Some locally optimal subset selection rules for comparison with a control

Let π₀,π₁,…,π_k be k+1 independent populations. For i=0,1,…,k,π_i has the densit f(x,θ_i), where the (unknown) parameter θ_i belongs to an interval of the real line. Our goal is to select from π₁,… π_k (experimental treatments) those populations, if any, that are better (suitably defined) than π₀ which is the control population. A locally optimal rule is derived in the class of rules for which Pr(π_i is selected)γ_i, i=1,…,k, when θ₀=θ₁=?=θ_k. The criterion used for local optimality amounts to maximizing the efficiency in a certain sense of the rule in picking out the superior populations for specific configurations of θ=(θ₀,…,θ_k) in a neighborhood of an equiparameter configuration. The general result is then applied to the following special cases: (a) normal means comparison — common known variance, (b) normal means comparison — common unknown variance, (c) gamma scale parameters comparison — known (unequal) shape parameters, and (d) comparison of regression slopes. In all these cases, the rule is obtained based on samples of unequal sizes.     

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.     

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.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.     

Some aspects of the theory of estimating equations

An estimating equation for a parameter θ, based on an observation ?, is an equation g(x,θ)=0 which can be solved for θ in terms of x. An estimating equation is unbiased if the funaction g has 0 mean for every θ. For the case when the form of the frequency function p(x,θ) is completely specified up to the unknown real parameter θ, the optimality of the m.1 equation $\text{?}\text{log}\text{p}\text{?θ}\text{=0}$ in the class of all unbiased estimating equations was established by Godambe (1960). In this paper we allow the form of the frequency function p to vary assuming that $\text{x=(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)?}\text{R}{}^{\mathrm{n}}$ and that under p, E(x_i)=θ. x₁,…, x_n are independent observations on a variate x, it is shown that among all the unbiased estimating equations for θ, $\text{x}\text{?}\text{?θ=0}$ is uniquely optimum up to a constant multiple.     

Minimizing the expected sample range

Let X_i be i.i.d. random variables with finite expectations, and θ_i arbitrary constants, i=1,…,n. Y_i=X_i+θ_i. The expected range of the Y's is R_n(θ₁,…,θ_n)=E(maxY_i-minY_i. It is shown that the expected range is minimized if and only if θ₁=?=θ_n. In the case where the X_i are independently and symmetrically distributed around the same constant, but not identically distributed, it is shown that θ₁=?=θ_n are not necessarily the only (θ₁,...,θ_n) minimizing R_n. Some lemmas which are applicable to more general problems of minimizing R_n are also given.     

Empirical Bayes testing for equivalence

For a fixed point θ₀ and a positive value c₀, this paper studies the problem of testing the hypotheses H₀:|θ−θ₀|≤c₀ against H₁:|θ−θ₀|>c₀ for the normal mean parameter θ using the empirical Bayes approach. With the accumulated past data, a monotone empirical Bayes test is constructed by mimicking the behavior of a monotone Bayes test. Such an empirical Bayes test is shown to be asymptotically optimal and its regret converges to zero at a rate (lnn)^2.5/n where n is the number of past data available, when the current testing problem is considered. A simulation study is also given, and the results show that the proposed empirical Bayes procedure has good performance for small to moderately large sample sizes. Our proposed method can be applied for testing close to a control problem or testing the therapeutic equivalence of one standard treatment compared to another in clinical trials.     

Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.     

Testing multiple slippages

A class of invariant Bayes rules is derived for testing homogeneity of k (≥2) different populations against (^k_t) slippage alternatives that some (unknown) subset of size t of the given populations has parameter larger than the remaining k-t, where t is a given integer between 1 and k-1. For a similar problem in nonparametric situations, locally best tests based on ranks are derived.     

