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.     

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.     

A method of unconstrained multiple comparisons with the best

There are three types of multiple comparisons: all-pairwise multiple comparisons (MCA), multiple comparisons with the best (MCB), and multiple comparisons with a control (MCC). There are also three levels of multiple comparisons inference: confidence sets, subset comparisons, test of homogeneity. In current practice, MCA procedures dominate. In correct attempts at more efficient comparisons, in the form of employing lower level MCA procedures for higher level inference, account for the most frequent abuses in multiple comparisons. A better strategy is to choose the correct type of inference at the level of inference desired. In particular, very often the simulataneous comparisons of each treatment with the best of the other treatments (MCB) suffice. Hsu (1984b) gave simultaneous confidence intervals for θ_i ? max_j≠iθ_j having the simple form [? (Y_i ?max_j≠i Y_j ? C) (Y_i?max_j≠i Y_j + C)⁺]. Those intervals were constrained, sothat even if a treatment is inferred to be the best, no positive bound on how much it is better thatn the rest is given, a somewhat undesirable property. In this article it is shown that by employing a slightly larger critical value, the nonpositivity constraint on the lower bound is removed.     

On asymptotic optimality of bayes empirical bayes estimators

In an empirical Bayes decision problem, a prior distribution ? is placed on a one-dimensfonal family G of priors G_w, wεΩ, to produce a Bayes empirical Bayes estimator, The asymptotic optimaiity of the Bayes estimator is established when the support of ? is Ω and the marginal distributions H_w have monotone likelihood ratio and continuous Kullback-Leibler information number.     

Parameter Estimation for Bivariate Shock Models with Singular Distribution for Censored Data with Concomitant Order Statistics

When two‐component parallel systems are tested, the data consist of Type‐II censored data X_(i), i= 1, n, from one component, and their concomitants Y _[i] randomly censored at X_(r), the stopping time of the experiment. Marshall & Olkin's (1967) bivariate exponential distribution is used to illustrate statistical inference procedures developed for this data type. Although this data type is motivated practically, the likelihood is complicated, and maximum likelihood estimation is difficult, especially in the case where the parameter space is a non‐open set. An iterative algorithm is proposed for finding maximum likelihood estimates. This article derives several properties of the maximum likelihood estimator (MLE) including existence, uniqueness, strong consistency and asymptotic distribution. It also develops an alternative estimation method with closed‐form expressions based on marginal distributions, and derives its asymptotic properties. Compared with variances of the MLEs in the finite and large sample situations, the alternative estimator performs very well, especially when the correlation between X and Y is small.     

Estimation of the scale parameter of the half-logistic distribution with multiply type II censored sample

In this paper, we consider the maximum likelihood and Bayes estimation of the scale parameter of the half-logistic distribution based on a multiply type II censored sample. However, the maximum likelihood estimator(MLE) and Bayes estimator do not exist in an explicit form for the scale parameter. We consider a simple method of deriving an explicit estimator by approximating the likelihood function and discuss the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney & Kadane, 1986) is used to obtain the Bayes estimator. In order to compare the MLE, approximate MLE and Bayes estimates of the scale parameter, Monte Carlo simulation is used.     

On the bayes risk incurred by using asymptotic shapes

Let X₁ X₂…denote Independent and Identically distributed random vectors whose common distributions form a multiparameter exponential family, and consider the problem of sequentially testing separated hypotheses. It is known that the sequential procedure which continues sampling until the likelihood ratio statistic for testing one of the hypotheses exceeds a given level approximates the optimal Bayesian procedure, under general conditions on the loss function and prior distribution. Here we ask whether the approximate procedure is Bayes risk efficient--that is, whether the ratio of the Bayes risk of the approximate procedure to the Bayes risk of the optimal procedure approaches one as the cost of samping approaches zero. We show that the answer depends on the choice of certain parameters in the approximation and the dimensions of the hypotheses.     

Exponential and polynomial tailbounds for change-point estimators

Let X_1n,…,X_nn be independent random elements with an unknown change point θ∈(0,1), that is X_in has a distribution ν₁ or ν₂, respectively, according to i⩽[nθ] or i>[nθ]. We propose an estimator θ_n of θ, which is defined as the maximizer of a weighted empirical process on (0,1). Finding upper bounds of polynomial and exponential type for the tails of nθ_n−[nθ], we are able to derive rates of almost sure convergence, of distributional convergence, of L_p-convergence and of convergence in the Ky-Fan- and in the Prokhorov-metric.     

