varwci {ConfIntVariance} R Documentation

## varwci

### Description

Surround the univariate variance estimator of the function var with a confidence interval, not assuming normality

### Usage

```varwci(x, conf.level=0.95)
```

### Arguments

 `x` A one-dimensional numeric vector `conf.level` The confidence level for the confidence interval. Defaults to 0.95

### Value

Returns a vector with two entries: the lower and the upper bound of the confidence interval, and the following attributes:

point.estimator

The usual sample variance at the center of the interval

conf.level

The confidence level used

var.SampleVariance

The estimated variance of the sample variance

### Warning

On very small sample sizes, the result is NA because there is insufficient information on the variance estimation

### Note

The underlying theory is that of U-statistics. See Hoeffding 1948.

Mathias Fuchs

### References

http://dx.doi.org/10.1080/15598608.2016.1158675 and https://mathiasfuchs.de/b3.html

### Examples

```##
## Example: throwing a dice
##

# throw a dice 100 times
s <- sample(6, 100, replace=TRUE)

# the standard point estimator for the variance
print(var(s))

# contains the true value 2.9166 with a probability of 95 percent.
print(varwci(s))

##
## Check the coverage probability of the confidence interval
##

# True quantities that do not depend on n
trueMeanOfDice <- mean(1:6)
trueVarianceOfDice <- mean((1:6)^2) - trueMeanOfDice^2

## see package description for more details
# number of times we draw a
# sample and compute a confidence interval
N <- 1e4
trueValueCovered <- rep(NA, N)
for (i in 1:N) {
if (i %% 1e3 == 0) print(i)
# throw a dice 100 times
x <- sample(6, 100, replace=TRUE)
# compute our confidence interval
ci <- varwci(x)
# We know that the true variance
# of the dice is 91/6 - 49/4 = 2.916666...
# did the confidence interval contain the correct value?
trueValueCovered[i] <- (trueVarianceOfDice > ci && trueVarianceOfDice < ci)
}

# Result of simulation study: should be close to 0.95
print(mean(trueValueCovered))
```

[Package ConfIntVariance version 1.0.2 Index]