Bayesian Point Estimation Using the Predictive Distribution

There is considerable question about how a Bayesian might provide a point estimate for a parameter when no loss function is specified. The mean, median, and mode of the posterior distribution have all been suggested. This article considers a natural Bayesian estimator based on the predictive distribution of future observations. It is shown that for the set of parameters that admit an unbiased estimate, this predictive estimate coincides with the posterior mean of the parameter. It is argued that this result provides some justification for use of the posterior mean as a Bayesian point estimate when there is no loss structure.     

On approximating point spread distributions

We discuss some properties of the point spread distribution, defined as the distribution of the difference of two independent binomial random variables with the same parameter n including exact and approximate probabilities and related optimization issues. We use various approximation techniques for different distributions, special functions, and analytic, combinatorial and symbolic methods, such as multi-summation techniques. We prove that in case of unequal success rates, if these rates change with their difference kept fixed and small, and n is appropriately bounded, then the point spread distribution only slightly changes for small point differences. We also prove that for equal success rates p, the probability of a tie is minimized if p=1/2. Numerical examples are included for the case with n=12.     

Object detection in multi-epoch data

In astronomy multiple images are frequently obtained at the same position of the sky for follow-up coaddition as it helps one go deeper and look for fainter objects. With large scale panchromatic synoptic surveys becoming more common, image co-addition has become even more necessary as new observations start to get compared with coadded fiducial sky in real time. The standard coaddition techniques have included straight averages, variance weighted averages, medians etc. A more sophisticated nonlinear response chi-square method is also used when it is known that the data are background noise limited and the point spread function is homogenized in all channels. A more robust object detection technique capable of detecting faint sources, even those not seen at all epochs which will normally be smoothed out in traditional methods, is described. The analysis at each pixel level is based on a formula similar to Mahalanobis distance.     

Analyzing point processes subjected to random deletions

Certain nonstationary point process data are viewed as having arisen through time dependent random deletions of a stationary point process. Initially the probability Of deletion is assumed known and estimates of the rate and autointensity function of the inherent stationary process are constructed. Then an estimate of the deletion probability function is developed for the case of the function depending on a finite dimensional parameter. An estimate is provided for the variance of the autointensity estimate.     

A generalized pivotal quantity approach to portfolio selection

The major problem of mean–variance portfolio optimization is parameter uncertainty. Many methods have been proposed to tackle this problem, including shrinkage methods, resampling techniques, and imposing constraints on the portfolio weights, etc. This paper suggests a new estimation method for mean–variance portfolio weights based on the concept of generalized pivotal quantity (GPQ) in the case when asset returns are multivariate normally distributed and serially independent. Both point and interval estimations of the portfolio weights are considered. Comparing with Markowitz's mean–variance model, resampling and shrinkage methods, we find that the proposed GPQ method typically yields the smallest mean-squared error for the point estimate of the portfolio weights and obtains a satisfactory coverage rate for their simultaneous confidence intervals. Finally, we apply the proposed methodology to address a portfolio rebalancing problem.     

Estimation Under Inequality Constraints: Semiparametric Estimation of Conditional Duration Models

This article proposes a semiparametric estimator of the parameter in a conditional duration model when there are inequality constraints on some parameters and the error distribution may be unknown. We propose to estimate the parameter by a constrained version of an unrestricted semiparametrically efficient estimator. The main requirement for applying this method is that the initial unrestricted estimator converges in distribution. Apart from this, additional regularity conditions on the data generating process or the likelihood function, are not required. Hence the method is applicable to a broad range of models where the parameter space is constrained by inequality constraints, such as the conditional duration models. In a simulation study involving conditional duration models, the overall performance of the constrained estimator was better than its competitors, in terms of mean squared error. A data example is used to illustrate the method.     

Testing and Confidence estimation of the Mean of a multidimensional Gaussian Process

From the sequential observation of a multidimensional continuous time Gaussian process, whose mean vector depends linearly of a multidimensional parameter, we consider the confidential estimation of the parameter value and the testing problem of a simple hypothesis about the parameter, in presence of a nuisance variance parameter. The method is based on a previously obtained [cf. 4] point estimate for the case of a known covariance structure. We first see that this estimate is, in fact, independent of the variance parameter. For the hypotheses testing problem, the invariance under certain groups of transformations and the partial sufficiency allows to construct optimal terminal tests. Furthermore we determine the observation time necessary to control its power function. These testing results may be translated in terms of most accurate confidence sets. If the observation is stopped according to the diameter of the confidence set, under some condition, the confidence level is preserved.     

