Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.     

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.     

Expansions of the coverage probabilities of prediction region based on a shrinkage estimator

An expansion formula for the coverage probability of prediction region based on a shrinkage estimator proposed by Joshi [Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal population. Ann. Math. Statist., 38, 1868–1875.] is obtained. Its error bound is evaluated in terms of a function of an unknown parameter. Applying this result, three types of asymptotic expansions are derived. These expansions show inadmissibility of the usual prediction region.     

Sufficient conditions for Bayesian consistency

We introduce the Hausdorff α$\alpha$-entropy to study the strong Hellinger consistency of posterior distributions. We obtain general Bayesian consistency theorems which extend the well-known results of Barron et al. [1999. The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27, 536–561] and Ghosal et al. [1999. Posterior consistency of Dirichlet mixtures in density estimation. Ann. Statist. 27, 143–158] and Walker [2004. New approaches to Bayesian consistency. Ann. Statist. 32, 2028–2043]. As an application we strengthen previous results on Bayesian consistency of the (normal) mixture models.     

General procedure for estimating the population mean using ranked set sampling

In this paper, we suggest a class of estimators for estimating the population mean ? of the study variable Y using information on X?, the population mean of the auxiliary variable X using ranked set sampling envisaged by McIntyre [A method of unbiased selective sampling using ranked sets, Aust. J. Agric. Res. 3 (1952), pp. 385–390] and developed by Takahasi and Wakimoto [On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 20 (1968), pp. 1–31]. The estimator reported by Kadilar et al. [Ratio estimator for the population mean using ranked set sampling, Statist. Papers 50 (2009), pp. 301–309] is identified as a member of the proposed class of estimators. The bias and the mean-squared error (MSE) of the proposed class of estimators are obtained. An asymptotically optimum estimator in the class is identified with its MSE formulae. To judge the merits of the suggested class of estimators over others, an empirical study is carried out.     

The marked empirical process to test nonlinear time series against a large class of alternatives when the random vectors are nonstationary and absolutely regular

In this paper, we propose a method for testing absolutely regular and possibly nonstationary nonlinear time-series, with application to general AR-ARCH models. Our test statistic is based on a marked empirical process of residuals which is shown to converge to a Gaussian process with respect to the Skohorod topology. This testing procedure was first introduced by Stute [Nonparametric model checks for regression, Ann. Statist. 25 (1997), pp. 613–641] and then widely developed by Ngatchou-Wandji [Weak convergence of some marked empirical processes: Application to testing heteroscedasticity, J. Nonparametr. Stat. 14 (2002), pp. 325–339; Checking nonlinear heteroscedastic time series models, J. Statist. Plann. Inference 133 (2005), pp. 33–68; Local power of a Cramer-von Mises type test for parametric autoregressive models of order one, Compt. Math. Appl. 56(4) (2008), pp. 918–929] under more general conditions. Applications to general AR-ARCH models are given.     

Sharp bounds on the expectations of second record values from symmetric populations

Combining the greatest convex minorant approximation (Moriguti, S. (1953). A modification of Schwarz's inequality with applications to distributions. Ann. Math. Statist., 24, 107–113.) with the Hölder inequality, we establish sharp bounds on the expectations of the second record statistics from symmetric populations. We also determine the distributions for which the bounds are attained. The optimal bounds are numerically evaluated and compared with other classical rough ones.     

On confidence regions of semiparametric nonlinear reproductive dispersion models

This article investigates the confidence regions for semiparametric nonlinear reproductive dispersion models (SNRDMs), which is an extension of nonlinear regression models. Based on local linear estimate of nonparametric component and generalized profile likelihood estimate of parameter in SNRDMs, a modified geometric framework of Bates and Wattes is proposed. Within this geometric framework, we present three kinds of improved approximate confidence regions for the parameters and parameter subsets in terms of curvatures. The work extends the previous results of Hamilton et al. [in Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions, Ann. Statist. 10, pp. 386–393, 1982], Hamilton [in Confidence regions for parameter subset in nonlinear regression, Biometrika, 73, pp. 57–64, 1986], Wei [in On confidence regions of embedded models in regular parameter families (a geometric approch), Austral. J. Statist. 36, pp. 327–338, 1994], Tang et al. [in Confidence regions in quasi-likelihood nonlinear models: a geometric approach, J. Biomath. 15, pp. 55–64, 2000b] and Zhu et al. [in On confidence regions of semiparametric nonlinear regression models, Acta. Math. Scient. 20, pp. 68–75, 2000].     

