CONTENTS... Preface to the English edition III... Preface to the original French edition V Summary (English, French, Russian, Spanish)... VII Tothereader... 1. Simple random character of series of observations. Test of hypothesis............................... 1 1.1 TEST OF HYPOTHESIS - SIGNIFICANCE TEST.........................,.............. 1 1.2 PARAMETRIC - NON-PARAMETRIC TESTS.......................................... 2 1.3 SIGNIFICANCE LEVEL OF A MULTIPLE TEST.....................................,... 3... 1.3.1 The Fisher test 3... 1.3.2 The test based on the binomial distribution 4... 1.3.3 Comparison between the Fisher test and the test based on the binomial 4 1.3.3.1 Example 1. Significance level of a series of twelve independent tests... 5 1.4 TEST FOR THE SIMPLE RANDOM CHARACTER OF A SERIES OF OBSERVATIONS................. 6... 1.4.1 Serial correlation test 7 1.4.1.1 Example 2. Application of the serial correlation test... 8 l.4.2trendtesr-s... 10... 1.4.2.1 Calculation of the Spearman coefficient rs 10 1.4.2.2 Calculation of the Kendall coefficient t (Mann test)... 11 1.4.2.3 Comparison between the rs and t statistics 11... 1.4.2.4 Example 3. Application of trend tests 11 1.4.2.5 Determination of the simple random character of the series of annual wind speed averages at Uccle... from 1933 to 1969 12 1.4.2.6 Progressive anaiysis of a series by means of the statistic u(t)... 12... 1.4.2.7 Case of series containing equal terms 14 XVI 1.4.3 Test for stability of the variance 1.5 REFERENCE WORKS CONSULTED 15 15 2. On statistical estimation 2.1 EMPIRICAL ESTIMATION..................................................... 17 2.1.l Distribution functions for the order statistics. Confidence intervals... 18 2.1.2 Central values, variances and covariances of the variables Ftn. Estimation of F(x J... 19 2.1.3 The empirical distribution function. The Kolmogorov test... 2 1 2.1.4 Empirical mean and variance... 22
XVIII CONTENTS 2.15 Case of scverai samples. Kruskai-Wallis homogeneity test.................................... 23 2.16 Graphical representation of a series of observations....................................... 24 2.1.7 Example 4. The monthly rainfall totais collected at Brussels-Uccie in % of the average for odd months......... 24 2.1.7.1 Homogeneity of the means of the six series of observations............................. 25 2.1.7.2 Homogeneity of the variances of the six series of observations............................ 26 2.1.7.3 CaIculation of the confidence intervals of the probabilities associated with the order statistics......... 27 2.1.7.4 Estimation of the vaiue of x for a given probability Fo. Estimation of the probability F for a given value x0. Associated two-sided confidence intetvals.................................. 28 2.1.7.5 Estimation of the probabilities in the case of equal values............,................. 29 2.1.7.6 Two-sided Koimogorov limits of the true distribution function........................... 30 2.1.7.7 Calculation of the empirical mean and variance. Confidence interval of the mean................. 3 1 2.1.7.8 Graphicai representation of the results......................................... 33 2.18 Case of discrete distributions.................................................... 33 2.1.9 The binomial distribution........................ _............................. 33 2.1.9.1 Confidence interval for a frequency..... _. _............................... _... 34 2.1.9.2 Example 5. Calculation of the two-sided confidence interval at the 0,95 level of the number of cases in in 100 years of an event whose probability is l/10 35 2.1.9.3 Confidence interval of a probability........... _......... _..................... 36 2.1.9.4 Example 6. Calculation of the confidence intervj of the probability associated with the frequency nl=10inatotalnumberofcasesn=60.. _.................................... 37 2.1.9.5 Themean recurrenceinterval...................... _................... _... _ 38 2.1.10 The asymptotic form of the multinomiai distribution. The Pearson test _....... _...........,...... 38 2.1.10.1 Example 7. Probabilities of the sequences of consecutive dry years at Brussels-Uccle............. 40 2.1-11 Reference works consulted........ _............... _.. _........................ 41 2.2 PARAMETRIC ESTIMATION.................................................... 41 2.2.1 Estimation by the method of moments. Fitting the normal distribution.......,... _..........,.... 43 2.2.1.1 Example 8. Estimation by fitting a normsl distribution.... _.. _........................ 