Alternatives to maximum likelihood estimation based on spacings and the Kullback–Leibler divergence

An alternative to the maximum likelihood (ML) method, the maximum spacing (MSP) method, is introduced in Cheng and Amin [1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403], and independently in Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112]. The method, as described by Ranneby [1984. The maximum spacing method. An estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112], is derived from an approximation of the Kullback–Leibler divergence. Since the introduction of the MSP method, several closely related methods have been suggested. This article is a survey of such methods based on spacings and the Kullback–Leibler divergence. These estimation methods possess good properties and they work in situations where the ML method does not. Important issues such as the handling of ties and incomplete data are discussed, and it is argued that by using Moran's [1951. The random division of an interval—Part II. J. Roy. Statist. Soc. Ser. B 13, 147–150] statistic, on which the MSP method is based, we can effectively combine: (a) a test on whether an assigned model of distribution functions is correct or not, (b) an asymptotically efficient estimation of an unknown parameter _θ0${\theta }_0$, and (c) a computation of a confidence region for _θ0${\theta }_0$.     

On the Estimation of Parameters of Kumaraswamy-G Distributions

Cordeiro and de Castro proposed a new family of generalized distributions based on the Kumaraswamy distribution (denoted as Kw-G). Nadarajah et al. showed that the density function of the new family of distributions can be expressed as a linear combination of the density of exponentiated family of distributions. They derived some properties of Kw-G distributions and discussed estimation of parameters using the maximum likelihood (ML) method. Cheng and Amin and Ranneby introduced a new method of estimating parameters based on Kullback–Leibler divergence (the maximum spacing (MSP) method). In this article, the estimates of parameters of Kw-G distributions are obtained using the MSP method. For some special Kw-G distributions, the new estimators are compared with ML estimators. It is shown by simulations and a real data application that MSP estimators have better properties than ML estimators.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

Analytical proofs of classical inequalities between Spearman's and Kendall's

Short analytical proofs are given for classical inequalities due to Daniels [1950. Rank correlation and population models. J. Roy. Statist. Soc. Ser. B 12, 171–181; 1951. Note on Durbin and Stuart's formula for E(r_s). J. Roy. Statist. Soc. Ser. B 13, 310] and Durbin and Stuart [1951. Inversions and rank correlation coefficients. J. Roy. Statist. Soc. Ser. B 13, 303–309] relating Spearman's ρ and Kendall's τ.     

Bayesian robustness in bidimensional models: Prior independence

When θ is a multidimensional parameter, the issue of prior dependence or independence of coordinates is a serious concern. This is especially true in robust Bayesian analysis; Lavine et al. (J. Amer. Statist. Assoc.86, 964–971 (1991)) show that allowing a wide range of prior dependencies among coordinates can result in near vacuous conclusions. It is sometimes possible, however, to make confidently the judgement that the coordinates of θ are independent a priori and, when this can be done, robust Bayesian conclusions improve dramatically. In this paper, it is shown how to incorporate the independence assumption into robust Bayesian analysis involving -contamination and density band classes of priors. Attention is restricted to the case θ = (θ₁, θ₂) for clarity, although the ideas generalize.     

Stationary mixture transition distribution (MTD) models via predictive distributions

This paper combines two ideas to construct autoregressive processes of arbitrary order. The first idea is the construction of first order stationary processes described in Pitt et al. [(2002). Constructing first order autoregressive models via latent processes. Scand. J. Statist.29, 657–663] and the second idea is the construction of higher order processes described in Raftery [(1985). A model for high order Markov chains. J. Roy. Statist. Soc. B.47, 528–539]. The resulting models provide appealing alternatives to model non-linear and non-Gaussian time series.     

Inference for Box–Cox Transformed Threshold GARCH Models with Nuisance Parameters

Abstract. Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time‐varying volatilities. To overcome the defect of the Gaussian quasi‐maximum likelihood estimator (QMLE) when the innovations follow either heavy‐tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two‐sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box–Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi‐likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

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].     

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.     

A Z-theorem with Estimated Nuisance Parameters and Correction Note for 'Weighted Likelihood for Semiparametric Models and Two-phase Stratified Samples, with Application to Cox Regression'

Abstract.  We state and prove a limit theorem for estimators of a general, possibly infinite dimensional parameter based on unbiased estimating equations containing estimated nuisance parameters. The theorem corrects a gap in the proof of one of the assertions of our paper 'Weighted likelihood for semiparametric models and two-phase stratified samples, with application to Cox regression' [ Scand. J. Statist . 34 (2007) 86–102].     

Rates of Posterior Convergence for iid Observations

By means of the Hausdorff α-entropy introduced by Xing and Ranneby (2009 Xing , Y. , Ranneby , B. ( 2009 ). Sufficient conditions for Bayesian consistency . J. Statist. Plann. Inference. 139 : 2479 – 2489 . [Google Scholar]), we give two theorems on rates of in-probability convergence of posterior distributions. The result is applied in study of the Bernstein polynomial priors.     

