The generalized leverage of an estimator is de®ned in regression models as a measure of the importance of individual observations. We derive a simple but powerful result, developing an explicit expression for leverage in a general M-estimation problem, of which the maximum likelihood problems are special cases. A variety of applications are considered, most notably to the exponential family non-linear models. The relationship between leverage and local in¯uence is also discussed. Numerical examples are given to illustrate our results.

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Sweet cherries are popular among consumers, with a recent explosion in sweet cherry production in China. However, the fragility of these fruits poses a challenge for expanding production and transport. With the aim of expanding the product categories of sweet cherries that can bypass these challenges, in this study, we prepared sweet cherry juice fermented by three different lactic acid bacteria (LAB; Lactobacillus acidophilus, Lactobacillus plantarum, and Lactobacillus rhamnosus GG), and evaluated the growth, physiochemical, and aroma characteristics. All three strains exhibited excellent growth potential in the sweet cherry juice; however, Lactobacillus acidophilus and Lactobacillus plantarum demonstrated more robust acid production capacity and higher microbial viability than Lactobacillus rhamnosus GG. Lactic acid was the primary fermentation product, and malic acid was significantly metabolized by LAB, indicating a transition in microbial metabolism from using carbohydrates to organic acids. The aroma profile was identified through integrated analysis of electronic nose (E-nose) and headspace gas chromatography–ion mobility spectrometry (HS-GC–IMS) data. A total of 50 volatile compounds characterized the aromatic profiles of the fermented juices by HS-GC–IMS. The flavor of sweet cherry juice changed after LAB fermentation and the fruity odor decreased overall. Lactobacillus acidophilus and Lactobacillus plantarum significantly increased 2-heptanone, ethyl acetate, and acetone contents, bringing about a creamy and rummy-like favor, whereas Lactobacillus rhamnosus GG significantly increased 2-heptanone, 3-hydroxybutan-2-one, and 2-pentanone contents, generating cheesy and buttery-like odors. Principal component analysis of GC–IMS data and linear discriminant analysis of E-nose results could effectively differentiate non-fermented sweet cherry juice and the sweet cherry juice separately inoculated with different LAB strains. Furthermore, there was a high correlation between the E-nose and GC–IMS results, providing a theoretical basis to identify different sweet cherry juice formulations and appropriate starter culture selection for fermentation. This study enables more extensive utilization of sweet cherry in the food industry and helps to improve the flavor of sweet cherry products.

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