SDNN: One-sided test of stochastic dominance against the null of...

In nilanjanalaha/SDNNtests: SHAPE CONSTRAINED TESTS OF STOCHASTIC DOMINANCE

Description

Calculates the p-values of one-sided tests of restricted stochastic dominance against the null of non-dominance. The concerned tests are the minimum t-statistic test and the two sample empirical process (TSEP) test of Laha et al. (2021). Each test can be either nonparametric, or semiparametric, ie. using unimodality or log-concavity assumption on the underlying densities.

Usage

SDNN(x, y, Method, t = 1, p1 = 0.05, p2 = 0.05)

Arguments

x	Vector of m independent and identically distributed random variables; corresponds to the first sample.
y	Vector of n independent and identically distributed random variables; corresponds to the second sample.
Method	Must be one among "NP" (nonparametric), "UM" (unimodal), and "LC" (log-concave). See 'Details'.
t	A positive real number. Only required when Method="UM", default value is 1. See Details.
p1	The proportion of combined data to be trimmed from the left prior to testing, should take value in [0,0.50), the default is set to 0.05.
p2	The proportion of combined data to be trimmed from the right prior to testing, should take value in [0,0.50), the default is set to 0.05.

Details

Suppose X_1,…, X_m and Y_1,…, y_n are independent random variables with distribution F and G, respectively. Denote by D_{p,m,n} the set

[H_n^{-1}(p), H_n^{-1}(1-p)]

where p\in[0,0.5) and H_n is the empirical distribution function of the combined sample

\{X_1,…, x_m,Y_1,…,Y_n\}.

The function SDNN tests H_0: F(x)≥q G(x) for some x\in D_{p,m,n} vs H_1: F(x)<G(x) for all x\in D_{p,m,n}. For more details, see Laha et al., 2021.

Method: "NP" corresponds to the nonparametric tests. "UM" corresponds to tests which use the function calc_mode to estimate the densities of X_i's and Y_i's. This function estimates the unimodal density estimator of Birge (1997). "LC" corresponds to tests which use the log-concave MLE (given by logConDens of R package “logcondens") of Dumbgen and Rufibatch (2009) to estimate the latter densities. For more detail, see Laha et al. (2021).

t: The parameter t corresponds to the parameter τ in Birge (1997). Higher values of t leads to more accurate estimation of the unimodal densities. This value ontrols the accuracy in unimodal density estimation upto the term (m+n)^{-t}. A value greater than or equal to one is recommended. See Birge (1997) for more details.

p: p corresponds to the set D_{p,m,n} in H_0 and H_1. To overcome the difficulties arising from the tail region, 100p percent of data is trimmed from both sides of the combined sample.

Value

A list of two numbers.

• T1 - The p-value of the test based on minimum T-statistic of Laha et al., (2021)

• T2 - The p-value of the TSEP test of Laha et al., (2021).

Author(s)

Nilanjana Laha (maintainer), [e-mail], Alex Luedtke, [e-mail].

References

Laha, N., Moodie, Z., Huang, Y., and Luedtke, A. (2021). Improved inference for vaccine-induced immune responses via shape-constrained methods. Submitted.

Dumbgen, L. and Rufibatch, K. (2009). Maximum likelihood estimation of a logconcave density and its distribution function: Basic properties and uniform consistency, Bernoulli, 15, 40–68.

Birge, L. (1997). Estimation of unimodal densities without smoothness assumptions, Ann. Statist., 25, 970–981.

See Also

calc_mode, logConDens

Examples

x <- rnorm(100); y <- rgamma(50, shape=1);
 SDNN(x, y, Method="UM", t=1, p1=0.01, p2=0.05)

nilanjanalaha/SDNNtests documentation built on June 15, 2021, 8:20 p.m.