Distributions minimizing fisher information for scale in kolmogorov neighbourhoods

We construct those distributions minimizing Fisher information for scale in Kolmogorov neighbourhoods K_?(G) = {F|sup_x|F(x) - G(x| ? ?} of d.f.'s G satisfying certain mild conditions. The theory is sufficiently general to include those cases in which G is normal, Laplace, logistic, Student's t, etc. As well, we consider G(x) = 1 - e^-x, ? 0, and correct some errors in the literature concerning this case.     

A NEW APPROXIMATION FOR FISHER'S Z

As the sample size increases, the coefficient of skewness of the Fisher's transformation z= tanh^-1r, of the correlation coefficient decreases much more rapidly than the excess of its kurtosis. Hence, the distribution of standardized z can be approximated more accurately in terms of the t distribution with matching kurtosis than by the unit normal distribution. This t distribution can, in turn be subjected to Wallace's approximation resulting in a new normal approximation for the Fisher's z transform. This approximation, which can be used to estimate the probabilities, as well as the percentiles, compares favorably in both accuracy and simplicity, with the two best earlier approximations, namely, those due to Ruben (1966) and Kraemer (1974). Fisher (1921) suggested approximating distribution of the variance stabilizing transform z=(1/2) log ((1 +r)/(1r)) of the correlation coefficient r by the normal distribution with mean = (1/2) log ((1 + p)/(lp)) and variance =l/(n3). This approximation is generally recognized as being remarkably accurate when ||Gr| is moderate but not so accurate when ||Gr| is large, even when n is not small (David (1938)). Among various alternatives to Fisher's approximation, the normalizing transformation due to Ruben (1966) and a t approximation due to Kraemer (1973), are interesting on the grounds of novelty, accuracy and/or aesthetics. If r?= r/√ (1r²) and r?|Gr = |Gr/√(1|Gr²), then Ruben (1966) showed that (1) g_n (r,|Gr) ={(2n5)/2}^1/2r?r{(2n3)/2}^1/2r?|GR, {1 + (1/2)(r?r²+r?|Gr²)}^1/2 is approximately unit normal. Kraemer (1973) suggests approximating (2) t_n (r, |Gr) = (r|GR¹) √ (n2), √(11r²) √(1|Gr²) by a Student's t variable with (n2) degrees of freedom, where after considering various valid choices for |Gr¹ she recommends taking |Gr¹= |Gr*, the median of r given n and |Gr.     

TESTS FOR DETECTING MORE NBU-NESS AT A SPECIFIC AGE

This paper proposes a class of non‐parametric test procedures for testing the null hypothesis that two distributions, F and G, are equal versus the alternative hypothesis that F is ‘more NBU (new better than used) at specified age t₀’ than G. Using Hoeffding's two‐sample U‐statistic theorem, it establishes the asymptotic normality of the test statistics and produces a class of asymptotically distribution‐free tests. Pitman asymptotic efficacies of the proposed tests are calculated with respect to the location and shape parameters. A numerical example is provided for illustrative purposes.     

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.     

McDonald's ωt,Cronbach's α, and Generalized θ for Composite Reliability of Common Factors Structures

In the common factor model for subtest scores, several reliability coefficients, including Cronbach's α, have been found to be biased. In this article, we introduce a new coefficient, θ_G, or Generalized θ, which is a generalized version of Armor's θ coefficient and is equal to the true reliability when the dimensions are orthogonal and the measures are parallel. We assessed the McDonald's ω_t, α, and θ_G in terms of mean bias, efficiency, and precision using a Monte Carlo simulation. θ_G outperformed ω_t when the factors were orthogonal or nearly orthogonal with low correlations between them.     

On the choice of transformations of the correlation coefficient with or without an outlier

Five transformations of the correlation coefficient, namely, Fisher's z, Nair's u, Sankaran's v, Ruben's y and Samiuddin's t are compared numerically using confidence intervals. Samiuddin's t_s transformation is close to the exact nominal confidence level for a small sample size ≤ 25 from a bivariate normal density. For a sample size > 25 both Samiuddin's t_s and Fisher's z can be used. In the presence of an outlier (on a minor axis), both Fisher's z and Samiuddin's t_s are not affected as long as |p| ≤ 0.3 but are seriously affected when |p&| > 0.3.     

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) _t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G _u /G _d /1, where G _u (G _d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G _u and G _d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ _y =inf{t≥0|X(t)?(0,y)}, y>0, and X(τ _y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.     

Probabilities and moments of generalized discrete distributfons using finite difference operators

The probabilities and factorial moments of the univar iate and multivariate generalized (or compound) discrete di st r-Lbut Lons with probability generating functions H(t)=F(G(t)) and H(t₁,…,t_k)=F(G(t₁,…,t_k))or H(t₁,…,t_k) = F(G₁(t₁),…, G_k( t_k)) are derived using finite difference operators.     

Asymptotic Tail Properties of Student's t-Distribution

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (t _ν for short) for ν tending to infinity when the argument x = x _ν of the pdf (cdf) depends on ν and tends to ± ∞ (?∞). To this end, we consider the ratio of the pdf's (cdf's) of the t _ν- and the standard normal distribution. Depending on the choice of the argument x _ν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when x _ν → ?∞.     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

Correctings tests for trend in proportions for skewness

We develop the score test for the hypothesis that a parameter of a Markov sequence is constant over time, against the alternatives that it varies over time, i.e., θ_t = θ + U_t; t = 1,2,…, where {U_t; t = 1,2,...} is a sequence of independently and identically distributed random variables with mean zero and variance σ^z _u and θ is a fixed constant. The asymptotic null distribution of the test statistic is proved to be normal. We illustrate our procedure by examples and a real life data analysis.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.     

RATES OF CONVERGENCE IN SEMI-PARAMETRIC MODELLING OF LONGITUDINAL DATA

We consider the problem of semi-parametric regression modelling when the data consist of a collection of short time series for which measurements within series are correlated. The objective is to estimate a regression function of the form E[Y(t) | x] =x'ß+μ(t), where μ(.) is an arbitrary, smooth function of time t, and x is a vector of explanatory variables which may or may not vary with t. For the non-parametric part of the estimation we use a kernel estimator with fixed bandwidth h. When h is chosen without reference to the data we give exact expressions for the bias and variance of the estimators for β and μ(t) and an asymptotic analysis of the case in which the number of series tends to infinity whilst the number of measurements per series is held fixed. We also report the results of a small-scale simulation study to indicate the extent to which the theoretical results continue to hold when h is chosen by a data-based cross-validation method.     

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

On random walks on affine group

Diaconis' presumption that the number of steps required to get close to uniform for a random walk on the affine group A _pis c(p)p ²with c(p) →ã is verified. We also discuss the random number generation associated with the random walk on the affine group. The number of steps to force the generated number to become random is improved. A modified version of Diacohis-Shahshahani's upper bound lemma is given and applied     

The Moment Bound is Tighter than Chernoff's Bound for Positive Tail Probabilities

Chernoff's bound on P[X ? t] is used almost universally when a tight bound on tail probabilities is required. In this article we show that for all positive t and for all distributions, the moment bound is tighter than Chernoff's bound. By way of example, we demonstrate that the improvement is often substantial.     

Upper bounds for the variances of certain random variables

Jacobson (1969) proved that the variance of a unimodal distribution on the unit interval is bounded above by 1/9. This paper generalizes Jacobson's result by deriving sharp upper bounds for the variances of α-unimodal distributions and for the variances of S-shaped distributions.