Semi- and non-parametric mixture of normal regression models are a flexible class of mixture of regression models. These models assume that the component mixing proportions, regression functions and/or variances are non-parametric functions of the covariates. Among this class of models, the semi-parametric mixture of partially linear models (SPMPLMs) combine the desirable interpretability of a parametric model and the flexibility of a non-parametric model. However, local-likelihood estimation of the non-parametric term poses a computational challenge. Traditional EM optimisation of the local-likelihood functions is not appropriate due to the label-switching problem. Separately applying the EM algorithm on each local-likelihood function will likely result in non-smooth function estimates. This is because the local responsibilities calculated at the E-step of each local EM are not guaranteed to be aligned. To prevent this, the EM algorithm must be modified so that the same (global) responsibilities are used at each local M-step. In this paper, we propose a one-step backfitting EM-type algorithm to estimate the SPMPLMs and effectively address the label-switching problem. The proposed algorithm estimates the non-parametric term using each set of local responsibilities in turn and then incorporates a smoothing step to obtain the smoothest estimate. In addition, to reduce the computational burden imposed by the use of the partial-residuals estimator of the parametric term, we propose a plug-in estimator. The performance and practical usefulness of the proposed methods was tested using a simulated dataset and two real datasets, respectively. Our finite sample analysis revealed that the proposed methods are effective at solving the label-switching problem and producing reasonable and interpretable results in a reasonable amount of time.

The non-parametric Gaussian mixture of regressions (NPGMRs) model serves as a flexible approach for the determination of latent heterogeneous regression relationships. This model assumes that the component means, variances and mixing proportions are smooth unknown functions of the covariates where the error distribution of each component is assumed to be Gaussian and hence symmetric. These functions are estimated over a set of grid points using the Expectation-Maximization (EM) algorithm to maximise the local-likelihood functions. However, maximizing each local-likelihood function separately does not guarantee that the local responsibilities and corresponding labels, obtained at the E-step of the EM algorithm, align at each grid point leading to a label-switching problem. This results in non-smooth estimated component regression functions. In this paper, we propose an estimation procedure to account for label switching by tracking the roughness of the estimated component regression functions. We use the local responsibilities to obtain a global estimate of the responsibilities which are then used to maximize each local-likelihood function. The performance of the proposed procedure is demonstrated using a simulation study and through an application using real world data. In the case of well-separated mixture regression components, the procedure gives similar results to competitive methods. However, in the case of poorly separated mixture regression components, the procedure outperforms competitive methods.

Background: In search of more entrepreneurs for economic development, academics and policy makers are continuously seeking ways in which the participation of potential entrepreneurs in the economy can be enhanced. Purpose: This study investigates whether entrepreneurial prototype factors could be identified to inform how entrepreneurs evaluate opportunities. Design/Methodology: In an experimental design, participants were requested to evaluate a single start-up opportunity. They completed a questionnaire exploring their thinking of the single case. Participants included 193 nascent and novice entrepreneurs that evaluated the same opportunity. The questionnaire was administered, leading to factor and regression analyses. Findings: The factor analysis identified four prototype factors for potential use in selection. Discrimination was possible between the prototype factors (cognitive frameworks) of novice (first-time) and repeat (experienced) entrepreneurs for “positive financial model”; “uniqueness of the idea”; “big markets”; and “intuition.” Significant differences for the identified factors were reported between those who decided for and against starting the venture. Regression analysis suggested further discriminatory value, with the prototype factors for the start-up decision contributing to a potential selection process by venture capitalists, as well as educators. Research limitations: The generalisability of the findings may be limited by the use of a single case evaluation. Originality/value: Firstly, support was found for the effectiveness of the methodology in identifying the prototypes. Secondly, the study contributes by informing educators of entrepreneurs about the relevancy of cognitive frameworks that could be developed to meaningfully enhance opportunity evaluation.

The mixture of generalised linear models (MGLM) requires knowledge about each mixture component’s specific exponential family (EF) distribution. This assumption is relaxed and a mixture of semi-parametric generalised linear models (MSPGLM) approach is proposed, which allows for unknown distributions of the EF for each mixture component while much of the parametric structure of the traditional MGLM is retained. Such an approach inherently allows for both symmetric and non-symmetric component distributions, frequently leading to non-symmetrical response variable distributions. It is assumed that the random component of each mixture component follows an unknown distribution of the EF. The specific member can either be from the standard class of distributions or from the broader set of admissible distributions of the EF which is accessible through the semi-parametric procedure. Since the inverse link functions of the mixture components are unknown, the MSPGLM estimates each mixture component’s inverse link function using a kernel smoother. The MSPGLM algorithm alternates the estimation of the regression parameters with the estimation of the inverse link functions. The properties of the proposed MSPGLM are illustrated through a simulation study on the separable individual components. The MSPGLM procedure is also applied on two data sets.