Uniform asymptotic linearity of a process based on a signed rank statistic

Consider the process with, cf. (1.2) on page 265 in B1, X₁, …, X_N a sample from a distribution F and, for i = 1, …, N, R${}_{\text{|x}{}_{\text{1}}\text{, - q}{}_{\text{1}}\text{ø|}}$ , the rank of |X₁ - q₁ø| among |X₁ - q₁ø|, …, |X_N - q_Nø|. It is shown that, under certain regularity conditions on F and on the constants p_i and q_i, T^ø_N(t) is asymptotically approximately a linear function of ø uniformly in t and in ø for |ø| ≤ C. The special case where the p_i and the q_i, are independent of i is considered.     

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.     

An asymptotic confidence interval for the process capability index Cpm

Abstract

The use of indices as an estimation tool of process capability is long-established among the statistical quality professionals. Numerous capability indices have been proposed in last few years. C_pm constitutes one of the most widely used capability indices and its estimation has attracted much interest. In this paper, we propose a new method for constructing an approximate confidence interval for the index C_pm. The proposed method is based on the asymptotic distribution of the index C_pm obtained by the Delta Method. Under some regularity conditions, the distribution of an estimator of the process capability index C_pm is asymptotically normal.     

On the invertibility conditions for moving average processes

Conditions for the general Moving Average process, of order q, to be invertible or borderline non-invertible are deduced. These are termed the acceptability conditions. It turns out that they depend on the magnitude of the final moving average parameter, θ _q . If ‖θ _q ‖ >1, the process is not acceptable. Should ‖θ _q ‖ = 1, the conditions, for any particular q, follow simply - if use is made of the remainder theorem. When ‖θ_q‖< 1, an appeal is made to ROUCH* E'S theorem, to establish the conditions. Analogous stationarity results immediately follow for autoregressive processes.     

Estimation of a Copula when a Covariate Affects only Marginal Distributions

This paper is concerned with studying the dependence structure between two random variables Y₁ and Y₂ in the presence of a covariate X, which affects both marginal distributions but not the dependence structure. This is reflected in the property that the conditional copula of Y₁ and Y₂ given X, does not depend on the value of X. This latter independence often appears as a simplifying assumption in pair‐copula constructions. We introduce a general estimator for the copula in this specific setting and establish its consistency. Moreover, we consider some special cases, such as parametric or nonparametric location‐scale models for the effect of the covariate X on the marginals of Y₁ and Y₂ and show that in these cases, weak convergence of the estimator, at ‐rate, holds. The theoretical results are illustrated by simulations and a real data example.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Asymptotic properties of a goodness-of-fit test based on maximum correlations

We study the efficiency properties of the goodness-of-fit test based on the Q _n statistic introduced in Fortiana and Grané [Goodness-of-fit tests based on maximum correlations and their orthogonal decompositions, J. R. Stat. Soc. B 65 (2003), pp. 115–126] using the concepts of Bahadur asymptotic relative efficiency and Bahadur asymptotic optimality. We compare the test based on this statistic with those based on the Kolmogorov–Smirnov, the Cramér-von Mises criterion and the Anderson–Darling statistics. We also describe the distribution families for which the test based on Q _n is locally asymptotically optimal in the Bahadur sense and, as an application, we use this test to detect the presence of hidden periodicities in a stationary time series.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

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.     

On testing exponentiality against new better than used of specified age

R.M. Hollander, D.H. Park and F. Proschan [A class of life distributions for aging, J. Amer. Statist. Assoc. 81 (1986) 91–95] introduced the concept of the larger class of life distributions called new better than used of specified age. In practice, one might be interested in the new better than used behaviour at an unknown but estimable age t₀. Here, we investigate the testing of new better than used of specified age t₀ (NBU-t₀) alternatives. A class of test statistics for testing NBU-t₀ (t₀ is known) based on a U-statistic whose kernel depends on sub-sample minima is proposed. A member of the class of tests proposed by N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] for this problem belongs to the class of tests proposed here. The distributional properties of the class of test statistics are studied. The performances of a few members of the proposed class of tests are studied in terms of Pitman asymptotic relative efficiency. The Pitman ARE values show that the members of the class perform well in comparison with the N. Ebrahimi and M. Habbibullah [Testing whether the survival distribution is new better than used of specified age, Biometrika 77 (1990) 212–215] tests. The proposed class of tests is shown to be consistent for NBU-t₀ alternatives.     

Exchangeability,predictive sufficiency and Bayesian bootstrap

Summary Let {X _n } be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X ₁,…,X _n ), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT _n (X ₁,…,X _n ) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with the results obtained following a common Bayesian approach.     

On Projection‐type Estimators of Multivariate Isotonic Functions

Abstract. Let M be an isotonic real‐valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection‐type estimators are r_n‐equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability‐weighted moment, quantile) and biometric (mean remaining lifetime) functions.     

Limiting distributions of MLE and UMVUE in the biparametric uniform distribution

In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ₁, θ₂) when sampling from a biparametric uniform distributionU(θ₁, θ₂). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution of UMVUE is a shift of the limiting distribution of MLE.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

The Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate

Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + μ _n )) _n∈?, where g is some smooth function, X is a random variable and (μ _n ) _n∈? is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace's type integral, randomized by the sequence (μ _n X) _n∈?, X being a Gamma distributed random variable.     

On testing more IFRA ordering-II

ABSTRACT

Suppose F and G are two life distribution functions. It is said that F is more IFRA (increasing failure rate average) than G (written by F ? _*G) if G^{? 1}F(x) is star-shaped on (0, ∞). In this paper, the problem of testing H₀: F = _*G against H₁: F ? _*G and F ≠ _*G is considered in both cases when G is known and when G is unknown. We propose a new test based on U-statistics and obtain the asymptotic distribution of the test statistics. The new test is compared with some well-known tests in the literature. In addition, we apply our test to a real data set in the context of reliability.     

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.