45 2.2.2 Sufficient statistics. Estimates with minimum variance....... _........................... 46 2.2.2.1 Case of the normal distribution........................... _. _............ 48 2.2.2.1-l Example9. Fittingof alognormal distribution......... _........ _.............. 49 2.2.2.2Thegammadistribution.... _............................................. 50 2.2.2.2-l Example 10. Fitting of a gamma distribution..,....... _.......................... 52 2.2.3 Estimation by the maximum likelihood method. Efficiency of an estimator................ _....... 53 2.2.3.1 The double exponential distibution (Fisher-Tippett Type I).........,..., _.............. 54 2.2.3.l.l Example 11. Fitting a double exponentiai distribution............................... 56 2.2.4 Estimation by least squares............ _............. _.......................... 58 2.2.4.1 Examples of estimation by the least squares...................................... 61 2.2.4.1.l Example 12. Calculation of the normai of the number of frost days in Belgium in January from the normal of the daily mean minimum air temperature................................ 61 2.2.4.1.2 Example 13. Normals of the monthly rainfall totals in January in Belgium. Caicuiation as a function of the altitude of the station....... _....... _...........................,... 65 2.2.4.2 Application to the distributions defined by a position parameter p and a scale parsmeter 0. Linear estimators of the parameters............................................... 68 2.2.4.3 Estimators with polynomial coefficients........................................ 69 2.2.4.4 Reference works consulted _.............. _..... _.......................... 70
CONTENTS XIX 2.2.5 On the choice of the distribution to be fitted.......................................... 70 2.2.5.1 The exponential distribution as a particular case of the gamma distribution... 71 2.2.5.1.l Example 14. The duration of glaze at Brussels... 72 2.2.5.2 Tests for normality... 73 2.2.5.2.1 Tests for normality based on calculation of the empirical moments... 74 2.2.5.2.1-l Example 15. Air temperature at 225 mb at Uccle in January. Distribution of daily averages. Test for normality based on the central empirical moments... 75 2.2.5.2.2 Tests for normality based on the order statistics or on the corresponding values of the normal distribution function... 77 2.2.5.2.2.1 Example 15 (continued). Application of tests for normality based on the order statistics... 79 2.2.5.2.3 On the choice of the test for normality... 80 2.2.5.3 On the choice of the alternative when the hypothesis of normality is rejected... 80 2.2.5.4 On graphical representation on probability paper. Final goodness-of-fit of a fitted distribution... 82 2.2.5.4.1 Example 15 (continued). Final fitting and graphical representation... 83 2.2.5.5 Reference works consulted... 86 2.2.6 1 The parametric estimation in the case of distributions of discrete variables... 86 2.2.6.1 Notation and generalities. The principle of minimum 2... 87 2.2.6.2 The binomial distribution... 90 2.2.6.2.1 Example 16. The frequency per year of months with low rainfall at Uccle... 91 2.2.6.3 The Poisson distribution... 93 2.2.6.3.1 Example 17. The frequency of storms in Belgium in January... 94 2.2.6.4 The negative binomial distribution... 95 2.2.6.4.1 Example 18. The frequency of storms in Belgium in March... 97 2.2.6.4.1.1 Calculation of the coefficient flof the negative binomial distribution by successive approximation... 99 2.2.6.5 The geometric distribution... 100 2.2.6.5.1 Example 19. Sequences of consecutive months with below-normal rainfall at Uccle... 101 2.2.6.6 The logarithmic distribution...................... d..... 102 2.2.6.6.1 Example 20. Sequences at Uccle of consecutive days with mean temperature <- 5 (70 years)... 104 2.2.6.7 Choice of the discrete distribution to be fitted... 105 2.2.6.7.1 Example 21. Number of sequences per year at Uccle of consecutive days with mean temperature <- 5O (70 years)... 107 2.2.6.8 Distribution of maximal sequences... 108 2.2.6.8.1 Example 22. The annual maximal sequence at Uccle of consecutive days with mean temperature <- 5 (70 years)... 109 2.2.6.9 The approximate continuous forms of discrete distributions... 111 2.2.6.9.1 Example 23. Number of days with measurable precipitation at Uccle in July (70 years)... 112 2.2.6.10 Reference works consulted... 113 2.3 PARAMETRIC ESTIMATION IN THE CASE OF CORRELATED SERIES... 114 2.3.1 Generalities and fundamental problems... 114 2.3.2 Optimal parametric estimation... 116 2.3.2.1 Case of a bivariate normal distribution... 116 2.3.2.1.1 Example 24. The annual rainfall on the drainage basin of the Senne at Vilvoorde (Belgium)... 117 2.3.2.2 Case in which the estimators of the location and scale parameters are linear... 119 2.3.2.2.1 Example 25. The annual maximum of the daily rainfall in the stations neighbouring the station at Uccle Estimation of the parameters of the distribution... 120 2.3.2.3 Estimation of missingvalues... 123 2.3.2.3.1 Example 25 (continued). The annual maximum of the daily rainfall in the stations neighbouring the station at Uccle. Estimation of missing values... 124
