Pooling multivariate data under W, LR and LM tests

Two independent random samples are drawn from two multivariate normal populations with mean vectors μ1 and μ2 and a common variance-covariance matrix Σ. Ahmed and Saleh (1990) considered preliminary test maximum likelihood estimator (PMLTE) for estimating μ1 based on the Hotelling's T _N ², when it is suspected that μ1=μ2. In this paper, the PTMLE based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. Using the quadratic risk function, the conditions of superiority of the proposed estimator for departure parameter are derived. A max-min rule for the size of the preliminary test of significance is presented. It is demonstrated that the PTMLE based on W test produces the highest minimum guaranteed efficiencies compared to UMLE among the three test procedures.     

ASYMPTOTIC LIKELIHOOD APPROXIMATIONS USING A PARTIAL LAPLACE APPROXIMATION

Elimination of a nuisance variable is often non‐trivial and may involve the evaluation of an intractable integral. One approach to evaluate these integrals is to use the Laplace approximation. This paper concentrates on a new approximation, called the partial Laplace approximation, that is useful when the integrand can be partitioned into two multiplicative disjoint functions. The technique is applied to the linear mixed model and shows that the approximate likelihood obtained can be partitioned to provide a conditional likelihood for the location parameters and a marginal likelihood for the scale parameters equivalent to restricted maximum likelihood (REML). Similarly, the partial Laplace approximation is applied to the t‐distribution to obtain an approximate REML for the scale parameter. A simulation study reveals that, in comparison to maximum likelihood, the scale parameter estimates of the t‐distribution obtained from the approximate REML show reduced bias.     

Evaluation of trace evidence in the form of multivariate data

Summary.  The evaluation of measurements on characteristics of trace evidence found at a crime scene and on a suspect is an important part of forensic science. Five methods of assessment for the value of the evidence for multivariate data are described. Two are based on significance tests and three on the evaluation of likelihood ratios. The likelihood ratio which compares the probability of the measurements on the evidence assuming a common source for the crime scene and suspect evidence with the probability of the measurements on the evidence assuming different sources for the crime scene and suspect evidence is a well-documented measure of the value of the evidence. One of the likelihood ratio approaches transforms the data to a univariate projection based on the first principal component. The other two versions of the likelihood ratio for multivariate data account for correlation among the variables and for two levels of variation: that between sources and that within sources. One version assumes that between-source variability is modelled by a multivariate normal distribution; the other version models the variability with a multivariate kernel density estimate. Results are compared from the analysis of measurements on the elemental composition of glass.     

A multistate approach for estimating the incidence of human immunodeficiency virus by using data from a prevalent cohort study

Summary.  Cohort studies of individuals infected with the human immunodeficiency virus (HIV) provide useful information on the past pattern of HIV diagnoses, progression of the disease and use of antiretroviral therapy. We propose a new method for using individual data from an open prevalent cohort study to estimate the incidence of HIV, by jointly modelling the HIV diagnosis, the inclusion in the cohort and the progression of the disease in a Markov model framework. The estimation procedure involves the construction of a likelihood function which takes into account the probability of observing the total number of subjects who are enrolled in the cohort and the probabilities of passage through the stages of disease for each observed subject conditionally on being included in the cohort. The estimator of the HIV infection rate is defined as the function which maximizes a penalized likelihood, and the solution of this maximization problem is approximated on a basis of cubic M -splines. The method is illustrated by using cohort data from a hospital-based surveillance system of HIV infection in Aquitaine, a region of south-western France. A simulation study is performed to study the ability of the model to reconstruct the incidence of HIV from prevalent cohort data.     

A structured variational learning approach for switching latent factor models

A data-driven approach for modeling volatility dynamics and co-movements in financial markets is introduced. Special emphasis is given to multivariate conditionally heteroscedastic factor models in which the volatilities of the latent factors depend on their past values, and the parameters are driven by regime switching in a latent state variable. We propose an innovative indirect estimation method based on the generalized EM algorithm principle combined with a structured variational approach that can handle models with large cross-sectional dimensions. Extensive Monte Carlo simulations and preliminary experiments with financial data show promising results.     

关于Gauss-Weierstrass算子线性组合的同时逼近

为了提高正线性算子 Gauss-Weierstrass 算子的逼近阶,往往采用线性组合的方法.本文主要研究了一类 Gauss-Weierstrass 算子线性组合的同时逼近问题,在一致逼近的意义下,给出了逼近的正定理、逆定理及特征刻划.即我们得到了如下结果:设 f∈C_(-∞,+∞),f~(m)(x)存在,W_(n,r)(f;x)表示 Gauss-Weierstrass 算子的一种线性组合,则当 a<2r 时,有(i)‖W_(n,r)~(m)(f;x)-f~(m)‖≤M[ω_(2r)(fn~(-1/2))+n~(-r)];(ii) k_(2r)(f~(m);n~(-r))≤‖W_(k,r)~(m)(f;x)-f_(x)~(m)‖+M(k/n)~rk_(2r)(f~(m);k~(-r));(iii)‖W_(n,r)~(m)(f;x)-f~(m)‖=O(n~(-(a/2))ω_(2r)(f~(m);h)=O(h~a).     

The effects of different choices of order for autoregressive approximation on the Gaussian likelihood estimates for ARMA models

This paper investigates, by means of Monte Carlo simulation, the effects of different choices of order for autoregressive approximation on the fully efficient parameter estimates for autoregressive moving average models. Four order selection criteria, AIC, BIC, HQ and PKK, were compared and different model structures with varying sample sizes were used to contrast the performance of the criteria. Some asymptotic results which provide a useful guide for assessing the performance of these criteria are presented. The results of this comparison show that there are marked differences in the accuracy implied using these alternative criteria in small sample situations and that it is preferable to apply BIC criterion, which leads to greater precision of Gaussian likelihood estimates, in such cases. Implications of the findings of this study for the estimation of time series models are highlighted.     

Bayesian Analysis of DNA Profiling Data in Forensic Identification Applications

The utilization of DNA evidence in cases of forensic identification has become widespread over the last few years. The strength of this evidence against an individual standing trial is typically presented in court in the form of a likelihood ratio (LR) or its reciprocal (the profile match probability). The value of this LR will vary according to the nature of the genetic relationship between the accused and other possible perpetrators of the crime in the population. This paper develops ideas and methods for analysing data and evaluating LRs when the evidence is based on short tandem repeat profiles, with special emphasis placed on a Bayesian approach. These are then applied in the context of a particular quadruplex profiling system used for routine case-work by the UK Forensic Science Service.     

Likelihood Estimation for Censored Random Vectors

This article shows how to construct a likelihood for a general class of censoring problems. This likelihood is proven to be valid, i.e. its maximizer is consistent and the respective root-n estimator is asymptotically efficient and normally distributed under regularity conditions. The method generalizes ordinary maximum likelihood estimation as well as several standard estimators for censoring problems (e.g. tobit type I-tobit type V).     

Confidence Intervals Based on Local Linear Smoother

Point-wise confidence intervals for a non-parametric regression function in conjunction with the popular local linear smoother are considered. The confidence intervals are based on the asymptotic normal distribution of the local linear smoother. Their coverage accuracy is evaluated by developing Edgeworth expansion for the coverage probability. It is found that the coverage error near the boundary of the support of the regression function is of a larger order than that in the interior, which implies that the local linear smoother is not adaptive to the boundary in terms of coverage. This is quite unexpected as the local linear smoother is adaptive to the boundary in terms of the mean squared error.