A statistical approach to the resolution of point sources

SUMMARY The Rayleigh criterion in optics states that two point sources of equal intensity are 'barely resolved' when the maximum of the diffraction pattern of one source overlaps the first minimum of the diffraction pattern of the second source. Although useful for rough comparisons of optical systems, such a criterion does not take into account the randomness in the detection process and does not tell whether sources can actually be distinguished. We present a statistical approach that addressed these issues. From quantum optics, the photon counts in the pixels are independent Poisson random variables with means that depend on the distance 2theta between the sources. Resolving the sources corresponds to testing H0: theta =0 vs Ha: theta >0, under conditions that make the information number zero at theta =0. We define resolution as the (asymptotic) power function of the likelihood ratio test rather than as a single number. The asymptotic distribution of the test statistic is derived under H0 and under contiguous alternatives. The results are illustrated by an application to a sky survey to detect binary stars using the Hubble space telescope.     

Component reduction in linear discriminant analysis

The linear discriminant function is transformed into a linear combination of independent random variables. It is shown that reducing dimensionality using the smallest distance criterion results in smaller increase in the error rate than using the smallest variance criterion. Three error rates are used to prove this.     

DENSITY FUNCTION APPROXIMATION WITH PARTICULAR REFERENCE TO THE DISTRIBUTION OF OVERLAP OF CIRCLES

Given that two circles overlap, the area in common is a function of the distance between their centres. This paper adopts a suitable random distribution for the intercentre distance and then derives the distribution of the area of overlap. An approximation is sought for the density function using a criterion which enables bounds to be placed on the difference between the moments of the density function and those of the approximation. This is an approach of general applicability. The importance of matching the end-point behaviour of the density and the approximation is stressed. It is shown that the distribution of the area of overlap may be well approximated by a mixture of beta distributions in which the parameters change smoothly with the ratio of radii.     

A Hellinger distance approach to MCMC diagnostics

Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.     

Testing hypotheses when there could be an initial effect

In many treatment-versus-control experiments, the observed random variables can be written as the product of a Bernoulli and a continuous random variable. The treatment can affect the distribution of the observations in two ways.

1. the probability that the observation is 0 could be altered.

2. the distribution of the nonzero observations could be changed.

We may also want to measure the combined effect of the treatment.

3. the expected value of control and treated units may differ.

A method is presented for testing for the presence of the combined effect when the general form of the distribution function of the continuous observations is known. For the case when this distribution function is from the family of gamma distributions, a previously proposed test criterion for the combined effect has poor power properties. In this paper, we discuss a test criterion that has improved power properties.     

Distance Between Random Points in Two Rectangular Cities

ABSTRACT

The travel distance between two cities of rectangular shape is considered. Two uniformly distributed random points, one from each city, are taken. Their straight path travel distance is then measured. The explicit forms for the probability density function of this distance and its expected value are obtained. Numerical results of calculating the exact expected distance and the estimated distance as well as computer simulation are given for various cases. The integer moments of the distance are also discussed.     

Some orthogonal expansions for the logistic distribution

We expand a continuous random variable as a sum of a sequence of un-correlated random variables. These variables are principal components of a Bernoulli process, as well as principal dimensions in continuous metric scaling on a particular distance function. We obtain expansions for the uniform, exponential and logistic distributions. A goodness-of-fit application is given.     

A certain locally most powerful test for one-way random effects model: full distribution and power considerations

We investigate the properties of the locally most powerful nonparametric criterion against logistic alternatives developed by Govindarajulu (1975) for testing one-way random effects modcls. We deduce the appropriate computational forms for the test criterion T and tabulate the critical values of T for α = .01, .05 and 0.10, and various sample sizes. Certain features of the computational methods are discussed. In the tables we retain only those sample sizes beyond which the asymptotic theory is meaningful. We also study the power comparison of the test for two populations with the classical F-test under a range of normal alternatives.     

On the LBI criterion for the multivariate one-way random effects model under non-normality

The asymptotic null distribution of the locally best invariant (LBI) test criterion for testing the random effect in the one-way multivariable analysis of variance model is derived under normality and non-normality. The error of the approximation is characterized as O(1/n). The non-null asymptotic distribution is also discussed. In addition to providing a way of obtaining percentage points and p-values, the results of this paper are useful in assessing the robustness of the LBI criterion. Numerical results are presented to illustrate the accuracy of the approximation.     

Asymptotic biases of information and cross-validation criteria under canonical parametrization

An asymptotic expansion of the cross-validation criterion (CVC) using the Kullback-Leibler distance is derived when the leave-k-out method is used and when parameters are estimated by the weighted score method. By this expansion, the asymptotic bias of the Takeuchi information criterion (TIC) is derived as well as that of the CVC. Under canonical parametrization in the exponential family of distributions when maximum likelihood estimation is used, the magnitudes of the asymptotic biases of the Akaike information criterion (AIC) and CVC are shown to be smaller than that of the TIC. Examples in typical statistical distributions are shown.     

Global sensitivity analysis with dependence measures

Global sensitivity analysis with variance-based measures suffers from several theoretical and practical limitations, since they focus only on the variance of the output and handle multivariate variables in a limited way. In this paper, we introduce a new class of sensitivity indices based on dependence measures which overcomes these insufficiencies. Our approach originates from the idea to compare the output distribution with its conditional counterpart when one of the input variables is fixed. We establish that this comparison yields previously proposed indices when it is performed with Csiszár f-divergences, as well as sensitivity indices which are well-known dependence measures between random variables. This leads us to investigate completely new sensitivity indices based on recent state-of-the-art dependence measures, such as distance correlation and the Hilbert–Schmidt independence criterion. We also emphasize the potential of feature selection techniques relying on such dependence measures as alternatives to screening in high dimension.     

