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Method of Construction of the Stochastic Integral with Respect to Fractional Brownian Motion
Diop Bou,
Ba Demba Bocar,
Thioune Moussa
Issue:
Volume 9, Issue 5, October 2021
Pages:
156-164
Received:
2 July 2021
Accepted:
6 August 2021
Published:
3 September 2021
Abstract: Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This process is called a (standard) fractional Brownian motion with Hurst parameter H. To simplify the presentation, it is always assumed that the fractional Brownian motion is 0 at t=0. If H = 1/2, then the corresponding fractional Brownian motion is the usual standard Brownian motion. If 1/2 < H < 1, Fractional Brownian motion (FBM) is neither a finite variation nor a semi-martingale. Consequently, the standard Ito calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. The classic methods (Itô and Stiliege) are excluted. The most studied case is that where H is between 0 and 1/2. Several attempts to define the stochastic integral are made. But so far some difficulties subjust. We give in this paper, several construction methods. So for the construction, we will use other tools to deal with such situations.
Abstract: Since the pioneering work of Hurst, and Mandelbrot, the fractional brownian motions have played and increasingly important role in many fields of application such as hydrology, economics and telecommunications. For every value of the Hurst index H ∈ (0,1) we define a stochastic integral with respect to fractional Brownian motion of index H. This pr...
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Proportional Subdistribution Hazards Model for Competing Risks in Case-Cohort Studies
Adane Fekadu Wogu,
Shanshan Zhao,
Hazel Bogan Nichols,
Jianwen Cai
Issue:
Volume 9, Issue 5, October 2021
Pages:
165-185
Received:
15 June 2021
Accepted:
6 July 2021
Published:
9 September 2021
Abstract: Competing risks refer to the situation where there are multiple causes of failure and the occurrence of one type of event prohibits the occurrence of the other types of event or alters the chance to observe them. In large cohort studies with long-term follow-up, there are often competing risks. When the failure events are rare, or the information on certain risk factors is difficult or costly to measure for the full cohort, a case-cohort study design can be a desirable approach. In this paper, we consider a semiparametric proportional subdistribution hazards model in the presence of competing risks in case-cohort studies. The subdistribution hazards function, unlike the cause-specific hazards function, gives the advantage of outlining the marginal probability of a particular type of event. We propose estimating equations based on inverse probability weighting techniques for the estimation of the model parameters. In the estimation methods, we considered a weighted availability indicator to properly account for the case-cohort sampling scheme. We also proposed a Breslow-type estimator for the cumulative baseline subdistribution hazard function. The resulting estimators are shown, using empirical processes and martingale properties, to be consistent and asymptotically normally distributed. The performance of the proposed methods in finite samples are examined through simulation studies by considering different levels of censoring and event of interest percentages. The simulation results from the different scenarios suggest that the parameter estimates are reasonably close to the true values of the respective parameters in the model. Finally, the proposed estimation methods are applied to a case-cohort sample from the Sister Study, in which we illustrated the proposed methods by studying the association between selected CpGs and invasive breast cancer in the presence of ductal carcinoma in situ as competing risk.
Abstract: Competing risks refer to the situation where there are multiple causes of failure and the occurrence of one type of event prohibits the occurrence of the other types of event or alters the chance to observe them. In large cohort studies with long-term follow-up, there are often competing risks. When the failure events are rare, or the information o...
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Prediction of Precipitation Rate Based on Stationary Extreme Value Theory
Issue:
Volume 9, Issue 5, October 2021
Pages:
186-191
Received:
23 September 2021
Accepted:
25 October 2021
Published:
30 October 2021
Abstract: For the determination of the effectiveness of weather forecasts such as temperature or flooding forecast, the single variable linear regression or simple exponential smoothing will not be an effective way. To accurately predict the effectiveness of such a trend, iterative and statistical methods that can determine the status of temperature or flooding in the United States were chosen in this paper. The task of modeling the pattern in a focused period and performing data analysis was performed. For the analysis, the extreme value theory was used to assess extreme events within probability distributions by quantifying tail behavior. The python tools are utilized for the analysis of big data. By analyzing the maximum values of samples, it was possible to determine probabilities for extreme events. A comparison was made with events previously observed and analyzed for authenticity. As evident in our observations, lower values of data have much shorter return periods. In other words, they are more likely to reoccur; however, as the values increase for higher precipitation values, the length of the return periods increase exponentially. Therefore, there is a tendency for precipitation values to remain in lower ranges. In this paper, the USGS geographical information and Gumbel distribution were used to find the return period corresponding to the exceedance probability. The Gumbel distribution is applied to Allegheny River, New York and Whetstone River, San Diego.
Abstract: For the determination of the effectiveness of weather forecasts such as temperature or flooding forecast, the single variable linear regression or simple exponential smoothing will not be an effective way. To accurately predict the effectiveness of such a trend, iterative and statistical methods that can determine the status of temperature or flood...
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