Nitis Mukhopadhyay

Suppose that we have independent observations X ₁,X ₂,… having a common exponential distribution with a probability density function f(x;λ) = λ⁻¹exp (− x/λ) if x > 0, and 0 elsewhere with unknown mean λ (> 0). Sequentially estimating the mean parameter λ is an important problem, and it has attracted attentions from many authors. The area was broadly reviewed by Mukhopadhyay ( 1988 Mukhopadhyay , N. ( 1988 ). Sequential Estimation Problems for Negative Exponential Populations , Communications in Statistics—Theory & Methods 17 : 2471 – 2506 . [INFOTRIEVE] [CSA] [Taylor & Francis Online], [Web of Science ®] [Google Scholar] 1995b Mukhopadhyay , N. ( 1995b ). Two-Stage and Multi-Stage Estimation , in The Exponential Distribution: Theory, Methods and Application , N. Balakrishnan and A. P. Basu , eds., pp. 429 – 452 , Amsterdam : Gordon and Breach . [Google Scholar] ). Having recorded observations X ₁,…,X _n , suppose that the loss function in estimating λ by the sample mean, is given by where A(> 0) is known. Given a preassigned number w(> 0), our goal is to make the associated risk for all fixed λ > 0. We refer to w as the preassigned risk-bound. The minimum required fixed sample size n would be the smallest integer ≥ n* = Aλ²/w, but its magnitude remains unknown. Hence, we propose a genuine two-stage sampling design, , where the pilot sample size is m(≥ 3). Here, B≡ B _m (> 0) is an appropriate design constant and ⟨ u⟩ stands for the largest integer < u. We finally estimate λ by , the sample mean based on the combined data from both stages. It is clear that cannot be expressed exclusively involving moments of N alone because I(N = n) and are dependent random variables, for all n ≥ m. The main result (Theorem 2.1) consists of explicitly determining B so that the risk associated with , namely , has the preassigned risk-bound w. The performances of the proposed methodology are first investigated with the help of simulations. Illustrations are included with data from a multicenter clinical trial involving patients with acute myeloctic leukemia (AML) prepared for transplantation with a radiation-free conditioning regimen.

A horticulturist was considering the number of days (X) each variety took from planting seeds to reach the stage when the first bud appeared for three local marigold varieties. The data X could be recorded with accuracy of one-half day. The primary interest was to estimate the maximum waiting time between “seeding” and “first budding” among three varieties under consideration. It was thought that a 99% confidence interval with width one day would suffice. The horticulturist felt comfortable to assume a normal distribution for the response variable. He provided positive lower bounds for the variances which forced pilot sample sizes to become unequal. We are not aware of any existing methodology with unequal pilot sample sizes that would readily apply here. Accordingly, a new two-stage sampling design was developed and implemented. The gathered data validated all assumptions made during the course of this investigation. Important exact as well as large-sample properties of the proposed methodology are highlighted (Theorem 1). This methodology is proven to be theoretically superior (Theorem 2) to the existing methodology for large sample sizes provided that the pilot sizes could be “chosen” equal. For the experimental data on hand, the superiority of the new methodology has also been indicated (Section 4.3). The solution to the primary estimation problem ultimately led to a natural and yet nontraditional selection problem involving identification of the “worst” marigold variety. For this selection problem, a practical approach is developed (Section 4.4) for evaluating the associated probability of correctly selecting the worst marigold variety.

A new type of accelerated sequential procedure is developed and its associated second-order characteristics are investigated. In the case of a few known sequential estimation problems, it is shown that the new methodology (i) is flexible in terms of what one plugs in as the estimator of the nuisance parameter in the boundary condition, and (ii) can economize at the same time on the starting sample size condition to provide second-order properties similar to those already available for the existing methodologies. The power and novelty of the newly constructed sampling techniques are also substantiated by the simplicity of derivation of the second-order expansions for the regret function associated with the estimator of the scale parameter of an exponential distribution, as well as in the context of one-sample and two-sample problems for estimating means when the distributions are unspecified.