get_bounds_one_sided {qqconf}

Calculates Rejection Region of One-Sided Equal Local Levels Test

Description

The context is that n i.i.d. observations are assumed to be drawn from some distribution on the unit interval with c.d.f. F(x), and it is desired to test the null hypothesis that F(x) = x for all x in (0,1), referred to as the "global null hypothesis," against the alternative F(x) > x for at least one x in (0, 1). An "equal local levels" test is used, in which each of the n order statistics is tested for significant deviation from its null distribution by a one-sided test with significance level \eta. The global null hypothesis is rejected if at least one of the order statistic tests is rejected at level eta, where eta is chosen so that the significance level of the global test is alpha. Given the size of the dataset n and the desired global significance level alpha, this function calculates the local level eta and the acceptance/rejection regions for the test. The result is a set of lower bounds, one for each order statistic. If at least one order statistic falls below the corresponding bound, the global test is rejected.

Usage

get_bounds_one_sided(alpha, n, tol = 1e-08, max_it = 100)

Arguments

alpha

Desired global significance level of the test.

n

Size of the dataset.

tol

(Optional) Relative tolerance of the alpha level of the simultaneous test. Defaults to 1e-8.

max_it

(Optional) Maximum number of iterations of Binary Search Algorithm used to find the bounds. Defaults to 100 which should be much larger than necessary for a reasonable tolerance.

Value

A list with components

• bound - Numeric vector of length n containing the lower bounds of the acceptance regions for the test of each order statistic.

• x - Numeric vector of length n containing the expectation of each order statistic. These are the x-coordinates for the bounds if used in a qq-plot. The value is c(1:n) / (n + 1).

• local_level - Significance level \eta of the local test on each individual order statistic. It is equal for all order statistics and will be less than alpha for all n > 1.

Examples

get_bounds_one_sided(alpha = .05, n = 10, max_it = 50)