The Loss of Efficiency Estimating Linear Functions under Restrictions

This article is motivated by the problem of estimating contrast in a one-way ANOVA model with restrictions in the parameter vector. We prove that when the restrictions are given by a tree order or a simple order the MLE of some contrast has greater MSE than the unrestricted estimator. A similar behaviour of the MLE is exhibited in a general restricted setting given by a multivariate normal distribution with mean vector constrained to belong to a circular cone. The approach we use focuses on the central direction of the cones. These directions appear to have the greater MSE when the dimension of the restricted cone is big enough.     

Changepoint Analysis as a Method for Isotonic Inference

Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric inputoutput analysis. The slope change and inflection point models are introduced correspondingly as the corners of the polyhedral cones defined by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justified for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate non-linear transformation, leads to an exact and very efficient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.     

Generalized orthogonal contrast tests for homogeneity of ordered means

Several procedures have been proposed for testing equality of ordered means. The best-known of these is the likelihood-ratio test introduced by Bartholomew, which possesses generally superior power characteristics to those of its competitors. Difficulties in implementing this test have led to the development of alternative approaches, such as tests based on single and multiple contrasts. Some recent approaches have utilized approximations to the polyhedral cone defining the restricted parameter space, including those of Akkerboom (circular cone) and Mudholkar & McDermott (orthant). This article proposes a class of tests based on an improved orthant approximation to the polyhedral cone. These tests may be viewed as generalizations of the orthogonal contrast test proposed by Mukerjee, Robertson & Wright. Studies of the power functions of several competing tests indicate that the generalized orthogonal contrast tests are effective alternatives to the likelihood-ratio test, especially when the latter is difficult to implement.     

Upper Bound for p-Value of the Test of Multivariate Normal Ordered Mean Vectors Against all Alternatives

Consider the problem of testing the isotonic of several p-variate normal mean vectors against all alternatives. It is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown. In the present paper, a test statistic is proposed for this problem of testing. A reformulation of the test statistic is given based on the orthogonal projections on the closed convex cones and then the upper bound for p-value of the test statistic is computed.     

The role of reversals in order‐restricted inference

A statistical model is said to be an order‐restricted statistical model when its parameter takes its values in a closed convex cone C of the Euclidean space. In recent years, order‐restricted likelihood ratio tests and maximum likelihood estimators have been criticized on the grounds that they may violate a cone order monotonicity (COM) property, and hence reverse the cone order induced by C. The authors argue here that these reversals occur only in the case that C is an obtuse cone, and that in this case COM is an inappropriate requirement for likelihood‐based estimates and tests. They conclude that these procedures thus remain perfectly reasonable procedures for order‐restricted inference.     

Test of equality of normal means in the absence of independent estimator of variance

Let X_l,…,X_n be normally and independently distributed with means θ_l,…,θ_nand a cornmorl variance. Thus there are n observations and n+i unknwon parameters. A test of the null hypothesis that, the θ_i's are all zero and the alternative that the vector (θ_l,…,θ_n) lies in a convex cone with its vertex a.t the origin is connsidered in this paper. It is shown that under a mild condition the likelihood ratio test is possible. The ordinary one sided t - test belongs to the class of tests considered in this paper. The hypothesis of equality of means against the simple order alternative can be tested in certain cases .     

An Exact Algorithm for Projection onto a Polyhedral Cone

Projection is an operation widely used in restricted statistical inferences. A polyhedral cone restriction includes many interesting problems in linear regression, order restricted inferences etc. This paper proposes an exact algorithm for the projection of a vector onto a polyhedral cone, and presents an application to second order polynomial regression subject to a non-negative, isotonic restriction.     

Directional monotonicity properties of the power functions of likelihood ratio tests for cone-restricted hypotheses of normal means

In some cone-restricted testing problems, it is known that the power function of the likelihood ratio test has a certain monotonicity property with respect to the direction of the alternatives. In this paper, we establish a unified result concerning such directional monotonicity by using an argument based on a cone ordering and a related probability inequality. From this result, we show that the power function is unimodal in direction when the cone is suitably symmetric.     

