Showing papers in "Statistics in 2009"

A new class of weighted exponential distributions

Abstract: Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized ...

Mixture inverse Gaussian distributions and its transformations, moments and applications

Abstract: Skewed models are important and necessary when parametric analyses are carried out on data. Mixture distributions produce widely flexible models with good statistical and probabilistic properties, and the mixture inverse Gaussian (MIG) model is one of those. Transformations of the MIG model also create new parametric distributions, which are useful in diverse situations. The aim of this paper is to discuss several aspects of the MIG distribution useful for modelling positive data. We specifically discuss transformations, the derivation of moments, fitting of models, and a shape analysis of the transformations. Finally, real examples from engineering, environment, insurance, and toxicology are presented for illustrating some of the results developed here. Three of the four data sets, which have arisen from the consulting work of the authors, are new and have not been previously analysed. All these examples display that the empirical fit of the MIG distribution to the data is very good.

A new method of imputation in survey sampling

Abstract: In this paper, an alternative estimator of population mean in the presence of non-response has been suggested which comes in the form of Walsh's estimator. The estimator of mean obtained from the proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean-squared error (MSE) of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optimum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. A suggestion to form ‘warm deck’ method of imputation has been suggested. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. The work has been extended to the case of multi-auxiliary information to be used for imputation. Numerical illustrations are also provided.

Advances on asymptotic normality in non-parametric functional time series analysis

Abstract: We consider a stationary process and wish to predict future values from previous ones. Instead of considering the process in its discretized form, we choose to see it as a sample of dependent curves. Then, we cut the process into N successive curves. Obviously, the N curves are not independent. The prediction issue can be translated into a non-parametric functional regression problem from dependent functional variables. This paper aims to revisit and complete two recent works on this topic. This article extends recent literature and provides asymptotic law with explicit constants under α-mixing assumptions. Then we establish pointwise confidence bands for the regression function. To conclude, we present how our results behave on a simulation and on a real time series.

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

Abstract: In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are...

On multiple constraint skewed models

Abstract: The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.

On kernel estimators of density ratio

Abstract: Let f(x) and g(x) denote two probability density functions and g(x)≠0. There are two ways to estimate the density ratio f(x)/g(x). One is to estimate f(x) and g(x) first and then the ratio, the other is to estimate f(x)/g(x) directly. In this paper, we derive asymptotic mean square errors and central limit theorems for both estimators.

Principal components regression estimator and a test for the restrictions

Abstract: In this article, we introduce restricted principal components regression (RPCR) estimator by combining the approaches followed in obtaining the restricted least squares estimator and the principal components regression estimator. The performance of the RPCR estimator with respect to the matrix and the generalized mean square error are examined. We also suggest a testing procedure for linear restrictions in principal components regression by using singly and doubly non-central F distribution.

Local linear density estimation for filtered survival data, with bias correction

Abstract: A class of local linear kernel density estimators based on weighted least-squares kernel estimation is considered within the framework of Aalen's multiplicative intensity model. This model includes the filtered data model that, in turn, allows for truncation and/or censoring in addition to accommodating unusual patterns of exposure as well as occurrence. It is shown that the local linear estimators corresponding to all different weightings have the same pointwise asymptotic properties. However, the weighting previously used in the literature in the i.i.d. case is seen to be far from optimal when it comes to exposure robustness, and a simple alternative weighting is to be preferred. Indeed, this weighting has, effectively, to be well chosen in a ‘pilot’ estimator of the survival function as well as in the main estimator itself. We also investigate multiplicative and additive bias-correction methods within our framework. The multiplicative bias-correction method proves to be the best in a simulation study c...

On robust Bayesian estimation under some asymmetric and bounded loss function

Abstract: The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.

