Bayesian inference for Gaussian excursion set generated Cox processes with set-marking

This work considers spatial Cox point processes where the random intensity is defined by a random closed set such that different point intensities appear in the two phases formed by the random set and its complement. The point pattern is observed as a set of point coordinates in a bounded region W?? ^d together with the information on the phase of the location of each point. This phase information, called set-marking, is not a representative sample from the random set, and hence it cannot be directly used for deducing properties of the random set. Excursion sets of continuous-parameter Gaussian random fields are applied as a flexible model for the random set. Fully Bayesian method and Markov chain Monte Carlo (MCMC) simulation is adopted for inferring the parameters of the model and estimating the random set. The performance of the new approach is studied by means of simulation experiments. Further, two forestry data sets on point patterns of saplings are analysed. The saplings grow in a clear-cut forest area where, before planting and natural seeding, the soil has been mounded forming a blotched soil structure. The tree densities tend to be different in the tilled patches and in the area outside the patches. The coordinates of each sapling have been measured and it is known whether this location is in a patch or outside. This example has been a motivation for the study.     

Random stopping sets in a sequential analysis of random measures and fields

A random stopping set is defined and some of its properties are proved. Namely, we prove the theorem on the absolute continuity of measures on a σ-algebra F_τ connected with a random stopping set τ, which can be applied to the sequential analysis of random measures and fields.     

The Muscatine children's obesity data reanalysed using pattern mixture models

A set of longitudinal binary, partially incomplete, data on obesity among children in the USA is reanalysed. The multivariate Bernoulli distribution is parameterized by the univariate marginal probabilities and dependence ratios of all orders, which together support maximum likelihood inference. The temporal association of obesity is strong and complex but stationary. We fit a saturated model for the distribution of response patterns and find that non-response is missing completely at random for boys but that the probability of obesity is consistently higher among girls who provided incomplete records than among girls who provided complete records. We discuss the statistical and substantive features of, respectively, pattern mixture and selection models for this data set.     

Estimation of the means of the bivariate normal using moving extreme ranked set sampling with concomitant variable

The estimation of the means of the bivariate normal distribution, based on a sample obtained using a modification of the moving extreme ranked set sampling technique (MERSS) is considered. The modification involves using a concomitant random variable. Nonparametric-type methods as well as the maximum likelihood estimation are considered. The estimators obtained are compared to their counterparts based on simple random sampling (SRS). It appears that the suggested estimators are more efficient. Also, MERSS with concomitant variable is easier to use in practice than the usual ranked set sampling (RSS) with concomitant variable. The issue of robustness of the procedure is addressed. Real trees data set is used for illustration.     

Bayesian mixture models for complex high dimensional count data in phage display experiments

Summary.  Phage display is a biological process that is used to screen random peptide libraries for ligands that bind to a target of interest with high affinity. On the basis of a count data set from an innovative multistage phage display experiment, we propose a class of Bayesian mixture models to cluster peptide counts into three groups that exhibit different display patterns across stages. Among the three groups, the investigators are particularly interested in that with an ascending display pattern in the counts, which implies that the peptides are likely to bind to the target with strong affinity. We apply a Bayesian false discovery rate approach to identify the peptides with the strongest affinity within the group. A list of peptides is obtained, among which important ones with meaningful functions are further validated by biologists. To examine the performance of the Bayesian model, we conduct a simulation study and obtain desirable results.     

Ranked set sampling with random subsamples

A ranked sampling procedure with random subsamples is proposed to estimate the population mean. Four methods of obtaining random subsamples are described. Several estimators of the mean of the population based on random subsamples in ranked set sampling are proposed. These estimators are compared with the mean of a simple random sample for estimating the mean of symmetric and skew distributions. Extensive simulation under several subsampling distributions, sample sizes, and symmetric and skew distributions shows that the estimators of the mean based on random subsamples are more accurate than existing methods.     

Estimation of population mean and variance in flock management: a ranked set sampling approach in a finite population setting

Ranked set sampling is a sampling technique that provides substantial cost efficiency in experiments where a quick, inexpensive ranking procedure is available to rank the units prior to formal, expensive and precise measurements. Although the theoretical properties and relative efficiencies of this approach with respect to simple random sampling have been extensively studied in the literature for the infinite population setting, the use of ranked set sampling methods has not yet been explored widely for finite populations. The purpose of this study is to use sheep population data from the Research Farm at Ataturk University, Erzurum, Turkey, to demonstrate the practical benefits of ranked set sampling procedures relative to the more commonly used simple random sampling estimation of the population mean and variance in a finite population. It is shown that the ranked set sample mean remains unbiased for the population mean as is the case for the infinite population, but the variance estimators are unbiased only with use of the finite population correction factor. Both mean and variance estimators provide substantial improvement over their simple random sample counterparts.     

Spatio-temporal analysis of a plant disease in a non-uniform crop: a Monte Carlo approach

Identification of the type of disease pattern and spread in a field is critical in epidemiological investigations of plant diseases. For example, an aggregation pattern of infected plants suggests that, at the time of observation, the pathogen is spreading from a proximal source. Conversely, a random pattern suggests a lack of spread from a proximal source. Most of the existing methods of spatial pattern analysis work with only one variety of plant at each location and with uniform genetic disease susceptibility across the field. Pecan orchards, used in this study, and other orchard crops are usually composed of different varieties with different levels of susceptibility to disease. A new measure is suggested to characterize the spatio-temporal transmission patterns of disease; a Monte Carlo test procedure is proposed to test whether the transmission of disease is random or aggregated. In addition, we propose a mixed-transmission model, which allows us to quantify the degree of aggregation effect.     

