Statistical inference for scale mixture models via Mellin transform approach

Abstract

This paper deals with statistical inference for the scale mixture models. We study an estimation approach based on the Mellin–Stieltjes transform that can be applied to both discrete and absolute continuous mixing distributions. The accuracy of the corresponding estimate is analysed in terms of its expected pointwise error. As an important technical result, we prove the analogue of the Berry–Esseen inequality for the Mellin transforms. The proposed statistical approach is illustrated by numerical examples.

Keywords:

• Mixture models
• scale mixtures
• multiplicative censoring model
• Mellin–Stieltjes transform
• Berry–Esseen inequality

2020 Mathematics Subject Classifications:

• 62G05
• 62G20
• 65C20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The asymptotic behaviour of the first term in (6(6) $\begin{array}{rl}{\rho }_{u^{\circ }}(\phi ,\psi )& \le \frac{b}{2}{\int }_{-T}^T\frac{|\mathcal{M}[\phi ](u^{\circ }+\mathrm{i}v)-\mathcal{M}[\psi ](u^{\circ }+\mathrm{i}v)|}{|v|}\mathrm{d}v\\ & +\mathrm{bT}{x}_0^{u^{\circ }-1}{\int }_0^{2c(b)/T}|\psi (x_0)-\psi \left(x_0{\mathrm{e}}^r\right)|\mathrm{d}r,\end{array}$(6) ) will be discussed later. Definitely, it depends on the closeness between the functions φ and ψ.

Additional information

Funding

The article was prepared in the framework of a research grant funded by the Ministry of Science and Higher Education of the Russian Federation (grant ID: 075-15-2022-325).