A note on the asymptotic efficiency of the sobel–wald test

Let X₁,X₂, … be iid random variables with the pdf f(x,θ)=exp(θx?b(θ)) relative to a σ-finite measure μ, and consider the problem of deciding among three simple hypotheses H_i:θ=θ_i (1?i?3) subject to P(acceptH_i|θ_i)=1?α (1?i?3). A procedure similar to Sobel–Wald procedure is discussed and its asymptotic efficiency as compared with the best nonsequential test is obtained by finding the limit lim_a→0(E_iN(a)/n(a)), where N (a) is the stopping time of the proposed procedure and n(a) is the sample size of the best non-sequential test. It is shown that the same asymptotic limit holds for the original Sobel–Wald procedure. Specializing to N(θ,1) distribution it is found that lim_a→0$\text{(E}{}_{\mathrm{i}}\text{N(α)/n(α))=}\text{1}\text{4}$ (i=1,2) and lim_a→0 (E₃N(α)n(α))=δ²₁/4δ, where δ_i=(θ_i+1?θ_i) with 0<δ₁?δ₂. Also, the asymptotic efficiency evaluated when the X's have an exponential distribution.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Bootstrap test of goodness of fit to a linear model when errors are correlated

Given the regression model Y_i = m(x_i) +ε_i (x_i ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {ε_i} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: H_o : m ε {mθ(.) = A^t(.)θ : θ ε ? ? R^q} with A a functional from R to R^q. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d²([mcirc]_n, A^t(.)θ_n), between [mcirc]_n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in H_o that is closest to m_n in terms of this distance. The consistency of the bootstrap distribution of D and θ_n is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d².     

Small-sample performance of Bernoulli two-armed bandit Bayesian strategies

In this paper we examine the small-sample performance of a number of strategies for Bernoulli two-armed bandit problems with independent arms. We first investigate strategies based on a one-armed bandit threshold value (an index analogous to the ‘Gittins index’) and on upper confidence bounds for θ_i. Using backward induction and the Bayesian viewpoint, we observe that these strategies improve on the myopic strategy and get much closer to optimal in terms of total expected reward, even though for very small samples, the myopic worth itself is already close to optimal. Second, we find that the myopic strategy and the strategy based on the one-armed threshold value dominate the Bayesian optimal strategy over a region in the parameter space that can have large probability under the assumed prior. Finally, through examples we show how this has an impact on robustness: small specifications of the prior can lead to the myopic strategy performing better than the optimal strategy in terms of Bayes worth.     

Subset selection based on likelihood from uniform and related popupulations

Let π_i (i=1,2,…, k) be charceterized by the uniform distribution on (a_i;b_i), where exactly one of a_i and b_i is unknown. With unequal sample sizes, suppose that from the k (>=2) given populations, we wish to select a random-size subset containing the one with the smllest value of θ_i= b_i - a_i. RuleR_i selects π if a likelihood-based k-dimensional confidence region for the unknown (θ₁,… θ_k) contains at least one point having θ_i as its smallest component. A second rule, R , is derived through a likelihood ratio and turns out to be that of Barr and prabhu whenthe sample sizes are equal. Numerical comparisons are made. The results apply to the larger class of densities g ( z ; θ_i) =M(z)Q(θ_i) if a(θ_i) < z <b(θ_i). Extensions to the cases when both a_i and b_i are unknown and when θ_j isof interest are indicated. 1<=j<=k     

Estimation of the scale parameter of the half-logistic distribution under progressively type II censored sample

Based on a progressively type II censored sample, the maximum likelihood and Bayes estimators of the scale parameter of the half-logistic distribution are derived. However, since the maximum likelihood estimator (MLE) and Bayes estimator do not exist in an explicit form for the scale parameter, we consider a simple method of deriving an explicit estimator by approximating the likelihood function and derive the asymptotic variances of MLE and approximate MLE. Also, an approximation based on the Laplace approximation (Tierney and Kadane in J Am Stat Assoc 81:82–86, 1986) and importance sampling methods are used for obtaining the Bayes estimator. In order to compare the performance of the MLE, approximate MLE and Bayes estimates of the scale parameter, we use Monte Carlo simulation.     

Empirical bayes rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

A Bayes-type estimator for the Bradley-Terry model for paired comparison

A Bayes-type estimator is proposed for the worth parameter π_i and for the treatment effect parameter ln π_i in the Bradley-Terry Model for paired comparison. In contrast to current Bayes estimators which require iterative numberical calculations, this estimator has a closed form expression. This estimation technique is also extended to obtain estimators for the Luce Multiple Comparison Model. An application of this technique to a 2³ factorial experiment with paired comparisons is presented.     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

Simultaneous estimation of several CDF’s: homogeneity constraint

Let {x_ij(1 ? j ? n_i)|i = 1, 2, …, k} be k independent samples of size n_j from respective distributions of functions F_j(x)(1 ? j ? k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ? 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.     

Central limit theorems for LS estimators in the EV regression model with dependent measurements

In this paper, we consider the simple linear errors-in-variables (EV) regression models: η_i=θ+βx_i+ε_i,ξ_i=x_i+δ_i,1≤i≤n, where θ,β,x₁,x₂,… are unknown constants (parameters), (ε₁,δ₁),(ε₂,δ₂),… are errors and ξ_i,η_i,i=1,2,… are observable. The asymptotic normality for the least square (LS) estimators of the unknown parameters β and θ in the model are established under the assumptions that the errors are m-dependent, martingale differences, ?-mixing, ρ-mixing and α-mixing.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.