Reference density for the test statistic between Same-source and Different-source averages using phase selection

Assuming an overall sample size of 36 (for 6 x 6 comparisons, n=6 independent objects), we are interested in the distribution of T defined as the difference between the average of the same source values and the median average (XXX see paper for definition) of the different source values: $$T = \bar{X}_{SS} - \tilde{X}_{DS}$$, where groups SS and DS are set by phase selection (after inspecting the set of $n$ values). For a (one-sided) hypothesis test of \(H_0\): no difference between the means and \(H_1:\): the same-source values are larger on average than the different source values, the test statistic T has under the null hypothesis and the group-finding strategy \(\cal G\) a distribution given as $$P(T < t \mid H_0, {\cal G}) = C \cdot \int_{-\infty}^{\infty} F^2(x) f(x) \left[ F(x + t) - F(x) \right]^3 dx$$ where $F$ is the (approximately normal) distribution of \(\bar{X}_{SS}\) under \(H_0\) with expected value of µ (\(=E[\bar{X}_{SS}] = E[\bar{X}_{DS}]\)) and standard deviation \(\sigma\) (standard deviation of a group of size n under the null hypothesis).

Usage

f_T(t, sigma = 1, n = 6)

Arguments

t

value of the observed test statistic, t is non-negative

sigma

(estimated) standard deviation of an average of size n of the (original) sample

n

integer value, number of values in the same-source group.

Examples

# example code
F_T(0.2, sigma = 0.1, lower.tail = FALSE) # only multitude of sigma informs the test
#> [1] 0.2066704
#> attr(,"abs.error")
#> [1] 1.602902e-07
#> attr(,"message")
#> [1] "OK"
F_T(2, lower.tail = FALSE)
#> [1] 0.2066704
#> attr(,"abs.error")
#> [1] 1.602902e-07
#> attr(,"message")
#> [1] "OK"
F_T(3, lower.tail = FALSE)
#> [1] 0.02398655
#> attr(,"abs.error")
#> [1] 1.413966e-07
#> attr(,"message")
#> [1] "OK"
F_T(4, lower.tail = FALSE)
#> [1] 0.001271115
#> attr(,"abs.error")
#> [1] 9.966644e-09
#> attr(,"message")
#> [1] "OK"
# critical values of t distribution and F_T (for sigma = 1):
F_T(qt(1-0.05, df=34), lower.tail = FALSE)
#> [1] 0.3381702
#> attr(,"abs.error")
#> [1] 1.176617e-07
#> attr(,"message")
#> [1] "OK"
F_T(qt(1-0.05/6, df=34), lower.tail = FALSE)
#> [1] 0.07501263
#> attr(,"abs.error")
#> [1] 2.498902e-07
#> attr(,"message")
#> [1] "OK"
F_T(qt(1-0.01/6, df=34), lower.tail = FALSE)
#> [1] 0.01590425
#> attr(,"abs.error")
#> [1] 8.744108e-08
#> attr(,"message")
#> [1] "OK"

x <- 10^-(1:4)
x <- sort(c(x, x/2))
t_q <- qt(1-x, df=34)
adjusted <- F_T(t_q, lower.tail=FALSE)