Non parametric estimation of the structural expectation of a stochastic increasing function

This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So, we define the structural expectation of the underlying stochastic function. Then, we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.     

Characteristic function estimation of non-Gaussian Ornstein–Uhlenbeck processes

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop efficient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

A non parametric CUSUM control chart based on the Mann–Whitney statistic

We consider a novel univariate non parametric cumulative sum (CUSUM) control chart for detecting the small shifts in the mean of a process, where the nominal value of the mean is unknown but some historical data are available. This chart is established based on the Mann–Whitney statistic as well as the change-point model, where any assumption for the underlying distribution of the process is not required. The performance comparisons based on simulations show that the proposed control chart is slightly more effective than some other related non parametric control charts.     

Nonparametric estimation of the multivariate survivor function: the multivariate Kaplan–Meier estimator

The Dabrowska (Ann Stat 16:1475–1489, 1988) product integral representation of the multivariate survivor function is extended, leading to a nonparametric survivor function estimator for an arbitrary number of failure time variates that has a simple recursive formula for its calculation. Empirical process methods are used to sketch proofs for this estimator’s strong consistency and weak convergence properties. Summary measures of pairwise and higher-order dependencies are also defined and nonparametrically estimated. Simulation evaluation is given for the special case of three failure time variates.     

Bimodal Birnbaum–Saunders distribution with applications to non negative measurements

In this article, we introduce a new extension of the Birnbaum–Saunders (BS) distribution as a follow-up to the family of skew-flexible-normal distributions. This extension produces a family of BS distributions including densities that can be unimodal as well as bimodal. This flexibility is important in dealing with positive bimodal data, given the difficulties experienced by the use of mixtures of distributions. Some basic properties of the new distribution are studied including moments. Parameter estimation is approached by the method of moments and also by maximum likelihood, including a derivation of the Fisher information matrix. Three real data illustrations indicate satisfactory performance of the proposed model.     

On the Gerber–Shiu function with random discount rate

In this paper, we study the Gerber–Shiu (G-S) function for the classical risk model, in which the discount rate is generalized from a constant to a random variable. The discounted interest force accumulated process is modeled by a Poisson process and a Gaussian process for the G-S function. In terms of the standard techniques in ruin theory, we derive the integro-differential equation and the defective renewal equation satisfied by the G-S function. Then, the asymptotic formula for the G-S function is obtained using the renewal theory.     

On estimation of linear transformation models with nested case–control sampling

Nested case–control (NCC) sampling is widely used in large epidemiological cohort studies for its cost effectiveness, but its data analysis primarily relies on the Cox proportional hazards model. In this paper, we consider a family of linear transformation models for analyzing NCC data and propose an inverse selection probability weighted estimating equation method for inference. Consistency and asymptotic normality of our estimators for regression coefficients are established. We show that the asymptotic variance has a closed analytic form and can be easily estimated. Numerical studies are conducted to support the theory and an application to the Wilms’ Tumor Study is also given to illustrate the methodology.     

Optimal Berry–Esseen bound for parameter estimation of SPDE with small noise

We investigate a rate of convergence on asymptotic normality of the maximum likelihood estimator (MLE) for parameter $\theta$ appearing in parabolic SPDEs of the form

$d{u}^{?}(t,x)=(A_0+\theta A_1)u^{?}(t,x)dt+?dW(t,x),$

where $A_0$ and$A_1$ are partial differential operators, $W$ is a cylindrical Brownian motion (CBM) and $?\downarrow 0$. We find an optimal Berry–Esseen bound for central limit theorem (CLT) of the MLE. It is proved by developing techniques based on combining Malliavin calculus and Stein’s method.     

