Asymptotics for tests on mean profiles,additional information and dimensionality under non-normality

We consider the comparison of mean vectors for k groups when k is large and sample size per group is fixed. The asymptotic null and non-null distributions of the normal theory likelihood ratio, Lawley–Hotelling and Bartlett–Nanda–Pillai statistics are derived under general conditions. We extend the results to tests on the profiles of the mean vectors, tests for additional information (provided by a sub-vector of the responses over and beyond the remaining sub-vector of responses in separating the groups) and tests on the dimension of the hyperplane formed by the mean vectors. Our techniques are based on perturbation expansions and limit theorems applied to independent but non-identically distributed sequences of quadratic forms in random matrices. In all these four MANOVA problems, the asymptotic null and non-null distributions are normal. Both the null and non-null distributions are asymptotically invariant to non-normality when the group sample sizes are equal. In the unbalanced case, a slight modification of the test statistics will lead to asymptotically robust tests. Based on the robustness results, some approaches for finite approximation are introduced. The numerical results provide strong support for the asymptotic results and finiteness approximations.     

Inference for functions of parameters in discrete distributions based on fiducial approach: Binomial and Poisson cases

In this article, we propose a simple method of constructing confidence intervals for a function of binomial success probabilities and for a function of Poisson means. The method involves finding an approximate fiducial quantity (FQ) for the parameters of interest. A FQ for a function of several parameters can be obtained by substitution. For the binomial case, the fiducial approach is illustrated for constructing confidence intervals for the relative risk and the ratio of odds. Fiducial inferential procedures are also provided for estimating functions of several Poisson parameters. In particular, fiducial inferential approach is illustrated for interval estimating the ratio of two Poisson means and for a weighted sum of several Poisson means. Simple approximations to the distributions of the FQs are also given for some problems. The merits of the procedures are evaluated by comparing them with those of existing asymptotic methods with respect to coverage probabilities, and in some cases, expected widths. Comparison studies indicate that the fiducial confidence intervals are very satisfactory, and they are comparable or better than some available asymptotic methods. The fiducial method is easy to use and is applicable to find confidence intervals for many commonly used summary indices. Some examples are used to illustrate and compare the results of fiducial approach with those of other available asymptotic methods.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

Compound patterns and generating functions: From basic waiting times to counts of occurrence

We derive an explicit, closed form expression for the double generating function of the corresponding counts of occurrence, within a finite time horizon, of the single patterns contained in a compound pattern. The expression is in terms of a basic single, and a basic joint, generating functions for which exact solutions exist in the literature. The single generating function is associated with the basic waiting time for the first occurrence of the compound pattern. The joint generating function is that for the waiting time to reach a given single pattern and the associated counts of occurrence, within that waiting time, of the single patterns contained in the compound pattern. The literature on patterns is huge. Also, there are results that establish links between generating functions for counts of occurrence of the single patterns contained in a compound pattern with generating functions of some more complex waiting times associated with that compound pattern. The latter waiting times are known in the literature with names such as sooner, or later waiting times, or generalisations of such. On the other hand, our result fills a gap in the literature by providing a neat link connecting the generating functions of the basic quantities associated with occurrence of compound patterns.     

A note on ‘curable’ shock processes

In most conventional shock models, the events caused by an external shock are initiated at the moments of its occurrence. Recently, Cha and Finkelstein (2012) had considered the case when each shock from a nonhomogeneous Poisson processes can trigger a failure of a system not immediately, as in the classical shock models, but with delay of some random time. In this paper, we suggest the new type of shock models, where each delayed failure can be cured (repaired) with certain probabilities. These shock processes have not been considered in the literature before. We derive and analyze the corresponding survival and failure rate functions and consider a meaningful reliability example of the stress–strength model.     

On Stein's method, smoothing estimates in total variation distance and mixture distributions

In many settings it is useful to have bounds on the total variation distance between some random variable Z and its shifted version Z+1. For example, such quantities are often needed when applying Stein's method for probability approximation. This note considers one way in which such bounds can be derived, in cases where Z is either the equilibrium distribution of some birth-death process or the mixture of such a distribution. Applications of these bounds are given to translated Poisson and compound Poisson approximations for Poisson mixtures and the Pólya distribution.     

Likelihood Asymptotics

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the usual asymptotic likelihood ratio tests, and for one-dimensional hypotheses the test statistic is known as r *, introduced by Barndorff-Nielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace-type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi-dimensional tests is developed. In the final part of the paper further problems and ideas are presented. Among these are linear models with non-normal error, non-parametric linear models obtained by estimation of the residual density in combination with the present results, and the generalization of the results to restricted maximum likelihood and similar structured models.     

