Dropping observations without affecting posterior and predictive distributions

Suppose in a distribution problem, the sample information W is split into two pieces W ₁ and W ₂, and the parameters involved are split into two sets, π containing the parameters of interest, and θ containing nuisance parameters. It is shown that, under certain conditions, the posterior distribution of π does not depend on the data W ₂, which can thus be ignored. This also has consequences for the predictive distribution of future (or missing) observations. In fact, under similar conditions, the predictive distributions using W or just W ₁ are identical.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Nuisance Parameters and the Use of Exploratory Graphical Methods in a Bayesian Analysis

Consider the problem of inference about a parameter θ in the presence of a nuisance parameter v. In a Bayesian framework, a number of posterior distributions may be of interest, including the joint posterior of (θ, ν), the marginal posterior of θ, and the posterior of θ conditional on different values of ν. The interpretation of these various posteriors is greatly simplified if a transformation (θ, h(θ, ν)) can be found so that θ and h(θ, v) are approximately independent. In this article, we consider a graphical method for finding this independence transformation, motivated by techniques from exploratory data analysis. Some simple examples of the use of this method are given and some of the implications of this approximate independence in a Bayesian analysis are discussed.     

A Bayesian inference of P(λ1<λ2) for two Poisson parameters

The statistical inference drawn from the difference between two independent Poisson parameters is often discussed in the medical literature. However, such discussions are usually based on the frequentist viewpoint rather than the Bayesian viewpoint. Here, we propose an index θ=P(λ_{1, post}<λ_{2, post}), where λ_{1, post} and λ_{2, post} denote Poisson parameters following posterior density. We provide an exact and an approximate expression for calculating θ using the conjugate gamma prior and compare the probabilities obtained using the approximate and the exact expressions. Moreover, we also show a relation between θ and the p-value. We also highlight the significance of θ by applying it to the result of actual clinical trials. Our findings suggest that θ may provide useful information in a clinical trial.     

some properties of posterior pitman closeness

This article presents some results on a Bayesian notion of Pitman Closeness, defined in Ghosh and Sen (1991) and called Posterior Pitman Closeness (PPC). Their criterion avoids some of the drawbacks of the well-known (frequentist) Pitman closeness criterion, as introduced by Pitman (1937). It is shown that, if two estimators have the same posterior distribution of the distance from θ, the posterior distribution of θ has to be symmetric. This implies, in particular, that the estimators are Posterior Pitman equivalent. It is also shown that the PPC criterion does not suffer from another paradoxical property illustrated by Blyth and Pathak (1985) - that of an estimator δ₁ being stochastically closer to a parameter θ than another estimator δ₂ and yet being Pitman closer to θ than δ₁. It turns out that, if δ₁ is stochastically closer to θ than δ₂, conditional on x, then it is also Posterior Pitman closer.

We show that the original multivariate concept of PPC is no longer transitive. We provide necessary and sufficient conditions for a Posterior Pitman closest estimator to exist, thus generalizing Theorems 2 and 3 of Ghosh and Sen (1991). We show that a Posterior Pitman closest estimator does not always exist in several dimensions.     

Asymptotics of Likelihood Ratio Tests for General One-sided Hypotheses in the Two-sample Normal Model

We consider the general one-sided hypotheses testing problem expressed as H₀: θ₁ ? h(θ₂) versus H₁: θ₁ < h(θ₂), where h( · ) is not necessary differentiable. The values of the right and the left differential coefficients, h′_?( · ) and h′₊( · ), at nondifferentiable points play an essential role in constructing the appropriate testing procedures with asymptotic size α on the basis of the likelihood ratio principle. The likelihood ratio testing procedure is related to an intersection–union testing procedure when h′_?(θ₂) ? h′₊(θ₂) for all θ₂, and to a union–intersection testing procedure when there exists a θ₂ such that h′_?(θ₂) < h′₊(θ₂).     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

ASYMPTOTIC AND SMALL SAMPLE STATISTICAL PROPERTIES OF RANDOM FRAILTY VARIANCE ESTIMATES FOR SHARED GAMMA FRAILTY MODELS

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean 1 and variance θ. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm, faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value θ₀ of θ is zero, the estimates of θ are asymptotically distributed as a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis. When θ₀ > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of θ are approximately distributed as an x ? (100 ? x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ₀ > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.     

On the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.     

