Tag Archives: average

Fairness and discrimination, PhD Course, #2 Insurance and risk classes

For the second course, we will get back a little bit on insurance pricing in a context of heterogeneous portfolio, and risk classification (slides are still online on the github repository). The starting point will be the pure premium.

See our online textbook, with Michel Denuit, Non Life Insurance Mathematics, for additional motivation. If we have some risk related variables \boldsymbol{x}=(x_1,\cdots,x_k), the pure premium will be the conditional expectation,

Here also, we have some law of numbers, for the conditional expected value,

This relationship, which defines the conditional expected value using the limiting value of a conditional frequency cannot be used to define properly \mathbb{P}[Y|\boldsymbol{X}=\boldsymbol{x}] and \mathbb{E}[Y|\boldsymbol{X}=\boldsymbol{x}]. One can consider a limit,\mathbb{P}\big(Y\in \mathcal{A}\big\vert X = x\big)=\lim_{\epsilon\to0}\frac{\mathbb{P}(\{Y\in \mathcal{A}\}\cap\{|X -x|\leq \epsilon\})}{\mathbb{P}(\{|X -x|\leq \epsilon\})}or\mathbb{P}\big(Y\in \mathcal{A}\big\vert X = x\big)=\lim_{\epsilon\to0}\mathbb{P}\big(Y\in \mathcal{A}\big\vert |X -x|\leq \epsilon\big)as in the law of the unconscious statistician or as Proschan and Presnell (1998) wrote it

statisticians make liberal use of conditioning arguments to shorten what would otherwise be long proofs

We can now compute conditional frequency, given some risk characteristics, for some quantity of interest y, such as the age of death, in life insurance contracts.

Demographic risk and heterogeneity

First, we will see some gender-based life tables, starting with the one obtained by Nicolaas Struyck (see e.g. Alberts et al. (2014))

More recently, in France, some wealth based life tables were obtained, with various quantiles

And finally, we will see some life tables obtained 50 years ago in the US, with racial distinction

Mean and variance decomposition

About pure premiums, an important property is the law of total expectations, and a desirable property, that we will name “balance property”

We will also mention variance and variance decomposition, depending if we take into heterogeneity, or not. With homogenous pricing, we have

If we use the “true” underlying risk factor, \Theta, we have the standard variance decomposition, also called law of total variance


And finally, if we do not observe \Theta, but we have a collection of covariates, \boldsymbol{X}=(X_1,\cdots,X_k),

Some historical perspectives

In the textbook, Insurance: Biases, Discrimination and Fairness, I have several paragraph about an historical perspective, starting with insurance as clubs, without segmentation. Then segmentation started, with risk classes and groups. For example, according to Issues And Needed Improvements In State Regulation Of The Insurance Business, by Harry Havens, in 1979,

The price which a person pays for automobile insurance depends on age, sex, marital status, place of residence and other factors. This risk classification system produces widely differing prices for the same coverage for different people. Questions have been raised about the fairness of this system, and especially about its reliability as a predictor of risk for a particular individual. While we have not tried to judge the propriety of these groupings, and the resulting price differences, we believe that the questions about them warrant careful consideration by the State insurance departments. In most States the authority to examine classification plans is based on the requirement that insurance rates are neither inadequate, excessive, nor unfairly discriminatory. The only criterion for approving classifications in most States is that the classifications be statistically justified — that is, that they reasonably reflect loss experience. Relative rates with respect to age, sex, and marital status are based on the analysis of national data. A youthful male driver, for example, is charged twice as much as an older driver all over the country} (…) t has also been claimed that insurance companies engage in redlining – the arbitrary denial of insurance to everyone living in a particular neighborhood. Community groups and others have complained that State regulators have not been diligent in preventing redlining and other forms of improper discrimination that make insurance unavailable in certain areas. In addition to outright refusals to insure, geographic discrimination can include such practices as: selective placement of agents to reduce business in some areas, terminating agents and not renewing their book of business, pricing insurance at un-affordable levels, and instructing agents to avoid certain areas. We reviewed what the State insurance departments were doing in response to these problem. To determine if redlining exists, it is necessary to collect data on a geographic oasis. Such data should include current insurance policies, new policies being written, cancellations, and non-renewals. It is also important to examine data on losses by neighborhoods within existing rating territories because marked discrepancies within territories would cast doubt on the validity of territorial boundaries. Yet, not even a fifth of the States collect anything other than loss data, and that data is gathered on a territory-wide basis.

