get_level_from_bounds_two_sided {qqconf}

Calculates Global Significance Level From Simultaneous Two-Sided Bounds for Rejection Region

Description

For a test of uniformity of i.i.d. observations on the unit interval, this function will determine the significance level as a function of the rejection region. Suppose n observations are drawn i.i.d. from some CDF F(x) on the unit interval, and it is desired to test the null hypothesis that F(x) = x for all x in (0, 1) against a two-sided alternative. Suppose the acceptance region for the test is described by a set of intervals, one for each order statistic. Given the bounds for these intervals, this function calculates the significance level of the test where the null hypothesis is rejected if at least one of the order statistics is outside its corresponding interval.

Usage

get_level_from_bounds_two_sided(lower_bounds, upper_bounds)

Arguments

lower_bounds

Numeric vector where the ith component is the lower bound for the acceptance interval for the ith order statistic. The components must lie in [0, 1], and each component must be greater than or equal to the previous one.

upper_bounds

Numeric vector of the same length as lower_bounds where the ith component is the upper bound for the acceptance interval for the ith order statistic. The components must lie in [0, 1], and each component must be greater than or equal to the previous one. In addition, the ith component of upper_bounds must be greater than or equal to the ith component of lower_bounds.

Details

Uses the method of Moscovich and Nadler (2016) as implemented in Crossprob (Moscovich 2020).

Value

Global Significance Level \alpha

References

• Moscovich, Amit, and Boaz Nadler. "Fast calculation of boundary crossing probabilities for Poisson processes." Statistics & Probability Letters 123 (2017): 177-182.

• Amit Moscovich (2020). Fast calculation of p-values for one-sided Kolmogorov-Smirnov type statistics. arXiv:2009.04954

Examples

# For X1, X2 iid unif(0,1), calculate 1 - P(.1 < min(X1, X2) < .6 and .5 < max(X1, X2) < .9)
get_level_from_bounds_two_sided(lower_bounds = c(.1, .5), upper_bounds = c(.6, .9))

# Finds the global significance level corresponding to the local level eta.
# Suppose we reject the null hypothesis that X1, ..., Xn are iid unif(0, 1) if and only if at least
# one of the order statistics X(i) is significantly different from
# its null distribution based on a level-eta
# two-sided test, i.e. we reject if and only if X(i) is outside the interval
# (qbeta(eta/2, i, n - i + 1), qbeta(1 - eta/2, i, n - i + 1)) for at least one i.
# The lines of code below calculate the global significance level of
# the test (which is necessarily larger than eta if n > 1).
n <- 100
eta <- .05
lb <- qbeta(eta / 2, c(1:n), c(n:1))
ub <- qbeta(1 - eta / 2, c(1:n), c(n:1))
get_level_from_bounds_two_sided(lower_bounds = lb, upper_bounds = ub)