561,
11.0268189, 9.251491708, 7.816018372, 6.652612096, 5.708038546,
4.940402436, 4.316681083, 3.810827712, 3.402311133, 3.074991259,
2.816254638, 2.616352869, 2.467900968, 2.365503721, 2.305486558,
2.285714286,
26.6578591, 21.70842882, 17.77156459, 14.62903647, 12.11197704,
10.08929945, 8.458929531, 7.141148429, 6.0735229, 5.20703381,
4.503111954, 3.931363246, 3.467819647, 3.093592715, 2.793836949,
2.556952971, 2.373977783, 2.238122517, 2.144428152, 2.089517571,
2.071428571,
24.97305857, 20.3259158, 16.63063154, 13.68187755, 11.32079857,
9.424102982, 7.895805196, 6.66095645, 5.6608713, 4.849482818,
4.190552338, 3.655528423, 3.221900887, 2.871933871, 2.591690565,
2.370283663, 2.199301874, 2.0723752, 1.984851205, 1.933561885,
1.916666667,
23.72267204, 19.29911891, 15.78260114, 12.97730357, 10.73175543,
8.928416104, 7.475793238, 6.302459321, 5.352496366, 4.582018088,
3.956507809, 3.448776016, 3.037390364, 2.705467401, 2.439742546,
2.22985541, 2.067803051, 1.947525575, 1.864597561, 1.816005865,
1.8,
22.75872054, 18.50712799, 15.12814181, 12.43324677, 10.27663975,
8.54519503, 7.150870282, 6.024942309, 5.113618989, 4.374689512,
3.774960568, 3.288289533, 2.89407277, 2.576082348, 2.321570752,
2.12058297, 1.965430277, 1.850291432, 1.770915745, 1.724409283,
1.709090909,
21.99337116, 17.87807412, 14.60812103, 12.00077286, 9.914711778,
8.24030648, 6.892245795, 5.803947699, 4.923303532, 4.209429498,
3.630180608, 3.160243679, 2.779671764, 2.472756467, 2.227160141,
2.033249152, 1.883583916, 1.772532444, 1.69598262, 1.651135061,
1.636363636
)
)
Msup <- c(1.453986705, 1.445928528, 1.444276978, 1.44268609, 1.444282713,
1.444054044, 1.446514826, 1.445421625, 1.44661582, 1.448280449
)
lst <- list(m.supLt = Msup[p],
varLtLs = c(Hvar[p, ], Hvar[p, 20:1])
)
# lst <- mt3_lst_param[[p]]
# This part costs memories, improve later.
# if (any(l != rep(1/sqrt(lT), lT))){
# tmp <- mt3_covLtLs(l= l, p = p, sTtTs = lst$l.sTtTs)
# lst$sLtLs <- tmp$sLtLs
# lst$m.supLt <- tmp$m.supLt
# lst$varLtLs <- diag(lst$sLtLs)
# }
}
v3 <- lapply(bigt/sqrt(p),function(u) d3hCGF(myt = rep(u, p), x = x))
L3 <- unlist(lapply(v3, function(v) sqrt(n) * sum(l*v)))
u <- sqrt(-2 *log(alpha))
band1 <- (u + lst$m.supLt)*sqrt(lst$varLtLs)
band2 <- -(u + lst$m.supLt)*sqrt(lst$varLtLs)
ylim <- max(max(abs(band1)), abs(L3))
plot(bigt, L3, ylim = c(-ylim, ylim), col = "blue",
lty = 1, type = "l", lwd = 2, ylab = bquote(L[3]), xlab = "t")
graphics::lines(bigt, band1, col = "orange", lty = 6, lwd = 2)
graphics::lines(bigt, band2, col = "orange", lty = 6, lwd = 2)
graphics::legend("top", c("Probability bands", expression(L[3])),
lty = c(6, 1), col = c("orange", "blue"), lwd = c(2, 2),
merge = TRUE, y.intersp = 1.5, text.width = .6)
graphics::title(main = bquote(MT[3] ~"plot"), adj = 0)
til.L <- L3/sqrt(lst$varLtLs)
deci <- ifelse(max(abs(til.L)) >= u + lst$m.supLt, "reject", "accept")
return(deci)
}
######################################################
#' Title
#' @rdname Multivariate_CGF_PLot
#'
#' @return \code{d4hCGF_plot} returns plot relying in forth derivatives.
