Ann Inst Stat Math Research Articles

Maximum leave-one-out likelihood method for the location parameter of variance gamma distribution with unbounded density

Maximum likelihood estimation of a location parameter fails when the density has unbounded mode. An alternative approach is considered by leaving out a data point to avoid the unbounded density in the full likelihood. This modification gives rise to the leave-one-out (LOO) likelihood. We propose an expectation-conditional maximisation (ECM) algorithm to estimate parameters of variance gamma (VG) distribution with unbounded density by maximising the LOO likelihood. VG distribution has normal mean-variance mixtures representation that facilitates ECM algorithm. K Podgórski, J Wallin [Maximizing leave-one-out likelihood for the location parameter of unbounded densities. Ann Inst Stat Math. 2015;67(1):19–38.] showed that the location parameter estimate which maximises the LOO likelihood is consistent and super-efficient. In the case of repeated data, the LOO likelihood is still unbounded and we extend it to Weighted LOO (WLOO) likelihood. We perform simulations to investigate the accuracy of ECM method using WLOO likelihood and compare it with other likelihood methods, including the alternating ECM with fixed and adaptive caps to the unbounded density, the ghyp package using multi-cycle ECM, and the ECM using LOO likelihood. Lastly, we explore the asymptotic properties of the location parameter estimate of VG distribution such as the optimal convergence rate and asymptotic distribution.

Functional Convergence of Berry’s Nodal Lengths: Approximate Tightness and Total Disorder

We consider Berry’s random planar wave model (J Phys A 10(12):2083–2092, 1977), and prove spatial functional limit theorems—in the high-energy limit—for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called second Wiener chaos, whose high-energy fluctuations are given by a Gaussian total disorder field indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by Davydov and Zitikis (Ann Inst Stat Math 60(2):345–365, 2008).

Limiting distributions of the likelihood ratio test statistics for independence of normal random vectors

Consider the likelihood ratio test (LRT) statistics for the independence of sub-vectors from a p-variate normal random vector. We are devoted to deriving the limiting distributions of the LRT statistics based on a random sample of size n. It is well known that the limit is chi-square distribution when the dimension of the data or the number of the parameters are fixed. In a recent work by Qi et al. (Ann Inst Stat Math 71:911–946, 2019), it was shown that the LRT statistics are asymptotically normal under condition that the lengths of the normal random sub-vectors are relatively balanced if the dimension p goes to infinity with the sample size n. In this paper, we investigate the limiting distributions of the LRT statistic under general conditions. We find out all types of limiting distributions and obtain the necessary and sufficient conditions for the LRT statistic to converge to a normal distribution when p goes to infinity. We also investigate the limiting distribution of the adjusted LRT test statistic proposed in Qi et al. (2019). Moreover, we present simulation results to compare the performance of classical chi-square approximation, normal and non-normal approximation to the LRT statistics, chi-square approximation to the adjusted test statistic, and some other test statistics.

Location invariant heavy tail index estimation with block method

Motivated by the work of Qi [On the tail index of a heavy tailed distribution. Ann Inst Stat Math. 2010;62:277–298] and Fraga Alves [A location invariant Hill-type estimator. Extremes. 2001b;4:199–217], a new class of location invariant Hill-type estimators for heavy tail index is proposed. The weak consistency of the estimator is studied, the asymptotic expansion and limit distribution of the estimator are derived under second-order regular varying conditions. Simulation studies are performed to compare the new estimator with closely related estimators.

Robust Wald-type test statistics based on minimum C-divergence estimators

ABSTRACT Maji et al. [Robust statistical inference based on the C-divergence family. Ann Inst Stat Math. 2019;71:1289–1322] introduced the minimum C-divergence estimators and plugging them in the C-divergence measures give test statistics for testing simple null and composite null hypotheses. One inconvenience of these test statistics is that their asymptotic distribution is not, in general, a chi-square distribution but a linear combination of chi-square distributions. To overcome this inconvenience, in this paper we consider Wald-type test statistics based on minimum C-divergence estimators. We establish that this family of test statistics is a chi-square distribution and we get an approximation of the power function under simple null hypothesis and composite null hypothesis. We have calculated both first order and second order influence function of the Wald-type test statistics and based on it we can see the robustness of the family of test statistics considered in this paper. Both simulated and real data examples have been shown as part of numerical results.