A new estimator of entropy and its application in testing normality

In this paper, we introduce a new estimator of entropy of a continuous random variable. We compare the proposed estimator with the existing estimators, namely, Vasicek [A test for normality based on sample entropy, J. Roy. Statist. Soc. Ser. B 38 (1976), pp. 54–59], van Es [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Correa [A new estimator of entropy, Commun. Statist. Theory and Methods 24 (1995), pp. 2439–2449] and Wieczorkowski-Grzegorewski [Entropy estimators improvements and comparisons, Commun. Statist. Simulation and Computation 28 (1999), pp. 541–567]. We next introduce a new test for normality. By simulation, the powers of the proposed test under various alternatives are compared with normality tests proposed by Vasicek (1976) and Esteban et al. [Monte Carlo comparison of four normality tests using different entropy estimates, Commun. Statist.–Simulation and Computation 30(4) (2001), pp. 761–785].     

A block bootstrap comparison for sparse chains

In this paper we apply the sequential bootstrap method proposed by Collet et al. [Bootstrap Central Limit theorem for chains of infinite order via Markov approximations, Markov Processes and Related Fields 11(3) (2005), pp. 443–464] to estimate the variance of the empirical mean of a special class of chains of infinite order called sparse chains. For this process, we show that we are able to compute numerically the true value of the standard error with any fixed error.

Our main goal is to present a comparison, for sparse chains, among sequential bootstrap, the block bootstrap method proposed by Künsch [The jackknife and the Bootstrap for general stationary observations, Ann. Statist. 17 (1989), pp. 1217–1241] and improved by Liu and Singh [Moving blocks jackknife and Bootstrap capture week dependence, in Exploring the limits of the Bootstrap, R. Lepage and L. Billard, eds., Wiley, New York, 1992, pp. 225–248] and the bootstrap method proposed by Bühlmann [Blockwise bootstrapped empirical process for stationary sequences, Ann. Statist. 22 (1994), pp. 995–1012].     

Transformations for Testing the Fit of the Inverse-Gaussian Distribution

Abstract

Use of the MVUE for the inverse-Gaussian distribution has been recently proposed by Nguyen and Dinh [Nguyen, T. T., Dinh, K. T. (2003). Exact EDF goodnes-of-fit tests for inverse Gaussian distributions. Comm. Statist. (Simulation and Computation) 32(2):505–516] where a sequential application based on Rosenblatt's transformation [Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23:470–472] led the authors to solve the composite goodness-of-fit problem by solving the surrogate simple goodness-of-fit problem, of testing uniformity of the independent transformed variables. In this note, we observe first that the proposal is not new since it was proposed in a rather general setting in O'Reilly and Quesenberry [O'Reilly, F., Quesenberry, C. P. (1973). The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests. Ann. Statist. I:74–83]. It is shown on the other hand that the results in the paper of Nguyen and Dinh (2003) are incorrect in their Sec. 4, specially the Monte Carlo figures reported. Power simulations are provided here comparing these corrected results with two previously reported goodness-of-fit tests for the inverse-Gaussian; the modified Kolmogorov–Smirnov test in Edgeman et al. [Edgeman, R. L., Scott, R. C., Pavur, R. J. (1988). A modified Kolmogorov-Smirnov test for inverse Gaussian distribution with unknown parameters. Comm. Statist. 17(B): 1203–1212] and the A ² based method in O'Reilly and Rueda [O'Reilly, F., Rueda, R. (1992). Goodness of fit for the inverse Gaussian distribution. T Can. J. Statist. 20(4):387–397]. The results show clearly that there is a large loss of power in the method explored in Nguyen and Dinh (2003) due to an implicit exogenous randomization.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.     

Bayes sequential estimation for a Poisson process under a LINEX loss function

In this paper, within the framework of a Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with a linear exponential (LINEX) loss function and a fixed cost per unit time. An asymptotically pointwise optimal (APO) rule is proposed. It is shown to be asymptotically optimal for the arbitrary priors and asymptotically non-deficient for the conjugate priors in a similar sense of Bickel and Yahav [Asymptotically pointwise optimal procedures in sequential analysis, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, CA, 1967, pp. 401–413; Asymptotically optimal Bayes and minimax procedures in sequential estimation, Ann. Math. Statist. 39 (1968), pp. 442–456] and Woodroofe [A.P.O. rules are asymptotically non-deficient for estimation with squared error loss, Z. Wahrsch. verw. Gebiete 58 (1981), pp. 331–341], respectively. The proposed APO rule is illustrated using a real data set.     