#' @export
d4hCGF_plot <- function(x,
# l = NULL,
alpha = 0.05){
bigt <- seq(-1, 1, by = 0.05)
p <- ncol(x); n <- nrow(x)
lT4 <- p + 3 *choose(p, 2) + 3*choose(p, 3) + choose(p, 4)
numt <- length(bigt)
l <- rep(1/sqrt(lT4), lT4)
# if (is.null(l)){
# l <- rep(1/sqrt(lT4), lT4)
# }
if (!(p %in% 1:10)){
lst <- mt4_get_param(p, l = l)
warning("Heavy computation")
} else {
Hvar <-
matrix(nrow = 10, ncol = 21, byrow = T,
data = c(
568.1209021, 448.0515896, 355.0182299, 282.6795124, 226.2374173,
182.0500981, 147.3446669, 120.0034577, 98.40450305, 81.30213209,
67.73735645, 56.97044429, 48.4300794, 41.67496479, 36.3648053,
32.23839726, 29.09714181, 26.79273777, 25.2181407, 24.30112719,
24,
280.078945, 220.4020183, 174.2856492, 138.5357285, 110.735366,
89.05053643, 72.08563482, 58.77581847, 48.30654298, 40.05326137,
33.53611834, 28.38583129, 24.31794483, 21.11337599, 18.60370546,
16.66006845, 15.18479424, 14.10516504, 13.36883132, 12.94054861,
12.8,
192.017125, 150.4754251, 118.5211313, 93.85864813, 74.76032964,
59.92208683, 48.35667067, 39.31457154, 32.22522209, 26.65317137,
22.26533269, 18.80644783, 16.08066803, 13.93770524, 12.2624126,
10.96695142, 9.984921882, 9.266999309, 8.777740015, 8.493314772,
8.4,
139.4637519, 109.4943184, 86.42252828, 68.59431806, 54.76680757,
44.00306309, 35.5945326, 29.00370386, 23.82158724, 19.73609938,
16.50848791, 13.95570699, 11.93721173, 10.34504666, 9.096400137,
8.128014685, 7.392003839, 6.85274504, 6.484607316, 6.270339886,
6.2,
104.9909838, 82.87394461, 65.77466418, 52.50044615, 42.15382226,
34.05709724, 27.69674951, 22.68233197, 18.71599557, 15.56982032,
13.0689038, 11.0787099, 9.495581318, 8.239611218, 7.249281821,
6.477434023, 5.888246794, 5.454990303, 5.15838038, 4.98541008,
4.928571429,
82.011931, 65.17873306, 52.08615944, 41.85866521, 33.83509445,
27.51431271, 22.51507366, 18.54629239, 15.38495284, 12.85963481,
10.83819206, 9.218508179, 7.921543093, 6.886091921, 6.064829952,
5.421329472, 4.927816535, 4.563496973, 4.313326754, 4.167136587,
4.119047619,
66.62374719, 53.30475074, 42.87967482, 34.68359842, 28.21195052,
23.08037733, 18.99500689, 15.73033808, 13.11274092, 11.00810133,
9.31253868, 7.945410092, 6.8440239, 5.959636079, 5.254414896,
4.699140766, 4.27146887, 3.954627172, 3.73645638, 3.608724244,
3.566666667,
56.33114824, 45.31277767, 36.64217968, 29.78935694, 24.35000213,
20.01455816, 16.54517401, 13.75847828, 11.51265495, 9.69771301,
8.228135973, 7.037312106, 6.073302562, 5.295619503, 4.67277072,
4.180389742, 3.799816819, 3.51703094, 3.321859343, 3.207411075,
3.16969697,
49.50809201, 39.95575422, 32.4136072, 26.4335349, 21.67217263,
17.86555133, 14.81010191, 12.34837317, 10.35825587, 8.744824767,
7.434142751, 6.36854146, 5.503017571, 4.802476072, 4.239619901,
