原文鏈接：http://tecdat.cn/?p=13963

在精算科學和保險費率制定中，考慮到風險敞口可能是一場噩夢。不知何故，簡單的結(jié)果是因為計算起來更加復雜，只是因為我們必須考慮到暴露是一個異構(gòu)變量這一事實。

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保險費率制定中的風險敞口可以看作是審查數(shù)據(jù)的問題（在我的數(shù)據(jù)集中，風險敞口始終小于1，因為觀察結(jié)果是合同，而不是保單持有人），利息變量是未觀察到的變量，因為我們必須為保險合同定價一年（整年）的保險期。因此，我們必須對保險索賠的年度頻率進行建模。

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在我們的數(shù)據(jù)集中，我們考慮索賠總數(shù)與總風險承擔比率。例如，如果我們考慮泊松過程，可能性是

即

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因此，我們有一個預期值的估算，一個自然估算?。

現(xiàn)在，我們需要估算方差，更準確地說是條件變量。

這可以用來檢驗泊松假設是否對頻率建模有效?？紤]以下數(shù)據(jù)集，

1. > ?nombre=rbind(nombre1,nombre2)

2. > ?baseFREQ = merge(contrat,nombre)

在這里，我們確實有兩個感興趣的變量，即每張合約的敞口，

> ?E <- baseFREQ$exposition

和（觀察到的）索賠數(shù)量（在該時間段內(nèi)）

> ?Y <- baseFREQ$nbre

無需協(xié)變量，可以計算每個合同的平均（每年）索賠數(shù)量以及相關(guān)的方差

1. > (mean=weighted.mean(Y/E,E))

2. [1] 0.07279295

3. > (variance=sum((Y-mean*E)^2)/sum(E))

4. [1] 0.08778567

看起來方差（略）大于平均值（我們將在幾周后看到如何更正式地對其進行測試）?？梢栽诒纬钟腥司幼〉牡貐^(qū)添加協(xié)變量，例如人口密度，

1. 


2. Density, zone 11 average = 0.07962411 ?variance = 0.08711477

3. Density, zone 21 average = 0.05294927 ?variance = 0.07378567

4. Density, zone 22 average = 0.09330982 ?variance = 0.09582698

5. Density, zone 23 average = 0.06918033 ?variance = 0.07641805

6. Density, zone 24 average = 0.06004009 ?variance = 0.06293811

7. Density, zone 25 average = 0.06577788 ?variance = 0.06726093

8. Density, zone 26 average = 0.0688496 ? variance = 0.07126078

9. Density, zone 31 average = 0.07725273 ?variance = 0.09067

10. Density, zone 41 average = 0.03649222 ?variance = 0.03914317

11. Density, zone 42 average = 0.08333333 ?variance = 0.1004027

12. Density, zone 43 average = 0.07304602 ?variance = 0.07209618

13. Density, zone 52 average = 0.06893741 ?variance = 0.07178091

14. Density, zone 53 average = 0.07725661 ?variance = 0.07811935

15. Density, zone 54 average = 0.07816105 ?variance = 0.08947993

16. Density, zone 72 average = 0.08579731 ?variance = 0.09693305

17. Density, zone 73 average = 0.04943033 ?variance = 0.04835521

18. Density, zone 74 average = 0.1188611 ? variance = 0.1221675

19. Density, zone 82 average = 0.09345635 ?variance = 0.09917425

20. Density, zone 83 average = 0.04299708 ?variance = 0.05259835

21. Density, zone 91 average = 0.07468126 ?variance = 0.3045718

22. Density, zone 93 average = 0.08197912 ?variance = 0.09350102

23. Density, zone 94 average = 0.03140971 ?variance = 0.04672329

可以可視化該信息

1. > plot(meani,variancei,cex=sqrt(Ei),col="grey",pch=19,

2. + xlab="Empirical average",ylab="Empirical variance")

3. > points(meani,variancei,cex=sqrt(Ei))

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圓圈的大小與組的大小有關(guān)（面積與組內(nèi)的總暴露量成正比）。第一個對角線對應于泊松模型，即方差應等于均值。也可以考慮其他協(xié)變量

