# R/Emvnorm.R In energy: E-Statistics: Multivariate Inference via the Energy of Data

#### Documented in mvnorm.emvnorm.etestmvnorm.testnormal.enormal.test

```mvnorm.test <- function(x, R) {
# parametric bootstrap E-test for multivariate normality
if (missing(R)) {
method = "Energy test of multivariate normality: (Specify R > 0 for MC test)"
R <- 0
} else {
method = "Energy test of multivariate normality: estimated parameters"
}

if (is.vector(x) || NCOL(x)==1) {
n <- NROW(x)
d <- 1
bootobj <- boot::boot(x, statistic = normal.e, R = R, sim = "parametric",
ran.gen = function(x, y) {
return(rnorm(n))
})
} else {
n <- nrow(x)
d <- ncol(x)
bootobj <- boot::boot(x, statistic = mvnorm.e, R = R, sim = "parametric",
ran.gen = function(x, y) {
return(matrix(rnorm(n * d), nrow = n, ncol = d))
})
}
if (R > 0)
p <- 1 - mean(bootobj\$t < bootobj\$t0) else p <- NA

names(bootobj\$t0) <- "E-statistic"
e <- list(statistic = bootobj\$t0, p.value = p,
method = method,
data.name = paste("x, sample size ", n, ", dimension ", d, ", replicates ",
R, sep = ""))
class(e) <- "htest"
e
}

mvnorm.etest <- function(x, R) {
return(mvnorm.test(x, R))
}

mvnorm.e <- function(x) {
# E-statistic for multivariate normality
if (is.vector(x) || NCOL(x)==1)
return(normal.e(x))
n <- nrow(x)
d <- ncol(x)
if (n < 2) {
warning("sample size must be at least 2")
return(NA)
}
# subtract column means and compute S^(-1/2)
z <- scale(x, scale = FALSE)
ev <- eigen(var(x), symmetric = TRUE)
P <- ev\$vectors
lambda <- ev\$values
D <- diag(d)
diag(D) <- 1 / sqrt(lambda)
y <- z %*% (P %*% D %*% t(P))
if (any(!is.finite(y))) {
warning("missing or non-finite y")
return(NA)
}

if (requireNamespace("gsl", quietly=TRUE)) {
const <- exp(lgamma((d+1)/2) - lgamma(d/2))
mean2 <- 2*const
ysq <- rowSums(y^2)
mean1 <- sqrt(2) * const *
mean(gsl::hyperg_1F1(-1/2, d/2, -ysq/2))
mean3 <- 2*sum(dist(y)) / n^2
return(n * (2*mean1 - mean2 - mean3))
} else {
return (NA)
}
}

normal.e <- function(x) {
## Case 4: unknown parameters
x <- as.vector(x)
n <- length(x)
s <- sd(x)
if (!is.finite(s) || !(s > 0)) {
warning("sd(x)>0 required")
return(NA)
}
y <- (x - mean(x)) / sd(x)
y <- sort(y)
K <- seq(1 - n, n - 1, 2)
return(2 * (sum(2 * y * pnorm(y) + 2 * dnorm(y)) -
n/sqrt(pi) - mean(K * y)))
}

normal.test <- function(x, method=c("mc", "limit"), R) {
## implements the test for for d=1
## Case 4: composite hypothesis
method <- match.arg(method)
estimate <- c(mean(x), sd(x))
names(estimate) <- c("mean", "sd")

if (method == "mc") {
## Monte Carlo approach
if (missing(R)) R <- 0
e <- energy::mvnorm.etest(x, R=R)
e\$method <- "Energy test of normality"
e\$method <- ifelse(R > 0,
paste0(e\$method,": estimated parameters"),
paste0(e\$method, "  (Specify R > 0 for MC test)"))
e\$estimate <- estimate
return(e)
}

## implement test using asymptotic distribution for p-value
if (!is.numeric(x) || (!is.vector(x) && NCOL(x) > 1)) {
warning("x must be a numeric vector")
return (NA)
} else {
x <- as.vector(x, mode="numeric")
}

n <- length(x)
t0 <- normal.e(x)
names(t0) <- "statistic"

EVnormal <- NULL
ev <- EVnormal[, "Case4"]

} else {
warning("limit distribution method requires CompQuadForm package for p-value")
p <- NA
}
estimate <- c(mean(x), sd(x))
names(estimate) <- c("mean", "sd")
e <- list(statistic = t0, p.value = p,
method = paste("Energy test of normality: limit distribution"),
estimate = estimate,
data.name = "Case 4: composite hypothesis, estimated parameters")
class(e) <- "htest"
e
}
```

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energy documentation built on Feb. 22, 2021, 5:08 p.m.