The star-shaped Λ-coalescent and Fleming–Viot process

The star-shaped Λ-coalescent and corresponding Λ-Fleming–Viot process, where the Λ measure has a single atom at unity, are studied in this article. The transition functions and stationary distribution of the Λ-Fleming–Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped Λ-coalescent is found. Extensions are made to a model with d types, either with parent-independent mutation or general Markov mutation, and an infinitely-many-types model, when d → ∞. An eigenfunction expansion for the transition functions is found, which has polynomial right eigenfunctions and left eigenfunctions described by hyperfunctions. A further star-shaped model with general frequency-dependent change is considered and the stationary distribution in the Fleming–Viot process derived. This model includes a star-shaped Λ-Fleming–Viot process with mutation and selection. In a general Λ-coalescent explicit formulae for the transition functions and stationary distribution, when there is mutation, are unknown. However, in this article, explicit formulae are derived in the star-shaped coalescent.     

Hypotheses testing for a multidimensional parameter of inhomogeneous Poisson processes

We observe a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown multidimensional parameter. We consider the asymptotic behaviour of the Rao score test for a simple null hypothesis against the multilateral alternative. By using the Edgeworth type expansion (under the null hypothesis) for a vector of stochastic integrals with respect to the Poisson process, we refine the (classic) threshold of the test (obtained by the central limit theorem), which improves the first type probability of error. The expansion allows us to describe the power of the test under the local alternative, i.e. a sequence of alternatives, which converge to the null hypothesis with a certain rate. The rates can be different for components of the parameter.     

A uniform central limit theorem for neural network-based autoregressive processes with applications to change-point analysis

We consider an autoregressive process with a nonlinear regression function that is modelled by a feedforward neural network. First, we derive a uniform central limit theorem which is useful in the context of change-point analysis. Then, we propose a test for a change in the autoregression function which – by the uniform central limit theorem – has asymptotic power one for a large class of alternatives including local alternatives not restricted to the correctly specified model.     

Optimal designs for estimating a choice hierarchy by a general nested multinomial logit model

Abstract

In choice experiments the process of decision-making can be more complex than the proposed by the Multinomial Logit Model (MNL). In these scenarios, models such as the Nested Multinomial Logit Model (NMNL) are often employed to model a more complex decision-making. Understanding the decision-making process is important in some fields such as marketing. Achieving a precise estimation of the models is crucial to the understanding of this process. To do this, optimal experimental designs are required. To construct an optimal design, information matrix is key. A previous research by others has developed the expression for the information matrix of the two-level NMNL model with two nests: Alternatives nest (J alternatives) and No-Choice nest (1 alternative). In this paper, we developed the likelihood function for a two-stage NMNL model for M nests and we present the expression for the information matrix for 2 nests with any amount of alternatives in them. We also show alternative D-optimal designs for No-Choice scenarios with similar relative efficiency but with less complex alternatives which can help to obtain more reliable answers and one application of these designs.     

A Nonparametric Test of the Predictive Regression Model

This article considers testing the significance of a regressor with a near unit root in a predictive regression model. The procedures discussed in this article are nonparametric, so one can test the significance of a regressor without specifying a functional form. The results are used to test the null hypothesis that the entire function takes the value of zero. We show that the standardized test has a normal distribution regardless of whether there is a near unit root in the regressor. This is in contrast to tests based on linear regression for this model where tests have a nonstandard limiting distribution that depends on nuisance parameters. Our results have practical implications in testing the significance of a regressor since there is no need to conduct pretests for a unit root in the regressor and the same procedure can be used if the regressor has a unit root or not. A Monte Carlo experiment explores the performance of the test for various levels of persistence of the regressors and for various linear and nonlinear alternatives. The test has superior performance against certain nonlinear alternatives. An application of the test applied to stock returns shows how the test can improve inference about predictability.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

On the asymptotic power and relative efficiency of the frequency X2 test

The asymptotic (Pitman) power of the X² test is investigated for particular classes of alternatives. A simple rule is introduced to identify ‘orthogonal alternatives’, for which the noncentrality parameter can be computed in a very simple way. In the sequel, restricted alternatives are considered and the ARE of the unrestricted test w.r.t. the restricted one is shown to depend only on the numbers of degrees of freedom. The concluding section discusses ‘undetectable alternatives’, i.e. alternatives for which the noncentrality vanishes.     

