The number of records within a random interval of the current record value

LetX ₁,X ₂, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ _n ⁻ , μ _n ⁺ be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ _n ⁻ , μ _n ⁺ are studied in the present paper. Some statistical applications connected with these variables are also provided.     

A Note on the Comparison between Simple Mean and Mean Based on Distinct Units in Sampling with Replacement

Basu (1958) and Raj and Khamis (1958) have shown that in sampling with replacement the average over distinct units possesses lower variance than the average over the entire sample, including repetitions. While Basu proves the result by the use of the Rao-Blackwell theorem, Raj and Khamis prove it by making use of the distribution of the number of distinct units u in the sample, and the proof is quite involved. Here, an elementary algebraic proof is given.     

Comparison of efficient L- and R-estimators of location

Scholz (1974) proved that the asymptotic variance of an R-estimator of location is no larger than that of an L-estimator when the observations come from a distribution G different from the distribution F for which the two estimators are efficient. This note extends this result to distributions F whose density has a first but no second derivative.     

On some generalized wishart expectations

In a recent paper Muirhead (1986) derived certain useful identities involving expectations taken with respect to the Wishart distribution. This note generalizes the above results by taking expectations with respect to a generalized version of the Wishart distribution, considered by Sutradhar and Ali (1989), based on a multivariate tdistribution.     

New approximations to the exact distribution of the kruskal-wallis test statistic

Several approximations to the exact distribution of the Kruskal-Wallis test' statistic presently exist. There approximations can roughly be grouped into two classes: (i) computationally difficult with good accuracy, and (ii) easy to compute but not as accurate as the first class. The purpose of this paper is to introduce two nev approximations (one in the latter class and one which is computationally more involved)y and to compare these with other popular approximations. These comparisons use exact probabilities where available and Monte Carlo simulation otherwise.     

Two new estimators for the proportion of true null hypotheses in multiple test

In multiple hypothesis test, an important problem is estimating the proportion of true null hypotheses. Existing methods are mainly based on the p-values of the single tests. In this paper, we propose two new estimations for this proportion. One is a natural extension of the commonly used methods based on p-values and the other is based on a mixed distribution. Simulations show that the first method is comparable with existing methods and performs better under some cases. And the method based on a mixed distribution can get accurate estimators even if the variance of data is large or the difference between the null hypothesis and alternative hypothesis is very small.     

Estimation of the Parameters of Type-I Generalized Logistic Distribution Using Order Statistics

In this work, we propose a technique of estimating the location parameter μ and scale parameter σ of Type-I generalized logistic distribution by U-statistics constructed by using best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

One-Sample One-Sided Student T-Test Under Contaminants

A paradoxical behavior of the t-test under ε-contamination is presented. The paradox consists in that under a fixed distribution of contaminants an increasing of the probability of the appearance of a contaminant may decrease the violation of the size of the test! A simple explanation of the phenomenon is given. It is revealed which contaminants make the test conservative and which make it liberal: it appears that, in spite of the established opinion, conservatism or liberalism of the test depends not so much on the tails of the contaminating distribution as on where its support is located.     

One‐Way anova for Functional Data via Globalizing the Pointwise F‐test

In this paper, we propose and study a new global test, namely, GPF test, for the one‐way anova problem for functional data, obtained via globalizing the usual pointwise F‐test. The asymptotic random expressions of the test statistic are derived, and its asymptotic power is investigated. The GPF test is shown to be root‐n consistent. It is much less computationally intensive than a parametric bootstrap test proposed in the literature for the one‐way anova for functional data. Via some simulation studies, it is found that in terms of size‐controlling and power, the GPF test is comparable with two existing tests adopted for the one‐way anova problem for functional data. A real data example illustrates the GPF test.     

