Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck (ROU) processes based on continuous observations. Both the cases with one-sided barrier and two-sided barriers are considered. We derive the explicit formulas for the estimators, and then prove their strong consistency and asymptotic normality. Moreover, the bias and mean square errors are represented in terms of the solutions to some PDEs with homogeneous Neumann boundary conditions. We also illustrate the asymptotic behavior of the estimators through a simulation study.     

Multivariate normal estimation: the case (n < p)

Estimation in the multivariate context when the number of observations available is less than the number of variables is a classical theoretical problem. In order to ensure estimability, one has to assume certain constraints on the parameters. A method for maximum likelihood estimation under constraints is proposed to solve this problem. Even in the extreme case where only a single multivariate observation is available, this may provide a feasible solution. It simultaneously provides a simple, straightforward methodology to allow for specific structures within and between covariance matrices of several populations. This methodology yields exact maximum likelihood estimates.     

Further identities for the Wishart distribution with applications in regression

Matrix analogues are given for a known scalar identity which relates certain expectations with respect to the Wishart distribution. (The scalar identity was independently derived by C. Stein and L. Haff.) The matrix analogues are more aptly called “matrix extensions.” They can be derived by using the scalar identity; nevertheless, they are seen (in quite elementary terms) to be more general than the latter. A method of doing multivariate calculations is developed from the identities, and several examples are worked in detail. We compute the first two moments of the regression coefficients and another matrix arising in regression analysis. Also, we give a new result for the matrix analogue of squared multiple correlation: the bias correction of Ezekiel (1930), a result often used in model building, is extended to the case of two or more dependent variables.     

Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution

Using the techniques developed by Subrahmaniam and Ching’anda (1978), we study the robustness to nonnormality of the linear discriminant functions. It is seen that the LDF procedure is quite robust against the likelihood ratio rule. The latter yields in all cases much smaller overall error rates; however, the disparity between the error rates of the LDF and LR procedures is not large enough to warrant the recommendation to use the more complicated LR procedure.     

Maximum likelihood estimation for arma models in the presence of ARMA errors

An ARMA(p, q) process observed with an ARMA(c, d) error has an ARMA (p + c, k) representation with k = max(c + q, p + d) whose parameters satisfy some nonlinear constraints. Identification of the model is discussed. We develop Newton-Raphson estimators for the ARMA(p + c, k) process which take advantage of the information contained in the nonlinear restrictions. Explicit expressions for the derivatives of the restrictions are derived.     

Maximum likelihood parameter estimation in the three-parameter gamma distribution with the use of Mathematica

The number of solutions of the system of the log-likelihood equations for the three-parameter case is still an open problem. Several methods have been developed for finding the solutions. In this article we present a program in Mathematica that can find all the solutions of the system of equations. Furthermore, we examine the case where the global maximum appears at the boundary of the domain of the log-likelihood function and we prove that any consistent estimators appear at the interior with probability tending to one.     

Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes with jumps

In this paper, we investigate the maximum likelihood estimation for the reflected Ornstein-Uhlenbeck processes with jumps based on continuous observations. We derive likelihood functions by using semimartingale theory. From this we get explicit formulas for estimators. The strong consistence and asymptotic normality of estimators are proved by using the method of stochastic integration.     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

Maximum likelihood estimation for directional conditionally autoregressive models

A spatial process observed over a lattice or a set of irregular regions is usually modeled using a conditionally autoregressive (CAR) model. The neighborhoods within a CAR model are generally formed using only the inter-distances or boundaries between the regions. To accommodate directional spatial variation, a new class of spatial models is proposed using different weights given to neighbors in different directions. The proposed model generalizes the usual CAR model by accounting for spatial anisotropy. Maximum likelihood estimators are derived and shown to be consistent under some regularity conditions. Simulation studies are presented to evaluate the finite sample performance of the new model as compared to the CAR model. Finally, the method is illustrated using a data set on the crime rates of Columbus, OH and on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Maximum likelihood estimation in the proportional hazards model of random censorship

The maximum likelihood estimator (MLE) for the survival function S_Tunder the proportional hazards model of censorship is derived and shown to differ from the Abdushukurov-Cheng-Lin estimator when the class of allowable distributions includes all continuous and discrete distributions. The estimators are compared via an example. The MLE is calculated using a Newton-Raphson iterative procedure and implemented via a FORTRAN algorithm.     

Maximum likelihood estimation for the generalized poisson distribution

The generalized Poisson distribution;containing two

parameters and studied by many researchers; describes the distribution of busy periods under a queueing system and has very interesting properties; The probabilities for successive classes depend upon the previous occurrences; The problem of admissible maximum likelihood estimators for for the parameters Is discussed and a necessary and sufficient condition is derived for which unique admissible maximum likelihood estimators exist; The first; order terms in the biases; variances and the covariance of these maximum likelihood estimators are obtained.     

