Berry–Esseen type bound of conditional mode estimation under truncation and strong mixing assumptions

ABSTRACT

In order to investigate the convergence rate of the asymptotic normality for the estimator of the conditional mode function for the left-truncation model, we derive a Berry–Esseen type bound of the estimator when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The finite sample performance of the estimator of the conditional mode function is explored through simulations.     

Testing hypotheses when there could be an initial effect

In many treatment-versus-control experiments, the observed random variables can be written as the product of a Bernoulli and a continuous random variable. The treatment can affect the distribution of the observations in two ways.

1. the probability that the observation is 0 could be altered.

2. the distribution of the nonzero observations could be changed.

We may also want to measure the combined effect of the treatment.

3. the expected value of control and treated units may differ.

A method is presented for testing for the presence of the combined effect when the general form of the distribution function of the continuous observations is known. For the case when this distribution function is from the family of gamma distributions, a previously proposed test criterion for the combined effect has poor power properties. In this paper, we discuss a test criterion that has improved power properties.     

Score tests in the two - way layout of counts

For the balanced two-way layout of a count response variable Y classified by fixed or random factors A and B, we address the problems of (i) testing for individual and interactive effects on Y of two fixed factors, and (ii) testing for the effect of a fixed factor in the presence of a random factor and conversely. In case (i), we assume independent Poisson responses with µ_ij= E(Y| A=i,B=j) = α_iβ_jγ_ij corresponding respectively to the multiplicative

interactive and non-interactive cases. For case (ii) with factor A random, we derive a multivariate gamma-Poisson model by mixing on the random variable associated with each level of A. In each case Neyman C(α) score tests are derived. We present simulation results,and apply the interaction test to a data set, to evaluate and compare the size and power of the score test for interaction between two fixed factors, the competing Poisson-based likelihood ratio test, and the F-tests based on the assumptions that √Y+1 or log(Y+1) are approximately normal. Our results provide strong evidence that the normal-theory based F-tests typically are very far from nominal size, and that the likelihood ratio test is somewhat more liberal than the score test.     

Small sample performance of large sample tests for tobit versus p -tobit

Large sample tests for the standard To bit model versus the ^p -Tobit model by Deaton and Irish (1984) are studied. The normalized one-tailed score test by Deaton and Irish (1984) is shown to be a version of Neyman's C(α) test that is valid for the non-standard problem of the null hypothesis lying on the boundary of the parameter space. Then, this paper reports the results of Monte Carlo experiments designed to study the small sample performance of large sample tests for the standard Tobit specification versus the ^p -Tobit specification.     

Multivariate semi-α-Laplace distributions

A multivariate semi-α-Laplace distribution (denoted by Ms-αLaplace) is introduced and studied in this paper. It is more general than the multivariate Linnik and Laplace distributions proposed by Sabu and Pillai (1991) or Anderson (1992). The Ms-αLaplace distribution has univariate semi-α-Laplace (Pillai, 1985) as marginal distribution. Various characterization theorems of the Ms-αLaplace distribution based on the closure property of the normalized geometric sum are proved.     

关于τ（α）函数在概率积分计算中的应用

本文通过τ（α）函数性质及τ（α）函数与B（P,q）函数的关系,得出了τ（α）函数的一系列特殊值,结合概率积分的特点,应用τ（α）函数与B（P,q）函数计算一些概率积分．     

Proper generalized neighbor designs in circular blocks

Generalized neighbor designs relax the condition of constancy on the number of times two treatments occur as neighbors in a design. In this paper, we constructed generalized neighbor designs in circular blocks of equal size. Catalogs of these designs for block size 5, 7 and 9 are also compiled. All the designs given here are binary.     

On a criterion for evaluating an estimator

A criterion for evaluating an estimator of the mean μ of a population is studied. The criterion considered is that the estimator be within a units of μ a large percentage (say, (1-β) 100%) of the time. This is referred to as the α -β criterion for estimating μ. A variation of this criterion is also studied.     

A new C(α) test for 2×2 tables

Two by two contingency tables which arise from experiments in which each subject serves as a self control are examined, Two types of experimental design are discussed: the confounded design and the counterbalanced design. In the case of the counterbalanced design, the locally asymptotically optimal C(α) test is compared with the binomial test for symmetry in a 2×2 contingency table. The binomial test is found to be less efficient than the C(α) test.     

On optimal tests for general interval-hypotheses

Optimal statistical tests, using the normality assumptions for general interval hypotheses including equivalence testing and testing for nonzero difference (or for non-unit) are presented. These tests are based on the decision theory for Polya Type distributions and are compared with usual confidence tests and with ’two one-sided tests’- procedures. A formal relationship between some optimal tests and the Anderson and Hauck procedure as well as a procedure recommended by Patel and Gupta is given. A new procedure for a generalisation of Student's test as well as for equivalence testing for thet-statistics is shown.