Admissible treatment rules for a risk-averse planner with experimental data on an innovation

Consider a planner choosing treatments for observationally identical persons who vary in their response to treatment. There are two treatments with binary outcomes. One is a status quo with known population success rate. The other is an innovation for which the data are the outcomes of an experiment. Karlin and Rubin [1956. The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27, 272–299] assumed that the objective is to maximize the population success rate and showed that the admissible rules are the KR-monotone   rules. These assign everyone to the status quo if the number of experimental successes is below a specified threshold and everyone to the innovation if experimental success exceeds the threshold. We assume that the objective is to maximize a concave-monotone function f(·)$f(·)$ of the success rate and show that admissibility depends on the curvature of f(·)$f(·)$. Let a fractional monotone   rule be one where the fraction of persons assigned to the innovation weakly increases with the number of experimental successes. We show that the class of fractional monotone rules is complete if f(·)$f(·)$ is concave and strictly monotone. Define an M-step monotone rule   to be a fractional monotone rule with an interior fractional treatment assignment for no more than M$M$ consecutive values of the number of experimental successes. The M$M$-step monotone rules form a complete class if f(·)$f(·)$ is differentiable and has sufficiently weak curvature. Bayes rules and the minimax-regret rule depend on the curvature of the welfare function.     

Goodness-of-fit tests for parametric regression with selection biased data

Consider the nonparametric location-scale regression model Y=m(X)+σ(X)ε$Y=m(X)+\sigma (X)\epsilon$, where the error ε$\epsilon$ is independent of the covariate X$X$, and m$m$ and σ$\sigma$ are smooth but unknown functions. The pair (X,Y)$(X,Y)$ is allowed to be subject to selection bias. We construct tests for the hypothesis that m(·)$m(·)$ belongs to some parametric family of regression functions. The proposed tests compare the nonparametric maximum likelihood estimator (NPMLE) based on the residuals obtained under the assumed parametric model, with the NPMLE based on the residuals obtained without using the parametric model assumption. The asymptotic distribution of the test statistics is obtained. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, the finite sample performance of the proposed tests is studied in a simulation study, and the developed tests are applied on environmental data.     

Representing a function of two variables as a superposition over angles of ridge functions with a given shape via linear programming

A ridge function with shape function g   in the horizontal direction is a function of the form g(x)h(y,0)$g(x)h(y,0)$. Along each horizontal line it has the shape g(x)$g(x)$, multiplied by a function h(y,0)$h(y,0)$ which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g   in the vertical direction has the form g(y)h(x,π/2)$g(y)h(x,\pi /2)$. For a given shape function g it may or may not be possible to represent an arbitrary   function f(x,y)$f(x,y)$ as a superposition over all angles of a ridge function with shape g   in each direction, where h=_hf=_hf,g$h=h_f=h_{f,g}$ depends on the functions f and g   and also on the direction, θ:h=_hf,g(·,θ)$\theta :h=h_{f,g}(·,\theta )$. We show that if g   is Gaussian centered at zero then this is always possible and we give the function _hf,g$h_{f,g}$ for a given f(x,y)$f(x,y)$. For highpass or for odd shapes g  , we show it is impossible to represent an arbitrary f(x,y)$f(x,y)$, i.e. in general there is no _hf,g$h_{f,g}$. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y)$f(x,y)$ is unknown and we find it by inverting the projections of f  ; here, f(x,y)$f(x,y)$ is known, g(z)$g(z)$ is known, and _hf(·,θ)=_hf,g(·,θ)$h_f(·,\theta )=h_{f,g}(·,\theta )$ is the unknown.     

Estimation of smooth regression functions in monotone response models

We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z,X)$(Z,X)$, where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z   is distributed according to the density p(·,ψ(Z))$p(·,\psi (Z))$, where p(·,θ)$p(·,\theta )$ is a one-parameter exponential family. The function ψ$\psi$ is a smooth monotone function. Under this formulation, the regression function E(X|Z)$E(X|Z)$ is monotone in the co-variate Z   (and can be expressed as a one–one function of ψ$\psi$); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ$\psi$ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ$\psi$ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.     

Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g  -matrix. This paper deals with the construction of D-optimal g(n,1)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

A marked point process perspective in fitting spatial point process models

This paper discusses a new perspective in fitting spatial point process models. Specifically the spatial point process of interest is treated as a marked point process where at each observed event x$\mathbf{x}$ a stochastic process M(x;t)$M(\mathbf{x};t)$, 0<t<r$0<t<r$, is defined. Each mark process M(x;t)$M(\mathbf{x};t)$ is compared with its expected value, say F(t;θ)$F(t;\theta )$, to produce a discrepancy measure at x$\mathbf{x}$, where θ$\theta$ is a set of unknown parameters. All individual discrepancy measures are combined to define an overall measure which will then be minimized to estimate the unknown parameters. The proposed approach can be easily applied to data with sample size commonly encountered in practice. Simulations and an application to a real data example demonstrate the efficacy of the proposed approach.     

Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA(N,k,s,t)$\mathrm{OA}(N,k,s,t)$'s is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of the system of equations. A branch-and-cut algorithm developed by Margot [2002. Pruning by isomorphism in branch-and-cut. Math. Programming Ser. A 94, 71–90; 2003a. Exploiting orbits in symmetric ILP. Math. Programming Ser. B 98, 3–21; 2003b. Small covering designs by branch-and-cut. Math. Programming Ser. B 94, 207–220; 2007. Symmetric ILP: coloring and small integers. Discrete Optim., 4, 40–62] for solving integer programming problems with large symmetry groups is used to find all non-isomorphic OA(24,7,2,2)$\mathrm{OA}(24,7,2,2)$'s, OA(24,k,2,3)$\mathrm{OA}(24,k,2,3)$'s for 6?k?11$6?k?11$, OA(32,k,2,3)$\mathrm{OA}(32,k,2,3)$'s for 6?k?11$6?k?11$, OA(40,k,2,3)$\mathrm{OA}(40,k,2,3)$'s for 6?k?10$6?k?10$, OA(48,k,2,3)$\mathrm{OA}(48,k,2,3)$'s for 6?k?8$6?k?8$, OA(56,k,2,3)$\mathrm{OA}(56,k,2,3)$'s, OA(80,k,2,4)$\mathrm{OA}(80,k,2,4)$'s, OA(112,k,2,4)$\mathrm{OA}(112,k,2,4)$'s, for k=6,7$k=6,7$, OA(64,k,2,4)$\mathrm{OA}(64,k,2,4)$'s, OA(96,k,2,4)$\mathrm{OA}(96,k,2,4)$'s for k=7,8$k=7,8$, and OA(144,k,2,4)$\mathrm{OA}(144,k,2,4)$'s for k=8,9$k=8,9$. Further applications to classifying covering arrays with the minimum number of runs and packing arrays with the maximum number of runs are presented.     

Analysis of one-way layout of count data in the presence of over or under dispersion

This article develops test statistics for the homogeneity of the means of several treatment groups of count data in the presence of over-dispersion or under-dispersion when there is no likelihood available. The C(α)$C(\alpha )$ or score type tests based on the models that are specified by only the first two moments of the counts are obtained using quasi-likelihood, extended quasi-likelihood, and double extended quasi-likelihood. Monte Carlo simulations are then used to study the comparative behavior of these C(α)$C(\alpha )$ statistics compared to the C(α)$C(\alpha )$ statistic based on a parametric model, namely, the negative binomial model, in terms of the following: size; power; robustness for departures from the data distribution as well as dispersion homogeneity. These simulations demonstrate that the C(α)$C(\alpha )$ statistic based on the double extended quasi-likelihood holds the nominal size at the 5% level well in all data situations, and it shows some edge in power over the other statistics, and, in particular, it performs much better than the commonly used statistic based on the quasi-likelihood. This C(α)$C(\alpha )$ statistic also shows robustness for moderate heterogeneity due to dispersion. Finally, applications to ecological, toxicological and biological data are given.     

On the joint distribution of placement statistics under progressive censoring and applications to precedence test

In reliability and survival analysis, comparison of two or more populations is an important problem. For example, while comparing a treatment group with a control group, one may be interested in determining whether the observations in the treatment group have a longer lifetime than those from the control group, that is, whether the treatment is effective or not. In such a study, it would be extremely valuable to make a decision based on early failures. In this paper, we consider independent progressively Type-II censored random samples from two populations with cumulative distribution function's (cdf) F(·)$F(·)$ and G(·)$G(·)$ respectively, and discuss a precedence test for testing the equality of the two distributions based on placements. For this purpose, we derive the joint distribution of the first l   placement statistics from the progressively censored sample from F(·)$F(·)$. We then derive the exact null distribution of the precedence test statistic which is simply the sum of the placements. We provide the rejection regions for fixed levels of significance and various sample sizes and different progressive censoring schemes.     