3.793336062, 3.447466197, 3.189878105, 3.011776472, 2.907207735,
2.872727273,
45.07928983, 36.41542436, 29.56901672, 24.13680211, 19.80897198,
16.34684669, 13.5660621, 11.32386571, 9.50948683, 8.036814083,
6.838811252, 5.863247519, 5.069425651, 4.425671118, 3.907404411,
3.495663057, 3.175973205, 2.937495809, 2.772391759, 2.675365267,
2.643356643
)
)
Msup <- c(1.478985651,
1.454432793,
1.448680764,
1.449313815,
1.453737199,
1.4561804,
1.463732497,
1.470303197,
1.476776159,
1.484990068)
lst <- list(m.supLt = Msup[p],
varLtLs = c(Hvar[p, ], Hvar[p, 20:1])
)
}
v4 <- lapply(bigt/sqrt(p),function(u) d4hCGF(myt = rep(u, p), x = x))
L4 <- unlist(lapply(v4, function(v) sqrt(n) * sum(l*v)))
u <- sqrt(-2 *log(alpha))
band1 <- (u + lst$m.supLt)*sqrt(lst$varLtLs)
band2 <- -(u + lst$m.supLt)*sqrt(lst$varLtLs)
ylim <- max(max(abs(band1)), abs(L4))
plot(bigt, L4, ylim = c(-ylim, ylim), col = "blue",
lty = 1, type = "l", lwd = 2, ylab = bquote(L[4]), xlab = "t")
graphics::lines(bigt, band1, col = "orange", lty = 6, lwd = 2)
graphics::lines(bigt, band2, col = "orange", lty = 6, lwd = 2)
graphics::legend("top", c("Probability bands", expression(L[4])),
lty = c(6, 1), col = c("orange", "blue"), lwd = c(2, 2),
merge = TRUE, y.intersp = 1.5, text.width = .6)
til.L <- L4/sqrt(lst$varLtLs)
deci <- ifelse(max(abs(til.L)) >= u + lst$m.supLt, "reject", "accept")
return(deci)
}
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Defines functions d4hCGF_plot d3hCGF_plot
Documented in d3hCGF_plot d4hCGF_plot
## -- Filename: hCGF_plot.R
#' @title Graphical plots to assess multivariate normality assumption.
#' @name Multivariate_CGF_PLot
#' @description Cumulant generating functions of normally distributed
#' random variables has derivatives of order higher than 3 are all 0.
#' Hence, plots of empirical third/fourth order derivatives with large value
#' or high slope gives indication of non-normality.
#' \code{Multivariate_CGF_PLot} estimates and provides confidence region for
#' average (or any linear combination) of third/fourth derivatives of empirical
#' cumulant function at the points \eqn{t = t^*1_p}. Plots for
#' \eqn{p = 2, 3, \dots, 10} will be faster to obtain, as confidence regions
#' and other necessary parameters are available in \code{mt3_lst_param.rda} and
#' \code{mt4_lst_param.rda}.
#' Higher dimension requires expensive computational cost.
#' @param x Data matrix of size \eqn{n \times p}
# @param l Vector of linear combination, having length is the number of
# unique third/fourth derivatives. Default is \code{NULL} where the algorithm
# will use the average of distinct derivatives.
#' @param alpha Significant level (default is \eqn{.05})
#' @return \code{d3hCGF_plot} returns plot relying in third derivatives.