A location- and scale-free goodness-of-fit statistic for the exponential distribution based on maximum correlations

Abstract: We propose a goodness-of-fit statistic, Q n , based on Hoeffding's maximum correlation (Fortiana and Grane 2003) to test the composite hypothesis that the data come from the two-parameter exponential family. We study its small and large sample properties, and we obtain tables of the critical values of Q n and some power curves. We compare our statistic with the Shapiro-Wilk statistic for exponentiality and the Gini statistic.

A generalization for two-sided power distributions and adjusted method of moments

Abstract: A new rich class of generalized two-sided power (TSP) distributions, where their density functions are expressed in terms of the Gauss hypergeometric functions, is introduced and studied. In this class, the symmetric distributions are supported by finite intervals and have normal shape densities. Our study on TSP distributions also leads us to a new class of discrete distributions on {0, 1, …, k}. In addition, a new numerical method for parameter estimation using moments is given.

Estimation of regression parameters in the presence of outliers in the response

Abstract: In this paper, we consider a regression model with non-spherical covariance structure and outliers in the response. The generalized least squares estimator obtained from the full data set is generally not used in the presence of outliers and an estimator based on only the non-outlying observations is preferred. Here we propose as an estimator a convex combination of the full set and the deleted set estimators and compare its performance with the other two.

A Bayesian Analysis For The Multivariate Point Null Testing Problem

Abstract: A Bayesian test for the point null testing problem in the multivariate case is developed A procedure to get the mixed distribution using the prior density is suggested For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence

Form-invariance under weighted sampling

Abstract: In this paper, form-invariant weighted distributions are considered in an exponential family. The class of bivariate distribution with invariant property is identified under exponential weight function. The class includes some of the custom bivariate models. The form-invariant multivariate normal distributions are obtained under a quadratic weight function.

Information theoretic results for circular distributions

Abstract: The broad class of generalized von Mises (GvM) circular distributions has certain optimal properties with respect to information theoretic quantities. It is shown that, under constraints on the trigonometric moments, and using the Kullback–Leibler information as the measure, the closest circular distribution to any other is of the GvM form. The lower bounds for the Kullback–Leibler information in this situation are also provided. The same problem is also considered using a modified version of the Kullback–Leibler information. Finally, series expansions are given for the entropy and the normalizing constants of the GvM distribution.

A binary control chart to detect small jumps

Abstract: The classic N p-chart gives a signal if the number of successes in a sequence of independent binary variables exceeds a control limit. Motivated by engineering applications in industrial image processing and, to some extent, financial statistics, we study a simple modification of this chart, which uses only the most recent observations. Our aim is to construct a control chart for detecting a shift of an unknown size, allowing for an unknown distribution of the error terms. Simulation studies indicate that the proposed chart is superior in terms of out-of-control average run length, when one is interested in the detection of very small shifts. We provide a (functional) central limit theorem under a change-point model with local alternatives, which explains that unexpected and interesting behaviour. Since real observations are often not independent, the question arises whether these results still hold true for the dependent case. Indeed, our asymptotic results work under the fairly general condition that th...

Bahadur representation for the nonparametric M-estimator under α-mixing dependence

Abstract: Under the condition that the observations, which come from a high-dimensional population (X, Y), are strongly stationary and strongly mixing, through using the local linear method, we investigate in this paper, the strong Bahadur representation of the nonparametric M-estimator for the unknown function m(x)=arg min a 𝔼(ρ (a, Y)|X=x), where the loss function ρ(a, y) is measurable. Furthermore, some related simulations are illustrated by using the cross-validation method for both bivariate linear and bivariate nonlinear time series contaminated by heavy-tailed errors. The M-estimator is applied to a series of S&P 500 index futures and spot prices to compare its performance in practice with the ‘usual’ squared-loss regression estimator.