Central limit theorems for LS estimators in the EV regression model with dependent measurements

In this paper, we consider the simple linear errors-in-variables (EV) regression models: η_i=θ+βx_i+ε_i,ξ_i=x_i+δ_i,1≤i≤n, where θ,β,x₁,x₂,… are unknown constants (parameters), (ε₁,δ₁),(ε₂,δ₂),… are errors and ξ_i,η_i,i=1,2,… are observable. The asymptotic normality for the least square (LS) estimators of the unknown parameters β and θ in the model are established under the assumptions that the errors are m-dependent, martingale differences, ?-mixing, ρ-mixing and α-mixing.     

The Use of Noncentral F Approximations for Calculation of Power and Sample Size

Approximations to the noncentral F distribution yield surprisingly accurate results for power and sample size problems arising from linear hypotheses about normal random variables. The approximations are easy to use with a desk (or hand-held) calculator that computes cumulative F probabilities. These approximations are particularly advantageous for testing the hypothesis that differences among the means are small against the alternative that the differences are large.     

Experience with a criterion for the classification of johnson distributions

We use simulation to investigate the choice of percentiles when using the criterion given by Bowman and Shenton (1989) to select between Johnson's SB and STJ distributions for a data set. We also look at the choice of a tolerance level in classifying distributions as Johnson's SL-     

A Note on Screening Regression Equations

Consider developing a regression model in a context where substantive theory is weak. To focus on an extreme case, suppose that in fact there is no relationship between the dependent variable and the explanatory variables. Even so, if there are many explanatory variables, the R ² will be high. If explanatory variables with small t statistics are dropped and the equation refitted, the R ² will stay high and the overall F will become highly significant. This is demonstrated by simulation and by asymptotic calculation.     

Comparison of two measures of association in two-way contingency tables

Stuart's (1953) measure of association in contingency tables, t_C, based on Kendall's (1962) t, is compared with Goodman and Kruskal's (1954, 1959, 1963, 1972) measure G. First, it is proved that |G| ≥ |t_C|; and then it is shown that the upper bound for the asymptotic variance of G is not necessarily always smaller than the upper bound for the asymptotic variance of t_C. It is proved, however, that the upper bound for the coefficient of variation of G cannot be larger in absolute value than the upper bound for the coefficient of variation of t_C. The asymptotic variance of t_C is also derived and hence we obtain an upper bound for this asymptotic variance which is sharper than Stuart's (1953) upper bound.     

The new robust two-sample test for randomly right-censored data

Using a minimum p-value principle, a new two-sample test MIN3 is proposed in the paper. The cumulative distribution function of the MIN3 test statistic is studied and approximated by the Beta distribution of the third kind. Lower percentage points of the distribution of the new test statistic under the null hypothesis are computed. Also the test power for a lot of types of alternative hypotheses (with 0, 1 and 2 point(-s) of the intersection(-s) of survival functions) is studied and we found that the usage of the MIN3 test is a preferred strategy by the Wald and Savage decision-making criteria under risk and uncertainty. The results of application of the MIN3 test are shown for two examples from lifetime data analysis.     

Dynamic Reliability Evaluation of Consecutive-k-Within-m-Out-of-n:F System

A consecutive k-within-m-out-of-n:F system consists of n linearly ordered components and fails if and only if there are m consecutive components which include among them at least k failed components. This system model generalizes both consecutive k-out-of-n:F and k-out-of-n:F systems. In this article, we study the dynamic reliability properties of consecutive k-within-m-out-of-n:F system consisting of exchangeable dependent components. We also obtain some stochastic ordering results and use them to get simple approximation formulae for the survival function and mean time to failure of this system.     

Generalized likelihood-ratio test of the number of components in finite mixture models

The number of components is an important feature in finite mixture models. Because of the irregularity of the parameter space, the log-likelihood-ratio statistic does not have a chi-square limit distribution. It is very difficult to find a test with a specified significance level, and this is especially true for testing k — 1 versus k components. Most of the existing work has concentrated on finding a comparable approximation to the limit distribution of the log-likelihood-ratio statistic. In this paper, we use a statistic similar to the usual log likelihood ratio, but its null distribution is asymptotically normal. A simulation study indicates that the method has good power at detecting extra components. We also discuss how to improve the power of the test, and some simulations are performed.