Robust estimation of multivariate location and shape

In this paper, we describe an overall strategy for robust estimation of multivariate location and shape, and the consequent identification of outliers and leverage points. Parts of this strategy have been described in a series of previous papers (Rocke, Ann. Statist., in press; Rocke and Woodruff, Statist. Neerlandica 47 (1993), 27–42, J. Amer. Statist. Assoc., in press; Woodruff and Rocke, J. Comput. Graphical Statist. 2 (1993), 69–95; J. Amer. Statist. Assoc. 89 (1994), 888–896) but the overall structure is presented here for the first time. After describing the first-level architecture of a class of algorithms for this problem, we review available information about possible tactics for each major step in the process. The major steps that we have found to be necessary are as follows: (1) partition the data into groups of perhaps five times the dimension; (2) for each group, search for the best available solution to a combinatorial estimator such as the Minimum Covariance Determinant (MCD) — these are the preliminary estimates; (3) for each preliminary estimate, iterate to the solution of a smooth estimator chosen for robustness and outlier resistance; and (4) choose among the final iterates based on a robust criterion, such as minimum volume. Use of this algorithm architecture can enable reliable, fast, robust estimation of heavily contaminated multivariate data in high (> 20) dimension even with large quantities of data. A computer program implementing the algorithm is available from the authors.     

On the analysis of circular balanced crossover designs

The paper considers Azaïs' (J. Roy. Statist. Soc. B, 49 (1987) 334–345) randomization procedure for circular balanced crossover designs. It is shown that this randomization does not justify the assumption of independent identically distributed errors when the estimates are corrected for carryover effects. This might lead to underestimation of the variance of treatment estimates. Similar to the results of Kunert (Biometrics, 43 (1987) 833–845) and Kunert and Utzig (J. Roy. Statist. Soc. B, 55 (1993) 919–927), we give constants, such that multiplication with this constant makes the usual estimate of variance conservative.     

Bounding convergence rates for Markov chains: An example of the use of computer algebra

Kolassa and Tanner (J. Am. Stat. Assoc. (1994) 89, 697–702) present the Gibbs-Skovgaard algorithm for approximate conditional inference. Kolassa (Ann Statist. (1999), 27, 129–142) gives conditions under which their Markov chain is known to converge. This paper calculates explicity bounds on convergence rates in terms calculable directly from chain transition operators. These results are useful in cases like those considered by Kolassa (1999).     

Semiparametric location mixtures with distinct components

We consider a two-component location mixture model with symmetric components, one of which is assumed to be known, the other is unknown. We show identifiability under assumptions on the tails of the characteristic function for the true underlying mixture, and also construct asymptotically normal estimates. The model is an extension of the contamination model in Bordes et al. [Semiparametric estimation of a two-component mixture model when a component is known, Scand. J. Statist. 33 (2006), pp. 733–752], and also related to a location mixture of one symmetric density as in Bordes et al. [Semiparametric estimation of a two component mixture model, Ann. Statist. 34 (2006), pp. 1204–1232]. We show by simulation that estimating the additional location parameter leads to a slight loss of efficiency as compared with the contamination model.     

Approximation to the distribution of LAD estimators for censored regression by random weighting method

Powell (J. Econometrics 25 (1984) 303) considered censored regression model, and established the asymptotic normality of the least absolute deviation (LAD) estimator. But the asymptotic covariance matrices depend on the error density and are therefore difficult to estimate reliably. In the earlier papers, this difficulty may be solved by applying the bootstrap method (see, e.g., Hahn (J. Econometric Theory 11 (1995) 105); Bilias et al. (J. Econometrics 99 (2000) 373). In this paper we propose a random weighting method to approximate the distribution of the LAD estimator. The random weighting method was developed by Rubin (Ann. Statist. 9 (1981) 130), Lo (Ann. Statist. 15 (1987) 360), Tu and Zheng (Chinese J. Appl. Probab. Statist. 3 (1987) 340) with reference to some statistics such as the sample mean. Rao and Zhao (Sankhya 54 (1992) 323) applied random weighting method to approximate asymptotic distribution of M-estimators in regression models. In this paper we extend this method to the censored regression model.     

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].     

Point and block prediction in log-Gaussian random fields: The non-constant mean case

This work considers the problems of point and block prediction in log-Gaussian random fields for the case when the mean of the log-process is not constant and depends linearly on unknown parameters. First, we propose a new point predictor that is optimal within a certain family of predictors, which extend a result in De Oliveira [2006. On optimal point and block prediction in log-Gaussian random fields. Scand. J. Statist. 33, 523–540.] that holds in the case when the mean of the log-process is constant. Second, we show that the results in De Oliveira [2006. On optimal point and block prediction in log-Gaussian random fields. Scand. J. Statist. 33, 523–540.] regarding optimal block prediction cannot be extended to the case when the mean of the log-process is not constant. Specifically, we show that the two families of block predictors considered by De Oliveira lack an optimal predictor. Finally, we numerically compare the predictive efficiency of the proposed point and block predictors.     

Robust estimation in simultaneous equations models

In this paper we review existing work on robust estimation for simultaneous equations models. Then we sketch three strategies for obtaining estimators with a high breakdown point and a controllable efficiency: (a) robustifying three-stage least squares, (b) robustifying the full information maximum likelihood method by minimizing the determinant of a robust covariance matrix of residuals, and (c) generalizing multivariate tau-estimators (Lopuhaä, 1992, Can. J. Statist., 19, 307–321) to these models. They have the same order of computational complexity as high breakdown point multivariate estimators. The latter seems the most promising approach.     

Consistency of Generalized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby & Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao & Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.