#' @seealso \code{\link[=dhCGF_plot1D]{dhCGF_plot1D()}}
#' @export
#' @examples
#' set.seed(1234)
#' p <- 3
#' x <- MASS::mvrnorm(500, rep(0, p), diag(p))
#' d3hCGF_plot(x)
#' d4hCGF_plot(x)
d3hCGF_plot <- function(x,
# l = NULL,
alpha = 0.05){
bigt <- seq(-1, 1, by = 0.05)
p <- ncol(x); n <- nrow(x)
lT <- p + choose(p, 2)*2 + choose(p, 3)
numt <- length(bigt)
l <- rep(1/sqrt(lT), lT)
# This part allows other values of l
# if (is.null(l)){
# l <- rep(1/sqrt(lT), lT)
# }
if (!(p %in% 1:10)){
lst <- mt3_get_param(p, l = l)
warning("Heavy computation")
} else {
Hvar <-
matrix(
nrow = 10, ncol = 21, byrow = T,
data = c(
92.42158217, 74.73877805, 60.7302487, 49.59500711, 40.7146832,
33.61041073, 27.91023887, 23.3244267, 19.62665994, 16.63973104,
14.22459407, 12.2719798, 10.69596014, 9.429002869, 8.418171,
7.622206539, 7.00930269, 6.555417571, 6.243020088, 6.060187813,
6,
51.64735474, 42.06049707, 34.43383781, 28.34512273, 23.46757024,
19.54747827, 16.3872706, 13.83262291, 11.76265797, 10.08245759,
8.717328805, 7.608403151, 6.70925268, 5.983284647, 5.401735469,
4.942128632, 4.58709455, 4.323475808, 4.141660713, 4.035103292,
4,
38.88149786, 31.69987493, 25.98318627, 21.41640716, 17.75565754,
14.81153012, 12.43646553, 10.51516324, 8.957276844, 7.691833972,
6.662962446, 5.826609195, 5.148016599, 4.599779034, 4.160346054,
3.812871459, 3.544332294, 3.344860732, 3.207246338, 3.126577521,
3.1,
32.68733899, 26.64565406, 21.8369797, 17.99606145, 14.91756688,
12.44204029, 10.4452675, 8.830198359, 7.520794105, 6.457327483,
5.592783363, 4.890095636, 4.320022099, 3.859508021, 3.490425881,
3.198606386, 2.973096793, 2.805598483, 2.690047987, 2.622315183,
2.6,
29.04678297, 23.66659881, 19.38571068, 15.96742076, 13.22853561,
11.0268189, 9.251491708, 7.816018372, 6.652612096, 5.708038546,
4.940402436, 4.316681083, 3.810827712, 3.402311133, 3.074991259,
2.816254638, 2.616352869, 2.467900968, 2.365503721, 2.305486558,
2.285714286,
26.6578591, 21.70842882, 17.77156459, 14.62903647, 12.11197704,
10.08929945, 8.458929531, 7.141148429, 6.0735229, 5.20703381,
4.503111954, 3.931363246, 3.467819647, 3.093592715, 2.793836949,
2.556952971, 2.373977783, 2.238122517, 2.144428152, 2.089517571,
2.071428571,
24.97305857, 20.3259158, 16.63063154, 13.68187755, 11.32079857,
9.424102982, 7.895805196, 6.66095645, 5.6608713, 4.849482818,
4.190552338, 3.655528423, 3.221900887, 2.871933871, 2.591690565,
2.370283663, 2.199301874, 2.0723752, 1.984851205, 1.933561885,
1.916666667,
23.72267204, 19.29911891, 15.78260114, 12.97730357, 10.73175543,
8.928416104, 7.475793238, 6.302459321, 5.352496366, 4.582018088,
3.956507809, 3.448776016, 3.037390364, 2.705467401, 2.439742546,
2.22985541, 2.067803051, 1.947525575, 1.864597561, 1.816005865,
1.8,
22.75872054, 18.50712799, 15.12814181, 12.43324677, 10.27663975,
8.54519503, 7.150870282, 6.024942309, 5.113618989, 4.374689512,
3.774960568, 3.288289533, 2.89407277, 2.576082348, 2.321570752,
2.12058297, 1.965430277, 1.850291432, 1.770915745, 1.724409283,
1.709090909,
21.99337116, 17.87807412, 14.60812103, 12.00077286, 9.914711778,
8.24030648, 6.892245795, 5.803947699, 4.923303532, 4.209429498,
3.630180608, 3.160243679, 2.779671764, 2.472756467, 2.227160141,
2.033249152, 1.883583916, 1.772532444, 1.69598262, 1.651135061,
1.636363636
)
)
Msup <- c(1.453986705, 1.445928528, 1.444276978, 1.44268609, 1.444282713,
1.444054044, 1.446514826, 1.445421625, 1.44661582, 1.448280449
)
lst <- list(m.supLt = Msup[p],
varLtLs = c(Hvar[p, ], Hvar[p, 20:1])
)
