Mitochondrial DNA sequence variation is now being used to study the history of our species. In this paper we discuss some aspects of estimation and inference that arise in the study of such variability, focusing in particular on the estimation of substitution rates and their use in calibrating estimates of the time since the most recent common ancestor of a sample of sequences. Observed DNA sequence variation is generated by superimposing the effects of mutation on the ancestral tree of the sequences. For data of the type studied here, this ancestral tree has to be modeled as a random process. Superimposing the effects of mutation produces complicated sampling distributions that form the basis of any statistical model for the data. Using such distributions--for example, for maximum likelihood estimation of rates--poses some difficult computational problems. We describe a Monte Carlo method, a cousin of the popular "Markov chain Monte Carlo," that has proved very useful in addressing some of these issues.

The polymerase chain reaction (PCR) makes possible rapid generation of a very large number of copies of a specific region of DNA. This enables the typing of quantities of DNA as small as a single molecule. PCR has led to the development of laboratory experiments which provide new approaches to many classic problems in genetics, such as estimation of linkage, marker ordering and genetic disease diagnosis. We describe some of these experiments and the statistical techniques that have been used to design them and to analyze the data they produce.

One of the great success stories of modern molecular genetics has been the ability of biologists to isolate and characterize the genes responsible for serious inherited diseases like Huntington's disease, cystic fibrosis and myotonic dystrophy. Instrumental in these efforts has been the construction of so-called physical maps of regions of human chromosomes. A major goal of the Human Genome Project is to construct physical maps of the entire human genome. Such maps will reduce the time and expense required to isolate and study interesting chromosomal regions by many orders of magnitude. This article describes what physical maps are and how they have been used, and it outlines some of the statistical issues involved in making them.

Monte Carlo likelihood is becoming increasingly used where exact likelihood analysis is computationally infeasible. One area in which such likelihoods arise is that of genetic mapping, where, increasingly, researchers wish to extract additional information from limited trait data through the use of multiple genetic markers. In the genetic analysis context, Monte Carlo likelihood is most conveniently considered as a latent variable problem. Markov chain Monte Carlo provides a method of obtaining realisations of underlying latent variables simulated under a genetic model, conditional upon observed data. Hence a Monte Carlo estimate of the likelihood surface can be formed. Choice of the latent variables can be as critical as choice of sampler. In the case of very few individuals observed in each pedigree structure, such as occurs in homozygosity mapping and affected relative pair methods of genetic mapping, multilocus segregation indicators are defined and proposed as the latent variables of choice. An example of five Werner's syndrome pedigrees is given; these are a subset of the 21 pedigrees on which homozygosity mapping has recently confirmed the location of the Werner's syndrome gene on chromosome 8. However, multilocus computations on these pedigrees are impractical with standard methods of exact likelihood computation.

The Chen-Stein method of Poisson approximation has been used to establish theorems about comparison of two DNA or protein sequences. The most useful result for sequence alignment applies to alignment scoring with no gaps. However, there has not been a valid method to assign statistical significance to alignment scores with gaps. In this paper we extend Poisson approximation techniques using the Aldous clumping heuristic to a practical method of estimating statistical significance.

A critical review is given of recent research activity on bootstrap and related procedures. Theoretical work has shown the bootstrap approach to be a potentially powerful addition to the statistician's toolkit. We consider its impact on statistical practice and argue that, measured against the hopes raised by theoretical advances, this has been until now fairly modest. We suggest that while this state of affairs is a consequence to be expected of the sophisticated character of the bootstrap procedures required to cope reliably in many of the settings of most interest, much theoretical work is not serving the immediate needs of statistical practice. Emerging lines of research are reviewed and important future research directions suggested. In particular, we appeal for greater focussing of research activity on practicalities.

Number-theoretic methods (NTM's) are a class of techniques by which representative points of the uniform distribution on the unit cube of $R^s$ can be generated. NTM have been widely used in numerical analysis, especially in evaluation of high-dimensional integrals. Recently, NTM's have been extended to generate representative points for many useful multivariate distributions and have been systematically applied in statistics. In this paper, we shall introduce NTM's and review their applications in statistics, such as evaluation of the expected value of a random vector, statistical inference, regression analysis, geometric probability and experimental design.

It has been noted that when the Book of Genesis is written as two-dimensional arrays, equidistant letter sequences spelling words with related meanings often appear in close proximity. Quantitative tools for measuring this phenomenon are developed. Randomization analysis shows that the effect is significant at the level of 0.00002.