The non-null limiting distribution of the generalized Baumgartner statistic based on the Fourier series approximation

Abstract

The non-null limiting distribution of the generalized Baumgartner statistic is approximated by applying the Fourier series approximation. Due to the development of computational power, the Fourier series approximation is readily utilized to approximate its probability density function. The infinite product part for a non-central parameter in the characteristic function is re-formulated by using a formula of the trigonometric function. The non-central parameter of the generalized Baumgartner statistic is formulated by the first moment of the generalized Baumgartner statistic under the alternative hypothesis. The non-central parameter is used to calculate the power of the generalized Baumgartner statistic.

Appendix

We derive that the characteristic function converges to Eq. (1). First, we introduce a famous formula for the partial fraction expansion of trigonometric function as follows:

$$\begin{aligned} \sum _{k=1}^{\infty } \frac{1}{x^2-(2k-1)^2} = - \frac{\pi }{4x}\tan \frac{\pi x}{2}. \end{aligned}$$

(5)

The characteristic function is rewritten as

$$\begin{aligned}&\prod _{j=1}^{\infty } \exp \left( \frac{\frac{\delta \mathcal {I} u}{j(j+1)}}{1-\frac{2 \mathcal {I} u}{j(j+1)}}\right) \left( 1-\frac{2 \mathcal {I} u}{j(j+1)}\right) ^{-\frac{k-1}{2}} \\= & {} \exp \left( \sum _{j=1}^{\infty } \frac{\delta \mathcal {I} u}{j(j+1)-2 \mathcal {I} u}\right) \prod _{j=1}^{\infty } \left( 1-\frac{2 \mathcal {I} u}{j(j+1)}\right) ^{-\frac{k-1}{2}}. \end{aligned}$$

Herein we focus on the first term. Then we have

$$\begin{aligned} \sum _{j=1}^{\infty } \frac{\delta \mathcal {I} u}{j(j+1)-2 \mathcal {I} u}= & {} \delta \mathcal {I} u \sum _{j=2}^{\infty } \frac{1}{j(j-1)-2 \mathcal {I} u} \nonumber \\= & {} 4\delta \mathcal {I} u \sum _{j=2}^{\infty } \frac{1}{4j^2 -4j-8 \mathcal {I} u +1-1} \nonumber \\= & {} -4\delta \mathcal {I} u \sum _{j=2}^{\infty } \frac{1}{1+8 \mathcal {I} u -(2j-1)^2} \nonumber \\= & {} -4\delta \mathcal {I} u \left[ \sum _{j=2}^{\infty } \frac{1}{1+8 \mathcal {I} u -(2j-1)^2} +\frac{1}{8\mathcal {I}u} -\frac{1}{8\mathcal {I}u}\right] \nonumber \\= & {} -4\delta \mathcal {I} u \left[ \sum _{j=1}^{\infty } \frac{1}{1+8 \mathcal {I} u -(2j-1)^2} -\frac{1}{8\mathcal {I}u}\right] . \end{aligned}$$

(6)

By applying (5)–(6), we obtain

$$\begin{aligned} -4\delta \mathcal {I} u \left[ -\frac{\pi }{4\sqrt{1+8\mathcal {I}u}} \tan \left( \frac{\pi }{2}\sqrt{1+8 \mathcal {I} u}\right) -\frac{1}{8\mathcal {I}u}\right] = \frac{\delta }{2}+\frac{ \pi \delta \mathcal {I} u\tan (\frac{\pi }{2}\sqrt{1+8 \mathcal {I} u}) }{\sqrt{1+8 \mathcal {I} u}}. \end{aligned}$$

Therefore, the characteristic function consists to (1).