Non-Gaussian limit distributions for U-statistics based on trimmed and Winsorized samples

Conditions ensuring the asymptotic normality of U-statistics based on either trimmed samples or Winsorized samples are well known [P. Janssen, R. Serfling, and N. Veraverbeke, Asymptotic normality of U-statistics based on trimmed samples, J. Statist. Plann. Inference 16 (1987), pp. 63–74; U-statistics on Winsorized and trimmed samples, Statist. Probab. Lett. 9 (1990), pp. 439–447]. However, the class of U-statistics has a much richer family of limiting distributions. This paper complements known results by providing general limit theorems for U-statistics based on trimmed or Winsorized samples where the limiting distribution is given in terms of multiple Ito–Wiener stochastic integrals.     

Full likelihood inference in the Cox model with LTRC data when covariates are discrete

For the regression parameter β in the Cox model, there have been several estimates based on different types of approximated likelihood. For right-censored data, Ren and Zhou [Full likelihood inferences in the Cox model: an empirical approach. Ann Inst Statist Math. 2011;63:1005–1018] derive the full likelihood function for (β, F₀), where F₀ is the baseline distribution function in the Cox model. In this article, we extend their results to left-truncated and right-censored data with discrete covariates. Using the empirical likelihood parameterization, we obtain the full-profile likelihood function for β when covariates are discrete. Simulation results indicate that the maximum likelihood estimator outperforms Cox's partial likelihood estimator in finite samples.     

Outlier detection for high dimensional data using the Comedian approach

The process of detection of outliers is an interesting and important aspect in the analysis of data, as it could impact the inference. There are various methods available in the literature for detection of outliers in multivariate data [V. Barnett and T. Lewis, Outliers in Statistical Data, John Wiley & Sons, Chichester, 1994] using the Mahalanobis distance measure. An attempt is made to propose an alternate method of outlier detection based on the comedian introduced by Falk [On MAD and Comedians, Ann. Inst. Statist. Math. 49 (1997), pp. 615–644]. The proposed method is computationally efficient with high breakdown value and low computation time. Further, important properties, namely, success rates (SR) and false detection rates (FDR) are studied and compared with some of the well-known outlier detection methods through a simulation study. The Comedian method has high SR and low FDR for all combination of parameters. On removal of the detected outliers or down weighing, the same, highly robust and approximately affine equivariant estimators of multivariate location and scatter can be obtained. Finally, the method is applied to well-known real data sets to evaluate its performance.     

A new proof of hsu's theorem in regression analysis– a coordinate-free approach

This paper provides a new proof of results by HSU (Statist. Res. Mem. 1938) and Drygas (Math. Operationsforsch. Statist. 1972) on estimating the variance σ²in HsU's regression model:By using a dispersion. mean-Correspondence the given problem in the theory of estimating the mean vector.     

Maximum variance of order statistics from symmetric populations revisited

We consider i.i.d. samples of size n with symmetric non-degenerate parent distributions and finite variances. Papadatos [A note on maximum variance of order statistics from symmetric populations, Ann. Inst. Statist. Math. 48 (1997), pp. 117–121] proved that the maximal variance of each non-extreme order statistic, expressed in the population variance units, is attained in a one-parametric family of symmetric two- and three-point distributions. The parameters of the extreme variance distributions coincide with the arguments maximizing some polynomials of degree 2n?1 over a finite interval. The bounds for variances are equal to the maximal values of the polynomials. We present a more precise solution to the problem by applying the variation diminishing property of Bernstein polynomials.     

Asymptotic properties of the score test for varying dispersion in exponential family nonlinear models

Wei et al. [B.C. Wei, J.Q. Shi, W.K. Fung, and Y.Q. Hu, Testing for varying dispersion in exponential family nonlinear models, Ann. Inst. Statist. Math. 50 (1998), pp. 277–294.] developed the score diagnostics for varying dispersion in exponential family nonlinear models, such as the normal, inverse Gaussian, and gamma models, and investigated the powers of these tests through Monte Carlo simulations. In this paper, the asymptotic behaviours, including asymptotic chi-square and approximate powers under local alternatives of the score tests, are studied and examined by Monte Carlo simulations. The methods to estimate local powers of the score tests are illustrated with Grass yield data [P. McCullagh, and J.A. Nelder, Generalized Linear Models, Chapman and Hall, London (1989).].     

Some aspects of statistical inference on the lagrange (generalized) poisson distribution

A generalization of the Poisson distribution was defined by Consul and Jain (Ann. Math. Statist., 41, (1970)) and was obtained as a particular family of Lagrange distributions by Consul and Shenton (SIAM. J. Appl. Math., 23, (1972)). The distribution is subsequently named the generalized Poisson distribution (GPD). This GPD reduces to the Poisson distribution for ? = 0. When the data have a one-way layout structure, the asymptotically locally optimal Neyman's C(d) test is constructed and compared with the conditional test on the hypothesis H_o? = 0. Within the framework of the generalized linear models an appropriate link function is given, and the asymptotic distributions of the estimated parameters are derived.