The seminal work of Morgan & Rubin (2012) considers rerandomization for all the units at one time.In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result that given the same number of rerandomizations, in expected value, under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time.

A combination of three psychiatric screening tests was used to uncover the extent of affective-psychotic symptoms, the indications of "chronic brain syndrome," and the perceptual limitations among two selected populations of elderly persons. It was found that 25 percent of 48 residents in a home for the aged, and 9 percent of 45 members in a social club, had four or more affective-psychotic com-

In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically prove Yule's 1926 empirical finding of "nonsense correlation" ([15]). We do so by analytically determining the second moment of the empirical correlation coefficient

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Classical coupling constructions arrange for copies of the same Markov process started at two different initial states to become equal as soon as possible. In this paper, we consider an alternative coupling framework in which one seeks to arrange for two different Markov (or other stochastic) processes to remain equal for as long as possible, when started in the same state. We refer to this "un-coupling" or "maximal agreement" construction as MEXIT, standing for "maximal exit". After highlighting the importance of un-coupling arguments in a few key statistical and probabilistic settings, we develop an explicit MEXIT construction for stochastic processes in discrete time with countable state-space. This construction is generalized to random processes on general state-space running in continuous time, and then exemplified by discussion of MEXIT for Brownian motions with two different constant drifts.MSC 2010: 60J05, 60J25; 60J60 copies of the same Markov process started at two different initial states in such a way that they become equal at a fast rate. Fastest possible rates are achieved by the maximal coupling constructions which were introduced and studied in Griffeath (1975), Pitman (1976), and Goldstein (1978. Our results and methods are closely related to the work of Goldstein (1978), who deals with rather general discrete-time random processes. Our situation is related to a time-reversal of the situation studied by Goldstein (1978). However our approach seems more direct.In the current work, we consider what might be viewed as the dual problem where coupling is used to try to construct two different Markov (or other stochastic) processes which remain equal for as long as possible, when they are started in the same state. That is, we move from consideration of the coupling time to focus on the un-coupling time at which the processes diverge, and try to make that as large as possible. We refer to this as MEXIT (standing for "maximal exit" time). While finalizing our current work, it came to our attention that this construction is the same as the maximal agreement coupling time of the August 2016 work of Völlering (2016), who additionally derives a lower bound on MEXIT . Nonetheless, we believe the current work complements Völlering (2016) well. It offers an explicit treatment of discrete-time countable-state-space, generalizes the continuous-time case, and discusses a number of significant applications of MEXIT . We note that the work of Völlering (2016) does not consider the continuous-time case.In addition to being a natural mathematical question, MEXIT has direct applications to many key statistical and probabilistic settings (see Section 2 below). In particular, couplings which are Markovian and faithful (Rosenthal (1997), i.e. couplings which preserve the marginal update distributions even when conditioning on both processes; alternatively "co-adapted" or "immersion", depending on the extent to which one wishes to emphasize the underlying filtration as in Burdzy and Kendall (2000) and Kendall (2015)) are ...

Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted S&P 500 portfolio as well as the often overlooked disadvantages of the market capitalization weighted Standard and Poor's (S&P 500) portfolio (see Bloomfield et al.

Using a bondholder who seeks to determine when to sell his bond as our motivating example, we revisit one of Larry Shepp's classical theorems on optimal stopping. We offer a novel proof of Theorem 1 from from [7]. Our approach is that of guessing the optimal control function and proving its optimality with martingales. Without martingale theory one could hardly prove our guess to be correct.2000 Mathematics Subject Classification. Primary 60G40; Secondary 91B70. Key words and phrases. Optimal stopping; bondholders; martingales. 1 For now, we ignore the fact that this gives an unrealistic model (at least for zero coupon bonds) because these are observed never to trade at a price higher than face value. Most consumers would never buy a zero coupon bond if the price is higher than the face value. But, if they did, then these models would be useful in setting the optimal selling price.