Although asymptotic nonparametric confidence bands have been constructed in the last decade in some inverse problems, like density deconvolution, inverse regression with a convolution operator, and regression with errors in variables, there seems to be no such construction for practically important inverse problems of stereology. Working with a kernel-type nonparametric estimator of the density of squared radii in the stereological Wicksell's problem, we partially fill this gap by constructing some corresponding asymptotic uniform confidence bands and an automatic bandwidth selection method, tuned to perform well in finite samples in terms of both area and coverage probability of the confidence bands. The performance of the new procedures is investigated in simulations and demonstrated with some astronomical data related to the M62 globular cluster.
Purpose: The aim of the study was to establish the relationship between the shape of the calcaneal tuberosity (flat, stepped, rounded, normal) and the probability that retrocalcaneal bursitis among people who train running regularly. Methods: The study included a group of 30 runners who suffered from retrocalcaneal bursitis in the past, and 30 people who never had symptoms of this disease. The study was based on a diagnostic survey, as well as on clinical examination. The surface of the calcaneal tuberosity and the slope of the calcaneus were assessed using X-rays. The mobility of the bursa, its surface size, the thickness of the Achilles tendon and its attachment rate were established during an ultrasound examination. Results: Flat surface of the calcaneal tuberosity increases fourfold the risk of suffering from retrocalcaneal bursitis (OR = 4.3). The people whose calcaneus slope is above 25° are at increased risk of suffering from such an inflammation compared with the people whose calcaneus bone is more horizontal (OR = 2.8). The analysis shows that the thickness of the Achilles tendon (p = 0.001), the surface size of the bursa (p = 0.009), as well as the flat surface of the calcaneal tuberosity (p = 0.008) are strongly associated with the occurrence of retrocalcaneal bursitis. Conclusions: The flat shape of the calcaneal tuberosity increases the risk of bursitis. The risk of inflammation is higher when the Achilles tendon is thicker and the surface of the bursa is smaller than normal.
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