Sequence-compound estimation in scale-exponential families and speed of convergence

This paper deals with a sequence-compound estimation. The component problem is the squared error loss estimation of θ?[a,b] based on an observation X whose p.d.f. is of the form $\text{u(x)c(θ)}\text{exp}\text{(?}\text{x}\text{θ}\text{)}$. For each $\text{0<t<}\text{1}\text{2}$ a class of sequence-compound estimators $\text{ψ}\text{?}\text{=}\text{ψ}\text{?}{}_1\text{,}\text{ψ}\text{?}{}_2\text{,…)}$ is exhibited whose compound risk (average of risks) up to stage n differs from the Bayes envelope (in the component problem) w.r.t. the empiric distribution G_n of the parameters involved up to stage n by a quantity of order O(n^?δt) for a δ>0. It is also shown that at any stage i the difference of the risk of ψ?_i and the risk of the Bayes response w.r.t. G_i?1 is O(i^?δt). Examples of the above type of families are given where δ is min{1,$\text{2a}\text{b}$} and t is arbitrarily close to $\text{1}\text{2}$. Here it may be worthwhile to mention that a rate $\text{O}\text{(n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$ or better has not yet been obtained even in a very special family of densities.     

An essentially complete class of multiple decision procedures

Let π₁,…, π_k represent k(?2) independent populations. The quality of the ith population π_i is characterized by a real-valued parameter θ_i, usually unknown. We define the best population in terms of a measure of separation between θ_i's. A selection of a subset containing the best population is called a correct selection (CS). We restrict attention to rules for which the size of the selected subset is controlled at a given point and the infimum of the probability of correct selection over the parameter space is maximized. The main theorem deals with construction of an essentially complete class of selection rules of the above type. Some classical subset selection rules are shown to belong to this class.     

Nonparametric statistical procedures for the changepoint problem

Let X₁,…,X_r?1,X_r,X_r+1,…,X_n be independent, continuous random variables such that X_i, i = 1,…,r, has distribution function F(x), and X_i, i = r+1,…,n, has distribution function F(x?Δ), with -∞ <Δ< ∞. When the integer r is unknown, this is refered to as a change point problem with at most one change. The unknown parameter Δ represents the magnitude of the change and r is called the changepoint. In this paper we present a general review discussion of several nonparametric approaches for making inferences about r and Δ.     

Consistency of Bayes estimators without the assumption that the model is correct

We examine a more general form of consistency which does not necessarily rely on the correct specification of the likelihood in the Bayesian setting, but we restrict the form of the likelihood to be in a minimal standard exponential family. First, we investigate the asymptotic behavior of the Bayes estimator of a parameter, and show that the Bayes estimator is consistent under the condition that the exponential family is full. However, we find that θ_i=θ_j and ∥θ_i−θ_j∥<ε, even for very small ε, behave differently, even in an asymptotic manner, when the model is not correct. We note that the distinction applies generally to Bayesian testing problems.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

Minimax estimation of a lower-bounded scale parameter of a gamma distribution for scale-invariant squared-error loss

Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θ_ρ|θ_ρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θ_ρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented.     

Robust designs and optimality of least squares for regression problems

Unbiased linear estimators are considered for the model

$\text{Y(}\text{x}{}_{\mathrm{i}}\text{)=θ}{}_0\text{+}\text{∑k}\text{j=1}\text{θ}{}_{\mathrm{j}}\text{x}{}_{\mathrm{ij}}\text{+ψ(}\text{x}{}_{\mathrm{i}}\text{)+ε}{}_{\mathrm{i}}\text{, i=1,2,…,n,}$

where ψ(x) is an unknown contamination. It is assumed that |ψ(x)|?φ(6x6) where φ is a convex function. Minimax analogues of Φ_p-optimality criteria are introduced. It is shown that, under certain (sufficient) conditions, the least squares estimators and corresponding designs are optimal in the class of all unbiased linear estimators and designs. It is also shown that, in the case when least squares estimators with symmetric design do not lead to an optimal solution, the relative efficiency of optimal least squares is not diminishing and has a uniform lower bound.     

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.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Comparison of rotatable designs for regression on balls,I (quadratic)

Designs for quadratic and cubic regression are considered when the possible choices of the controlable variable are points x=( x₁,x₂,…,x_q) in the q-dimensional. Full of radius R, B_q(R) ={x:Σ⁴_ix²_i?R²}. The designs that are optimum among rotatable designs with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria, including extrapolation. Additionally, the performance of a design optimum for one value of R, when it is implemented for a different value of R, is investigated. Some of the results are developed algebraically; others, numerically. For example, in quadratic regression the A-optimum design appears to be fairly robust in its efficiency, under variation of criterion.