Optimal consumption and portfolio selection with negative wealth constraints,subsistence consumption constraints,and CARA utility

We consider the optimal consumption and portfolio selection problem with constant absolute risk aversion (CARA) utility. The economic agent in this model receives constant labor income, and her economic behavior is restricted on consumption and wealth, which are called the subsistence consumption constraint and the negative wealth constraint. We use the convex duality method to derive the value function and the optimal policies in closed-form solutions. Also we illustrate some numerical examples.     

A modified least-failures sampling procedure for bernoulli subset selection

We restrict attention to a class of Bernoulli subset selection procedures which take observations one-at-a-time and can be compared directly to the Gupta-Sobel single-stage procedure. For the criterion of minimizing the expected total number of observations required to terminate experimentation, we show that optimal sampling rules within this class are not of practical interest. We thus turn to procedures which, although not optimal, exhibit desirable behavior with regard to this criterion. A procedure which employs a modification of the so-called least-failures sampling rule is proposed, and is shown to possess many desirable properties among a restricted class of Bernoulli subset selection procedures. Within this class, it is optimal for minimizing the number of observations taken from populations excluded from consideration following a subset selection experiment, and asymptotically optimal for minimizing the expected total number of observations required. In addition, it can result in substantial savings in the expected total num¬ber of observations required as compared to a single-stage procedure, thus it may be de¬sirable to a practitioner if sampling is costly or the sample size is limited.     

Nonparametric regression for functional data: Automatic smoothing parameter selection

We study regression estimation when the explanatory variable is functional. Nonparametric estimates of the regression operator have been recently introduced. They depend on a smoothing factor which controls its behavior, and the aim of our work is to construct some data-driven criterion for choosing this smoothing parameter. The criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown regression operator, it is seen that this rule is asymptotically optimal. As by-products of this result, we state some asymptotic equivalences for several measures of accuracy for nonparametric estimate of the regression operator. We also present general inequalities for bounding moments of random sums involving functional variables. Finally, a short simulation study is carried out to illustrate the behavior of our method for finite samples.     

Response improvement in complex experiments by co-information composite likelihood optimization

We propose an adaptive procedure for improving the response outcomes of complex combinatorial experiments. New experiment batches are chosen by minimizing the co-information composite likelihood (COIL) objective function, which is derived by coupling importance sampling and composite likelihood principles. We show convergence of the best experiment within each batch to the globally optimal experiment in finite time, and carry out simulations to assess the convergence behavior as the design space size increases. The procedure is tested as a new enzyme engineering protocol in an experiment with a design space size of order 10⁷.     

Multinomial Probability Estimation by Wavelet Thresholding

In this article, we introduce a wavelet threshold estimator to estimate multinomial probabilities. The advantages of the estimator are its adaptability to the roughness and sparseness of the data. The asymptotic behavior of the estimator is investigated through an often-used criteria: the mean sum of squared error (MSSE). We show that the MSSE of the estimator achieves the optimal rate of convergence. Its performance on finite samples is examined through simulation studies which show favorable results for the new estimator over the commonly used kernel estimator.     

Sub-optimal investment for insurers

Abstract

We consider the investment problem for a non-life insurance company seeking to minimize the ruin probability. Its reserve is described by a perturbed risk process possibly correlated with the financial market. Assuming exponential claim size, the Hamilton-Jacobi-Bellman equation reduces to a first order nonlinear ordinary differential equation, which seems hard to solve explicitly. We study the qualitative behavior of its solution and determine the Cramér-Lundberg approximation. Moreover, our approach enables to find very naturally that the optimal investment strategy is not constant. Then, we analyze how much the company looses by adopting sub-optimal constant (amount) investment strategies.     

Optimal design for discrete choice experiments

In this paper we derive locally optimal designs for discrete choice experiments. As in Kanninen (2002) we consider a multinomial logistic model, which contains various qualitative attributes as well as a quantitative one, which may range over a sufficiently large interval. The derived optimal designs improve upon those given in the literature, but have the feature that every choice set contains alternatives, which coincide in all but the quantitative attributes. The multinomial logistic model will then lead to a response behavior, which is apparently unrealistic.     

Efficient volatility estimation in a two-factor model

We statistically analyze a multivariate Heath-Jarrow-Morton diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense of constructing an estimator that achieves an optimal information bound in a semiparametric setting. We also nonparametrically identify the paths of the volatility processes and achieve minimax bounds. We address the problem of degeneracy that occurs when the dimension of the process is greater than two, and give in particular optimal limit theorems under suitable regularity assumptions on the drift process. We consistently analyze the numerical behavior of our estimators on simulated and real datasets of prices of forward contracts on electricity markets.     

Estimation and Test for Multi-Dimensional Regression Models

This work is concerned with the estimation of multi-dimensional regression and the asymptotic behavior of the test involved in selecting models. The main problem with such models is that we need to know the covariance matrix of the noise to get an optimal estimator. We show in this article that if we choose to minimize the logarithm of the determinant of the empirical error covariance matrix, then we get an asymptotically optimal estimator. Moreover, under suitable assumptions, we show that this cost function leads to a very simple asymptotic law for testing the number of parameters of an identifiable and regular regression model. Numerical experiments confirm the theoretical results.     

