## List of Tables

1.1 Main types of climate archives, covered time ranges and absolute dating methods

1.2 Climate archives and variables studied in this book (selection)

1.3 Measurement and proxy errors in selected climate time series

2.1 Result of DFA study, estimated power-law exponents a

3.1 Monte Carlo experiment, mean estimation of a Gaussian

3.2 Monte Carlo experiment, standard deviation estimation of a Gaussian purely random process 71

3.3 Monte Carlo experiment, mean and median estimation of a lognormal purely random process 72

3.4 Estimation settings (theoretical and practical) and approaches (classical and bootstrap) to solve practical problems 75

3.5 Monte Carlo experiment, mean estimation of AR(1) noise processes with uneven spacing, normal and lognormal shape 90

3.6 Notation 95 3.6 Notation (continued) 96 3.6 Notation (continued) 97 3.6 Notation (continued) 98 3.6 Notation (continued) 99

purely random process

3.7 Monte Carlo experiment, moving block bootstrap adaption to uneven spacing 102

4.1 Monte Carlo experiment, linear OLS regression with AR(1) noise of normal shape, even spacing:

CI coverage performance 125

4.2 Monte Carlo experiment, linear OLS regression with AR(1) noise of normal shape, even spacing:

average CI length 126

4.3 Monte Carlo experiment, linear OLS regression with

AR(1) noise of lognormal shape, even spacing 126

4.4 Monte Carlo experiment, linear OLS regression with

AR(2) noise of normal shape, even spacing 127

4.5 Monte Carlo experiment, linear OLS regression with ARFIMA(0, รถ, 0) noise of normal shape, even spacing 128

4.6 Errors and spread of time values for dated proxy time series 130

4.7 Monte Carlo experiment, linear OLS regression with timescale errors and AR(1) noise of normal shape:

CI coverage performance, slope 137

4.8 Monte Carlo experiment, linear OLS regression with timescale errors and AR(1) noise of normal shape:

RMSE and average CI length, slope 138

4.9 Monte Carlo experiment, linear OLS regression with timescale errors and AR(1) noise of normal shape:

CI coverage performance, intercept 139

4.10 Monte Carlo experiment, linear OLS regression with timescale errors and AR(1) noise of lognormal shape:

CI coverage performance 139

4.11 Monte Carlo experiment, linear OLS regression with timescale errors and AR(2) noise of normal shape:

CI coverage performance 140

4.12 Monte Carlo experiment, linear OLS regression with AR(2) noise of normal shape: dependence on size of timescale errors 140

4.13 Monte Carlo experiment, ramp regression with timescale errors and AR(1) noise of normal shape:

CI coverage performance 147

4.14 Monte Carlo experiment, break regression with timescale errors and AR(1) noise of normal shape:

CI coverage performance 153

6.1 Monte Carlo experiment, hypothesis tests for trends in occurrence of extremes 261

6.2 Monte Carlo experiment, hypothesis tests for trends in occurrence of extremes (continued) 262

6.3 Monte Carlo experiment, hypothesis tests for trends in occurrence of extremes (continued) 263

6.4 Monte Carlo experiment, hypothesis tests for trends in occurrence of extremes (continued) 264

6.5 Notation for Section 6.4 267

6.6 GEV distribution, parameter notations 270

7.1 Monte Carlo experiment, Spearman's correlation coefficient with Fisher's z-transformation for bi-

variate lognormal AR(1) processes 303

7.2 Monte Carlo experiment, Spearman's correlation coefficient with Fisher's z-transformation for bi-variate lognormal AR(1) processes: influence of block length selection 304

7.3 Monte Carlo experiment, Spearman's correlation coefficient without Fisher's z-transformation for bi-

variate lognormal AR(1) processes 305

7.4 Monte Carlo experiment, Pearson's correlation coefficient with Fisher's z-transformation for bivari-

ate lognormal AR(1) processes 306

7.5 Monte Carlo experiment, Pearson's correlation coefficient with Fisher's z-transformation for binormal AR(1) processes 307

7.6 Monte Carlo experiment, Pearson's and Spearman's correlation coefficients with Fisher's z-transformation for bivariate lognormal AR(1) processes: calibrated

### CI coverage performance 308

7.7 Monte Carlo experiment, Pearson's and Spearman's correlation coefficients with Fisher's z-transformation for bivariate lognormal AR(1) processes: average calibrated CI length 308

7.8 Grade correlation coefficient, bivariate lognormal distribution 327

8.1 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape and complete prior knowledge: CI coverage performance 351

8.2 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape and complete prior knowledge: CI coverage performance (continued) 352

8.3 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape and complete prior knowledge: RMSE 353

8.4 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal/lognormal shape and incomplete prior knowledge: CI coverage performance 354

8.5 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape: influence of accuracy of prior knowledge on CI coverage performance 356

8.6 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape: influence of accuracy of prior knowledge on RMSE 357

8.7 Monte Carlo experiment, linear errors-in-variables regression with AR(1) noise of normal shape: influence of mis-specified prior knowledge on CI coverage performance 358

8.8 Estimates of the effective climate sensitivity 377

Part I

Fundamental Concepts

Chapter 1

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