The study of extremes has attracted the attention of scientists, engineers, actuaries, policy makers, and statisticians for many years. Extreme value theory (EVT) deals with the extreme deviations from the median of probability distributions and is used to study rare but extreme events. EVT’s main results characterize the distribution of the sample maximum or the distribution of values above a given threshold. In this study, EVT has been used to construct a model on the extreme and rare earthquakes that have happened in the United States from 1700 to 2011.The primary goal of fitting such a model is to estimate the amount of losses due to those extreme events and the probabilities of such events. Several diagnostic methods (for example, QQ plot and Mean Excess Plot) have been used to justify that the data set follows generalized Pareto distribution (GPD). Three estimation techniques have been employed to estimate parameters. The consistency and reliability of estimated parameters have been observed for different threshold values. The purpose of this study is manifold: first, we investigate whether the data set follows GPD, by using graphical interpretation and hypothesis testing. Second, we estimate GPD parameters using three different estimation techniques. Third, we compare consistency and reliability of estimated parameters for different threshold values. Last, we investigate the bias of estimated parameters using a simulation study. The result is particularly useful because it can be used in many applications (for example, disaster management, engineering design, insurance industry, hydrology, ocean engineering, and traffic management) with a minimal set of assumptions about the true underlying distribution of a data set. KEYWORDS: Extreme Value Theory; QQ Plot; Mean Excess Plot; Mean Residual Plot; Peak Over Threshold; Generalized Pareto Distribution; Maximum Likelihood Method; Method of Moments; Probability-Weighted Moments; Shapiro-Wilk test; Anderson- Darling Test
Background: Previous research has documented the proportion of “tall and fall” (TF) and “drop and drive” (DD) pitching styles among Major League Baseball (MLB) pitchers who underwent ulnar collateral ligament reconstruction (UCLR). The proportion of these 2 styles among all MLB pitchers remains unknown. Purpose: To determine the proportion of the TF and DD pitching styles in all rostered MLB pitchers during a single season as well as the proportion of TF and DD pitchers who sustained an upper extremity (UE) injury and those who underwent UCLR. Study Design: Cross-sectional study; Level of evidence, 3. Methods: Pitcher demographic characteristics from the 2019 MLB season and pitching information were obtained via open-access sources. Two-dimensional video analysis was used to categorize the included pitchers into TF and DD groups. Statistical comparisons and contrasts were made using 2-tailed t tests, chi-square tests, and Pearson correlation analyses as appropriate. Results: Of the 660 MLB rostered pitchers in 2019 (age, 27.39 ± 3.51 years; body mass index, 26.34 ± 2.47 kg/m2; fastball velocity, 150.49 ± 3.99 kph [93.51 ± 2.48 mph]), 412 (62.4%) pitchers used the TF style and 248 (37.6%) pitchers used the DD style. Significantly more UE injuries were seen in the TF group compared with the DD group (112 vs 38 injuries, respectively; P < .001). Twelve pitchers underwent UCLR (TF, 10; DD, 2), representing a 1.8% UCLR rate among all pitchers. This was a second surgery for 2 pitchers, both of whom used the TF pitching style. Significantly more pitchers in the TF group than the DD group had undergone UCLR before 2019 (135 vs 56 pitchers, respectively; P = .005). Conclusion: The results of the present study demonstrated a higher prevalence of both UE injury and prior UCLR in TF pitchers. Further research is needed to explore the potential association between pitching style and UE injury.
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