On the nonperiodic cyclic equivalence classes of Hamming codes and BCH codes

The number of nonperiodic cyclic equivalence classes of binary cyclic Hamming codes and BCH codes is counted in this paper.     

Empirical Bayes testing for equivalence

For a fixed point θ₀ and a positive value c₀, this paper studies the problem of testing the hypotheses H₀:|θ−θ₀|≤c₀ against H₁:|θ−θ₀|>c₀ for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate (lnn)^2.5/n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.     

Signed directed acyclic graphs for causal inference

Summary.  Formal rules governing signed edges on causal directed acyclic graphs are described and it is shown how these rules can be useful in reasoning about causality. Specifically, the notions of a monotonic effect, a weak monotonic effect and a signed edge are introduced. Results are developed relating these monotonic effects and signed edges to the sign of the causal effect of an intervention in the presence of intermediate variables. The incorporation of signed edges in the directed acyclic graph causal framework furthermore allows for the development of rules governing the relationship between monotonic effects and the sign of the covariance between two variables. It is shown that when certain assumptions about monotonic effects can be made then these results can be used to draw conclusions about the presence of causal effects even when data are missing on confounding variables.     

Compatible prior distributions for directed acyclic graph models

Summary.  The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given directed acyclic graph model, using a conditioning approach. We define a class of parameterizations that is consistent with the modular structure of the directed acyclic graph and derive a procedure, that is invariant within this class, which we name reference conditioning.     

Small sample equivalence tests for exponentiality

We consider small sample equivalence tests for exponentialy. Statistical inference in this setting is particularly challenging since equivalence testing procedures typically require much larger sample sizes, in comparison with classical “difference tests,” to perform well. We make use of Butler's marginal likelihood for the shape parameter of a gamma distribution in our development of small sample equivalence tests for exponentiality. We consider two procedures using the principle of confidence interval inclusion, four Bayesian methods, and the uniformly most powerful unbiased (UMPU) test where a saddlepoint approximation to the intractable distribution of a canonical sufficient statistic is used. We perform small sample simulation studies to assess the bias of our various tests and show that all of the Bayes posteriors we consider are integrable. Our simulation studies show that the saddlepoint-approximated UMPU method performs remarkably well for small sample sizes and is the only method that consistently exhibits an empirical significance level close to the nominal 5% level.     

An exact multinomial test for equivalence

Existing equivalence tests for multinomial data are valid asymptotically, but the level is not properly controlled for small and moderate sample sizes. We resolve this difficulty by developing an exact multinomial test for equivalence and an associated confidence interval procedure. We also derive a conservative version of the test that is easy to implement even for very large sample sizes. Both tests use a notion of equivalence that is based on the cumulative distribution function, with two probability vectors being considered equivalent if their partial sums never differ by more than some specified constant. We illustrate the methods by applying them to Weldon's dice data, to data on the digits of , and to data collected by Mendel. The Canadian Journal of Statistics 37: 47–59; © 2009 Statistical Society of Canada     

Parameter estimation for switched counting processes

The paper investigates parameter estimation problems in special Markov modulated counting processes. The events occuring at any state of an underlying Markov chain can be equipped with marks performing additional information on the events. Specifying the model to the case of two-state Markov chain modulation, the so-called switched counting process, some statistical problems are studied:maximum likelihood estimators, Rao-Blackwell optimal estimators, test of equality of the counting intensities of the two states and minimax estimation procedures. Tne consideration could be applied in various practical problems, in particular, in queueing and in reliability models, for example in failure-repair processes with alternatively operating repair systems.     

Formulas for the Exact LMS and LQS Estimators

The author presents the derivation of formulas for the calculation of critical values of the median function or the general version of it, namely, the quantile functions. In statistics, these functions are used to detect outliers in the data set and to make predictions that are resistant to outliers. Therefore, these formulas can also be used as estimators for these regressions. The fact that these formulas are able to calculate the global optimum gives the exact least median squares or the exact least quantile of squares estimators. The author provides the theoretical background for deriving these estimator formulas and derives the estimator formulas for regression models up to three parameters. In addition, the author provides guides for the derivation of formulas for other models, illustrates the use of these formulas, and emphasizes their properties that are useful for future works. One important conclusion is that each regression model has its own set of formulas.     