Marginal information for expectation parameters

The authors consider a special case of inference in the presence of nuisance parameters. They show that when the orthogonalized score function is a function of a statistic S, no Fisher information for the interest parameter is lost by using the marginal distribution of S rather than the full distribution of the observations. Therefore, no information for the interest parameter is recovered by conditioning on an ancillary statistic, and information will be lost by conditioning on an approximate ancillary statistic. This is the case for regular multivariate exponential families when the interest parameter is a subvector of the expectation parameter and the statistic is the maximum likelihood estimate of the interest parameter. Several examples are considered, including the 2 × 2 table.     

Communication in Statistics-Theory and methods improved GQL estimation method for the generalised BINMA(1) model

In a recent research, the quasi-likelihood estimation methodology was developed to estimate the regression effects in the Generalized BINMA(1) (GBINMA(1)) process. The method provides consistent parameter estimates but, in the intermediate computations, moment estimating equations were used to estimate the serial- and cross-correlation parameters. This procedure may not result optimal parameter estimates, in particular, for the regression effects. This paper provides an alternative simpler GBINMA(1) process based on multivariate thinning properties where the main effects are estimated via a robust generalized quasi-likelihood (GQL) estimation approach. The two techniques are compared through some simulation experiments. A real-life data application is studied.     

Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Modelling micro-texture

There are many instances when texture contains valuable information in images, and various methods have been used for texture analysis. We distinguish between micro-textures and macro-textures. The paper models micro-texture using the general spin Ising model from statistical mechanics. This model allows for any number of grey levels and any set of pair interactions. For a given texture, we select an appropriate set of pair interactions and estimate the correspomding parameter values, using linked cluster expansions of the auto-covariances and the partition function. The series expansions are valid for parameters smaller than the critical parameters for which an infinite system would exhibit a phase transition. Hence, sufficiently small-grained micro-textures may be modelled. To ensure that the data meet this requirement, we simulate the model using the Markov chain Meet Carlo method and estimate its critical parameters using the series expansions. We demonstrate these methods on both real and simulated images.     

Tracking multiple moving objects in images using Markov Chain Monte Carlo

A new Bayesian state and parameter learning algorithm for multiple target tracking models with image observations are proposed. Specifically, a Markov chain Monte Carlo algorithm is designed to sample from the posterior distribution of the unknown time-varying number of targets, their birth, death times and states as well as the model parameters, which constitutes the complete solution to the specific tracking problem we consider. The conventional approach is to pre-process the images to extract point observations and then perform tracking, i.e. infer the target trajectories. We model the image generation process directly to avoid any potential loss of information when extracting point observations using a pre-processing step that is decoupled from the inference algorithm. Numerical examples show that our algorithm has improved tracking performance over commonly used techniques, for both synthetic examples and real florescent microscopy data, especially in the case of dim targets with overlapping illuminated regions.     

Some results on estimating a change-point using non-parametric type statistics

A Mann-Whitney type statistic is used to estimate a change-point when a change, at an unknown point in a sequence of random variables, has taken place. This estimate is compared, using Monte Carlo techniques, with the normal theory maximum likelihood estimate, when a location change has occurred, for different underlying distributions ranging from the normal to the long tailed “normal over uniform” distribution. The distribution of the Mann-Whitney type estimate remains fairly constant over the various distributions. Two generalisations of the statistic are considered and investigated.     

Non-parametric Estimation for NHPP Software Reliability Models

The non-homogeneous Poisson process (NHPP) model is a very important class of software reliability models and is widely used in software reliability engineering. NHPPs are characterized by their intensity functions. In the literature it is usually assumed that the functional forms of the intensity functions are known and only some parameters in intensity functions are unknown. The parametric statistical methods can then be applied to estimate or to test the unknown reliability models. However, in realistic situations it is often the case that the functional form of the failure intensity is not very well known or is completely unknown. In this case we have to use functional (non-parametric) estimation methods. The non-parametric techniques do not require any preliminary assumption on the software models and then can reduce the parameter modeling bias. The existing non-parametric methods in the statistical methods are usually not applicable to software reliability data. In this paper we construct some non-parametric methods to estimate the failure intensity function of the NHPP model, taking the particularities of the software failure data into consideration.     

