Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean–Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of \(\mathrm {TOL}\), is Open image in new window when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of Open image in new window . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.     

The Frisch–Waugh–Lovell theorem for the lasso and the ridge regression

The Frisch–Waugh–Lovell (FWL) (partitioned regression) theorem is essential in regression analysis. This is partly because it is quite useful to derive theoretical results. The lasso regression and the ridge regression, both of which are penalized least-squares regressions, have become popular statistical techniques. This article describes that the FWL theorem remains valid for these penalized least-squares regressions. More precisely, we demonstrate that the covariates corresponding to unpenalized regression parameters in these penalized least-squares regression can be projected out. Some other results related to the FWL theorem in such penalized least-squares regressions are also presented.     

An improved method of estimation for the parameters of the Birnbaum–Saunders distribution

The Birnbaum–Saunders (BS) distribution is a positively skewed distribution, frequently used for analysing lifetime data. In this paper, we propose a simple method of estimation for the parameters of the two-parameter BS distribution by making use of some key properties of the distribution. Compared with the maximum likelihood estimators and the modified moment estimators, the proposed method has smaller bias, but having the same mean square errors as these two estimators. We also discuss some methods of construction of confidence intervals. The performance of the estimators is then assessed by means of Monte Carlo simulations. Finally, an example is used to illustrate the method of estimation developed here.     

A simple method for estimating parameters of the location–scale distribution family

The purpose of this paper is to estimate the parameters of the location–scale distribution family. As a special case, the method is used for estimating the parameters of the normal distribution and Cauchy distribution. For the Cauchy distribution, neither the moment estimation method nor the maximum likelihood estimation method works properly for estimating the parameters. The quantiles for obtaining confidence intervals and point estimates for the parameters of the two-parameter Cauchy distribution are given in the paper. It is shown that the estimators obtained in this paper are unbiased with respect to the median and possess some optimal properties.     

A modification of the Box–Meyer method for finding the active factors in screening experiments

Screening experiments are conducted to identify a few active factors among a large number of factors. For the objective of identifying active factors, Box and Meyer provided an innovative approach, the Box–Meyer method (BMM). With the use of means models, we propose a modification of the BMM in this paper. Compared with the original BMM, the modified BMM (MBMM) can circumvent the problem that the original BMM runs into, namely that it may fail to identify some active factors due to the ignorance of higher order interactions. Furthermore, the number of explanatory variables in the MBMM is smaller. Therefore, the computational complexity is reduced. Finally, three examples with different types of designs are used to demonstrate the wide applicability of the MBMM.     

The Anderson–Darling Goodness-of-Fit Test Statistic for the Three-Parameter Lognormal Distribution

Anderson–Darling goodness-of-fit test percentage points are given for the three-parameter lognormal distribution for both the cases of positive skewness and a lower bound and negative skewness and an upper bound. The focus is on the most practical case when all parameters are unknown and must be estimated from the sample data. Fitted response functions for the critical values based on the shape parameter and sample size are reported to avoid using a vast array of tables.     

A modification of the Whittaker–Henderson method of graduation

This paper presents a modified Whittaker–Henderson (WH) Method of Graduation. After giving a closed-form solution, we show that it is of practical use because it provides not only a smoothed series identical to that of the WH graduation, but also an extrapolation beyond the sample limit of current data. In addition, we introduce two other penalized least squares problems and show that they provide the same results as those of the modified WH graduation.     

A general reduction method for n–variate normal orthant probability

Let X₁,X₂,…,X_n be n normal variates with zero means, unit variances and correlation matrix {p_ij). The orthant probability is the probability that all of the X₁'s are simultaneously positive. This paper presents a general reduction method by extending the method of Childs (1967), and shows that the probability can be represented by a linear combination of some multivariate integrals of order([n/2]?1). As illustrations, we apply the proposed method to the quadrivariate and six–variate cases. Some numerical results are also given.     

On the effect of item nonresponse on the estimation of a two–panel–waves wage equation

Summary: In this paper, we present results of the estimation of a two–panel–waves wage equation based on completely observed units and on a multiply imputed data set. In addition to the survey information, reliable income data is available from the register. These external data are used to assess the reliability of wage regressions that suffer from item nonresponse. The findings reveal marked differences between the complete case analyses and both versions of multiple imputation analyses. We argue that the results based on the multiply imputed data sets are more reliable than those based on the complete case analysis.* We would like to thank Statistics Finland for providing the data. We are also very grateful to Susanna Sandström and Marjo Pyy–Martikainen for their helpful advice using the Finnish data. Helpful comments from Joachim Winter and participants of the Workshop on Item Nonresponse and Data Quality in Large Social Surveys, Basel, October, 2003, on an earlier version of the paper are greatfully acknowledged. Further, we would like to thank three anonymous referees and the editor for helpful comments and suggestions.     

A simple generalization of the Box–Muller method for obtaining a pair of correlated standard normal variables

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.     

A pathwise inference method for the parameters of diffusion terms

We consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.     

A penalized likelihood method for nonseparable space–time generalized additive models

In this paper, we study space–time generalized additive models. We apply the penalyzed likelihood method to fit generalized additive models (GAMs) for nonseparable spatio-temporal correlated data in order to improve the estimation of the response and smooth terms of GAMs. The results show that our space–time generalized additive models estimated response and smooth terms reasonable well, and in addition, the mean squared error, mean absolute deviation and coverage intervals improved considerably compared to the classic GAM. An application on particulate matter concentration in the North-Italian region of Piemonte is also presented.     

S–S method for stochastically ordered multinomial populations with missing data

Stochastic ordering is a useful concept in order restricted inferences. In this paper, we propose a new estimation technique for the parameters in two multinomial populations under stochastic orderings when missing data are present. In comparison with traditional maximum likelihood estimation method, our new method can guarantee the uniqueness of the maximum of the likelihood function. Furthermore, it does not depend on the choice of initial values for the parameters in contrast to the EM algorithm. Finally, we give the asymptotic distributions of the likelihood ratio statistics based on the new estimation method.     

The wild bootstrap for multivariate Nelson–Aalen estimators

Lifetime Data Analysis - We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson–Aalen estimator under Aalen’s multiplicative intensity...     

The asymptotic size and power of the augmented Dickey–Fuller test for a unit root

It is shown that the limiting distribution of the augmented Dickey–Fuller (ADF) test under the null hypothesis of a unit root is valid under a very general set of assumptions that goes far beyond the linear AR(∞) process assumption typically imposed. In essence, all that is required is that the error process driving the random walk possesses a continuous spectral density that is strictly positive. Furthermore, under the same weak assumptions, the limiting distribution of the ADF test is derived under the alternative of stationarity, and a theoretical explanation is given for the well-known empirical fact that the test's power is a decreasing function of the chosen autoregressive order p. The intuitive reason for the reduced power of the ADF test is that, as p tends to infinity, the p regressors become asymptotically collinear.     

Robust inference for the stress–strength reliability

We address the problem of robust inference about the stress–strength reliability parameter R = P(X < Y), where X and Y are taken to be independent random variables. Indeed, although classical likelihood based procedures for inference on R are available, it is well-known that they can be badly affected by mild departures from model assumptions, regarding both stress and strength data. The proposed robust method relies on the theory of bounded influence M-estimators. We obtain large-sample test statistics with the standard asymptotic distribution by means of delta-method asymptotics. The finite sample behavior of these tests is investigated by some numerical studies, when both X and Y are independent exponential or normal random variables. An illustrative application in a regression setting is also discussed.