Regression modelling of the flows in an input–output table with accounting constraints

This paper details a method for estimating the unknown parameters of a regression model when the estimates of the dependent variable should be embedded in an input–output table with accounting constraints. Since in regression modelling the dependent variable is usually transformed either to achieve homoscedasticity of the residuals or for a better interpretation of the model, the estimating procedure becomes an optimization problem of an opportunely defined Lagrangian function with non-linear constraints. After detailing the algorithm and deriving the asymptotic distribution of the restricted estimator, the methodology is applied to estimate the flows of tourism within and between Italian regions with a gravity model. The procedure can be seen as an extension of Byron’s (J R Stat Soc Ser A 141:359–367, 1978) balancing method.     

Linear sufficiency and completeness in the context of estimating the parametric function in the general Gauss–Markov model

In this paper we consider linear sufficiency and linear completeness in the context of estimating the estimable parametric function K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$ under the general Gauss–Markov model {y,Xβ,σ²V}$\{\mathbf{y},\mathbf{X}\mathit{\beta },{\sigma }^2\mathbf{V}\}$. We give new characterizations for linear sufficiency, and define and characterize linear completeness in a case of estimation of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$. Also, we consider a predictive approach for obtaining the best linear unbiased estimator of K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$, and subsequently, we give the linear analogues of the Rao–Blackwell and Lehmann–Scheffé Theorems in the context of estimating K^′β${\mathbf{K}}^{\prime }\mathit{\beta }$.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

Umvu estimation for the inverse gaussian distribution I(μ,cμ2) with known c

The uniformly minimum variance unbiased estimator and the maximum likelihood estimator of μ for the inverse Gaussian distribution I(μc,μ ) with known c are constructed, and they are shown to be asymptoti- cally equivalent.     

Point estimation of the stress–strength reliability parameter for parallel system with independent and non-identical components

In this article, the estimation problem of the multicomponent stress–strength reliability parameter is considered where the stress and the strength systems have arbitrary fixed numbers of independent and non-identical parallel components. It is assumed that the distribution functions of the stress and the strength components satisfy the proportional reversed hazard rate model. The study is done in more details when the baseline distributions are exponential. Maximum likelihood and uniformly minimum variance unbiased estimators are obtained and compared. Also, Bayes and empirical Bayes estimators are discussed and Monte Carlo simulations are carried out to compare their performances.     

On the dependent competing risks using Marshall–Olkin bivariate Weibull model: Parameter estimation with different methods

Competing risks models are of great importance in reliability and survival analysis. They are often assumed to have independent causes of failure in literature, which may be unreasonable. In this article, dependent causes of failure are considered by using the Marshall–Olkin bivariate Weibull distribution. After deriving some useful results for the model, we use ML, fiducial inference, and Bayesian methods to estimate the unknown model parameters with a parameter transformation. Simulation studies are carried out to assess the performances of the three methods. Compared with the maximum likelihood method, the fiducial and Bayesian methods could provide better parameter estimation.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.     

A fast Monte Carlo expectation–maximization algorithm for estimation in latent class model analysis with an application to assess diagnostic accuracy for cervical neoplasia in women with atypical glandular cells

In this article, we use a latent class model (LCM) with prevalence modeled as a function of covariates to assess diagnostic test accuracy in situations where the true disease status is not observed, but observations on three or more conditionally independent diagnostic tests are available. A fast Monte Carlo expectation–maximization (MCEM) algorithm with binary (disease) diagnostic data is implemented to estimate parameters of interest; namely, sensitivity, specificity, and prevalence of the disease as a function of covariates. To obtain standard errors for confidence interval construction of estimated parameters, the missing information principle is applied to adjust information matrix estimates. We compare the adjusted information matrix-based standard error estimates with the bootstrap standard error estimates both obtained using the fast MCEM algorithm through an extensive Monte Carlo study. Simulation demonstrates that the adjusted information matrix approach estimates the standard error similarly with the bootstrap methods under certain scenarios. The bootstrap percentile intervals have satisfactory coverage probabilities. We then apply the LCM analysis to a real data set of 122 subjects from a Gynecologic Oncology Group study of significant cervical lesion diagnosis in women with atypical glandular cells of undetermined significance to compare the diagnostic accuracy of a histology-based evaluation, a carbonic anhydrase-IX biomarker-based test and a human papillomavirus DNA test.