Estimation of the Location and Scale Parameters of Generalized Pareto Distribution Based on Progressively Type-II Censored Order Statistics

Based on progressively type-II right censored order statistics, we establish several recurrence relations for the single and product moments from the generalized Pareto distribution due to Pikands (1975). Further, recursive computational algorithm is provided which enable us to compute all the means, variances and covariances for all sample sizes n, effective sample sizes m, and all progressive censoring schemes $$(R_{1},\ldots , R_{m})$$. These relations generalize the results given by Aggarwala and Balakrishnan (Ann Inst Stat Math 48:757–771, 1996) and Joshi (Sankhyd Set B 39:362–371, 1978) for standard exponential distribution. Besides, these moments are then utilized to derive best linear unbiased estimators (BLUEs) of the scale and location parameters of the generalized Pareto distribution. Next, we obtain the maximum likelihood estimators of the unknown parameters of the model under progressively type-II right censored order statistics. Monte Carlo simulations are performed to compare the performances of the proposed method, and a numerical example is presented to illustrate the method developed here to obtain BLUEs.

Tail dependence and smoothness of time series

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence of these in time. The extremal index \(\theta \) (Leadbetter in Z Wahrscheinlichkeitstheor Verw Geb 65:291–306, 1983) allows to infer the tendency for clustering of high values, but does not allow to evaluate the greater or less amount of oscillations in a cluster. The estimation of \(\theta \) entails the validation of local dependence conditions regulating the distance between high levels oscillations of the process, which is difficult to implement in practice. In this work, we propose a smoothness coefficient to evaluate the degree of smoothness/oscillation in the trajectory of a process, with an intuitive reading and simple estimation. Application in some examples will be provided. We will see that, in a stationary sequence, it coincides with the tail dependence coefficient \(\lambda \) (Sibuya in Ann Inst Stat Math 11:195–210, 1960; Joe in Multivariate models and dependence concepts. Monographs on statistics and applied probability, vol 73. Chapman and Hall, London, 1997), providing a new interpretation of the latter. This relationship will inspire a new estimator for \(\lambda \), and its performance will be evaluated based on a simulation study. We illustrate with an application to financial series.

Some properties of local weighted second-order statistics for spatio-temporal point processes

Diagnostics of goodness-of-fit in the theory of point processes are often considered through the transformation of data into residuals as a result of a thinning or a rescaling procedure. We alternatively consider here second-order statistics coming from weighted measures. Motivated by Adelfio and Schoenberg (Ann Inst Stat Math 61(4):929–948, 2009) for the temporal and spatial cases, we consider an extension to the spatio-temporal context in addition to focussing on local characteristics. In particular, our proposed method assesses goodness-of-fit of spatio-temporal models by using local weighted second-order statistics, computed after weighting the contribution of each observed point by the inverse of the conditional intensity function that identifies the process. Weighted second-order statistics directly apply to data without assuming homogeneity nor transforming the data into residuals, eliminating thus the sampling variability due to the use of a transforming procedure. We provide some characterisations and show a number of simulation studies.

Residual-based CUSUM of squares test for Poisson integer-valued GARCH models

ABSTRACTThis study considers the problem of testing for a parameter change in integer-valued time series models in which the conditional density of current observations is assumed to follow a Poisson distribution. As a test, we consider the CUSUM of the squares test based on the residuals from INGARCH models and find that the test converges weakly to the supremum of a Brownian bridge. A simulation study demonstrates its superiority to the residual and standardized residual-based CUSUM tests of Kang and Lee [Parameter change test for Poisson autoregressive models. Scand J Statist. 2014;41:1136–1152] and Lee and Lee [CUSUM tests for general nonlinear inter-valued GARCH models: comparison study. Ann Inst Stat Math. 2019;71:1033–1057.] as well as the CUSUM of squares test based on standardized residuals.

Taking correlations in GPS least squares adjustments into account with a diagonal covariance matrix

