Showing papers in "Annals of the Institute of Statistical Mathematics in 1991"

Bayesian image restoration, with two applications in spatial statistics

Abstract: There has been much recent interest in Bayesian image analysis, including such topics as removal of blur and noise, detection of object boundaries, classification of textures, and reconstruction of two- or three-dimensional scenes from noisy lower-dimensional views. Perhaps the most straightforward task is that of image restoration, though it is often suggested that this is an area of relatively minor practical importance. The present paper argues the contrary, since many problems in the analysis of spatial data can be interpreted as problems of image restoration. Furthermore, the amounts of data involved allow routine use of computer intensive methods, such as the Gibbs sampler, that are not yet practicable for conventional images. Two examples are given, one in archeology, the other in epidemiology. These are preceded by a partial review of pixel-based Bayesian image analysis.

Rejoinder (Bayesian image restoration,with two applications in spatial statistics)

Abstract: There has been much recent interest in Bayesian image analysis, including such topics as removal of blur and noise, detection of object boundaries, classification of textures, and reconstruction of two- or three-dimensional scenes from noisy lower-dimensional views. Perhaps the most straightforward task is that of image restoration, though it is often suggested that this is an area of relatively minor practical importance. The present paper argues the contrary, since many problems in the analysis of spatial data can be interpreted as problems of image restoration. Furthermore, the amounts of data involved allow routine use of computer intensive methods, such as the Gibbs sampler, that are not yet practicable for conventional images. Two examples are given, one in archeology, the other in epidemiology. These are preceded by a partial review of pixel-based Bayesian image analysis.

Estimation of the arrival times of seismic waves by multivariate time series model

Abstract: A computationally efficient procedure was developed for the fitting of many multivariate locally stationary autoregressive models. The details of the Householder method for fitting multivariate autoregressive model and multivariate locally stationary autoregressive model (MLSAR model) are shown. The proposed procedure is quite efficient in both accuracy and computation. The amount of computation is bounded by a multiple of Nm 2 with N being the data length and m the highest model order, and does not depend on the number of models checked. This facilitates the precise estimation of the change point of the AR model. Based on the AICs' of the fitted MLSAR models and Akaike's definition of the likelihood of the models, a method of evaluating the posterior distribution of the change point of the AR model is also presented. The proposed procedure is, in particular, useful for the estimation of the arrival time of the S wave of a microearthquake. To illustrate the usefulness of the proposed procedure, the seismograms of the foreshocks of the 1982 Urakawa-Oki Earthquake were analyzed. These data sets have been registered to AISM Data Library and the readers of this Journal can access to them by the method described in this issue.

A class of consistent tests for exponentiality based on the empirical Laplace transform

Abstract: The Laplace transform ψ(t=E[exp(−tX)]) of a random variable with exponential density λ exp(−λx), x≥0, satisfies the differential equation (λ+t)ψ′(t)+ψ(t=0, t≥0). We study the behaviour of a class of consistent (“omnibus”) tests for exponentiality based on a suitably weighted integral of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGBbGaaiikai% qbeU7aSzaajaWaaSbaaSqaaGqaciaa-5gaaeqaaOGaey4kaSIaamiD% aiaacMcacqaHipqEcaWFNaWaaSbaaSqaaiaad6gaaeqaaOGaaiikai% aadshacaGGPaGaey4kaSIaeqiYdK3aaSbaaSqaaiaad6gaaeqaaOGa% aiikaiaadshacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaaaaa!4C69!\[[(\hat \lambda _n + t)\psi '_n (t) + \psi _n (t)]^2 \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqcam% aaBaaaleaaieGacaWFUbaabeaaaaa!3A66!\[\hat \lambda _n \] is the maximum-likelihood-estimate of λ and ψn is the empirical Laplace transform, each based on an i.i.d. sample X1,...,Xn.

Counterexamples to parsimony and bic

Abstract: Suppose that the log-likelihood-ratio sequence of two models with different numbers of estimated parameters is bounded in probability, without necessarily having a chi-square limiting distribution. Then BIC and all other related “consistent” model selection criteria, meaning those which penalize the number of estimated parameters with a weight which becomes infinite with the sample size, will, with asymptotic probability 1, select the model having fewer parameters. This note presents examples of nested and non-nested regression model pairs for which the likelihood-ratio sequence is bounded in probability and which have the property that the model in each pair with more estimated parameters has better predictive properties, for an independent replicate of the observed data, than the model with fewer parameters. Our second example also shows how a one-dimensional regressor can overfit the data used for estimation in comparison to the fit of a two-dimensional regressor.