Estimating the distribution function of a transformed random vector

Frequently a random vector Y with known distribution function is readily observed. However, the random variable of interest is a transformation of Y say h(Y), and sample values of h are expensive to evaluate. The objective is to estimate the distribution function of using only a small sample on Y. Four estimators are proposed for use when Y is discrete. A Monte Carlo study of the estimators is presented This estimation problem frequently arises when Y is a parameter in a mathematical programming problem and h(Y) is the optimal objective function value. Two examples of this type are presented.     

Bayesian inference for pairwise interacting point processes

Pairwise interacting point processes are commonly used to model spatial point patterns. To perform inference, the established frequentist methods can produce good point estimates when the interaction in the data is moderate, but some methods may produce severely biased estimates when the interaction in strong. Furthermore, because the sampling distributions of the estimates are unclear, interval estimates are typically obtained by parametric bootstrap methods. In the current setting however, the behavior of such estimates is not well understood. In this article we propose Bayesian methods for obtaining inferences in pairwise interacting point processes. The requisite application of Markov chain Monte Carlo (MCMC) techniques is complicated by an intractable function of the parameters in the likelihood. The acceptance probability in a Metropolis-Hastings algorithm involves the ratio of two likelihoods evaluated at differing parameter values. The intractable functions do not cancel, and hence an intractable ratio r must be estimated within each iteration of a Metropolis-Hastings sampler. We propose the use of importance sampling techniques within MCMC to address this problem. While r may be estimated by other methods, these, in general, are not readily applied in a Bayesian setting. We demonstrate the validity of our importance sampling approach with a small simulation study. Finally, we analyze the Swedish pine sapling dataset (Strand 1972) and contrast the results with those in the literature.     

The Performance of a Robust Multistage Estimator in Nonlinear Regression with Heteroscedastic Errors

In this article, a robust multistage parameter estimator is proposed for nonlinear regression with heteroscedastic variance, where the residual variances are considered as a general parametric function of predictors. The motivation is based on considering the chi-square distribution for the calculated sample variance of the data. It is shown that outliers that are influential in nonlinear regression parameter estimates are not necessarily influential in calculating the sample variance. This matter persuades us, not only to robustify the estimate of the parameters of the models for both the regression function and the variance, but also to replace the sample variance of the data by a robust scale estimate.     

Diagnostics for glms with linear inequality parameter constraints

This paper proposes an iterative process, that can be implemented using GLIM, for fitting generalized linear models with linear inequality parameter constraints, when the maximum likelihood estimates exist and are unique. A one-step estimate is also introduced and some diagnostic measures are obtained. Finally an example is given for illustration.     

Semiparametric Analysis of Isotonic Errors-in-Variables Regression Models with Missing Response

This article is concerned with the estimation problem in the semiparametric isotonic regression model when the covariates are measured with additive errors and the response is missing at random. An inverse marginal probability weighted imputation approach is developed to estimate the regression parameters and a least-square approach under monotone constraint is employed to estimate the functional component. We show that the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent. A simulation study is conducted to examine the finite-sample properties of the proposed estimators. A data set is used to demonstrate the proposed approach.     

On Sensitivity of Inverse Response Plot Estimation and the Benefits of a Robust Estimation Approach

Abstract. Inverse response plots are a useful tool in determining a response transformation function for response linearization in regression. Under some mild conditions it is possible to seek such transformations by plotting ordinary least squares fits versus the responses. A common approach is then to use nonlinear least squares to estimate a transformation by modelling the fits on the transformed response where the transformation function depends on an unknown parameter to be estimated. We provide insight into this approach by considering sensitivity of the estimation via the influence function. For example, estimation is insensitive to the method chosen to estimate the fits in the initial step. Additionally, the inverse response plot does not provide direct information on how well the transformation parameter is being estimated and poor inverse response plots may still result in good estimates. We also introduce a simple robustified process that can vastly improve estimation.