Semiparametric lower bounds for tail index estimation

We consider estimation of the tail index parameter from i.i.d. observations in Pareto and Weibull type models, using a local and asymptotic approach. The slowly varying function describing the non-tail behavior of the distribution is considered as an infinite dimensional nuisance parameter. Without further regularity conditions, we derive a local asymptotic normality (LAN) result for suitably chosen parametric submodels of the full semiparametric model. From this result, we immediately obtain the optimal rate of convergence of tail index parameter estimators for more specific models previously studied. On top of the optimal rate of convergence, our LAN result also gives the minimal limiting variance of estimators (regular for our parametric model) through the convolution theorem. We show that the classical Hill estimator is regular for the submodels introduced with limiting variance equal to the induced convolution theorem bound. We also discuss the Weibull model in this respect.     

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.     

Common threshold in quantile regressions with an application to pricing for reputation

The paper develops a systematic estimation and inference procedure for quantile regression models where there may exist a common threshold effect across different quantile indices. We first propose a sup-Wald test for the existence of a threshold effect, and then study the asymptotic properties of the estimators in a threshold quantile regression model under the shrinking threshold effect framework. We consider several tests for the presence of a common threshold value across different quantile indices and obtain their limiting distributions. We apply our methodology to study the pricing strategy for reputation through the use of a data set from Taobao.com. In our economic model, an online seller maximizes the sum of the profit from current sales and the possible future gain from a targeted higher reputation level. We show that the model can predict a jump in optimal pricing behavior, which is considered as “reputation effect” in this paper. The use of threshold quantile regression model allows us to identify and explore the reputation effect and its heterogeneity in data. We find both reputation effects and common thresholds for a range of quantile indices in seller’s pricing strategy in our application.     

Risk efficient estimation of fully dependent random coefficient autoregressive models of general order

We consider a stochastic dynamic model with autoregressive progression. The drift coefficients of the autoregressive model are random where the randomness in the coefficients can have any dependence structure. We propose a two-step sequential estimator and study the asymptotic behavior of few important properties. Paradigm of sequential estimation has its own advantage in reducing sample size and plugging estimates of nuisance parameters while inferring about the main parameters. Our proposed estimator is asymptotically optimal as the predictive risk of the proposed estimator attains the risk of the oracle that assumes known nuisance parameters. Extensive simulation confirms our results.     

Clinical Trials with Exponential Survival Times

We consider a bandit process with delayed responses which are exponentially distributed survival times. The objective is to maximize the expected value of the total response from all selections. We formulate the problem and show that the optimal strategy is characterized by a sequence of break-even values. A monotonicity property of this sequence is derived, which implies the non-optimality of the myopic strategy and a special optimal stopping solution. An example is included to illustrate a possible application of the main results.     

Binary discrimination methods for high-dimensional data with a geometric representation

Four binary discrimination methods are studied in the context of high-dimension, low sample size data with an asymptotic geometric representation, when the dimension increases while the sample sizes of the classes are fixed. We show that the methods support vector machine, mean difference, distance-weighted discrimination, and maximal data piling have the same asymptotic behavior as the dimension increases. We study the consistent, inconsistent, and strongly inconsistent cases in terms of angles between the normal vectors of the separating hyperplanes of the methods and the optimal direction for classification. A simulation study is done to assess the theoretical results.     

Statistical inference for a single-index varying-coefficient model

We investigate the estimators of parameters of interest for a single-index varying-coefficient model. To estimate the unknown parameter efficiently, we first estimate the nonparametric component using local linear smoothing, then construct an estimator of parametric component by using estimating equations. Our estimator for the parametric component is asymptotically efficient, and the estimator of nonparametric component has asymptotic normality and optimal uniform convergence rate. Our results provide ways to construct confidence regions for the involved unknown parameters. The finite-sample behavior of the new estimators is evaluated through simulation studies, and applications to two real data are illustrated.     

Bayesian optimal design for changepoint problems

We investigate Bayesian optimal designs for changepoint problems. We find robust optimal designs which allow for arbitrary distributions before and after the change, arbitrary prior densities on the parameters before and after the change, and any log‐concave prior density on the changepoint. We define a new design measure for Bayesian optimal design problems as a means of finding the optimal design. Our results apply to any design criterion function concave in the design measure. We illustrate our results by finding the optimal design in a problem motivated by a previous clinical trial. The Canadian Journal of Statistics 37: 495–513; 2009 © 2009 Statistical Society of Canada     

On the efficiency of affine minimax rules in estimating a bounded multivariate normal mean

Problems involving bounded parameter spaces, for example T-minimax and minimax esyimation of bounded parameters, have received much attention in recent years. The existing literature is rich. In this paper we consider T-minimax estimation of a multivariate bounded normal mean by affine rules, and discuss the loss of efficiency due to the use of such rules instead of optimal, unrestricted rules. We also investigate the behavior of 'probability restricted' affine rules, i.e., rules that have a guaranteed large probability of being in the bounded parameter space of the problem.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Optimal portfolio of safety-first models

The purpose of this article is to study Kataoka's safety-first (KSF) model, which is a representative of safety-first models of most popular models in portfolio selection of modern finance. We obtain conditions that guarantee that the KSF model has a finite optimal solution without normality assumption. When short-sell is allowed, we provide an explicit analytical solution of the KSF model in two cases. When short-sell is not allowed, we propose an iterating algorithm for finding the optimal portfolios of the KSF model. We also investigate a KSF model with constraint of mean return and obtain the explicit analytical expression of the optimal portfolio.