On the power of durbin-watson statistic against fractionally integrated processes

This paper provides the theoretical explanation and Monte Carlo experiments of using a modified version of Durbin-Watson ( D W ) statistic to test an 1 ( 1 ) process against I ( d ) alternatives, that is, integrated process of order d, where d is a fractional number. We provide the exact order of magnitude of the modified D W test when the data generating process is an I ( d ) process with d E (0. 1.5). Moreover, the consistency of the modified DW statistic as a unit root test against I ( d ) alternatives with d E ( 0 , l ) U ( 1 , 1.5) is proved in this paper. In addition to the theoretical analysis, Monte Carlo experiments show that the performance of the modified D W statistic reveals that it can be used as a unit root test against I ( d ) alternatives.     

A Test for Constant Hazard Against a Change-Point Alternative

In this article, we consider the change-point hazard rate model which arises quite commonly in mechanical or biological systems, which experience a high hazard rate early in their lifetime due to infant mortality and then a constant or steady hazard rate after the threshold time. We first derive the corresponding mean residual life function (MRLF) and observe that the MRLF is initially increasing and then constant. Here, we derive a test statistic for exponentiality against Increasing Initially then Constant Mean Residual Life (ICMRL). We also derive the asymptotic distribution of the test statistic and compare the power of the test with other existing tests such as likelihood ratio, Weibull, and Log gamma tests considered in the literature. The test performs quite well as compared to other alternatives studied.     

Tests for the Proportional Intensity Assumption Based on the Score Process

Proportional intensity models are widely used for describing the relationship between the intensity of a counting process and associated covariates. A basic assumption in this model is the proportionality, that each covariate has a multiplicative effect on the intensity. We present and study tests for this assumption based on a score process which is equivalent to cumulative sums of the Schoenfeld residuals. Tests within principle power against any type of departure from proportionality can be constructed based on this score process. Among the tests studied, in particular an Anderson-Darling type test turns out to be very useful by having good power properties against general alternatives. A simulation study comparing various tests for proportionality indicates that this test seems to be a good choice for an omnibus test for proportionality.     

Testing for No Effect by Cosine Series Methods

The properties of three lack-of-fit tests that are related to non-parametric cosine regression analysis are examined in the context of testing for a constant mean function. Analytic power comparisons of these tests vs a most powerful test are made using intermediate asymptotic relative efficiency. In particular, a data-driven test is produced which is asymptotically as efficient as the most powerful test over a class of alternatives. A small scale simulation experiment is conducted to ascertain the extent that the large sample comparisons are applicable to finite samples.     

Comparison of Two Distribution-Free Procedures for Multiple Comparisons With a Control

Chakraborti and Desu (1988) presented a distribution-free procedure for testing that k (≥1) distributions are equal to a control distribution. They compared their procedure, a generalization of the test proposed by Mathisen (1943), to the procedure proposed by Slivka (1970). They asserted that their procedure has shorter expected duration than Slivka's procedure in life-testing experiments where observations become available in an ordered manner. Here it is proven that, in fact, Slivka's procedure has shorter duration in such circumstances. Normal approximations are presented which indicate that their procedure requires a smaller sample size to guarantee a specified power for Lehmann alternatives and proportional hazard alternatives when all observations are to be observed.     

A cramer - von mises type goodness of fit test with asymmetric weight function

A goodness of fit test of the Cramer - von Mises type, which gives more weight to the upper (or to the lower) tail of the distribution, is proposed and studied. It is found the orthogonal representation of the test for the case of a simple null hypothesis. The characteristic function of the asymptotic null distribution is found and inverted to get percentage points. The asymptotic power of the test is obtained for the normal null hypothesis, against mean and variance shifts and more asymmetric alternatives.

Also the case of the exponential null hypothesis is studied. It is found that the test, which emphasizes the upper tail, has more power than those of Anderson - Darling and Cramer - von Mises, against alternatives which differ from the null hypothesis mainly in the upper tail, and less power when the main difference is in the lower tail of the distribution.     

