BG {gofIG}R Documentation

The Baringhaus-Gaigall test statistic

Description

This function computes the test statistic of the goodness-of-fit test for the inverse Gaussian family due to Baringhaus and Gaigall (2015).

Usage

BG(data)

Arguments

data

a vector of positive numbers.

Details

The test statistic of the Baringhaus-Gaigall test is defined as:

BG_{n} = \frac{n}{(n(n-1))^5} \sum_{\mu, \nu = 1, \mu \neq \nu}^{n} \left( N_1(\mu, \nu)N_4(\mu, \nu) - N_2(\mu, \nu)N_3(\mu, \nu) \right)^2,

where

N_1(\mu, \nu) = \sum_{i,j = 1, i \neq j}^{n} \mathbf{1} \left\{ \tilde{Y}_{i,j} \leq \tilde{Y}_{\mu, \nu}, \tilde{Z}_{i,j} \leq \tilde{Z}_{\mu, \nu} \right\},

N_2(\mu, \nu) = \sum_{i,j = 1, i \neq j}^{n} \mathbf{1} \left\{ \tilde{Y}_{i,j} \leq \tilde{Y}_{\mu, \nu}, \tilde{Z}_{i,j} > \tilde{Z}_{\mu, \nu} \right\},

N_3(\mu, \nu) = \sum_{i,j = 1, i \neq j}^{n} \mathbf{1} \left\{ \tilde{Y}_{i,j} > \tilde{Y}_{\mu, \nu}, \tilde{Z}_{i,j} \leq \tilde{Z}_{\mu, \nu} \right\},

N_4(\mu, \nu) = \sum_{i,j = 1, i \neq j}^{n} \mathbf{1} \left\{ \tilde{Y}_{i,j} > \tilde{Y}_{\mu, \nu}, \tilde{Z}_{i,j} > \tilde{Z}_{\mu, \nu} \right\},

with \mathbf{1} being the indicator function. Let f(X_i,X_j) = (X_i + X_j)/2 and g(X_i,X_j) = (X_i^{-1} + X_j^{-1})/2 - f(X_i,X_j)^{-1}, with X_1,...,X_n positive, independent and identically distributed random variables with finite moments \mathbb{E}[X_1^2] and \mathbb{E}[X_1^{-1}]. Then (\tilde{Y}_{i,j}, \tilde{Z}_{i,j}) = (f(X_i,X_j), g(X_i,X_j)), 1 \leq i,j \leq n, i \neq j. Note that \tilde{Y}_{i,j} and \tilde{Z}_{i,j} are independent if, and only if X_1,...,X_n are realized from an inverse Gaussian distribution.

Value

value of the test statistic.

References

Baringhaus, L. Gaigall, D. (2015). "On an independence test approach to the goodness-of-fit problem", Journal of Multivariate Analysis, 140, 193-208. doi:10.1016/j.jmva.2015.05.013

Examples

BG(rmutil::rinvgauss(20,2,1))


[Package gofIG version 1.0 Index]