Estimation of the Selected Treatment Mean in Two-Stage Drop-the-Losers Design

A common problem faced in clinical studies is that of estimating the effect of the most effective (e.g., the one having the largest mean) treatment among $k~(\geq2)$ available treatments. The most effective treatment is adjudged based on numerical values of some statistic corresponding to the $k$ treatments. A proper design for such problems is the so-called "Drop-the-Losers Design (DLD)". We consider two treatments whose effects are described by independent Gaussian distributions having different unknown means and a common known variance. To select the more effective treatment, the two treatments are independently administered to $n_1$ subjects each and the treatment corresponding to the larger sample mean is selected. To study the effect of the adjudged more effective treatment (i.e., estimating its mean), we consider the two-stage DLD in which $n_2$ subjects are further administered the adjudged more effective treatment in the second stage of the design. We obtain some admissibility and minimaxity results for estimating the mean effect of the adjudged more effective treatment. The maximum likelihood estimator is shown to be minimax and admissible. We show that the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the selected treatment mean is inadmissible and obtain an improved estimator. In this process, we also derive a sufficient condition for inadmissibility of an arbitrary location and permutation equivariant estimator and provide dominating estimators in cases where this sufficient condition is satisfied. The mean squared error and the bias performances of various competing estimators are compared via a simulation study. A real data example is also provided for illustration purposes

On Estimating the Selected Treatment Mean under a Two-Stage Adaptive Design

Adaptive designs are commonly used in clinical and drug development studies for optimum utilization of available resources. In this article, we consider the problem of estimating the effect of the selected (better) treatment using a two-stage adaptive design. Consider two treatments with their effectiveness characterized by two normal distributions having different unknown means and a common unknown variance. The treatment associated with the larger mean effect is labeled as the better treatment. In the first stage of the design, each of the two treatments is independently administered to different sets of $n_1$ subjects, and the treatment with the larger sample mean is chosen as the better treatment. In the second stage, the selected treatment is further administered to $n_2$ additional subjects. In this article, we deal with the problem of estimating the mean of the selected treatment using the above adaptive design. We extend the result of \cite{cohen1989two} by obtaining the uniformly minimum variance conditionally unbiased estimator (UMVCUE) of the mean effect of the selected treatment when multiple observations are available in the second stage. We show that the maximum likelihood estimator (a weighted sample average based on the first and the second stage data) is minimax and admissible for estimating the mean effect of the selected treatment. We also propose some plug-in estimators obtained by plugging in the pooled sample variance in place of the common variance $\sigma^2$, in some of the estimators proposed by \cite{misra2022estimation} for the situations where $\sigma^2$ is known. The performances of various estimators of the mean effect of the selected treatment are compared via a simulation study. For the illustration purpose, we also provide a real-data application

Estimation of Order Restricted Location/Scale Parameters of a General Bivariate Distribution Under General Loss function: Some Unified results

We consider component-wise equivariant estimation of order restricted location/scale parameters of a general bivariate distribution under quite general conditions on underlying distributions and the loss function. This paper unifies various results in the literature dealing with sufficient conditions for finding improvments over arbitrary location/scale equivariant estimators. The usefulness of these results is illustrated through various examples. A simulation study is considered to compare risk performances of various estimators under bivariate normal and independent gamma probability models. A real-life data analysis is also performed to demonstrate applicability of the derived results

A Note On Simultaneous Estimation of Order Restricted Location Parameters of a General Bivariate Symmetric Distribution Under a General Loss Function

The problem of simultaneous estimation of order restricted location parameters $\theta_1$ and $\theta_2$ ($-\infty<\theta_1\leq \theta_2<\infty$) of a bivariate location symmetric distribution, under a general loss function, is being considered. In the literature, many authors have studied this problem for specific probability models and specific loss functions. In this paper, we unify these results by considering a general bivariate symmetric model and a quite general loss function. We use the Stein and the Kubokawa (or IERD) techniques to derive improved estimators over any location equivariant estimator under a general loss function. We see that the improved Stein type estimator is robust with respect to the choice of a bivariate symmetric distribution and the loss function, as it only requires the loss function to satisfy some generic conditions. A simulation study is carried out to validate the findings of the paper. A real-life data analysis is also provided

U-statistic based on overlapping sample spacings

For testing goodness of fit, we consider a class of U-statistics of overlapping spacings of order two, and investigate their asymptotic properties. The standard U-statistic theory is not directly applicable here as the overlapping spacings form a dependent random sequence. The asymptotic distribution of the statistics under the null hypothesis and under a sequence of local alternatives are derived. In terms of the Pitman ARE, the U-statistic based on Gini's mean square difference of overlapping spacings is found to be the asymptotically locally most powerful. Interestingly, this test has the same efficacy as the Greenwood test based on overlapping spacings.Comment: 15 page