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或汽車品牌，

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也可以將駕駛員的年齡視為分類變量

讓我們更仔細地看一下不同年齡段的人，

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在右邊，我們可以觀察到年輕的（沒有經(jīng)驗的）駕駛員。那是預料之中的。但是有些類別??低于??第一個對角線：期望的頻率很大，但方差不大。也就是說，我們??可以肯定的??是，年輕的駕駛員會發(fā)生更多的車禍。相反，它不是一個異類：年輕的駕駛員可以看作是一個相對同質(zhì)的類，發(fā)生車禍的頻率很高。

使用原始數(shù)據(jù)集（在這里，我僅使用具有50,000個客戶的子集），我們確實獲得了以下圖形：

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由于圈正在從18歲下降到25歲，因此具有明顯的經(jīng)驗影響。

同時我們可以發(fā)現(xiàn)有可能將曝光量視為標準變量，并查看系數(shù)實際上是否等于1。如果沒有任何協(xié)變量，

1. 


2. 


3. Call:

4. glm(formula = Y ~ log(E), family = poisson("log"))

5. 


6. Deviance Residuals:

7. Min ? ? ? 1Q ? Median ? ? ? 3Q ? ? ?Max

8. -0.3988 ?-0.3388 ?-0.2786 ?-0.1981 ?12.9036

9. 


10. Coefficients:

11. Estimate Std. Error z value Pr(>|z|)

12. (Intercept) -2.83045 ? ?0.02822 -100.31 ? <2e-16 ***

13. log(E) ? ? ? 0.53950 ? ?0.02905 ? 18.57 ? <2e-16 ***

14. ---

15. Signif. codes: ?0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

16. 


17. (Dispersion parameter for poisson family taken to be 1)

18. 


19. Null deviance: 12931 ?on 49999 ?degrees of freedom

20. Residual deviance: 12475 ?on 49998 ?degrees of freedom

21. AIC: 16150

22. 


23. Number of Fisher Scoring iterations: 6

也就是說，該參數(shù)顯然嚴格小于1。它與重要性均不相關(guān)，

1. Linear hypothesis test

2. 


3. Hypothesis:

4. log(E) = 1

5. 


6. Model 1: restricted model

7. Model 2: Y ~ log(E)

8. 


9. Res.Df Df ?Chisq Pr(>Chisq)

10. 1 ?49999

11. 2 ?49998 ?1 251.19 ?< 2.2e-16 ***

12. ---

13. Signif. codes: ?0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

我也沒有考慮協(xié)變量，

1. 


2. 


3. Deviance Residuals:

4. Min ? ? ? 1Q ? Median ? ? ? 3Q ? ? ?Max

5. -0.7114 ?-0.3200 ?-0.2637 ?-0.1896 ?12.7104

6. 


7. Coefficients:

8. Estimate Std. Error z value Pr(>|z|)

9. (Intercept) ? ? ? ? ? ? ? ? ?-14.07321 ?181.04892 ?-0.078 0.938042

10. log(exposition) ? ? ? ? ? ? ? ?0.56781 ? ?0.03029 ?18.744 ?< 2e-16 ***

11. carburantE ? ? ? ? ? ? ? ? ? ?-0.17979 ? ?0.04630 ?-3.883 0.000103 ***

12. as.factor(ageconducteur)19 ? ?12.18354 ?181.04915 ? 0.067 0.946348

13. as.factor(ageconducteur)20 ? ?12.48752 ?181.04902 ? 0.069 0.945011

因此，假設暴露是此處的外生變量可能是一個過強的假設。

接下來我們開始討論建模索賠頻率時的過度分散。在前面，我討論了具有不同暴露程度的經(jīng)驗方差的計算。但是我只使用一個因素來計算類。當然，可以使用更多的因素。例如，使用因子的笛卡爾積，

1. 