An asymptotic sequential test based on confidence sequences

In the context of a translation parameter family of distributions F₀(x) = F(x-θ) an asymptotic sequential test of H₀: θ ≤ -△ versus H₁: θ ≥ △ developed. The test is based on confidence sequences. In the special case where F is a specified normal distribution the proposed test is uniformly at least as efficient (in the sense of Rechanter (1960)) as the Wald sequention probibilty ratio test.     

Estimators for the Drift of Subfractional Brownian Motion

In this paper, we consider, using technique based on Girsanov theorem, the problem of efficient estimation for the drift of subfractional Brownian motion S^H ? (S^H_t)_{t ∈ [0, T]}. We also construct a class of biased estimators of James-Stein type which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.     

Some Pitman closeness properties pertinent to symmetric populations

In this paper, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ?₁ and θ?₂ and obtain necessary and sufficient conditions for θ?₁ to be Pitman closer to the common median θ than θ?₂. We then establish some properties in the context of estimation under the Pitman closeness criterion. We define Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other independent and symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

Analysing truncated data with semiparametric transformation models

Left-truncation often arises when patient information, such as time of diagnosis, is gathered retrospectively. In some cases, the distribution function, say G(x), of left-truncated variables can be parameterized as G(x; θ), where θ∈Θ?R^q and θ is a q-dimensional vector. Under semiparametric transformation models, we demonstrated that the approach of Chen et al. (Semiparametric analysis of transformation models with censored data. Biometrika. 2002;89:659–668) can be used to analyse this type of data. The asymptotic properties of the proposed estimators are derived. A simulation study is conducted to investigate the performance of the proposed estimators.     

THE LIMITING FORM OF THE NON-CENTRAL WISHART DISTRIBUTION1,2

Let S (p×p) have a Wishart distribution -with v degrees of freedom and non-centrality matrix θ= [θ_jK] (p×p). Define θ₀= min {| θ_jk |}, let θ₀→∞, and suppose that | θ_jK | = 0(θ_o). Then the limiting form of the standardized non-central distribution, as θ while n? remains fixed, is a multivariate Gaussian distribution. This result in turn is used to obtain known asymptotic properties of multivariate chi-square and Rayleigh distributions under somewhat weaker conditions.     

Tests based on equivariant estimators

Let ?⁽¹⁾ and ?⁽²⁾ be location-equivariant estimators of an unknown location parameter μ. It is shown that the test for H₀: μ ≤ μ₀ versus H_A : μ > μ₀ that rejects H₀ if ?⁽¹⁾ is large is uniformly more powerful than the one that rejects H₀ if ?⁽²⁾ is large if and only if ?⁽²⁾ is “more dispersed” than ?⁽¹⁾. A similar result is obtained for tests on scale using the star-shaped ordering. Examples are given.     

Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X₁,X₂, … ,X_p)^τ be a random vector, the X₁ being independently distributed as N(θ₁,σ²) random variables (σ² known), while the θ₁ are thought to be exchangeable, modelled as i.i.d, N(μ,τ²). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ² is assumed to be independent of µ having a distribution g(τ²) lying in a certain class of distributions g. For several g's, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ₁ as g ranges over g.     

Tests for the mean of a multivariate normal population

We study the problem of testing: H₀ : μ ∈ P against H₁ : μ ? P, based on a random sample of N observations from a p-dimensional normal distribution N_p(μ, Σ) with Σ > 0 and P a closed convex positively homogeneous set. We develop the likelihood-ratio test (LRT) for this problem. We show that the union-intersection principle leads to a test equivalent to the LRT. It also gives a large class of tests which are shown to be admissible by Stein's theorem (1956). Finally, we give the α-level cutoff points for the LRT.     

Teaching Decision Theory Proof Strategies Using a Crowdsourcing Problem

Teaching how to derive minimax decision rules can be challenging because of the lack of examples that are simple enough to be used in the classroom. Motivated by this challenge, we provide a new example that illustrates the use of standard techniques in the derivation of optimal decision rules under the Bayes and minimax approaches. We discuss how to predict the value of an unknown quantity, θ ∈ {0, 1}, given the opinions of n experts. An important example of such crowdsourcing problem occurs in modern cosmology, where θ indicates whether a given galaxy is merging or not, and Y₁, …, Y_n are the opinions from n astronomers regarding θ. We use the obtained prediction rules to discuss advantages and disadvantages of the Bayes and minimax approaches to decision theory. The material presented here is intended to be taught to first-year graduate students.