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) offer a potentially useful way to investigate many factors with only few experiments in the preliminary stages of experimentation. This paper explores how to construct E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs using k-cyclic generators. The necessary and sufficient conditions for the existence of mixed-level k-circulant SSDs with the equal occurrence property are provided. Properties of the mixed-level k  -circulant SSDs are investigated, in particular, the sufficient condition under which the generator vector produces an E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal SSD is obtained. Moreover, many new E(_fNOD)$E(f_{\mathrm{NOD}})$-optimal mixed-level SSDs are constructed and listed. The method here generalizes the one proposed by Liu and Dean [2004. k$k$-circulant supersaturated designs. Technometrics 46, 32–43] for two-level SSDs and the one due to Georgiou and Koukouvinos [2006. Multi-level k-circulant supersaturated designs. Metrika 64, 209–220] for the multi-level case.     

On asymptotic properties of the parameter estimator for a type of SPDE

In this paper, we study a random field _U?(t,x)$U_{?}(t,x)$ governed by some type of stochastic partial differential equations with an unknown parameter θ$\theta$ and a small noise ?$?$. We construct an estimator of θ$\theta$ based on the continuous observation of N   Fourier coefficients of _U?(t,x)$U_{?}(t,x)$, and prove the strong convergence and asymptotic normality of the estimator when the noise ?$?$ tends to zero.     

Super-simple cyclic designs with small values

Super-simple cyclic designs are useful on constructing codes and designs such as superimposed codes, perfect hash families and optical orthogonal codes with index two. In this paper, we show that there exists a super-simple cyclic (v,4,λ)$(v,4,\lambda )$ for 7?v?41$7?v?41$ and all admissible λ$\lambda$ with two definite exceptions of (v,λ)=(9,3),(13,5)$(v,\lambda )=(9,3),(13,5)$ and one possible exception of (v,λ)=(39,18)$(v,\lambda )=(39,18)$. Some useful algorithms are explained for computer search and new designs are displayed.     

Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(_β1,…,_βp)$\beta =({\beta }_1,\dots ,{\beta }_p)$ and independent and identically N(0,σ²)$N(0,{\sigma }^2)$ distributed errors. Suppose that the parameter of interest is θ=a^Tβ$\theta =a^T\beta$ where a$a$ is a specified vector. Define the parameter τ=c^Tβ-t$\tau =c^T\beta -t$ where the vector c$c$ and the number t$t$ are specified and a$a$ and c$c$ are linearly independent. Also suppose that we have uncertain prior information that τ=0$\tau =0$. We present a new frequentist 1-α$1-\alpha$ confidence interval for θ$\theta$ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α$1-\alpha$ confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0$\tau =0$. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ$\tau$ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2$2\times 2$ factorial experiment with 20 replicates. Suppose that the parameter of interest θ$\theta$ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ$\theta$ that utilizes this prior information.     

Skew Dyck paths

In this paper we study the class S$\mathbb{S}$ of skew Dyck paths, i.e. of those lattice paths that are in the first quadrant, begin at the origin, end on the x-axis, consist of up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, and such that up steps never overlap with left steps. In particular, we show that these paths are equinumerous with several other combinatorial objects, we describe some involutions on this class, and finally we consider several statistics on S$\mathbb{S}$.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Optimally robust credible sets with a class of priors

We determine a credible set A   that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ$\Gamma$ of the starting prior _π0(θ)${\pi }_0(\theta )$. Among the class of sets with credibility γ$\gamma$ under _π0${\pi }_0$, the “optimally robust” set will be the one which maximizes the minimum probability of including θ$\theta$ as the prior varies over Γ$\Gamma$. This procedure is also Γ-minimax$\Gamma \text{-minimax}$ with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ$\Gamma$, the ε-contamination$\epsilon \text{-contamination}$ class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective.     

Minimum Hellinger distance estimation in a nonparametric mixture model

In this paper, we investigate the estimation problem of the mixture proportion λ$\lambda$ in a nonparametric mixture model of the form λF(x)+(1-λ)G(x)$\lambda F(x)+(1-\lambda )G(x)$ using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G   as well as from the mixture distribution λF+(1-λ)G$\lambda F+(1-\lambda )G$ are available. We construct a minimum Hellinger distance estimator of λ$\lambda$ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ$\lambda$. Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.