Maximum likelihood estimation for the proportional odds model with mixed interval-censored failure time data

This article discusses regression analysis of mixed interval-censored failure time data. Such data frequently occur across a variety of settings, including clinical trials, epidemiologic investigations, and many other biomedical studies with a follow-up component. For example, mixed failure times are commonly found in the two largest studies of long-term survivorship after childhood cancer, the datasets that motivated this work. However, most existing methods for failure time data consider only right-censored or only interval-censored failure times, not the more general case where times may be mixed. Additionally, among regression models developed for mixed interval-censored failure times, the proportional hazards formulation is generally assumed. It is well-known that the proportional hazards model may be inappropriate in certain situations, and alternatives are needed to analyze mixed failure time data in such cases. To fill this need, we develop a maximum likelihood estimation procedure for the proportional odds regression model with mixed interval-censored data. We show that the resulting estimators are consistent and asymptotically Gaussian. An extensive simulation study is performed to assess the finite-sample properties of the method, and this investigation indicates that the proposed method works well for many practical situations. We then apply our approach to examine the impact of age at cranial radiation therapy on risk of growth hormone deficiency in long-term survivors of childhood cancer.     

Maximum likelihood estimation for generalized conditionally autoregressive models of spatial data

Conditionally autoregressive (CAR) models are often used to analyze a spatial process observed over a lattice or a set of irregular regions. The neighborhoods within a CAR model are generally formed deterministically using the inter-distances or boundaries between the regions. To accommodate directional and inherent anisotropy variation, a new class of spatial models is proposed that adaptively determines neighbors based on a bivariate kernel using the distances and angles between the centroid of the regions. The newly proposed model generalizes the usual CAR model in a sense of accounting for adaptively determined weights. Maximum likelihood estimators are derived and simulation studies are presented for the sampling properties of the estimates on the new model, which is compared to the CAR model. Finally the method is illustrated using a data set on the elevated blood lead levels of children under the age of 72 months observed in Virginia in the year of 2000.     

Maximum penalized likelihood estimation for skew-normal and skew-t distributions

The skew-normal and the skew-t distributions are parametric families which are currently under intense investigation since they provide a more flexible formulation compared to the classical normal and t distributions by introducing a parameter which regulates their skewness. While these families enjoy attractive formal properties from the probability viewpoint, a practical problem with their usage in applications is the possibility that the maximum likelihood estimate of the parameter which regulates skewness diverges. This situation has vanishing probability for increasing sample size, but for finite samples it occurs with non-negligible probability, and its occurrence has unpleasant effects on the inferential process. Methods for overcoming this problem have been put forward both in the classical and in the Bayesian formulation, but their applicability is restricted to simple situations. We formulate a proposal based on the idea of penalized likelihood, which has connections with some of the existing methods, but it applies more generally, including the multivariate case.     

Penalized maximum likelihood estimation in the modified extended Weibull distribution

We address the issue of performing inference on the parameters that index the modified extended Weibull (MEW) distribution. We show that numerical maximization of the MEW log-likelihood function can be problematic. It is even possible to encounter maximum likelihood estimates that are not finite, that is, it is possible to encounter monotonic likelihood functions. We consider different penalization schemes to improve maximum likelihood point estimation. A penalization scheme based on the Jeffreys’ invariant prior is shown to be particularly useful. Simulation results on point estimation, interval estimation, and hypothesis testing inference are presented. Two empirical applications are presented and discussed.     

Maximum likelihood estimation in the presence of outiliers

This paper deals with the maximum likelihood estimation of parameters when the sample (x₁…x_n ) may heve k spuriously generated observations from another distribution, say G≠F, where F is the distribution of the target population. If G is stochastically larger than F, then these k observations may give rise to k extreme observations or ‘outliers’. This situation is often described by a so-called ‘k-outlier model’ in which in addition to the parameters involved in F and G, the set ν={ν₁,…,ν_k} of indices, for which x_νj , j=1,…,k, come from G, is also unknow.     

Unimodality of the pareto distribution likelihood function for multicensored samples and implications for estimation

This paper discusses maximum likelihood parameter estimation in the Pareto distribution for multicensored samples. In particu-

lar, the modality of the associated conditional log-likelihood function is investigated in order to resolve questions concerninc

the existence and uniqurneas of the lnarimum likelihood estimates.For the cases with one parameter known, the maximum likelihood

estimates of the remaining unknown parameters are shown to exist and to be unique. When both parameters are unknown, the maximum likelihood estimates may or may not exist and be unique. That is, their existence and uniqueness would seem to depend solely upon the information inherent in the sample data. In viav of the possible nonexistence and/or non-uniqueness of the maximum likelihood estimates when both parameters are unknown, alternatives to standard iterative numerical methods are explored.     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Maximum likelihood estimation of dirichlet distributions

Global concavity of the likelihood function is proved by means of an inequality involving the trigamma function. The computation of maximum likelihood estimates is discussed.