# lst <- mt3_lst_param[[p]]
# This part costs memories, improve later.
# if (any(l != rep(1/sqrt(lT), lT))){
# tmp <- mt3_covLtLs(l= l, p = p, sTtTs = lst$l.sTtTs)
# lst$sLtLs <- tmp$sLtLs
# lst$m.supLt <- tmp$m.supLt
# lst$varLtLs <- diag(lst$sLtLs)
# }
}
v3 <- lapply(bigt/sqrt(p),function(u) d3hCGF(myt = rep(u, p), x = x))
L3 <- unlist(lapply(v3, function(v) sqrt(n) * sum(l*v)))
u <- sqrt(-2 *log(alpha))
band1 <- (u + lst$m.supLt)*sqrt(lst$varLtLs)
band2 <- -(u + lst$m.supLt)*sqrt(lst$varLtLs)
ylim <- max(max(abs(band1)), abs(L3))
plot(bigt, L3, ylim = c(-ylim, ylim), col = "blue",
lty = 1, type = "l", lwd = 2, ylab = bquote(L[3]), xlab = "t")
graphics::lines(bigt, band1, col = "orange", lty = 6, lwd = 2)
graphics::lines(bigt, band2, col = "orange", lty = 6, lwd = 2)
graphics::legend("top", c("Probability bands", expression(L[3])),
lty = c(6, 1), col = c("orange", "blue"), lwd = c(2, 2),
merge = TRUE, y.intersp = 1.5, text.width = .6)
graphics::title(main = bquote(MT[3] ~"plot"), adj = 0)
til.L <- L3/sqrt(lst$varLtLs)
deci <- ifelse(max(abs(til.L)) >= u + lst$m.supLt, "reject", "accept")
return(deci)
}
######################################################
#' Title
#' @rdname Multivariate_CGF_PLot
#'
#' @return \code{d4hCGF_plot} returns plot relying in forth derivatives.
#' @export
d4hCGF_plot <- function(x,
# l = NULL,
alpha = 0.05){
bigt <- seq(-1, 1, by = 0.05)
p <- ncol(x); n <- nrow(x)
lT4 <- p + 3 *choose(p, 2) + 3*choose(p, 3) + choose(p, 4)
numt <- length(bigt)
l <- rep(1/sqrt(lT4), lT4)
# if (is.null(l)){
# l <- rep(1/sqrt(lT4), lT4)
# }
if (!(p %in% 1:10)){
lst <- mt4_get_param(p, l = l)
warning("Heavy computation")
} else {
Hvar <-
matrix(nrow = 10, ncol = 21, byrow = T,
data = c(
568.1209021, 448.0515896, 355.0182299, 282.6795124, 226.2374173,
182.0500981, 147.3446669, 120.0034577, 98.40450305, 81.30213209,
67.73735645, 56.97044429, 48.4300794, 41.67496479, 36.3648053,
32.23839726, 29.09714181, 26.79273777, 25.2181407, 24.30112719,
24,
280.078945, 220.4020183, 174.2856492, 138.5357285, 110.735366,
89.05053643, 72.08563482, 58.77581847, 48.30654298, 40.05326137,
33.53611834, 28.38583129, 24.31794483, 21.11337599, 18.60370546,
16.66006845, 15.18479424, 14.10516504, 13.36883132, 12.94054861,
12.8,
192.017125, 150.4754251, 118.5211313, 93.85864813, 74.76032964,
59.92208683, 48.35667067, 39.31457154, 32.22522209, 26.65317137,
22.26533269, 18.80644783, 16.08066803, 13.93770524, 12.2624126,
10.96695142, 9.984921882, 9.266999309, 8.777740015, 8.493314772,
8.4,
139.4637519, 109.4943184, 86.42252828, 68.59431806, 54.76680757,
44.00306309, 35.5945326, 29.00370386, 23.82158724, 19.73609938,
16.50848791, 13.95570699, 11.93721173, 10.34504666, 9.096400137,