On the estimation of ordered means of two exponential populations

Abstract: Let random samples of equal sizes be drawn from two exponential distributions with ordered means λ i . The maximum likelihood estimator λ i * of λ i is shown to have a smaller mean square error than that of the usual estimator Xi, for each i=1,2. The asymptotic efficiency of λ i * relative to Xi has also been found.

On characterizations of distributions by mean absolute deviation and variance bounds

Abstract: In this paper we present a bound for the mean absolute deviation of an arbitrary real-valued function of a discrete random variable. Using this bound we characterize a mixture of two Waring (hence geometric) distributions by linearity of a function involved in the bound. A double Lomax distribution is characterized by linearity of the same function involved in the analogous bound for a continuous distribution. Finally, we characterize the Pearson system of distributions and the generalized hypergeometric distributions by a quadratic function involved in a similar bound for the variance of a function of a random variable.

A nonlinear time series model and estimation of missing observations

Abstract: This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.

A note on estimating eigenvalues of scale matrix of the multivariate F-distribution

Abstract: Let F pxp have the multivariate F-distribution with a scale matrix Δ and degrees of freedom n 1and n 2. In this paper the problem of estimating eigenvalues of Δ is considered. By constructing the improved orthogonally invariant estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] of Δ, which are analogous to Haff-type estimators of a normal covariance matrix, new estimators of eigenvalues of Δ are given. This is because the eigenvalues of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqqHuoaraSqabeaacaqGEbaaaOGaaiikaiaadAeacaGG% Paaaaa!402A!\[\mathop \Delta \limits^{\rm{\^}} (F)\] are taken as estimates of the eigenvalues of Δ.

A kernel approximation to the kriging predictor of a spatial process

Abstract: Suppose a two-dimensional spatial process z(x) with generalized covariance function G(x, x′) α |x − x′|2 log |x − x′| (Matheron, 1973, Adv. in Appl. Probab., 5, 439–468) is observed with error at a number of locations. This paper gives a kernel approximation to the optimal linear predictor, or kriging predictor, of z(x) under this model as the observations get increasingly dense. The approximation is in terms of a Kelvin function which itself can be easily approximated by series expansions. This generalized covariance function is of particular interest because the predictions it yields are identical to an order 2 thin plate smoothing spline. For moderate sample sizes, the kernel approximation is seen to work very well when the observations are on a square grid and fairly well when the observations come from a uniform random sample. This manuscript was prepared using computer facilities supported in part by National Science Foundation Grants No. DMS-8601732 and DMS-8404941 to the Department of Statistics at The University of Chicago.

Yokes and tensors derived from yokes

Abstract: A yoke on a differentiable manifold M gives rise to a whole family of derivative strings. Various elemental properties of a yoke are discussed in terms of these strings. In particular, using the concept of intertwining from the theory of derivative strings it is shown that a yoke induces a family of tensors on M. Finally, the expected and observed a-geometries of a statistical model and related tensors are shown to be derivable from particular yokes.

A prediction of individual growth of height according to an empirical bayesian approach

Abstract: An empirical Bayesian approach is applied to a prediction of an individual growth in height at an early stage of life. The sample has 548 normal growth of Japanese girls whose measurements are available on request. The prior distribution of estimator of the growth parameter vector in a lifetime growth model is obtained conventionally from the least squares estimates of the growth parameters. The choice of prior distributions is discussed from a practical point of view. It is possible to obtain a relevant prediction of growth based upon only measurements during the first six years of life. The lifetime prediction of individual growth at the age of 6 is enough approximation of real measurements obtained. This report deals with the comparison between the least squares estimates and an empirical Bayes estimates of the growth parameters and the characteristic points of the growth curve. We discuss the mean-constant growth curves of the groups classified by the height intervals at the age of 6.

Recent developments in the stereological analysis of particles

Abstract: Recent developments in the stereological analysis of particles are reviewed. The trend has been towards methods which are applicable without specific assumptions about particle shape. Geometric samples of a local 3-d character are used. Stereological estimators of particle intensity, particle size distribution and particle interaction are presented and discussed.

