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Penalized likelihood models are widely used to simultaneously select variables and estimate model parameters. However, the existence of weak signals can lead to inaccurate variable selection, biased parameter estimation, and invalid inference. Thus, identifying weak signals accurately and making valid inferences are crucial in penalized likelihood models. We develop a unified approach to identify weak signals and make inferences in penalized likelihood models, including the special case when the responses are categorical. To identify weak signals, we use the estimated selection probability of each covariate as a measure of the signal strength and formulate a signal identification criterion. To construct confidence intervals, we propose a two-step inference procedure. Extensive simulation studies show that the proposed procedure outperforms several existing methods. We illustrate the proposed method by applying it to the Practice Fusion diabetes data set.

Generating accurate and reliable sales forecasts is crucial in the E-commerce business. The current state-of-the-art techniques are typically univariate methods, which produce forecasts considering only the historical sales data of a single product. However, in a situation where large quantities of related time series are available, conditioning the forecast of an individual time series on past behaviour of similar, related time series can be beneficial. Since the product assortment hierarchy in an E-commerce platform contains large numbers of related products, in which the sales demand patterns can be correlated, our attempt is to incorporate this cross-series information in a unified model. We achieve this by globally training a Long Short-Term Memory network (LSTM) that exploits the non-linear demand relationships available in an E-commerce product assortment hierarchy. Aside from the forecasting framework, we also propose a systematic pre-processing framework to overcome the challenges in the E-commerce business. We also introduce several product grouping strategies to supplement the LSTM learning schemes, in situations where sales patterns in a product portfolio are disparate. We empirically evaluate the proposed forecasting framework on a real-world online marketplace dataset from Walmart.com. Our method achieves competitive results on category level and super-departmental level datasets, outperforming state-of-the-art techniques.

Weak signal identification and inference are very important in the area of penalized model selection, yet they are under-developed and not well-studied. Existing inference procedures for penalized estimators are mainly focused on strong signals. In this paper, we propose an identification procedure for weak signals in finite samples, and pro- vide a transition phase in-between noise and strong signal strengths. We also introduce a new two-step inferential method to construct better confidence intervals for the identified weak signals. Our theory development assumes that variables are orthogonally designed. Both theory and numerical studies indicate that the proposed method leads to better confidence coverage for weak signals, compared with those using asymptotic inference. In addition, the proposed method out- performs the perturbation and bootstrap resampling approaches. We illustrate our method for HIV antiretroviral drug susceptibility data to identify genetic mutations associated with HIV drug resistance.