2. Class D A (17,24] ?average = 0.06274415 ?variance = 0.06174966

3. Class D A (24,40] ?average = 0.07271905 ?variance = 0.07675049

4. Class D A (40,65] ?average = 0.05432262 ?variance = 0.06556844

5. Class D A (65,101] average = 0.03026999 ?variance = 0.02960885

6. Class D B (17,24] ?average = 0.2383109 ? variance = 0.2442396

7. Class D B (24,40] ?average = 0.06662015 ?variance = 0.07121064

8. Class D B (40,65] ?average = 0.05551854 ?variance = 0.05543831

9. Class D B (65,101] average = 0.0556386 ? variance = 0.0540786

10. Class D C (17,24] ?average = 0.1524552 ? variance = 0.1592623

11. Class D C (24,40] ?average = 0.0795852 ? variance = 0.09091435

12. Class D C (40,65] ?average = 0.07554481 ?variance = 0.08263404

13. Class D C (65,101] average = 0.06936605 ?variance = 0.06684982

14. Class D D (17,24] ?average = 0.1584052 ? variance = 0.1552583

15. Class D D (24,40] ?average = 0.1079038 ? variance = 0.121747

16. Class D D (40,65] ?average = 0.06989518 ?variance = 0.07780811

17. Class D D (65,101] average = 0.0470501 ? variance = 0.04575461

18. Class D E (17,24] ?average = 0.2007164 ? variance = 0.2647663

19. Class D E (24,40] ?average = 0.1121569 ? variance = 0.1172205

20. Class D E (40,65] ?average = 0.106563 ? ?variance = 0.1068348

21. Class D E (65,101] average = 0.1572701 ? variance = 0.2126338

22. Class D F (17,24] ?average = 0.2314815 ? variance = 0.1616788

23. Class D F (24,40] ?average = 0.1690485 ? variance = 0.1443094

24. Class D F (40,65] ?average = 0.08496827 ?variance = 0.07914423

25. Class D F (65,101] average = 0.1547769 ? variance = 0.1442915

26. Class E A (17,24] ?average = 0.1275345 ? variance = 0.1171678

27. Class E A (24,40] ?average = 0.04523504 ?variance = 0.04741449

28. Class E A (40,65] ?average = 0.05402834 ?variance = 0.05427582

29. Class E A (65,101] average = 0.04176129 ?variance = 0.04539265

30. Class E B (17,24] ?average = 0.1114712 ? variance = 0.1059153

31. Class E B (24,40] ?average = 0.04211314 ?variance = 0.04068724

32. Class E B (40,65] ?average = 0.04987117 ?variance = 0.05096601

33. Class E B (65,101] average = 0.03123003 ?variance = 0.03041192

34. Class E C (17,24] ?average = 0.1256302 ? variance = 0.1310862

35. Class E C (24,40] ?average = 0.05118006 ?variance = 0.05122782

36. Class E C (40,65] ?average = 0.05394576 ?variance = 0.05594004

37. Class E C (65,101] average = 0.04570239 ?variance = 0.04422991

38. Class E D (17,24] ?average = 0.1777142 ? variance = 0.1917696

39. Class E D (24,40] ?average = 0.06293331 ?variance = 0.06738658

40. Class E D (40,65] ?average = 0.08532688 ?variance = 0.2378571

41. Class E D (65,101] average = 0.05442916 ?variance = 0.05724951

42. Class E E (17,24] ?average = 0.1826558 ? variance = 0.2085505

43. Class E E (24,40] ?average = 0.07804062 ?variance = 0.09637156

44. Class E E (40,65] ?average = 0.08191469 ?variance = 0.08791804

45. Class E E (65,101] average = 0.1017367 ? variance = 0.1141004

46. Class E F (17,24] ?average = 0 ? ? ? ? ? variance = 0

47. Class E F (24,40] ?average = 0.07731177 ?variance = 0.07415932

48. Class E F (40,65] ?average = 0.1081142 ? variance = 0.1074324

49. Class E F (65,101] average = 0.09071118 ?variance = 0.1170159

同樣，可以將方差與平均值作圖，

1. > plot(vm,vv,cex=sqrt(ve),col="grey",pch=19,

2. + xlab="Empirical average",ylab="Empirical variance")

3. > points(vm,vv,cex=sqrt(ve))

4. > abline(a=0,b=1,lty=2)

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一種替代方法是使用樹。樹可以從其他變量獲得，但它應該是相當接近我們理想的模型。在這里，我確實使用了整個數(shù)據(jù)庫（超過60萬行）

樹如下

1. > plot(T)

2. > text(T)

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現(xiàn)在，每個分支都定義了一個類，可以使用它來定義一個類。應該被認為是同質(zhì)的。

1. 


