Multistage point estimation methodologies for a negative exponential location under a modified linex loss function: Illustrations with infant mortality and bone marrow data

ABSTRACT

We have designed Stein-type (Stein, 1945, Annals of Mathematical Statistics) two-stage, modified two-stage (Mukhopadhyay and Duggan, 1997, Sankhya, Series A), and purely sequential strategies (Chow and Robbins, 1965, Annals of Mathematical Statistics) to estimate an unknown location parameter of a negative exponential distribution having an unknown scale parameter under a newly defined and modified Linex loss function. We aim at controlling the associated risk function per unit cost by bounding it from above with a fixed preassigned positive number, ω, and we emphasize both asymptotic first-order and asymptotic second-order properties for the modified two-stage and purely sequential estimation strategies. In developing asymptotic second-order properties for the modified two-stage methodology, we have heavily relied upon basic ideas rooted in Mukhopadhyay and Duggan (1997). In developing asymptotic second-order properties for the purely sequential methodology, however, we have heavily relied upon nonlinear renewal theory (Lai and Siegmund, 1977, 1979, Annals of Statistics; Woodroofe, 1977, Annals of Statistics). Then, we take to extensive data analysis carried out via computer simulations when requisite sample sizes range from small to moderate to large. We find that the Stein-type two-stage estimation methodology oversamples significantly and yet the achieved risk is not close to preset goal ω. On the other hand, both modified two-stage and purely sequential estimation strategies perform remarkably well. We have validated their main theoretical first-order and second-order properties through simulated data. The latter methodologies have been illustrated and implemented using two real data sets from health studies, namely, infant mortality data and bone marrow data.

KEYWORDS:

• Bone marrow data
• cancer research
• first-order properties
• infant mortality data
• Linex loss
• location parameter
• modified linex loss
• modified two-stage
• negative exponential
• nonlinear renewal theory
• purely sequential
• real data
• risk
• risk per unit cost
• scale parameter
• second-order properties
• simulations
• two-stage

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L12
• 62L05
• 62G20
• 62F10
• 62P10
• 62P30

Acknowledgments

Professor Stanley Wasserman and Professor John Klein gave us permission to use infant mortality data from Leinhardt and Wasserman (1979) and to use bone marrow data from Klein-Moeschberger (2003) book respectively. We express our sincere gratitude to these two colleagues. We also thank the Associate Editor and a referee for their encouraging comments.