Statistical estimation of quadratic Rényi entropy for a stationary m-dependent sequence

The Rényi entropy is a generalisation of the Shannon entropy and is widely used in mathematical statistics and applied sciences for quantifying the uncertainty in a probability distribution. We consider estimation of the quadratic Rényi entropy and related functionals for the marginal distribution of a stationary m-dependent sequence. The U-statistic estimators under study are based on the number of ε-close vector observations in the corresponding sample. A variety of asymptotic properties for these estimators are obtained (e.g. consistency, asymptotic normality, and Poisson convergence). The results can be used in diverse statistical and computer science problems whenever the conventional independence assumption is too strong (e.g. ε-keys in time series databases and distribution identification problems for dependent samples).     

Functional convergence of quantile-type frontiers with application to parametric approximations

Nonparametric estimators of the upper boundary of the support of a multivariate distribution are very appealing because they rely on very few assumptions. But in productivity and efficiency analysis, this upper boundary is a production (or a cost) frontier and a parametric form for it allows for a richer economic interpretation of the production process under analysis. On the other hand, most of the parametric approaches rely on often too restrictive assumptions on the stochastic part of the model and are based on standard regression techniques fitting the shape of the center of the cloud of points rather than its boundary. To overcome these limitations, Florens and Simar [2005. Parametric approximations of nonparametric frontiers. J. Econometrics 124 (1), 91–116] propose a two-stage approach which tries to capture the shape of the cloud of points near its frontier by providing parametric approximations of a nonparametric frontier. In this paper we propose an alternative method using the nonparametric quantile-type frontiers introduced in Aragon, Daouia and Thomas-Agnan [2005. Nonparametric frontier estimation: a conditional quantile-based approach. Econometric Theory 21, 358–389] for the nonparametric part of our model. These quantile-type frontiers have the superiority of being more robust to extremes. Our main result concerns the functional convergence of the quantile-type frontier process. Then we provide convergence and asymptotic normality of the resulting estimators of the parametric approximation. The approach is illustrated through simulated and real data sets.     

Likelihood and higher-order approximations to tail areas: A review and annotated bibliography

Recent developments in higher-order asymptotic theory for statistical inference have emphasized the fundamental role of the likelihood function in providing accurate approximations to cumulative distribution functions. This paper summarizes the main results, with an emphasis on classes of problems for which relatively easily implemented solutions exist. A survey of the literature indicates the large number of problems solved and solvable by this method. Generalizations and extensions, with suggestions for further development, are considered.     

The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this article, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide range of parameters. It is proposed that the probability distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this article.     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.     

AN INFINITE-PHASE QUASI-BIRTH-AND-DEATH MODEL FOR THE NON-PREEMPTIVE PRIORITY M/PH/1 QUEUE

This paper considers a single server queue that handles arrivals from N classes of customers on a non-preemptive priority basis. Each of the N classes of customers features arrivals from a Poisson process at rate λ _i and class-dependent phase type service. To analyze the queue length and waiting time processes of this queue, we derive a matrix geometric solution for the stationary distribution of the underlying Markov chain. A defining characteristic of the paper is the fact that the number of distinct states represented within the sub-level is countably infinite, rather than finite as is usually assumed. Among the results we obtain in the two-priority case are tractable algorithms for the computation of both the joint distribution for the number of customers present and the marginal distribution of low-priority customers, and an explicit solution for the marginal distribution of the number of high-priority customers. This explicit solution can be expressed completely in terms of the arrival rates and parameters of the two service time distributions. These results are followed by algorithms for the stationary waiting time distributions for high- and low-priority customers. We then address the case of an arbitrary number of priority classes, which we solve by relating it to an equivalent three-priority queue. Numerical examples are also presented.     

Approximating probabilities of first passage in a particular gaussian process

Shepp (1971) derives the distribution of waiting times of first passage for a particular Gaussian process. However, Shepp notes that for moderate to large waiting tines the expressions for the probability cannot be evaluated either numerically or by asymptotic estimation. Me present a useful approximation for the distribution and expected waiting time for the conditional and unconditional versions of this first passage problem. The probabilities play a role in bounds by Adler (1984) for the probability distribution of the supremum of a particular two-parameter Gaussian field, a detection problem (Lai, 197 3) and the study of signal shape problems in radars (Zakai Ziv, 1969).     