Sample Size Formulas for Normal Theory T Tests

Formulas that yield minimum sample size for standard T tests are presented. Although the results are approximations, they usually yield the exact solution. Involving only standard normal quantiles, they could be used in an elementary course.     

An equivalence test for comparing DNA sequences

The practice of sequence alignment is constantly oscillating between the risk of overlooking important structure and that of discovering any arbitrarily defined kind of structure anywhere. On the other hand, the use of a condensed consensus sequence may lead to a substantial loss in valuable information. While adopting a Mahalanobis‐type index we allow for a certain degree of uncertainty in the measurements. This uncertainty may be caused by inaccurate measurements or ambiguity. In this paper, we test the similarity between DNA sequences within the framework of equivalence testing, accounting for both variances and covariances between frequencies of nucleotides. Statistical methods for testing equivalence were first developed in the context of pharmacokinetics and later extended to the field of clinical trials. Nowadays, (bio)equivalence tests seem to be less frequently used outside the drug testing field, including statistical genetics. Copyright © 2005 John Wiley & Sons, Ltd.     

Regeneration-based bootstrap for Markov chains

We consider a bootstrap method for Markov chains where the original chain is broken into a (random) number of cycles based on an atom (regeneration point) and the bootstrap scheme resamples from these cycles. We investigate the asymptotic accuracy of this method for the case of a sum (or a sample mean) related to the Markov chain. Under some standard moment conditions, the method is shown to be at least as good as the normal approximation, and better (second-order accurate) in the case of nonlattice summands. We give three examples to illustrate the applicability of our results.     

Formulas for the Design of CUSUM Quality Control Charts

ABSTRACT

In the design of CUSUM control charts, it is common to use charts, tables, or software to find an appropriate critical threshold (h). This article provides an approximate formula to calculate the threshold directly from prespecified values of the reference value (k) and the in-control average run length (ARL₀). Formulas are also provided for choosing k and h from prespecified values of the in-control and out-of-control average run lengths.     

Formulas for the second-order analysis of marked point processes

Following Ripley, for the second moment measure of stationary isotropic marked point processes in R^destimators are given. Two examples are considered.     

Testing for equivalence of variances using Hartley's ratio

Hartley's test for homogeneity of k normal‐distribution variances is based on the ratio between the maximum sample variance and the minimum sample variance. In this paper, the author uses the same statistic to test for equivalence of k variances. Equivalence is defined in terms of the ratio between the maximum and minimum population variances, and one concludes equivalence when Hartley's ratio is small. Exact critical values for this test are obtained by using an integral expression for the power function and some theoretical results about the power function. These exact critical values are available both when sample sizes are equal and when sample sizes are unequal. One related result in the paper is that Hartley's test for homogeneity of variances is no longer unbiased when the sample sizes are unequal. The Canadian Journal of Statistics 38: 647–664; 2010 © 2010 Statistical Society of Canada     

A unified approach to permutation testing for equivalence

Statistical Methods & Applications - The notion of testing for equivalence of two treatments is widely used in clinical trials, pharmaceutical experiments, bioequivalence and quality control....     

Estimation for hidden Markov random fields

A finite state Markov random field is noisily observed via a second finite state process. The parameters of the model are estimated, as well as the most likely signal given the observations.     

Sparse Markov Chains for Sequence Data

Finite memory sources and variable‐length Markov chains have recently gained popularity in data compression and mining, in particular, for applications in bioinformatics and language modelling. Here, we consider denser data compression and prediction with a family of sparse Bayesian predictive models for Markov chains in finite state spaces. Our approach lumps transition probabilities into classes composed of invariant probabilities, such that the resulting models need not have a hierarchical structure as in context tree‐based approaches. This can lead to a substantially higher rate of data compression, and such non‐hierarchical sparse models can be motivated for instance by data dependence structures existing in the bioinformatics context. We describe a Bayesian inference algorithm for learning sparse Markov models through clustering of transition probabilities. Experiments with DNA sequence and protein data show that our approach is competitive in both prediction and classification when compared with several alternative methods on the basis of variable memory length.