Maximum likelihood estimators: Nonparametric approach

In the paper, the maximum likelihood estimator for an unknown density is investigated. We deal with censored observations and construct a confidence region for this density. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 220–228.     

Comparison of estimators in stable models

In this paper, several estimators of the index and scale parameter of a symmetric univariate stable distribution are analyzed and compared. It turns out that minimum distance estimators with a suitable choice of the weight function have good theoretical and empirical performance. Also some modifications of the moment method are reliable in large parts of the parameter space. The classical Hill estimator, however, works well only for a small stability index and for an extremely large number of observations. One can construct tail-estimators with a considerably improved behaviour compared to the Hill estimator if the scale is known.     

Nonparametric estimators of the survival function with twice censored data

Patilea and Rolin (Ann Stat 34(2):925–938, 2006) proposed a product-limit estimator of the survival function for twice censored data. In this article, based on a modified self-consistent (MSC) approach, we propose an alternative estimator, the MSC estimator. The asymptotic properties of the MSC estimator are derived. A simulation study is conducted to compare the performance between the two estimators. Simulation results indicate that the MSC estimator outperforms the product-limit estimator and its advantage over the product-limit estimator can be very significant when right censoring is heavy.     

Nonparametric maximum likelihood density estimation and simulation-based minimum distance estimators

Indirect inference estimators (i.e., simulation-based minimum distance estimators) in a parametric model that are based on auxiliary nonparametric maximum likelihood density estimators are shown to be asymptotically normal. If the parametricmodel is correctly specified, it is furthermore shown that the asymptotic variance-covariance matrix equals the inverse of the Fisher-information matrix. These results are based on uniform-in-parameters convergence rates and a uniform-inparameters Donsker-type theorem for nonparametric maximum likelihood density estimators.     

Nonparametric significance tests for imperfect repair

Given the failure history for $K\ge 2$ independent and identical repairable systems, a nonparametric procedure is presented to test the null hypothesis of minimal repair (MR) against the alternative of imperfect repair. The main idea is that, under non-harmful (harmful) repair, systems that failed later (earlier) are more reliable than those that have failed (later). This fact allows one, at any moment in time, to rank the systems from more to less reliable and, hence, to define a vector that counts how many times the system ranked $r$ failed. When all the systems are time-truncated at the same time $T$, it is shown that, under the null hypothesis of MR, the vector of counts follows a multinomial distribution with class probabilities $p_r=K^{-1}$ for $r=1,\dots ,K$. The test proceeds by computing a chi-bar squared test statistic similar to the one used to test one-sided alternatives in the multinomial setup, which allows us to compute $p$-values using either asymptotic theory or a straightforward Monte Carlo simulation using the null multinomial distribution. Extension to the case of different truncation times is also discussed. The procedure is applied to two real datasets regarding equipment used in the mining industry.     

Confidence models of incomplete preferences

This paper introduces a new class of representations for incomplete preferences called confidence models. Confidence models describe decision makers who behave as if they have probabilistic uncertainty over their true preferences, and are only willing to express a binary preference if it is sufficiently likely to hold. Confidence models provide a natural way to connect incomplete preferences with stochastic choice. This connection is characterized by a simple joint condition on an incomplete preference relation and a random choice rule.     

Admissible linear estimators in restricted linear models

Necessary and sufficient conditions are established for a linear estimator to be admissible among the set of all homogeneous and inhomogeneous, linear estimators under a linear model with the vector of parameters subject to linear restrictions. These conditions are then utilized to characterize influence that restrictions involved in a linear model have on the class of admissible linear estimators.     

Thresholding projection estimators in functional linear models

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows us to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits us to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove that these estimators are minimax and rates of convergence are given for some particular cases.     

Semiparametric efficient estimators in heteroscedastic error models

In the mean regression context, this study considers several frequently encountered heteroscedastic error models where the regression mean and variance functions are specified up to certain parameters. An important point we note through a series of analyses is that different assumptions on standardized regression errors yield quite different efficiency bounds for the corresponding estimators. Consequently, all aspects of the assumptions need to be specifically taken into account in constructing their corresponding efficient estimators. This study clarifies the relation between the regression error assumptions and their, respectively, efficiency bounds under the general regression framework with heteroscedastic errors. Our simulation results support our findings; we carry out a real data analysis using the proposed methods where the Cobb–Douglas cost model is the regression mean.

