Chi-squared test

A chi-squared test (also chi-square or χ2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables (two dimensions of the contingency table) are independent in influencing the test statistic (values within the table). The test is valid when the test statistic is chi-squared distributed under the null hypothesis, specifically Pearson's chi-squared test and variants thereof. Pearson's chi-squared test is used to determine whether there is a statistically significant difference between the expected frequencies and the observed frequencies in one or more categories of a contingency table. For contingency tables with smaller sample sizes, a Fisher's exact test is used instead. In the standard applications of this test, the observations are classified into mutually exclusive classes. If the null hypothesis that there are no differences betwe

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

In this thesis, we treat robust estimation for the parameters of the Ornstein–Uhlenbeck process, which are the mean, the variance, and the friction. We start by considering classical maximum likelihood estimation. For the simulation study, where we also investigate the choice of the time lag, we use the method of moment (MoM) estimator as initial estimator for the friction parameter of the maximum likelihood estimator (MLE). However, in several aspects the MLE is not robust. For robustification, we first derive elementary M-estimates by extending the method of M-estimation from Huber (1981). We use an intuitively robustified MoM estimate as initial estimate and compare by means of simulation the M-estimate with the MLE. This approach is, however, only ad-hoc since Huber’s minimum Fisher information and minimax asymptotic variance theory remains incomplete for simultaneous location and scale, and does not cover more general models (as for example the Ornstein–Uhlenbeck process). A more general robustness concept due to Kohl et al. (2010), Rieder (1994), and Staab (1984) is based on local asymptotic normality (LAN), asymptotically linear (AL) estimates, and shrinking neighborhoods. We then apply this concept to the Ornstein–Uhlenbeck process. As a measure of robustness, we consider the maximum asymptotic mean square error (maxasyMSE), which is determined by the influence curve (IC) of AL estimates. The IC represents the standardized influence of an individual observation on the estimator given the past. For two kind of neighborhoods (average and average square neighborhoods) we obtain optimally robust ICs. In case of average neighborhoods, their graph exhibits surprising, redescending behavior. For average square neighborhoods the graph is between the one of the elementary M-estimates and the MLE. Finally, we discuss the estimator construction, that is, the problem of constructing an estimator from the family of optimal ICs. We carry out in our context the One-Step construction dating back to LeCam and use both an intuitively robustified MoM estimate and the elementary M-estimate as initial estimate. This results in optimally AL estimates (for average and average square neighborhoods). By means of simulation we then compare the different estimators: MLE, elementary M-estimates, and optimally AL estimates. In addition, we give an application to electricity prices.

EPFL2012

NovelT2Mapping for Evaluating Cervical Cancer Features by Providing QuantitativeT2Maps and Synthetic Morphologic Images: A Preliminary Study

Tom Hilbert, Tobias Kober, Jie Liu, Meng Yang, Yong Zhang

Background The application value of T(2)mapping in evaluating cervical cancer (CC) features remains unclear. Purpose To investigate the role of T(2)values in evaluating CC classification, grade, and lymphovascular space invasion (LVSI) in comparison to apparent diffusion coefficient (ADC), and to compare synthetic T-2-weighted (T2W) images calculated from T(2)values to conventional T2W images for CC staging. Study Type Retrospective. Population Sixty-three patients with histopathologically confirmed CC. Field Strength/Sequence 3T, conventional T2W turbo spin-echo, diffusion-weighted echo-planar, and accelerated T(2)mapping sequence. Assessment T(2)and ADC values between different pathological features of CC were compared. The diagnostic accuracies of conventional and synthetic T2W images in staging were also compared. Statistical Tests Parameters were compared using an independentt-test, Wilcoxon signed-rank test, and the chi-square test. Receiver operating characteristic analysis was performed. Results The T(2)values varied significantly between well/moderately differentiated and poorly differentiated tumors ([92.8 +/- 9.5 msec] vs. [83.8 +/- 9.5 msec],P < 0.05) and between LVSI-positive and LVSI-negative CC ([82.2 +/- 8.2 msec] vs. [93.9 +/- 9.1 msec],P < 0.05). The ADC values showed a significant difference for grade ([0.76 +/- 0.10 x 10(-3)mm(2)/s] vs. [0.65 +/- 0.11 x 10(-3)mm(2)/s],P < 0.05) and no difference for LVSI status ([0.71 +/- 0.11x 10(-3)mm(2)/s] vs. [0.73 +/- 0.12x 10(-3)mm(2)/s],P= 0.472). There was no significant difference in T(2)and ADC values between squamous cell carcinoma and adenocarcinoma (P= 0.378 andP= 0.661, respectively). In MRI staging, the conventional and synthetic T2W images resulted in a similar accuracy (71% vs. 68%,P= 0.698). Data Conclusion The accelerated T(2)mapping sequence may facilitate grading and staging of CC by providing quantitative T(2)maps and synthetic T2W images in one acquisition. T(2)values may be superior to ADC in predicting LVSI. Level of Evidence 2 Technical Efficacy Stage 2

WILEY2020

NovelT2Mapping for Evaluating Cervical Cancer Features by Providing QuantitativeT2Maps and Synthetic Morphologic Images: A Preliminary Study

Tom Hilbert, Tobias Kober, Jie Liu, Meng Yang, Yong Zhang

WILEY2020

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

Sonja Rieder

EPFL2012

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process

NovelT2Mapping for Evaluating Cervical Cancer Features by Providing QuantitativeT2Maps and Synthetic Morphologic Images: A Preliminary Study

A bootstrap adaptive test for two-way analysis of variance

A bootstrap adaptive test for two-way analysis of variance

NovelT2Mapping for Evaluating Cervical Cancer Features by Providing QuantitativeT2Maps and Synthetic Morphologic Images: A Preliminary Study

Robust Parameter Estimation for the Ornstein-Uhlenbeck Process