Modern Stochastics: Theory and Applications

Consistent estimators of the baseline hazard rate and the regression parameter are constructed in the Cox proportional hazards model with heteroscedastic measurement errors, assuming that the baseline hazard function belongs to a certain class of functions with bounded Lipschitz constants.

In this paper, a sample estimator of the tangency portfolio (TP) weights is considered. The focus is on the situation where the number of observations is smaller than the number of assets in the portfolio and the returns are i.i.d. normally distributed. Under these assumptions, the sample covariance matrix follows a singular Wishart distribution and, therefore, the regular inverse cannot be taken. In the paper, bounds and approximations for the first two moments of the estimated TP weights are derived, as well as exact results are obtained when the population covariance matrix is equal to the identity matrix, employing the Moore–Penrose inverse. Moreover, exact moments based on the reflexive generalized inverse are provided. The properties of the bounds are investigated in a simulation study, where they are compared to the sample moments. The difference between the moments based on the reflexive generalized inverse and the sample moments based on the Moore–Penrose inverse is also studied.

A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag τ. The main contributions are asymptotic results that can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. This class is extended, and the unmixing estimator is considered under stronger dependency structures. In particular, the asymptotic behavior of the unmixing estimator is estimated for both long- and short-range dependent complex-valued processes. Consequently, this theory covers unmixing estimators that converge slower than the usual $\sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful preprocessing tool and highly applicable in several fields of statistics.

A linear structural regression model is studied, where the covariate is observed with a mixture of the classical and Berkson measurement errors. Both variances of the classical and Berkson errors are assumed known. Without normality assumptions, consistent estimators of model parameters are constructed and conditions for their asymptotic normality are given. The estimators are divided into two asymptotically independent groups.

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.

A bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with copula-joint innovations is studied. Different parameter estimation methods are analyzed and compared via Monte Carlo simulations with emphasis on estimation of the copula dependence parameter. An empirical application on defaulted and non-defaulted loan data is carried out using different combinations of copula functions and marginal distribution functions covering the cases where both marginal distributions are from the same family, as well as the case where they are from different distribution families.

This paper deals with a homoskedastic errors-in-variables linear regression model and properties of the total least squares (TLS) estimator. We partly revise the consistency results for the TLS estimator previously obtained by the author [18]. We present complete and comprehensive proofs of consistency theorems. A theoretical foundation for construction of the TLS estimator and its relation to the generalized eigenvalue problem is explained. Particularly, the uniqueness of the estimate is proved. The Frobenius norm in the definition of the estimator can be substituted by the spectral norm, or by any other unitarily invariant norm; then the consistency results are still valid.

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of X, which, in the case of normal errors, coincides with the maximum likelihood one. We give conditions for asymptotic normality of the estimator when the number of rows in A is increasing. Under mild assumptions, the covariance structure of the limit Gaussian random matrix is nonsingular. For normal errors, the results can be used to construct an asymptotic confidence interval for a linear functional of X.

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated.

In this paper we define the consistent criteria of hypotheses such as the probability of any kind of errors is zero for given criteria. We prove necessary and sufficient conditions for the existence of such criteria.