Based on the results of Luati and Proietti (Ann Inst Stat Math 63:673–686, 2011) on an equivalence for a certain class of polynomial regressions between the diagonally weighted least squares (DWLS) and the generalized least squares (GLS) estimator, an alternative way to take correlations into account thanks to a diagonal covariance matrix is presented. The equivalent covariance matrix is much easier to compute than a diagonalization of the covariance matrix via eigenvalue decomposition which also implies a change of the least squares equations. This condensed matrix, for use in the least squares adjustment, can be seen as a diagonal or reduced version of the original matrix, its elements being simply the sums of the rows elements of the weighting matrix. The least squares results obtained with the equivalent diagonal matrices and those given by the fully populated covariance matrix are mathematically strictly equivalent for the mean estimator in terms of estimate and its a priori cofactor matrix. It is shown that this equivalence can be empirically extended to further classes of design matrices such as those used in GPS positioning (single point positioning, precise point positioning or relative positioning with double differences). Applying this new model to simulated time series of correlated observations, a significant reduction of the coordinate differences compared with the solutions computed with the commonly used diagonal elevation-dependent model was reached for the GPS relative positioning with double differences, single point positioning as well as precise point positioning cases. The estimate differences between the equivalent and classical model with fully populated covariance matrix were below the mm for all simulated GPS cases and below the sub-mm for the relative positioning with double differences. These results were confirmed by analyzing real data. Consequently, the equivalent diagonal covariance matrices, compared with the often used elevation-dependent diagonal covariance matrix is appropriate to take correlations in GPS least squares adjustment into account, yielding more accurate cofactor matrices of the unknown.

Bootstrapping sample quantiles of discrete data

Sample quantiles are consistent estimators for the true quantile and satisfy central limit theorems (CLTs) if the underlying distribution is continuous. If the distribution is discrete, the situation is much more delicate. In this case, sample quantiles are known to be not even consistent in general for the population quantiles. In a motivating example, we show that Efron’s bootstrap does not consistently mimic the distribution of sample quantiles even in the discrete independent and identically distributed (i.i.d.) data case. To overcome this bootstrap inconsistency, we provide two different and complementing strategies. In the first part of this paper, we prove that $$m$$ -out-of- $$n$$ -type bootstraps do consistently mimic the distribution of sample quantiles in the discrete data case. As the corresponding bootstrap confidence intervals tend to be conservative due to the discreteness of the true distribution, we propose randomization techniques to construct bootstrap confidence sets of asymptotically correct size. In the second part, we consider a continuous modification of the cumulative distribution function and make use of mid-quantiles studied in Ma et al. (Ann Inst Stat Math 63:227–243, 2011). Contrary to ordinary quantiles and due to continuity, mid-quantiles lose their discrete nature and can be estimated consistently. Moreover, Ma et al. (Ann Inst Stat Math 63:227–243, 2011) proved (non-)central limit theorems for i.i.d. data, which we generalize to the time series case. However, as the mid-quantile function fails to be differentiable, classical i.i.d. or block bootstrap methods do not lead to completely satisfactory results and $$m$$ -out-of- $$n$$ variants are required here as well. The finite sample performances of both approaches are illustrated in a simulation study by comparing coverage rates of bootstrap confidence intervals.

Empirical likelihood for semivarying coefficient model with measurement error in the nonparametric part

A semivarying coefficient model with measurement error in the nonparametric part was proposed by Feng and Xue (Ann Inst Stat Math 66:121–140, 2014), but its inferences have not been systematically studied. This paper applies empirical likelihood method to construct confidence regions/intervals for the regression parameter and coefficient function. When some auxiliary information about the parametric part is available, the empirical log-likelihood ratio statistic for the regression parameter is introduced based on the corrected local linear estimator of the coefficient function. Furthermore, corrected empirical log-likelihood ratio statistic for coefficient function is also investigated with the use of auxiliary information. The limiting distributions of the resulting statistics both for the regression parameter and coefficient function are shown to have standard Chi-squared distribution. Simulation experiments and a real data set are presented to evaluate the finite sample performance of our proposed method.

Interval estimation for the Gini index of shifted exponential distribution under Type-II double censoring

Given a shifted exponential distribution, the exact sampling distribution of the Gini index has been derived by Moothathu (Ann Inst Stat Math 37: 473–479, 1985). However, direct derivation of confidence interval of the Gini index via inverting the sampling distribution requires highly intensive computational power. In this note, we propose an exact interval estimation procedure through the concept of generalized confidence interval introduced by Weerahandi (J Am Stat Assoc 88: 899–905, 1993). The proposed method is much more computationally efficient and can be readily extended for Type-II doubly censored data. The idea is further developed for the interval estimation of the difference of Gini indexes from two shifted exponential distributions. Numerical studies show that the generalized interval estimator is better than commonly used asymptotic approximation and bootstrap in term of coverage probability, particularly for small to median sample size. The proposed approaches are illustrated via an application to a previously published data set.