Existence theorems for some group testing strategies

Group testing problems are considered as examples of discrete search problems. Existence theorems for optimal nonsequential designs developed for the general discrete search problems in O'Geran et al. (Acta Appl. Math. 25 (1991) 241–276) are applied for construction of upper bounds for the length of optimal group testing strategies in the case of additive model. The key point in the study is derivation of analytic expressions for the so-called Renyi coefficients. In addition, some asymptotic results are obtained and an asymptotic design problem is considered. The results particularly imply that if the number of significant factors is relatively small compared to the total number of factors then the choice of the test collections all containing a half of the total number of factors is asymptotically optimal in a proper sense.     

Continuity corrected approximations for and ‘exact’ inference with Pearson's X2

The classical adjustments for the inadequacy of the asymptotic distribution of Pearson's X² statistic, when some cells are sparse or the cell expectations are small, use continuity corrections and exact moments; the recent approach is to use computer based ‘exact inference’. In this paper we observe that the original exact test due to Freeman and Halton (Biometrika 38 (1951), 141–149) and its computer implementation are theoretically unsound. Furthermore, the corrected algorithmic version for the exact p-value in StatXact is practically useful in very few cases, and the results of its present version which includes Monte Carlo estimates can be highly variable. We then derive asymptotic expansions for the moments of the null distribution of Pearson's X², introduce a new method of correcting for discreteness and finite range of Pearson's X² as an alternative to the classical continuity correction, and use them to construct new and improved approximations for the null distribution. We also offer diagnostic criteria applicable to the tables for selecting an appropriate approximation. The exact methods and the competing approximations are studied and compared using thirteen test cases from the literature. It is concluded that the accuracy of the appropriate approximation is comparable with the truly exact method whenever it is available. The use of approximations is therefore preferable if the truly exact computer intensive solutions are unavailable or infeasible.     

Methods for high-dimensional multivariate and multi-group repeated measures data under non-normality

Asymptotic tests for multivariate repeated measures are derived under non-normality and unspecified dependence structure. Notwithstanding their broader scope of application, the methods are particularly useful when a random vector of large number of repeated measurements are collected from each subject but the number of subjects per treatment group is limited. In some experimental situations, replicating the experiment large number of times could be expensive or infeasible. Although taking large number of repeated measurements could be relatively cheaper, due to within subject dependence the number of parameters involved could get large pretty quickly. Under mild conditions on the persistence of the dependence, we have derived asymptotic multivariate tests for the three testing problems in repeated measures analysis. The simulation results provide evidence in favour of the accuracy of the approximations to the null distributions.     

On Wrapped Version of Some Life Testing Models

ABSTRACT

In this paper we primarily consider waiting time problems under three different sampling rules. SR₁ is the usual sampling with replacement, SR₂ is without replacement, and SR₃ is also with replacement, but uses no repetitions. We develop a new methodology for solving a wide variety of waiting time problems under each of the three sampling rules. A connection between waiting time problems under SR₂ and SR₃ is established which enables one to simultaneously solve waiting time problems under both of these sampling rules. The methods are illustrated with a large number of examples.     

General Saddlepoint Approximation for Testing Separate Location-Scale Families of Hypotheses

For testing separate families of hypotheses, the likelihood ratio test does not have the usual asymptotic properties. This paper considers the asymptotic distribution of the ratio of maximized likelihoods (RML) statistic in the special case of testing separate scale or location-scale families of distributions. We derive saddlepoint approximations to the density and tail probabilities of the log of the RML statistic. These approximations are based on the expansion of the log of the RML statistic up to the second order, which is shown not to depend on the location and scale parameters. The resulting approximations are applied in several cases, including normal versus Laplace, normal versus Cauchy, and Weibull versus log-normal. Our results show that the saddlepoint approximations are satisfactory, even for fairly small sample sizes, and are more accurate than normal approximations and Edgeworth approximations, especially for tail probabilities that are the values of main interest in hypothesis testing problems.     

What is the effective sample size of a spatial point process?

Point process models are a natural approach for modelling data that arise as point events. In the case of Poisson counts, these may be fitted easily as a weighted Poisson regression. Point processes lack the notion of sample size. This is problematic for model selection, because various classical criteria such as the Bayesian information criterion (BIC) are a function of the sample size, n, and are derived in an asymptotic framework where n tends to infinity. In this paper, we develop an asymptotic result for Poisson point process models in which the observed number of point events, m, plays the role that sample size does in the classical regression context. Following from this result, we derive a version of BIC for point process models, and when fitted via penalised likelihood, conditions for the LASSO penalty that ensure consistency in estimation and the oracle property. We discuss challenges extending these results to the wider class of Gibbs models, of which the Poisson point process model is a special case.