8.128014685, 7.392003839, 6.85274504, 6.484607316, 6.270339886,
6.2,
104.9909838, 82.87394461, 65.77466418, 52.50044615, 42.15382226,
34.05709724, 27.69674951, 22.68233197, 18.71599557, 15.56982032,
13.0689038, 11.0787099, 9.495581318, 8.239611218, 7.249281821,
6.477434023, 5.888246794, 5.454990303, 5.15838038, 4.98541008,
4.928571429,
82.011931, 65.17873306, 52.08615944, 41.85866521, 33.83509445,
27.51431271, 22.51507366, 18.54629239, 15.38495284, 12.85963481,
10.83819206, 9.218508179, 7.921543093, 6.886091921, 6.064829952,
5.421329472, 4.927816535, 4.563496973, 4.313326754, 4.167136587,
4.119047619,
66.62374719, 53.30475074, 42.87967482, 34.68359842, 28.21195052,
23.08037733, 18.99500689, 15.73033808, 13.11274092, 11.00810133,
9.31253868, 7.945410092, 6.8440239, 5.959636079, 5.254414896,
4.699140766, 4.27146887, 3.954627172, 3.73645638, 3.608724244,
3.566666667,
56.33114824, 45.31277767, 36.64217968, 29.78935694, 24.35000213,
20.01455816, 16.54517401, 13.75847828, 11.51265495, 9.69771301,
8.228135973, 7.037312106, 6.073302562, 5.295619503, 4.67277072,
4.180389742, 3.799816819, 3.51703094, 3.321859343, 3.207411075,
3.16969697,
49.50809201, 39.95575422, 32.4136072, 26.4335349, 21.67217263,
17.86555133, 14.81010191, 12.34837317, 10.35825587, 8.744824767,
7.434142751, 6.36854146, 5.503017571, 4.802476072, 4.239619901,
3.793336062, 3.447466197, 3.189878105, 3.011776472, 2.907207735,
2.872727273,
45.07928983, 36.41542436, 29.56901672, 24.13680211, 19.80897198,
16.34684669, 13.5660621, 11.32386571, 9.50948683, 8.036814083,
6.838811252, 5.863247519, 5.069425651, 4.425671118, 3.907404411,
3.495663057, 3.175973205, 2.937495809, 2.772391759, 2.675365267,
2.643356643
)
)
Msup <- c(1.478985651,
1.454432793,
1.448680764,
1.449313815,
1.453737199,
1.4561804,
1.463732497,
1.470303197,
1.476776159,
1.484990068)
lst <- list(m.supLt = Msup[p],
varLtLs = c(Hvar[p, ], Hvar[p, 20:1])
)
}
v4 <- lapply(bigt/sqrt(p),function(u) d4hCGF(myt = rep(u, p), x = x))
L4 <- unlist(lapply(v4, function(v) sqrt(n) * sum(l*v)))
u <- sqrt(-2 *log(alpha))
band1 <- (u + lst$m.supLt)*sqrt(lst$varLtLs)
band2 <- -(u + lst$m.supLt)*sqrt(lst$varLtLs)
ylim <- max(max(abs(band1)), abs(L4))
plot(bigt, L4, ylim = c(-ylim, ylim), col = "blue",
lty = 1, type = "l", lwd = 2, ylab = bquote(L[4]), xlab = "t")
graphics::lines(bigt, band1, col = "orange", lty = 6, lwd = 2)
graphics::lines(bigt, band2, col = "orange", lty = 6, lwd = 2)
graphics::legend("top", c("Probability bands", expression(L[4])),
lty = c(6, 1), col = c("orange", "blue"), lwd = c(2, 2),
merge = TRUE, y.intersp = 1.5, text.width = .6)
til.L <- L4/sqrt(lst$varLtLs)
deci <- ifelse(max(abs(til.L)) >= u + lst$m.supLt, "reject", "accept")
return(deci)
}
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