Maximal Expectations of Extreme Order Statistics from Increasing Density and Failure Rate Populations

We determine sharp bounds for the expectations of the first and second and last order statistics for the i.i.d. samples from the increasing density and failure rate populations. These bounds are expressed by means of the population mean and standard deviation. The results are determined by use of the projections of elements of Hilbert spaces onto convex cones.     

Flocks and partial flocks of quadrics: a survey

If O is an ovoid of PG(3,q), then a partition of all but two points of O into q−1 disjoint ovals is called a flock of O. A partition of a nonsingular hyperbolic quadric Q⁺(3,q) into q+1 disjoint irreducible conics is called a flock of Q⁺(3,q). Further, if O is either an oval or a hyperoval of PG(2,q) and if K is the cone with vertex a point x of PG(3,q)⧹PG(2,q) and base O, then a partition of K⧹{x} into q disjoint ovals or hyperovals in the respective cases is called a flock of K. The theory of flocks has applications to projective planes, generalized quadrangles, hyperovals, inversive planes; using flocks new translation planes, hyperovals and generalized quadrangles were discovered. Let Q be an elliptic quadric, a hyperbolic quadric or a quadratic cone of PG(3,q). A partial flock of Q is a set P consisting of β disjoint irreducible conics of Q. Partial flocks which are no flocks, have applications to k-arcs of PG(2,q), to translation planes and to partial line spreads of PG(3,q). Recently, the definition and many properties of flocks of quadratic cones in PG(3,q) were generalized to partial flocks of quadratic cones with vertex a point in PG(n,q), for n⩾3 odd.     

On cone representations of translation planes

Let F be a field of order q. It is known that an orthogonal array of the same order q has rank n over F if and only if it is represented as a cone cut by hyperplanes in n-dimensional space over F. Here we show that translation planes have a cone representation in (n + 1)-dimensional space over F, where n is the dimension of the plane over its kernel. If the plane is a semifield plane then the representation takes a particularly nice form. Rank 3 representations of Moulton planes are also briefly discussed.     

Simultaneous estimation of restricted location parameters based on permutation and sign-change

The problem of simultaneously estimating location parameters is addressed, where the vector of location parameters belongs to a polyhedral cone including simple order, tree order and positive orthant restrictions and so forth. This paper proposes modified estimators based on orthogonal transformations such as sign-change and permutation and proves that, in a multivariate location family, the modified estimators are minimax under quadratic loss. Shrinkage minimax estimators improving on the modified estimators are obtained for a restricted mean vector of spherically symmetric distribution. An application of sign-change transformation is also given in estimation of a bounded normal mean.     

Weighted Logrank Permutation Tests for Randomly Right Censored Life Science Data

In biomedical research, weighted logrank tests are frequently applied to compare two samples of randomly right censored survival times. We address the question how to combine a number of weighted logrank statistics to achieve good power of the corresponding survival test for a whole linear space or cone of alternatives, which are given by hazard rates. This leads to a new class of semiparametric projection tests that are motivated by likelihood ratio tests for an asymptotic model. We show that these tests can be carried out as permutation tests and discuss their asymptotic properties. A simulation study together with the analysis of a classical data set illustrates the advantages.     

Estimation of weighted multinomial probabilities under log-convex constraints

Many problems in Statistics involve maximizing a multinomial likelihood over a restricted region. In this paper, we consider instead maximizing a weighted multinomial likelihood. We show that a dual problem always exits which is frequently more tractable and that a solution to the dual problem leads directly to a solution of the primal problem. Moreover, the form of the dual problem suggests an iterative algorithm for solving the MLE problem when the constraint region can be written as a finite intersection of cones. We show that this iterative algorithm is guaranteed to converge to the true solution and show that when the cones are isotonic, this algorithm is a version of Dykstra's algorithm (Dykstra, J. Amer. Statist. Assoc. 78 (1983) 837–842) for the special case of least squares projection onto the intersection of isotonic cones. We give several meaningful examples to illustrate our results. In particular, we obtain the nonparametric maximum likelihood estimator of a monotone density function in the presence of selection bias.     