Non-positive upper bounds on expectations of low rank order statistics from DFR populations

Abstract: Danielak (Sharp upper mean-variance bounds on trimmed means from restricled families, Statistics 37 (2003), pp. 305–324) established strictly positive optimal upper mean-variance bounds on the expectations of order statistics with relatively large ranks coming from decreasing failure rate populations. She also proved that the respective evaluations of low rank order statistics cannot be positive. Here, we show that the zero bounds on the deviation of the expected small order statistic from the population mean expressed in the scale units based on the pth absolute central moments with p>1 cannot be improved, and determine strictly negative sharp bounds in terms of the mean absolute deviation units.

On asymptotic posterior normality for controlled branching processes

Abstract: For a controlled branching process (CBP) with offspring distribution belonging to the power series family, the asymptotic normality of the posterior distribution of the basic parameter and the offspring mean is proved. As practical applications, we calculate asymptotic high probability density credibility sets for the offspring mean and we provide a rule to make inference about the value of this parameter. Moreover, the asymptotic posterior normality of the respective parameters of two classical branching models, namely the standard Galton–Watson process and the Galton–Watson process with immigration, is derived as particular cases of the CBP.

On the construction of stationary AR(1) models via random distributions

Abstract: We explore a method for constructing first-order stationary autoregressive-type models with given marginal distributions. We impose the underlying dependence structure in the model using Bayesian non-parametric predictive distributions. This approach allows for nonlinear dependency and at the same time works for any choice of marginal distribution. In particular, we look at the case of discrete-valued models; that is the marginal distributions are supported on the non-negative integers.

Mean Squared Errors of Small Area Estimators under a Unit-Level Multivariate Model

Abstract: This work deals with estimating the vector of means of certain characteristics of small areas. In this context, a unit level multivariate model with correlated sampling errors is considered. An approximation is obtained for the mean-squared and cross-product errors of the empirical best linear unbiased predictors of the means, when model parameters are estimated either by maximum likelihood (ML) or by restricted ML. This approach has been implemented on a Monte Carlo study using social and labour data from the Spanish Labour Force Survey.

Asymptotic variance–covariance matrix of sample autocorrelations for threshold-asymmetric GARCH processes

Abstract: In the field of financial time series, threshold-asymmetric conditional variance models can be used to explain asymmetric volatilities [C.W. Li and W.K. Li, On a double-threshold autoregressive heteroscedastic time series model, J. Appl. Econometrics 11 (1996), pp. 253–274]. In this paper, we consider a broad class of threshold-asymmetric GARCH processes (TAGARCH, hereafter) including standard ARCH and GARCH models as special cases. Since sample autocorrelation function provides a useful information to identify an appropriate time-series model for the data, we derive asymptotic distributions of sample autocorrelations both for original process and for squared process. It is verified that standard errors of sample autocorrelations for TAGARCH models are significantly different from unity for lower lags and they are exponentially converging to unity for higher lags. Furthermore they are shown to be asymptotically dependent while being independent of standard GARCH models. These results will be interesting i...

Bayesian and frequentist confidence intervals via adjusted likelihoods under prior specification on the interest parameter

Abstract: Suppose a prior is specified only on the interest parameter and a posterior distribution, free from nuisance parameters, is considered on the basis of the profile likelihood or an adjusted version thereof. In this setup, we derive higher order asymptotic results on the construction of confidence intervals that have approximately correct posterior as well as frequentist coverage. Apart from meeting both Bayesian and frequentist objectives under prior specification on the interest parameter alone, these results allow a comparison with their counterpart arising when the nuisance parameters are known, and hence provide additional justification for the Cox and Reid adjustment from a Bayesian-cum-frequentist perspective, with regard to neutralization of unknown nuisance parameters.

On testing the bioequivalence of several treatments using the measure of distance

Abstract: A studentized range test is proposed to test the hypothesis of bioequivalence of normal means in terms of a standardized distance among means. A least favourable configuration (LFC) of means to guarantee the maximum level at a null hypothesis and an LFC of means to guarantee the minimum power at an alternative hypothesis are obtained. This level and power of the test are fully independent of the unknown means and variances. For a given level, the critical value of the test under a null hypothesis can be determined. Furthermore, if the power under an alternative is also required at a given level, then both the critical value and the required sample size for an experiment can be simultaneously determined. In situations where the common population variance is unknown and the bioequivalence is the actual distance between means without standardization, a two-stage sampling procedure can be employed to find these solutions.

