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Interaction between short repeats may be a source of genomic rearrangements and deletions. We investigated possible interactions between short (9 base pairs) direct repeats in yeast by using our previously described system for analyzing bacterial transposon TnS excision in yeast. Mutations of either POL3 or POLI, the proposed structural genes for polymerases 8 and a, respectively, yield high levels of excision at semipermissive temperatures. poi2 (correspond to polymerase E) and po12 pol3 double mutants do not exhibit enhanced excision. A majority of excision events involve direct repeats and are precise; the remaining imprecise excisions occur within or in the vicinity of the repeats. The three DNA repair pathways identified by radi, rad6 and radl8, radSO and rad52 mutations were examined for their possible role in TnS excision; no enhancement was observed in mutants. However, the pol3-stimulated TnS excision was reduced in rad52 and radSO mutants. This suggests the potential for interaction between the systems for DNA double-strand break/recombi- polymerases 8 and a are responsible for lagging-strand synthesis and that polymerase e is responsible for leading-strand synthesis, we suggest that TnS excision is stimulated under conditions of altered lagging-strand synthesis, possibly due to extended opportunities for single-strand interactions between the inverted insertion sequence 1550 repeats of TnS.Short repeats of DNA sequence are a source of genomic rearrangements, duplications, and deletions. In bacteria, precise excision occurs between short, 9-base-pair (bp) direct repeats of a transposon target sequence. Excision is stimulated by the long, internal, inverted insertion sequences (IS). Molecular and genetic studies suggest that such deletion may result from "illegitimate" recombination between short repeats. The most likely mechanism is the "slippage" of a replicating strand (1-3). The same mechanism was proposed for formation of deletions between short direct repeats (4). TnS and TnOO excision in Escherichia coli is influenced by genes involved in recombination (recB, recC), mismatch correction (dam, mutH, mutL, mutS, uvrD), single-stranded DNA-binding protein (ssb), and the proofreading exonuclease subunit of DNA polymerase III (dnaQ) (5,6). Precise excision of a small, 50-bp palindromic TnJO remnant is increased in polA (DNA polymerase I) mutants (6).Based on results from bacteria, transposon excision may provide a means for studying various DNA metabolic processes, including DNA repair, recombination, and replication. We therefore introduced TnM into the yeast Saccharomyces cerevisiae and examined excision from the L YS2 gene (7). The low excision rate (10-9) was increased 100-fold in tex mutants (8). These and several other yeast DNA repair and metabolism mutants have been examined in order to understand the TnS excision process. Mutations in the structural genes for two of three yeast DNA polymerases (polymerase a and 5, but not polymerase E) greatly enhance the excision frequency. These obse...

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We investigate the behavior of the nonparametric maximum likelihood estimatorfn for a decreasing density f near the boundaries of the support of f . We establish the limiting distribution offn(n −α ), where we need to distinguish between different values of 0 < α < 1. Similar results are obtained for the upper endpoint of the support, in the case it is finite. This yields consistent estimators for the values of f at the boundaries of the support. The limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator.

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with that of the supremum distance. It turns out that in the uniform case, the measures of deviation are of greater order and their limit distributions are different.

Abstract:We consider the processF n −F n , being the difference between the empirical distribution function F n and its least concave majorantF n , corresponding to a sample from a decreasing density. We extent Wang's result on pointwise convergence ofF n − F n and prove that this difference converges as a process in distribution to the corresponding process for two-sided Brownian motion with parabolic drift.

We investigate the limit behavior of the L k -distance between a decreasing density f and its nonparametric maximum likelihood estimatorfn for k ≥ 1. Due to the inconsistency offn at zero, the case k = 2.5 turns out to be a kind of transition point. We extend asymptotic normality of the L1-distance to the L k -distance for 1 ≤ k < 2.5, and obtain the analogous limiting result for a modification of the L kdistance for k ≥ 2.5. Since the L1-distance is the area between f and fn, which is also the area between the inverse g of f and the more tractable inverse Un offn, the problem can be reduced immediately to deriving asymptotic normality of the L1-distance between Un and g. Although we lose this easy correspondence for k > 1, we show that the L k -distance between f andfn is asymptotically equivalent to the L k -distance between Un and g.

Let f be a nonincreasing function defined on [0,1]. Under standard regularity conditions, we derive the asymptotic distribution of the supremum norm of the difference between f and its Grenandertype estimator on sub-intervals of [0, 1]. The rate of convergence is found to be of order (n/ log n) −1/3 and the limiting distribution to be Gumbel.1. Introduction. After the derivation of the nonparametric maximum likelihood estimator (NPMLE) of a monotone density and a monotone failure rate by Grenander [10], and the least squares estimator of a monotone regression function by Brunk [4], it has taken some time before the distribution theory for such estimators entered the literature. The limiting distribution of the NPMLE of a decreasing density on [0, ∞) at a fixed point in the interior of the support, has been established by Prakasa Rao [27]. Similar results were obtained for the NPMLE of a monotone failure rate in [28] and for an estimator of a monotone regression function in [5]. Woodroofe and Sun [32] showed that the NPMLE of a decreasing density is inconsistent at zero. The behavior at the boundary has been further investigated in [2,23].Smooth estimation has been studied in [25], for monotone regression curves, and in [31] for monotone densities; see also [9] and [1]. The limit distribution of the NPMLE of a decreasing function in the Gaussian white noise model was obtained in [33]. Related likelihood ratio based techniques have been investigated in [3,26].Groeneboom [11] reproved the result in [27] by introducing a new approach based on inverses. This approach has become a cornerstone in deriving pointwise asymptotics of several shape constrained nonparametric estimators, for example, for the distribution function of interval censored