2. Class ?6 average = ? 0.04010406 ?variance = 0.04424163

3. Class ?8 average = ? 0.05191127 ?variance = 0.05948133

4. Class ?9 average = ? 0.07442635 ?variance = 0.08694552

5. Class ?10 average = ?0.4143646 ? variance = 0.4494002

6. Class ?11 average = ?0.1917445 ? variance = 0.1744355

7. Class ?15 average = ?0.04754595 ?variance = 0.05389675

8. Class ?20 average = ?0.08129577 ?variance = 0.0906322

9. Class ?22 average = ?0.05813419 ?variance = 0.07089811

10. Class ?23 average = ?0.06123807 ?variance = 0.07010473

11. Class ?24 average = ?0.06707301 ?variance = 0.07270995

12. Class ?25 average = ?0.3164557 ? variance = 0.2026906

13. Class ?26 average = ?0.08705041 ?variance = 0.108456

14. Class ?27 average = ?0.06705214 ?variance = 0.07174673

15. Class ?30 average = ?0.05292652 ?variance = 0.06127301

16. Class ?31 average = ?0.07195285 ?variance = 0.08620593

17. Class ?32 average = ?0.08133722 ?variance = 0.08960552

18. Class ?34 average = ?0.1831559 ? variance = 0.2010849

19. Class ?39 average = ?0.06173885 ?variance = 0.06573939

20. Class ?41 average = ?0.07089419 ?variance = 0.07102932

21. Class ?44 average = ?0.09426152 ?variance = 0.1032255

22. Class ?47 average = ?0.03641669 ?variance = 0.03869702

23. Class ?49 average = ?0.0506601 ? variance = 0.05089276

24. Class ?50 average = ?0.06373107 ?variance = 0.06536792

25. Class ?51 average = ?0.06762947 ?variance = 0.06926191

26. Class ?56 average = ?0.06771764 ?variance = 0.07122379

27. Class ?57 average = ?0.04949142 ?variance = 0.05086885

28. Class ?58 average = ?0.2459016 ? variance = 0.2451116

29. Class ?59 average = ?0.05996851 ?variance = 0.0615773

30. Class ?61 average = ?0.07458053 ?variance = 0.0818608

31. Class ?63 average = ?0.06203737 ?variance = 0.06249892

32. Class ?64 average = ?0.07321618 ?variance = 0.07603106

33. Class ?66 average = ?0.07332127 ?variance = 0.07262425

34. Class ?68 average = ?0.07478147 ?variance = 0.07884597

35. Class ?70 average = ?0.06566728 ?variance = 0.06749411

36. Class ?71 average = ?0.09159605 ?variance = 0.09434413

37. Class ?75 average = ?0.03228927 ?variance = 0.03403198

38. Class ?76 average = ?0.04630848 ?variance = 0.04861813

39. Class ?78 average = ?0.05342351 ?variance = 0.05626653

40. Class ?79 average = ?0.05778622 ?variance = 0.05987139

41. Class ?80 average = ?0.0374993 ? variance = 0.0385351

42. Class ?83 average = ?0.06721729 ?variance = 0.07295168

43. Class ?86 average = ?0.09888492 ?variance = 0.1131409

44. Class ?87 average = ?0.1019186 ? variance = 0.2051122

45. Class ?88 average = ?0.05281703 ?variance = 0.0635244

46. Class ?91 average = ?0.08332136 ?variance = 0.09067632

47. Class ?96 average = ?0.07682093 ?variance = 0.08144446

48. Class ?97 average = ?0.0792268 ? variance = 0.08092019

49. Class ?99 average = ?0.1019089 ? variance = 0.1072126

50. Class ?100 average = 0.1018262 ? variance = 0.1081117

51. Class ?101 average = 0.1106647 ? variance = 0.1151819

52. Class ?103 average = 0.08147644 ?variance = 0.08411685

53. Class ?104 average = 0.06456508 ?variance = 0.06801061

54. Class ?107 average = 0.1197225 ? variance = 0.1250056

55. Class ?108 average = 0.0924619 ? variance = 0.09845582

56. Class ?109 average = 0.1198932 ? variance = 0.1209162

在這里，當根據(jù)索賠的經(jīng)驗平均值繪制經(jīng)驗方差時，我們得到

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在這里，我們可以識別剩余異質(zhì)性的類。

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