This paper analyzes customer product-choice behavior based on the recency and frequency of each customer's page views on e-commerce sites. Recently, we devised an optimization model for estimating product-choice probabilities that satisfy monotonicity, convexity, and concavity constraints with respect to recency and frequency. This shape-restricted model delivered high predictive performance even when there were few training samples. However, typical e-commerce sites deal in many different varieties of products, so the predictive performance of the model can be further improved by integration of such product heterogeneity. For this purpose, we develop a novel latent-class shape-restricted model for estimating product-choice probabilities for each latent class of products. We also give a tailored expectation-maximization algorithm for parameter estimation. Computational results demonstrate that higher predictive performance is achieved with our latent-class model than with the previous shape-restricted model and common latent-class logistic regression.
Shape-restricted regression, which has its origin in earlier work @cite_5 @cite_24 @cite_33 , fits a nonparametric function to given data points under shape restrictions such as monotonicity, convexity, and concavity. Some popular examples include the estimation of utility production cost profit functions in economics @cite_34 @cite_41 and option pricing functions in finance @cite_6 . Various algorithms have been developed for shape-restricted regression @cite_19 @cite_25 @cite_37 @cite_39 @cite_0 @cite_7 , a special case of which is isotonic regression. This has many applications in statistics, operations research, and image processing @cite_13 . @cite_23 recently used the maximum likelihood method to estimate product-choice probabilities subject to the monotonicity, convexity, and concavity constraints with respect to recency and frequency. Their shape-restricted model was a new effective application of shape-restricted regression to the analysis of clickstream data.
@cite_37 @cite_33 @cite_7 @cite_41 @cite_6 @cite_39 @cite_24 @cite_19 @cite_0 @cite_23 @cite_5 @cite_34 @cite_13 @cite_25
2494296723 2079313454 1967819130 2064922317 2151447307 2086688853 1975116882 2033062730 2795929191 2257459631 2127441697 2006560229 1995237252
In some statistical non-parametric models the mean of the random variable y has to satisfy specific constraints. We consider the case where the set defined by the constraints is a closed polyhedral cone K in Rk. For example, when the mean is required to be concave in x, the set of acceptable means is a closed convex cone defined by k-2 linear inequalities in Rk. The least squares estimate of the mean is then the projection of the data point y on the cone K. In this paper, we present an algorithm to find the least square estimate of the mean in a finite number of steps. Other algorithms to solve this problem have been given before. The successive approximations in such algorithms are usually points on the faces of K. The solution here is reached by following a fixed line joining an abitrary but suitably chosen initial point y0 to the data point y. The 1- dimensional subspace spanned by the generators of the cone K is divided into 2' regions which can be described as the set of points with non-negative coordinates in mixed primal-dual bases relative to the cone K. Any point y belongs to one and only one of these regions SJ with corresponding basis 3j. The projection of y on K is immediately obtained from the expression of y in Oj by dropping the dual component of y. Abstract A method is developed for obtaining maximum likelihood estimates of points on a surface of unspecified algebraic form when ordinates of the points are required to satisfy a set of linear inequalities. A production function with one variable input is considered in some detail. In this case the restrictions follow from the assumption of non-increasing returns. An illustrative computation is worked out using a procedure based on equivalence between the estimation problem and a certain saddle point problem. Alternative procedures for production functions with two variable inputs are sketched. The objective of this article is to develop a computationally efficient estimator of the regression function subject to various shape constraints. In particular, nonparametric estimators of monotone and or convex (concave) regression functions are obtained by using a nested sequence of Bernstein polynomials. One of the key distinguishing features of the proposed estimator is that a given shape constraint (e.g., monotonicity and or convexity) is maintained for any finite sample size and satisfied over the entire support of the predictor space. Moreover, it is shown that the Bernstein polynomial based regression estimator can be obtained as a solution of a constrained least squares method and hence the estimator can be computed efficiently using a quadratic programming algorithm. Finally, the asymptotic properties (e.g., strong uniform consistency) of the estimator are established under very mild conditions, and finite sample properties are explored using several simulation studies and real data analysis. The predictive performances are compared with some of the existing methods. Empirical economists using flexible functional forms