Geometry of f-divergence

Abstract: Amari's ±1-divergences and geometries provide an important description of statistical inference. The ±1-divergences are constructed so that they are compatible with a metric that is defined by the Fisher information. In many cases, the ±1-divergences are but two in a family of divergences, called the f-divergences, that are compatible with the metric. We study the geometries induced by these divergences. Minimizing the f-divergence provides geometric estimators that are naturally described using certain curvatures. These curvatures are related to asymptotic bias and efficiency loss. Under special but important restrictions, the geometry of f-divergence is closely related to the α-geometry, Amari's extension of the ±1-geometries. One application of these results is illustrated in an example.

Detection of heterogeneity and estimation of population characteristics from the field survey data: 1987/88 Japanese feasibility study of the southern hemisphere Minke whales

Abstract: To get reliable information of the age structure of whale population, Japan conducted a feasibility study of scientific research in the Antarctic in 1987/88. Though the sample was not large enough, it was the first data free from the problem of selectivity and whaling ground bias. From the analysis, it was found that the biological characteristics are highly heterogeneous spatially or other ways. Considering this, we recognize that the survey should be designed to collect the sample from the whole research area uniformly to obtain unbiased estimates of population characteristics. However, in an actual biological field survey, it is difficult to keep the sampling fractions thecisely the same for each sampling units. Therefore, it is important to detect the heterogeneity in the sample, and poststratify the data corresponding to the heterogeneity. The methodology of the estimation and model evaluation presented here will be useful for the development of biological field survey in general.

Bonferroni-type inequalities ; Chebyshev-type inequalities for the distributions on [0, n]

Abstract: An elementary “majorant-minorant method” to construct the most stringent Bonferroni-type inequalities is presented. These are essentially Chebyshev-type inequalities for discrete probability distributions on the set {0, 1,..., n}, where n is the number of concerned events, and polynomials with specific properties on the set lead to the inequalities. All the known results are proved easily by this method. Further, the inequalities in terms of all the lower moments are completely solved by the method. As examples, the most stringent new inequalities of degrees three and four are obtained. Simpler expressions of Margaritescu's inequality (1987, Stud. Cerc. Mat., 39, 246–251), improving Galambos' inequality, are given.

Bayesian detection of structural changes

Abstract: A Bayesian solution is given to the problem of making inferences about an unknown number of structural changes in a sequence of observations. Inferences are based on the posterior distribution of the number of change points and on the posterior probabilities of possible change points. Detailed analyses are given for binomial data and some regression problems, and nu- merical illustrations are provided. In addition, an approximation procedure to compute the posterior probabilities is presented.

On the determination and construction of optimal row-column designs having unequal row and column sizes

Abstract: In this paper we consider experimental situations requiring usage of a row-column design where v treatments are to be applied to experimental units arranged in b 1 rows and b 2 columns where row i has size k 1i , i=1,..., b 1 and column j has size k 2j , j=1,..., b 2. Conditions analogous to those given in Kunert (1983, Ann. Statis., 11, 247–257) and Cheng (1978, Ann. Statist., 6, 1262–1272) are given which can often be used to establish the optimality of a given row-column design from the optimality of an associated block design. In addition, sufficient conditions are derived which guarantee the existence of an optimal row-column design which can be constructed by appropriately arranging treatments within blocks of an optimal block design.

Frequency domain characteristics of linear operator to decompose a time series into the multi-components

Abstract: Frequency domain properties of the operators to decompose a time series into the multi-components along the Akaike's Bayesian model (Akaike (1980, Bayesian Statistics, 143–165, University Press, Valencia, Spain)) are shown. In that analysis a normal disturbance-linear-stochastic regression prior model is applied to the time series. A prior distribution, characterized by a small number of hyperparameters, is specified for model parameters. The posterior distribution is a linear function (filter) of observations. Here we use frequency domain analysis or filter characteristics of several prior models parametrically as a function of the hyperparameters.

Stein-type improvements of confidence intervals for the generalized variance

Abstract: Based on independent random matices X: p×m and S: p×p distributed, respectively, as N pm (μ, ∑ ⊗ I m ) and W p (n, ∑) with μ unknown and n≥p, the problem of obtaining confidence interval for |∑| is considered. Stein's idea of improving the best affine equivariant point estimator of |∑| has been adapted to the interval estimation problem. It is shown that an interval estimator of the form |S|(b −1, a −1) can be improved by min{|S|, c|S +XX'|}(b −1, a −1) for a certain constant c depending on (a, b).