A skew INAR(1) process on $${\mathbb {Z}}$$ Z

Integer-valued time series models have been a recurrent theme considered in many papers in the last three decades, but only a few of them have dealt with models on $${\mathbb {Z}}$$ (that is, including both negative and positive integers). Our aim in this paper is to introduce a first-order, integer-valued autoregressive process on $${\mathbb {Z}}$$ with skew discrete Laplace marginals (Kozubowski and Inusah, Ann Inst Stat Math 58:555–571, 2006). For this, we define a new operator that acts on two independent latent processes, similarly as made by Freeland (Adv Stat Anal 94:217–229, 2010). We derive some joint and conditional basic properties of the proposed process such as characteristic function, moments, higher-order moments and jumps. Estimators for the parameters of our model are proposed and their asymptotic normality is established. We run a Monte Carlo simulation to evaluate the finite-sample performance of these estimators. In order to illustrate the potential for practice of our process we apply it to a real data set about stock market.

Second-order least-squares estimation for regression models with autocorrelated errors

In their recent paper, Wang and Leblanc (Ann Inst Stat Math 60:883---900, 2008) have shown that the second-order least squares estimator (SLSE) is more efficient than the ordinary least squares estimator (OLSE) when the errors are independent and identically distributed with non zero third moments. In this paper, we generalize the theory of SLSE to regression models with autocorrelated errors. Under certain regularity conditions, we establish the consistency and asymptotic normality of the proposed estimator and provide a simulation study to compare its performance with the corresponding OLSE and generalized least square estimator (GLSE). It is shown that the SLSE performs well giving relatively small standard error and bias (or the mean square error) in estimating parameters of such regression models with autocorrelated errors. Based on our study, we conjecture that for less correlated data, the standard errors of SLSE lie between those of the OLSE and GLSE which can be interpreted as adding the second moment information can improve the performance of an estimator.

Understanding sufficiency in one-way random effects model

In this paper we consider one-way random effects analysis of variance model and examine the concept of sufficiency while comparing different allocation designs for the observations under the model. We explicitly provide the transformation leading to sufficiency of ‘optimal’ allocations as characterized in DeGroot (Ann Inst Stat Math 18:13–28, 1966) and Stepniak (Ann Inst Stat Math 34:175–180, 1982).

Robust second-order least-squares estimator for regression models

The second-order least-squares estimator (SLSE) was proposed by Wang (Statistica Sinica 13:1201–1210, 2003) for measurement error models. It was extended and applied to linear and nonlinear regression models by Abarin and Wang (Far East J Theor Stat 20:179–196, 2006) and Wang and Leblanc (Ann Inst Stat Math 60:883–900, 2008). The SLSE is asymptotically more efficient than the ordinary least-squares estimator if the error distribution has a nonzero third moment. However, it lacks robustness against outliers in the data. In this paper, we propose a robust second-order least squares estimator (RSLSE) against X-outliers. The RSLSE is highly efficient with high breakdown point and is asymptotically normally distributed. We compare the RSLSE with other estimators through a simulation study. Our results show that the RSLSE performs very well.

Testing the independence of maxima: from bivariate vectors to spatial extreme fields

Characterizing the behaviour of multivariate or spatial extreme values is of fundamental interest to understand how extreme events tend to occur. In this paper we propose to test for the asymptotic independence of bivariate maxima vectors. Our test statistic is derived from a madogram, a notion classically used in geostatistics to capture spatial structures. The test can be applied to bivariate vectors, and a generalization to the spatial context is proposed. For bivariate vectors, a comparison to the test by Falk and Michel (Ann Inst Stat Math 58:261–290, 2006) is conducted through a simulation study. In the spatial case, special attention is paid to pairwise dependence. A multiple test procedure is designed to determine at which lag asymptotic independence takes place. This new procedure is based on the bootstrap distribution of the number of times the null hypothesis is rejected. It is then tested on maxima of three classical spatial models and finally applied to two climate datasets.

Chernoff distance for truncated distributions

In the present paper we extend the definition of Chernoff distance considered in Akahira (Ann Inst Stat Math 48:349–364, 1996) for truncated distributions and examine its properties. The relationship of this measure with other discrimination measures is examined. We study Chernoff distance between the original and weighted distributions. We also provide a characterization result for the proportional hazards model using the functional form of Chernoff distance.

Mixtures of power series distributions: identifiability via uniqueness in problems of moments

We treat the identifiability problem for mixtures involving power series distributions. Applying an idea of Sapatinas (Ann Inst Stat Math 47:447–459, 1995) we prove and elaborate that a mixture distribution is identifiable if a certain Stieltjes problem of moments has a unique solution while a non-uniqueness leads to a non- identifiable mixture. We describe explicitly models of identifiable mixtures and models of non-identifiable mixtures. Illustrative examples and comments on related questions are also given.