Stein-type improved estimation of standard error under asymmetric LINEX loss function

Abstract: This paper considers the estimation of standard error. More than 40 years ago, Stein [C. Stein, Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Institute Statist. Math. 16 (1964), pp. 155–160] proposed a classical improved estimator over the minimum risk equivariant estimator under quadratic loss. This is a textbook result. A generalization of quadratic loss is LINEX loss which considers asymmetric penalty of overestimation and underestimation. What is the corresponding version to Stein's improved estimator under LINEX loss? The problem has not been solved yet. This paper gives us an answer. Our method also applies to some other loss functions such as quadratic loss and entropy loss.

Second moment formulae for geometric sampling with test probes

Abstract: First moment formulae for the intersection between a fixed geometric object and one moving according to the kinematic density constitute the basis of stereology and they are essentially well known. However, the corresponding second moment formulae are less known; some of them are scattered in the literature and their proofs are often absent. Such formulae may be useful to compute exact variances when the objects involved have a known geometric shape on the one hand, and to obtain practical predictors of the estimation variance on the other. The purpose of this paper is to present a partly new, coherent set of second moment formulae for a fixed compact set intersected by mobile flats, bounded probes, or test systems endowed with the corresponding invariant measures, or by a stationary manifold process. We thereby contribute to a better understanding of the so-called variance ‘paradoxes’, arising when a probe performs better than a higher-dimensional one. For completeness, we include the necessary tools in ...

Properties of a class of binary ARMA models

Abstract: We investigate a class of ARMA-type models for stationary binary time series developed in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371–375, E. McKenzie, Extending the correlation structure of exponential autoregressive-moving-average processes, J. Appl. Prob. 18 (1981), pp. 181–189.], which we shall refer to as BinARMA models. This sparsely parameterized model family is even able to deal with negative autocorrelations, which occur in language modelling, for instance. While the autocorrelation structure of the BinAR(p) models has been studied before in [M. Kanter, Autoregression for discrete processes mod 2, J. Appl. Probabil. 12 (1975), pp. 371–375], we shall present new results on the autocorrelation structure of general BinARMA models. These results simplify in the BinMA(q) case, while the known results concerning BinAR(p) models are included as a special case. A real-data example indicates possible fields of application of these models.

On approximating the distribution of indefinite quadratic forms

Abstract: This paper provides a simple methodology for approximating the distribution of indefinite quadratic forms in normal random variables. It is shown that the density function of a positive definite quadratic form can be approximated in terms of the product of a gamma density function and a polynomial. An extension which makes use of a generalized gamma density function is also considered. Such representations are based on the moments of a quadratic form, which can be determined from its cumulants by means of a recursive formula. After expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, one can obtain an approximation to its density function by means of the transformation of variable technique. An explicit representation of the resulting density approximant is given in terms of a degenerate hypergeometric function. An easily implementable algorithm is provided. The proposed approximants produce very accurate percentiles over the entire range of the distribution....

Consistency of maximum likelihood estimators for the regime-switching GARCH model

Abstract: The regime-switching GARCH (generalized autoregressive conditionally heteroscedastic) model incorporates the idea of Markov switching into the more restrictive GARCH model, which significantly extends the GARCH model. However, the statistical inference for such an extended model is rather difficult because observations at any time point then depend on the whole regime path and the likelihood becomes intractable quickly as the length of observations increases. In this paper, by transforming it into an infinite order ARCH model, we obtain the possibility of writing a likelihood which can be handled directly and the consistency of the maximum likelihood estimators is proved. Simulation studies to illustrate the consistency and asymptotic normality of the estimators (for both Gaussian and non-Gaussian innovations) and a model specification problem are presented.