On the almost everywhere properties of the kernel regression estimate

Abstract: The regression m(x)=E{Y|X=x} is estimated by the kernel regression estimate % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x) calculated from a sequence (X(1, Y1), ..., (Xn, Yn) of independent identically distributed random vectors from Rd×R. The second order asymptotic expansions for E% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x) and var % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x)} are derived. The expansions hold for almost all (μ) x∈Rd, μ is the probability measure of X. No smoothing conditions on μ and m are imposed. As a result, the asymptotic distribution-free normality for a stochastic component of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGTbGbambaaa% a!3888!\[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over m} \](x)} is established. Also some bandwidth-selection rule is suggested and bias adjustment is proposed.

The 3/2th and 2nd order asymptotic efficiency of maximum probability estimators in non-regular cases*

Abstract: In this paper we consider the estimation problem on independent and identically distributed observations from a location parameter family generated by a density which is positive and symmetric on a finite interval, with a jump and a nonnegative right differential coefficient at the left endpoit. It is shown that the maximum probability estimator (MPE) is 3/2th order two-sided asymptotically efficient at a point in the sense that it has the most concentration probability around the true parameter at the point in the class of 3/2th order asymptotically median unbiased (AMU) estimators only when the right differential coefficient vanishes at the left endpoint. The second order upper bound for the concentration probability of second order AMU estimators is also given. Further, it is shown that the MPE is second order two-sided asymptotically efficient at a point in the above case only.

Estimation of a smooth quantile function under the proportional hazards model

Abstract: The problem of estimating a smooth quantile function, Q(·), at a fixed point p, 0 < p < 1, is treated under a nonparametric smoothness condition on Q. The asymptotic relative deficiency of the sample quantile based on the maximum likelihood estimate of the survival function under the proportional hazards model with respect to kernel type estimators of the quantile is evaluated. The comparison is based on the mean square errors of the estimators. It is shown that the relative deficiency tends to infinity as the sample size, n, tends to infinity.

Bayes estimation of number of signals

Abstract: Bayes estimation of the number of signals, q, based on a binomial prior distribution is studied. It is found that the Bayes estimate depends on the eigenvalues of the sample covariance matrix S for white-noise case and the eigenvalues of the matrix S2 (S1+A)−1 for the colored-noise case, where S1 is the sample covariance matrix of observations consisting only noise, S2 the sample covariance matrix of observations consisting both noise and signals and A is some positive definite matrix. Posterior distributions for both the cases are derived by expanding zonal polynomial in terms of monomial symmetric functions and using some of the important formulae of James (1964, Ann. Math. Statist., 35, 475–501).

Admissibility of unbiased tests for a composite hypothesis with a restricted alternative

Abstract: This paper discusses α-admissiblility and d-admissiblity which are important concepts in studying the performance of statistical tests for composite hypotheses. A sufficient condition for α-admissibility is presented. When α=1/m, the Nomakuchi-Sakata test, which is uniformly more powerful than the likelihood ratio test for hypotheses min (θ1, θ1) = 0 versus min (θ1, θ1) > 0, is generalized for a class of distributions in an exponential family, and its unbiasedness and α-admissibility are shown. Finally, the case of α≠ 1/m is discussed in brief.

A geometric look at nuisance parameter effect of local powers in testing hypothesis

Abstract: This paper is concerned with the theory of testing hypothesis with composite null hypothesis or with nuisance parameters. The asymptotic be- haviour of the likelihood ratio and the associated test statistics are investigated. Under a class of local alternatives with local orthogonality relative to the nui- sance parameter vector~ a unique decomposition of local power is presented. The decomposition consists of two parts; one is the influence of nuisance pa- rameters and the other is the power corresponding to the simple case where the nuisance parameters are known. The decomposition formula is applied to some examples, including the gamma, Weibull and location-scale family.

A note on characterizations of multivariate stable distributions

Abstract: Several characterizations of multivariate stable distributions together with a characterization of multivariate normal distributions and multivariate stable distributions with Cauchy marginals are given. These are related to some standard characterizations of marcinkiewicz.

Estimation of a common mean of several univariate inverse gaussian populations

Abstract: The problem of estimating the common mean μ of k independent and univariate inverse Gaussian populations IG(μ, λ i ), i=1,..., k with unknown and unequal λ's is considered. The difficulty with the maximum likelihood estimator of μ is pointed out, and a natural estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] of μ along the lines of Graybill and Deal is proposed. Various finite sample properties and some decision-theoretic properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] are discussed.