Bootstrap tests for multivariate directional alternatives

Tests on multivariate means that are hypothesized to be in a specified direction have received attention from both theoretical and applied points of view. One of the most common procedures used to test this cone alternative is the likelihood ratio test (LRT) assuming a multivariate normal model for the data. However, the resulting test for an ordered alternative is biased in that the only usable critical values are bounds on the null distribution. The present paper provides empirical evidence that bootstrapping the null distribution of the likelihood ratio statistic results in a bootstrap test (BT) with comparable power properties without the additional burden of assuming multivariate normality. Additionally, the tests based on the LRT statistic can reject the null hypothesis in favor of the alternative even though the true means are far from the alternative region. The BT also has similar properties for normal and nonnormal data. This anomalous behavior is due to the formulation of the null hypothesis and a possible remedy is to reformulate the null to be the complement of the alternative hypothesis. We discuss properties of a BT for the modified set of hypotheses (MBT) based on a simulation study. The resulting test is conservative in general and in some specific cases has power estimates comparable to those for existing methods. The BT has higher sensitivity but relatively lower specificity, whereas the MBT has higher specificity but relatively lower sensitivity.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

An exact decomposition for a class of exponential dispersion models: Application to sequential testing

We define the Wishart distribution on the cone of positive definite matrices and an exponential distribution on the Lorentz cone as exponential dispersion models. We show that these two distributions possess a property of exact decomposition, and we use this property to solve the following problem: given q samples (y_il,… y_iNj), i = l,…,q, from a N(μ_i,Σ_i,) distribution, test H:Σ₁ = Σ₂ = … = σ_q. Using the exact decomposition property, the classical test statistic for H, involving q parameters p_i = (N_i, - l)/2, i = 1,…,q, is replaced by a sequence of q - l test statistics for the sequence of tests H_i,:σ₁ =σ₂ = … =σ_i given that H_i-1 is true, i = 2,…,q. Each one of these test statistics involves two parameters only, p._i-1 = p₁ + … + p_i-1 and p_i. We also use the exact decomposition property to test equality of the “direction parameters” for q sample points from the exponential distribution on the Lorentz cone. We give a table of critical values for the distribution on the three-dimensional Lorentz cone. Tables of critical values in higher dimensions can easily be computed following the same method as in dimension three.     

A pseudo restricted MLE under multivariate order restrictions and its algorithm

Abstract

It is shown in this paper that a quasi order for the vectors in R^p is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.     

Parametric and permutation testing for multivariate monotonic alternatives

We are firstly interested in testing the homogeneity of k mean vectors against two-sided restricted alternatives separately in multivariate normal distributions. This problem is a multivariate extension of Bartholomew (in Biometrica 46:328–335, 1959b) and an extension of Sasabuchi et al. (in Biometrica 70:465–472, 1983) and Kulatunga and Sasabuchi (in Mem. Fac. Sci., Kyushu Univ. Ser. A: Mathematica 38:151–161, 1984) to two-sided ordered hypotheses. We examine the problem of testing under two separate cases. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but common. For the general case that covariance matrices are known the test statistic is obtained using the likelihood ratio method. When the known covariance matrices are common and diagonal, the null distribution of test statistic is derived and its critical values are computed at different significance levels. A Monte Carlo study is also presented to estimate the power of the test. A test statistic is proposed for the case when the common covariance matrices are unknown. Since it is difficult to compute the exact p-value for this problem of testing with the classical method when the covariance matrices are completely unknown, we first present a reformulation of the test statistic based on the orthogonal projections on the closed convex cones and then determine the upper bounds for its p-values. Also we provide a general nonparametric solution based on the permutation approach and nonparametric combination of dependent tests.