Exact sample size determination for binomial experiments

In experiments designed to estimate a binomial parameter, sample sizes are often calculated to ensure that the point estimate will be within a desired distance from the true value with sufficiently high probability. Since exact calculations resulting from the standard formulation of this problem can be difficult, “conservative” and/or normal approximations are frequently used. In this paper, some problems with the current formulation are given, and a modified criterion that leads to some improvement is provided. A simple algorithm that calculates the exact sample sizes under the modified criterion is provided, and these sample sizes are compared to those given by the standard approximate criterion, as well as to an exact conservative Bayesian criterion.     

Renyi Entropy Properties of Order Statistics

In this article, we discuss some properties of Renyi entropy and Renyi information of order statistics. Some bounds for Renyi entropy of order statistics are obtained. Also, we relate Renyi entropy ordering of order statistics to Renyi entropy ordering and other well known orderings of parent random variables. Then it is proved that the Renyi information between order statistics and parent random variable is distribution free, and it is shown, as expected, the distance is minimum for the median.     

Mean shift and influence measures in linear measurement error models with stochastic linear restrictions

We present influence diagnostics for linear measurement error models with stochastic linear restrictions using the corrected likelihood of Nakamura in 1990. The case deletion and mean shift outlier models are developed to identify outlying and influential observations. We derive a corrected score test statistic for outlier detection based on mean shift outlier models. The analogs of Cook's distance and likelihood distance are proposed to determine influential observations based on case deletion models. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics and a simulation study has been used to evaluate the performance of the proposed estimators based on the mean squares error criterion and the score test statistic. Finally, a numerical example is given to illustrate the theoretical results.     

Minimum variance rectangular designs for U-statistics

We consider a certain class of rectangular designs for incomplete U-statistics based on Latin squares and show it to be optimal with respect to the minimal variance criterion. We also show it to be asymptotically efficient when compared with the corresponding complete statistics, as well as uniformly more efficient than the random subset selection. We provide the necessary and sufficient conditions for the existence of our design and give some examples of applications.     

Non nested model selection for spatial count regression models with application to health insurance

In this paper we consider spatial regression models for count data. We examine not only the Poisson distribution but also the generalized Poisson capable of modeling over-dispersion, the negative Binomial as well as the zero-inflated Poisson distribution which allows for excess zeros as possible response distribution. We add random spatial effects for modeling spatial dependency and develop and implement MCMC algorithms in $R$ for Bayesian estimation. The corresponding R library ‘spatcounts’ is available on CRAN. In an application the presented models are used to analyze the number of benefits received per patient in a German private health insurance company. Since the deviance information criterion is only appropriate for exponential family models, we use in addition the Vuong and Clarke test with a Schwarz correction to compare possibly non nested models. We illustrate how they can be used in a Bayesian context.     

Confidence Intervals for the Median of a Finite Population Based on the Sign Test

This article presents methods for constructing confidence intervals for the median of a finite population under simple random sampling without replacement, stratified random sampling, and cluster sampling. The confidence intervals, as well as point estimates and test statistics, are derived from sign estimating functions which are based on the well-known sign test. Therefore, a unified approach for inference about the median of a finite population is given.     

Model Selection for Linear Mixed Models Using Predictive Criteria

Predictive criteria, including the adjusted squared multiple correlation coefficient, the adjusted concordance correlation coefficient, and the predictive error sum of squares, are available for model selection in the linear mixed model. These criteria all involve some sort of comparison of observed values and predicted values, adjusted for the complexity of the model. The predicted values can be conditional on the random effects or marginal, i.e., based on averages over the random effects. These criteria have not been investigated for model selection success.

We used simulations to investigate selection success rates for several versions of these predictive criteria as well as several versions of Akaike's information criterion and the Bayesian information criterion, and the pseudo F-test. The simulations involved the simple scenario of selection of a fixed parameter when the covariance structure is known.

Several variance–covariance structures were used. For compound symmetry structures, higher success rates for the predictive criteria were obtained when marginal rather than conditional predicted values were used. Information criteria had higher success rates when a certain term (normally left out in SAS MIXED computations) was included in the criteria. Various penalty functions were used in the information criteria, but these had little effect on success rates. The pseudo F-test performed as expected. For the autoregressive with random effects structure, the results were the same except that success rates were higher for the conditional version of the predictive error sum of squares.

Characteristics of the data, such as the covariance structure, parameter values, and sample size, greatly impacted performance of various model selection criteria. No one criterion was consistently better than the others.     

Inference for Random Coefficient INAR(1) Process Based on Frequency Domain Analysis

In this article, we present the explicit expressions for the higher-order moments and cumulants of the first-order random coefficient integer-valued autoregressive (RCINAR(1)) process. The spectral and bispectral density functions are also obtained, which can characterize the RCINAR(1) process in the frequency domain. We use a frequency domain approach which is named Whittle criterion to estimate the parameters of the process. We propose a test statistic which is based on the frequency domain approach for the hypothesis test, H₀: α = 0?H₁: 0 < α < 1, where α is the mean of the random coefficient in the process. The asymptotic distribution of the test statistic is obtained. We compare the proposed test statistic with other statistics that can test serial dependence in time series of count via a typically numerical simulation, which indicates that our proposed test statistic has a good power.