A sequential design for a clinical trial with a linear prognostic factor

We study a randomized adaptive design to assign one of the L$L$ treatments to patients who arrive sequentially by means of an urn model. At each stage n$n$, a reward is distributed between treatments. The treatment applied is rewarded according to its response, 0?_Yn?1$0?Y_n?1$, and 1-_Yn$1-Y_n$ is distributed among the other treatments according to their performance until stage n-1$n-1$. Patients can be classified in K+1$K+1$ levels and we assume that the effect of this level in the response to the treatments is linear. We study the asymptotic behavior of the design when the ordinary least square estimators are used as a measure of performance until stage n-1$n-1$.     

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y$y$ on x$x$ given by a pair of mean and variance functions with a parameter vector θ$\theta$ to be estimated that also appears in the distribution of the regressor variable x$x$. The estimation of θ$\theta$ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y$y$ unbiased estimating functions. Of special interest is the case where the distribution of x$x$ depends only on a subvector α$\alpha$ of θ$\theta$, which may be considered a nuisance parameter. In general, α$\alpha$ must be estimated simultaneously together with the rest of θ$\theta$, but there are cases where α$\alpha$ can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.     

The use of an identity in Anderson for multivariate multiple testing

Consider the model where there are I$I$ independent multivariate normal treatment populations with p×1$p\times 1$ mean vectors _μi${\mathit{\mu }}_i$, i=1,…,I$i=1,\dots ,I$, and covariance matrix Σ$\Sigma$. Independently the (I+1)$(I+1)$st population corresponds to a control and it too is multivariate normal with mean vector _μI+1${\mathit{\mu }}_{I+1}$ and covariance matrix Σ$\Sigma$. Now consider the following two multiple testing problems.     

Characterizations based on record values and order statistics

Consider a sequence of independent and identically distributed random variables {_Xi,i?1}$\{X_i,i?1\}$ with a common absolutely continuous distribution function F  . Let _X1:n?_X2:n???_Xn:n$X_{1:n}?X_{2:n}???X_{n:n}$ be the order statistics of {_X1,_X2,…,_Xn}$\{X_1,X_2,\dots ,X_n\}$ and {_Yl,l?1}$\{Y_l,l?1\}$ be the sequence of record values generated by {_Xi,i?1}$\{X_i,i?1\}$. In this work, the conditional distribution of _Yl$Y_l$ given _Xn:n$X_{n:n}$ is established. Some characterizations of F   based on record values and _Xn:n$X_{n:n}$ are then given.     

Distribution of record statistics in a geometrically increasing population

The probability function and binomial moments of the number _Nn$N_n$ of (upper) records up to time (index) n in a geometrically increasing population are obtained in terms of the signless q-Stirling numbers of the first kind, with q   being the inverse of the proportion λ$\lambda$ of the geometric progression. Further, a strong law of large numbers and a central limit theorem for the sequence of random variables _Nn$N_n$, n=1,2,…,$n=1,2,\dots ,$ are deduced. As a corollary the probability function of the time _Tk$T_k$ of the kth record is also expressed in terms of the signless q  -Stirling numbers of the first kind. The mean of _Tk$T_k$ is obtained as a q  -series with terms of alternating sign. Finally, the probability function of the inter-record time _Wk=_Tk-_Tk-1$W_k=T_k-T_{k-1}$ is obtained as a sum of a finite number of terms of q  -numbers. The mean of _Wk$W_k$ is expressed by a q-series. As k   increases to infinity the distribution of _Wk$W_k$ converges to a geometric distribution with failure probability q. Additional properties of the q-Stirling numbers of the first kind, which facilitate the present study, are derived.     

Lattice and Schröder paths with periodic boundaries

We consider paths in the plane with (1,0$1,0$), (0,1$0,1$), and (a,b$a,b$)-steps that start at the origin, end at height n$n$, and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b/a$b/a$, then the ordinary generating function for the number of such paths ending at height n   is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power zⁿ$z^n$ is replaced by a power series of the form zⁿ_φn(z),$z^n{\phi }_n(z),$ where _φn(0)=1.${\phi }_n(0)=1.$ Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem.     

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.     

Optimal global rates of convergence for nonparametric regression with unbounded data

Estimation of regression functions from independent and identically distributed data is considered. The _L2$L_2$ error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates a boundedness assumption on the explanatory variable X$X$ is made besides smoothness assumptions on the regression function and moment conditions on the response variable Y$Y$. In this article we consider the kernel estimate and show that by replacing the boundedness assumption on X$X$ by a proper moment condition the same (optimal) rate of convergence can be shown as for bounded data. This answers Question 1 in Stone [1982. Optimal global rates of convergence for nonparametric regression. Ann. Statist., 10, 1040–1053].     

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.     

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.     

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.     

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.     

Estimation of extreme quantiles from heavy and light tailed distributions

In Gardes et al. (2011), a new family of distributions is introduced, depending on two parameters τ$\tau$ and θ$\theta$, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for θ$\theta$ and extreme quantiles are also proposed, but they both depend on the unknown parameter τ$\tau$, making them useless in practical situations. In this paper, we propose an estimator of τ$\tau$ which is independent of θ$\theta$. Plugging our estimator of τ$\tau$ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.     

United statistics,confidence quantiles,Bayesian statistics

A survey of research by Emanuel Parzen on how quantile functions provide elegant and applicable formulas that unify many statistical methods, especially frequentist and Bayesian confidence intervals and prediction distributions. Section 0: In honor of Ted Anderson's 90th birthday; Section 1: Quantile functions, endpoints of prediction intervals; Section 2: Extreme value limit distributions; Sections 3, 4: Confidence and prediction endpoint function: Uniform(0,θ)$(0,\theta )$, exponential; Sections: 5, 6: Confidence quantile and Bayesian inference normal parameters μ$\mu$, σ$\sigma$; Section 7: Two independent samples confidence quantiles; Section 8: Confidence quantiles for proportions, Wilson's formula. We propose ways that Bayesians and frequentists can be friends!     

Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1$q+1$ of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞$p\to \infty$, m=n-p+q→∞$m=n-p+q\to \infty$, and p/n→c∈(0,1)$p/n\to c\in (0,1)$, where n=N-q-1$n=N-q-1$. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)$(n,p,q)$.