xx CONTENTS 2.3.3 The meihodof differences................................... i................. 126 2.3.4 Reference works consulted..................................................... 126 3. Estimation in the case of recurrent series 3.1 PERIODIC SERIES.......................................................... 129 3.1.l Estimation of the deterministic component of a periodic series when the period is known and the covariance matrix of the random component is unknown......................,................... 130 3.1.1.l Example 26. The mean daily value of rainfall at Ucclc, for days with measurable precipitation.......... 134 3.1.1.2 Example 27. Estimation of the lnre mean values of the monthly averages of the air temperature at Uccle, from a series of monthly averages over two years (1971 and 1972).......................... 136 3.1.2 Estimation of the deterministic component of a periodic series when the period of the deterrniniitic component andthecovariancematrixoftherandompartareknown....,............................... 138 3.1.2.1 Example 28. Daily averafs of the air temperature at Uccle. Estimation of the true mean values on the basis of 12 equally spaced daily normals (period 1901-1970)............................. 141 3.1.3 Reference works consulted................... _................................. 145 3;1.4 Estimation of the deterministic component of a periodic series when the period of the deterministic component is known but has an arbitrary value, the covariance matrix of the random part being unknown.... _......... 145 3.1.4.1 Example 29. Estimation of a simulated periodic component introduced in advance into the total annual rainfall series for Uccle................................................... 148 3-1.5 Estimation of the deterministic component of a periodic series when the period of the component is known only approximately or is unknown, the varianceof the random component also being unknown... _... 151 3.1.5.l Least squares estimation in the non-linear case... ;... 152 3.1.5.2 Empirical autocovariances of a time series with a periodic deterministic component... 153 3.1.5.3 Determination of the periodic deterministic component when the period is unknown... 154 3.1.5.4 Example 30. Estimate of a periodic deterministic component introduced in advance into the total annual rainfall series for Uccle, when the period itself is subject to estimation... 155 3.2 RECURRENT SERIES........................................................ 158 3.2.1 General properties of recurrent series... 158 3.2.2 Estimation and determination of the recurrence... 160 3.2.3 Estimation of the mean value, the variance and the autocmariances of a recurrent series... 162 3.2.3.1 Example 3 1. Determination and estimation of the recurrence between the daily averages of the atmospheric pressure at Uccle (two years of observations)... 164 3.2.4 Extreme value of a recurrent series. Bartels equivalent number of repetitions...,... 168 3.2.4.1 Example 32. The maximum values of the daily average of the atmospheric pressure at Uccle, in April... 169 3.2.5 Application to the determination of the periodic deterministic components of a time series................ 171 3.2.5.1 Example 33. Determination of any periodic deterministic component in the (unmodified) total annuai rainfall series for BrusselsUccle (137 years)....................................... 172
CONTENTS XXI 3.3 PERSISTENT SERIES... 174 3.3.1 Example 34. The daily maxima of the air temperature recorded at 8 a.m. at Uccle in July... 175 3.4 MOVING AVERAGES. WEIGHTED AVERAGES. FILTERS... 177 3.4.1 Example 35. Cumulative sums of the rainfall at Uccle over 3,6 and 12 consecutive months. Application to the case of the droughts of 1921.1949 and 1864... 179 3.5 CONCLUSIONS.....18 1 3.6 REFERENCE WORKS CONSULTED... 181 4. Acknowledgements Annex.....................................................................185 Bibliography................................................................. 191