often face the disturbing choice of drawing inferences from an approximation violating properties dictated by theory or imposing global restrictions that greatly restrict the flexibility of the functional form. Focusing on the cost function, this paper presents an alternative approach which imposes monotonicity and concavity properties only over the set of prices where inferences will be drawn. An application investigating elasticities for Berndt-Wood data set using the translog, generalized Leontief, and symmetric generalized McFadden flexible functional forms illustrates the technique. Copyright 1996 by John Wiley & Sons, Ltd. Frequently, economic theory places shape restrictions on functional relationships between economic variables. This paper develops a method to constrain the values of the first and second derivatives of nonparametric locally polynomial estimators. We apply this technique to estimate the state price density (SPD), or risk-neutral density, implicit in the market prices of options. The option pricing function must be monotonic and convex. Simulations demonstrate that nonparametric estimates can be quite feasible in the small samples relevant for day-to-day option pricing, once appropriate theory-motivated shape restrictions are imposed. Using S&P500 option prices, we show that unconstrained nonparametric estimators violate the constraints during more than half the trading days in 1999, unlike the constrained estimator we propose. The additive isotonic least-squares regression model has been fit using a sequential pooled adjacent violators algorithm, estimating each isotonic component in turn, and looping until convergence. However, the individual components are not, in general, estimable. The sum of the components, i.e. the expected value of the response, has a unique estimate, which can be found using a single cone projection. Estimators for the individual components are then easily obtained, which are unique if the conditions for estimability hold. Parametrically modelled covariates are easily included in the cone projection specification. The cone structure also provides information about the degrees of freedom of the fit, which can be used in inference methods, variable selection, and estimation of the model variance. Simulations show that these methods can compare favourably to standard parametric methods, even when the parametric assumptions are correct. The estimation and inference methods can be extended to other constraints such as convex regression or isotonic regression on partial orderings. Abstract 2.1. Limitations of the parametric methods. In the previous sections we have studied the efficiency of various methods of estimating the force of mortality. The most efficient of these is, at least for large samples, the one given by the maximum likelihood method, and the rest of them have to be compared to this best estimate. However in this discussion the notion of efficiency is based on the assumption that the mortality intensity can be expressed by Makeham’s formula. How realistic is this in practice? This paper investigates the relationship between customers' page views (PVs) and the probabilities of their product choices on e-commerce sites. For this purpose, we create a probability table consisting of product-choice probabilities for all recency and frequency combinations of each customers' previous PVs. To reduce the estimation error when there are few training samples, we develop optimization models for estimating the product-choice probabilities that satisfy monotonicity, convexity and concavity constraints with respect to recency and frequency. Computational results demonstrate that our method has clear advantages over logistic regression and kernel-based support vector machine. The isotonic regression problem has applications in statistics, operations research, and image processing. In this paper a general framework for the isotonic regression algorithm is proposed. Under this framework, we discuss the isotonic regression problem in the case where the directed graph specifying the order restriction is a directed tree with n vertices. A new algorithm is presented for this case, which can be regarded as a generalization of the PAV algorithm of Using a simple tree structure such as the binomial heap, the algorithm can be implemented in O(n log n) time, improving the previously best known O(n 2 ) time algorithm. We also present linear time algorithms for special cases where the directed graph is a path or a star. Abstract We consider the efficient estimation in the semiparametric additive isotonic regression model where each additive nonparametric component is assumed to be a monotone function. We show that the least-square estimator of the finite-dimensional regression coefficient is root- n consistent and asymptotically normal. Moreover, the isotonic estimator of each additive functional component is proved to have the oracle property, which means the additive component can be estimated with the highest asymptotic accuracy as if the other components were known. A fast algorithm is developed by iterating between a cyclic pool adjacent violators procedure and solving a standard ordinary least squares problem. Simulations are used to illustrate the performance of the proposed procedure and verify the oracle property.