Testing equality of correlation coefficients in two populations via permutation methods

The present paper investigates the asymptotic behaviour of a studentized permutation test for testing equality of (Pearson) correlation coefficients in two populations. It is shown that this test is asymptotically of exact level and has the same power for contiguous alternatives as the corresponding asymptotic test. As a by-product we specify the assumptions needed for the validity of the permutation test suggested in Sakaori (2002). A small simulation study compares the finite sample properties of the considered tests.     

Testing the equality of the variances of two linear models

Testing the equality of variances of two linear models with common β-parameter is considered. A test based on least squares residuals (ASR test) is proposed, and it is shown that this test is invariant under the group of scale and translation changes. For some special cases, it is also proved that this test has a monotone power function. Finding the exact critical values of this test is not easy; an approximation is given to facilitate the computation of these. The powers of the BLUS test, the F-test and the new test are computed for various alternatives and compared in a particular case. The proposed test seems to be locally more powerful than the alternative tests.     

Asymptotic deficiency of score test for inhomogeneous Poisson processes

In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown real parameter, we test a simple null hypothesis against a sequence of close (contiguous) one-sided alternatives. The main object is to obtain the asymptotic deficiency of the score test with respect to the Neyman–Pearson test.     

A class of robust stepwise alternatives to Hotelling's T 2 tests

Hotelling's T 2 test is known to be optimal under multivariate normality and is reasonably validity-robust when the assumption fails. However, some recently introduced robust test procedures have superior power properties and reasonable type I error control with non-normal populations. These, including the tests due to Tiku & Singh (1982), Tiku & Balakrishnan (1988) and Mudholkar & Srivastava (1999b, c), are asymptotically valid but are useful with moderate size samples only if the population dimension is small. A class of B-optimal modifications of the stepwise alternatives to Hotellings T 2 introduced by Mudholkar & Subbaiah (1980) are simple to implement and essentially equivalent to the T 2 test even with small samples. In this paper we construct and study the robust versions of these modified stepwise tests using trimmed means instead of sample means. We use the robust one- and two-sample trimmed- t procedures as in Mudholkar et al. (1991) and propose statistics based on combining them. The results of an extensive Monte Carlo experiment show that the robust alternatives provide excellent type I error control and a substantial gain in power.     

A rank test for stochastically ordered alternatives

A rank test based on the number of ‘near-matches’ among within-block rankings is proposed for stochastically ordered alternatives in a randomized block design with t treatments and b blocks. The asymptotic relative efficiency of this test with respect to the Page test is computed as number of blocks increases to infinity. A sequential analog of the above test procedure is also considered. A repeated significance test procedure is developed and average sample number is computed asymptotically under the null hypothesis as well as under a sequence of contiguous alternatives.     

On the theory of modified randomization tests for nonparametric hypotheses

A general randomization test for nonparametric hypotheses which is a modification of permutation tests in proposed. The exact level of the test is derived and under mild gegularity conditions, a general result on the consistency of the power function is obtained. Applications to several testing problems are considered. Asymptotic expansions of the power of this test are derived with respect to contiguous alternatives thus test are derived with respect to contiguous alternatives thus enabling us to make deficiency comparisons with permutation tests. The paper concludes with some Monte Carlo simulations verifying the theoretical results derived.     

A TEST OF HOMOGENEITY AGAINST SCALE ALTERNATIVES USING SUBSAMPLE EXTREMA1

In this paper a new class of non-parametric tests for testing homogeneity of several populations against scale alternatives is proposed. For this, independent samples of fixed sizes are drawn from each population and from these samples, all possible sub-samples of the same size are drawn and their maxima and minima are computed. Using these extreme the class of tests is obtained. Tests of this type have been offered for the two-sample slippage problem by Kochar (1978). Under certain conditions, this class of tests is shown to be consistent against ‘difference in scale’ alternatives. The test has been compared with Bhapkar's V-test (1961), Deshpande's D-test (1965), Sugiura's D_rs-test (1965) and with a classical test given by Lehmann (1959, pp. 273–275). It is shown that some members of this proposed class of tests are more efficient than the first three tests in the case of uniform, Laplace and normal distributions, when the number of populations compared is small.