cutoff.edgeworth {L2DensityGoFtest}

Critical value based on Edgeworth expansion of the size function for the density goodness-of-fit test \hat{S}_n(h) of Bagkavos, Patil and Wood (2021)

Description

Implements the critical value for the density goodness-of-fit test S.n, approximating via an Edgeworth expansion the size function of the test statistic S.n.

Usage

cutoff.edgeworth(xin, dist, kfun, p1, p2, sig.lev)

Arguments

xin

A vector of data points - the available sample.

dist

The null distribution.

kfun

The kernel to use in the density estimates used in the bandwidth expression.

p1

Parameter 1 (vector or object) for the null distribution.

p2

Parameter 2 (vector or object) for the null distribution.

sig.lev

Significance level of the hypothesis test.

Details

Implements the critical value for the density goodness-of-fit test S.n, approximating via an Edgeworth expansion the size function of the test statistic S.n, given by

l_\alpha = z_\alpha + d_0 \sqrt{h} + d_2(n \sqrt{h})^{-1}

where z_\alpha is the 1-\alpha quantile of the normal distribution and d_0 = d_1 - C_{ H_0} and

d_j = (z_\alpha^2 - 1)c_j, j=1,2

with

c_1 = \frac{4K^{(3)}(0)\mu_2^3 \nu_3}{3\sigma^3}, \; c_2 = \frac{\mu_3^2K^2(0)}{\sigma^3}, \; \mu_i =\int K^i(x)\,dx, i=1,\dots.

and

C_{H_0} = 2\left (E f_0'( \theta_0) \right )^2 \Delta^{-1}, \; \nu_i = E \left \{f^{i}(x)\right \} = \int f^{i+1}(x)\,dx, i=1,\dots

This critical value is the density function equivalent to the critical value estimate obtained in the closely relatated regression setting in Gao and Gijbels (2008) and is suitable for finite sample implementations of the test.

Value

A scalar, the estimate of the critical value at the given significance level.

Author(s)

Dimitrios Bagkavos

R implementation and documentation: Dimitrios Bagkavos <[e-mail]>

References

Gao and Gijbels, Bandwidth selection in nonparametric kernel testing, pp. 1584-1594, JASA (2008)

See Also

cutoff.asymptotic, cutoff.bootstrap