     

Strong convergence of estimators in nonlinear autoregressive models

In the paper we prove rates of strong convergence of M-estimators for the parameters in a general nonlinear autoregressive model. In the proofs we utilize a variational principle from stochastic optimization theory which was proved by Shapiro (Ann. Oper. Res. 30 (1991) 169). The application of the general theory is illustrated in the case of continuous threshold models.     

Weak consistency of least-squares estimators in linear models

Let Y_n, n≥1, be a sequence of integrable random variables with EY_n = x_n1β₁ + x_n2β₂ + … + x_npβ_p, where the x_ij's are known and β^T = (β₁, β₂,…, β_p) unknown. Let b_n be the least-squares estimator of β based on Y₁, Y₂,…, Y_n. Weak consistency of b_n, n≥1, has been considered in the literature under the assumption that each Y_n is square integrable. In this paper, we study weak consistency of b_n, n≥1, and associated rates of convergence under the minimal assumption that each Y_n is integrable.     

Sensitivity of GLS estimators in random effects models

This paper studies the sensitivity of random effects estimators in the one-way error component regression model. Maddala and Mount (1973) [6] give simulation evidence that in random effects models the properties of the feasible GLS estimator are not affected by the choice of the first-step estimator used for the covariance matrix. Taylor (1980) [8] gives a theoretical example of this effect. This paper provides a reason for this in terms of sensitivity. The properties of are transferred via an uncorrelated (and independent under normality) link, called sensitivity. The sensitivity statistic counteracts the improvement in . A Monte Carlo experiment illustrates the theoretical findings.     

Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.      

James-Stein estimators for time series regression models

The least squares (LS) estimator seems the natural estimator of the coefficients of a Gaussian linear regression model. However, if the dimension of the vector of coefficients is greater than 2 and the residuals are independent and identically distributed, this conventional estimator is not admissible. James and Stein [Estimation with quadratic loss, Proceedings of the Fourth Berkely Symposium vol. 1, 1961, pp. 361-379] proposed a shrinkage estimator (James-Stein estimator) which improves the least squares estimator with respect to the mean squares error loss function. In this paper, we investigate the mean squares error of the James-Stein (JS) estimator for the regression coefficients when the residuals are generated from a Gaussian stationary process. Then, sufficient conditions for the JS to improve the LS are given. It is important to know the influence of the dependence on the JS. Also numerical studies illuminate some interesting features of the improvement. The results have potential applications to economics, engineering, and natural sciences.     

Nonparametric regression estimation in models with weak error's structure

In this paper we propose nonparametric estimates of the regression function and its derivative when it is only assumed a weak error's structure. We study their local and global asymptotic behaviour when we observe dependent trajectories.     

Nonparametric estimation of competing risks models with covariates

In competing risks model, several failure times arise potentially. The smallest failure time and its index only are observed. Without specific assumptions, the joint or even the marginal distribution functions of the underlying failure times are not identifiable (A. Tsiatis, Proc. Natl. Acad. Sci. USA 72 (1975) 20). Nonetheless, if each individual is characterized by a “sufficiently informative” set of covariates, these distributions are identifiable under some conditions of regularity (J.J. Heckman and B. Honoré, Biometrika 76 (1989) 325). In this paper, nonparametric kernel estimators of the joint distribution function of failure times conditional on the covariates are proposed. Their weak and strong consistency are discussed.     

Efficient estimators and LAN in canonical bivariate POT models

Bivariate generalized Pareto distributions (GPs) with uniform margins are introduced and elementary properties such as peaks-over-threshold (POT) stability are discussed. A unified parameterization with parameter ?∈[0,1] of the GPs is provided by their canonical parameterization. We derive efficient estimators of ? and of the dependence function of the GP in various models and establish local asymptotic normality (LAN) of the loglikelihood function of a 2×2 table sorting of the observations. From this result we can deduce that the estimator of ? suggested by Falk and Reiss (2001, Statist. Probab. Lett. 52, 233-242) is not efficient, whereas a modification actually is.