Random distances and edge correction

Two classes of random distances (generated as contact distances or free path lengths) external to a stationary random closed set in Euclidean space are introduced. The censored distance distributions within bounded region are obtained. Unbiased estimators of the random distance distribution functions using only the information from inside the bounded region are constructed.     

Random vs. systematic non-response in group-level research

This study investigates whether, in the context of group-level research, the effect of random non-response pattern, in biasing group-level scores, would be more salient than the effect of systematic non-response pattern with all other factors (i.e. group size, within-group variability, between-group variability and response rate) held constant. The findings of the Monte Carlo simulation study are counter-intuitive and reveal that the amount of error in the group-level variable is indeed the largest under the conditions of random non-response pattern.     

Interlitter Response Variability in a Threshold Dose-Response Model

In this article, we extend a previously formulated threshold dose-response model with random litter effects that was applied to a data set from a developmental toxicity study. The dose-response pattern of the data indicates that a threshold dose level may exist. Additionally, there is noticeable variation between the responses across the dose levels. With threshold estimation being critical, the assumed variability structure should adequately model the variation while not taking away from the estimation of the threshold as well as the other parameters directly involved in the dose-response relationship. In the prior formulation, the random effect was modeled assuming identical variation in the interlitter response probabilities across all dose levels, that is, the model had a single parameter to account for the interlitter variability. In this new model, the random effect is modeled as having different response variability across dose levels, that is, multiple interlitter variability parameters. We performed the likelihood ratio test (LRT) to compare our extended model to the previous model. We conducted a simulation study to compare the bias of each model when fit to data generated with the underlying parametric structure of the opposing model. The extended threshold dose-response model with multiple response variation was less biased.     

Multiresolution analysis of linearly oriented spatial point patterns

The assumption of direction invariance, i.e. isotropy, is often made in the practical analysis of spatial point patterns due to simpler interpretation and ease of analysis. However, this assumption is many times hard to find in real applications. We propose a wavelet-based approach to test for isotropy in spatial patterns based on the logarithm of the directional scalogram. Under the null hypothesis of isotropy, a random isotropic point pattern should be expected to have the same value of the directional scalogram for any possible direction. Monte Carlo simulations of the logarithm of the directional scalogram over all directions are used to approximate the test distribution and the critical values. We demonstrate the efficacy of the approach through simulation studies and an application to a desert plant data set, where our approach confirms suspected directional effects in the spatial distribution of the desert plant species.     

Asymptotic relative efficiency of the linear discriminant function under partial nonrandom classification of the training data

This paper considers the problem where the linear discriminant rule is formed from training data that are only partially classified with respect to the two groups of origin. A further complication is that the data of unknown origin do not constitute an observed random sample from a mixture of the two under- lying groups. Under the assumption of a homoscedastic normal model, the overall error rate of the sample linear discriminant rule formed by maximum likelihood from the partially classified training data is derived up to and including terms of the first order in the case of univariate feature data. This first- order expansion of the sample rule so formed is used to define its asymptotic efficiency relative to the rule formed from a completely classified random training set and also to the rule formed from a completely unclassified random set.     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.     

Likelihood Inference for Unions of Interacting Discs

Abstract. This is probably the first paper which discusses likelihood inference for a random set using a germ‐grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation‐based maximum likelihood inference and the effect of specifying different reference Poisson models.     

Information content of partially rank-ordered set samples

Partially rank-ordered set (PROS) sampling is a generalization of ranked set sampling in which rankers are not required to fully rank the sampling units in each set, hence having more flexibility to perform the necessary judgemental ranking process. The PROS sampling has a wide range of applications in different fields ranging from environmental and ecological studies to medical research and it has been shown to be superior over ranked set sampling and simple random sampling for estimating the population mean. We study Fisher information content and uncertainty structure of the PROS samples and compare them with those of simple random sample (SRS) and ranked set sample (RSS) counterparts of the same size from the underlying population. We study uncertainty structure in terms of the Shannon entropy, Rényi entropy and Kullback–Leibler (KL) discrimination measures.     

Sharp bounds for L-statistics from dependent samples of random length

Several distribution-free bounds on expected values of L-statistics based on the sample of possibly dependent and nonidentically distributed random variables are given in the case when the sample size is a random variable, possibly dependent on the observations, with values in the set {1,2,…}. Some bounds extend the results of Papadatos (2001a) to the case of random sample size. The others provide new evaluations even if the sample size is nonrandom. Some applications of the presented bounds are also indicated.     

A goodness-of-fit test using randomly selected cell boundaries

This paper describes a technique for testing for the goodness-of-fit of a set of observations on a random variable to a hypothesized distribution. The cell boundaries needed for partitioning the range of the random variable are randomly chosen using the computer, thus eliminating an arbitrary choice by the analyst. The power of this test procedure is compared to the test procedure using equal expected cell frequencies