Central limit theorems for four new types of U-designs

ABSTRACT

Orthogonal array (OA)-based Latin hypercube designs, also called U-designs, have been popularly adopted in designing a computer experiment. Nested U-designs, sliced U-designs, strong OA-based U-designs and correlation controlled U-designs are four types of extensions of U-designs for different applications in computer experiments. Their elaborate multi-layer structure or multi-dimensional uniformity, which makes them desirable for different applications, brings difficulty in analysing the related statistical properties. In this paper, we derive central limit theorems for these four types of designs by introducing a newly constructed discrete function. It is shown that the means of the four samples generated from these four types of designs asymptotically follow the same normal distribution. These results are useful in assessing the confidence intervals of the gross mean. Two examples are presented to illustrate the closeness of the simulated density plots to the corresponding normal distributions.

KEYWORDS:

• Central limit theorem
• correlation controlled U-design
• nested U-design
• sliced U-design
• strong orthogonal array-based U-design

AMS SUBJECT CLASSIFICATION:

• 62K15
• 62K20

Acknowledgments

The authors thank the editor, the associate editor and two referees for their valuable comments and suggestions that improved the presentation of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work is supported by NSFC [grant number 11331011], [grant number 11671019], BCMIIS and LMEQF, as well as grants from IIM Calcutta and J.C. Bose National Fellowship, Government of India.