According to The Role of Risk Classification in Property and Casualty Insurance: A Study of the Risk Assessment Process : Final Report, by Barbara Casey, Jacques Pezier and Carl Spetzler, in 1976,

On the other hand, the opinion that distinctions based on sex, or any other group variable, necessarily violate individual rights reflects ignorance of the basic rules of logical inference in that it would arbitrarily forbid the use of relevant information. It would be equally fallacious to reject a classification system based on socially acceptable variables because the results appear discriminatory. For example, a classification system may be built on use of car, mileage, merit rating, and other variables, excluding sex. However, when verifying the average rates according to sex one may discover significant differences between males and females. Refusing to allow such differences would be attempting to distort reality by choosing to be selectively blind. The use of rating territories is a case in point. Geographical divisions, however designed, are often correlated with socio-demographic factors such as income level and race because of natural aggregation or forced segregation according to these factors. Again we conclude that insurance companies should be free to delineate territories and assess territorial differences as well as they can. At the same time, insurance companies should recognize that it is in their best interest to be objective and use clearly relevant factors to define territories lest they be accused of invidious discrimination by the public. (…) One possible standard does exist for exception to the counsel that particular rating variables should not be proscribed. What we have called `equal treatment’ standard of fairness may precipitate a societal decision that the process of differentiating among individuals on the basis of certain variables is discriminatory and intolerable. This type of decision should be made on a specific, statutory basis. Once taken, it must be adhered to in private and public transactions alike and enforced by the insurance regulator. This is, in effect, a standard for conduct that by design transcends and preempts economic considerations. Because it is not applied without economic cost, however, insurance regulators and the industry should participate in and inform legislative deliberations that would ban the, use of particular rating variables as discriminatory.

And then, more recently, we started to talk about personalization, as in Barry and Charpentier (2020). And next week, we will start talking about predictive modeling, and machine learning.

Classification with Categorical Variables (the fuzzy side)

The Gaussian and the (log) Poisson regressions share a very interesting property,

i.e. the average predicted value is the empirical mean of our sample.

> mean(predict(lm(dist~speed,data=cars)))
[1] 42.98
> mean(cars$dist)
[1] 42.98

One can prove that it is also the prediction for the average individual in our sample

> predict(lm(dist~speed,data=cars),
+ newdata=data.frame(speed=mean(cars$speed))) 

The geometric interpretation is that the regression line passes through the centroid,

> plot(cars)
> abline(lm(dist~speed,data=cars),col="red")
> abline(h=mean(cars$dist),col="blue")
> abline(v=mean(cars$speed),col="blue")
> points(mean(cars$speed),mean(cars$dist))

But in all other cases, it is no longer the case. Consider for instance the case of a logistic regression. And to ask for something even more complicated, consider the case where we have only categorical explanatory variables. In that context, it is more difficult to get a prediction for the “average individual”. Unless we consider some fuzzy interpretation of the regression.

Continue reading Classification with Categorical Variables (the fuzzy side)

Variance of the Average of a Sequence

In the case where  are i.i.d. random variables, then

Now, what if  are identically distributed, but no longer independent. What if we have an autoregressive process? Assume that


can be written

Here, we will express the variance as a function of  and , but it is possible to use also , since, in the context of an ,

Now, since  we get

which can be simplified, since


So, the variance of the mean can be writen as

Observe that if  is large enough,

This asymptotic relationship is well known actually. A simple way to get it is the following. One can can write

or equivalently

But actually, the first relationship is probably more interesting to get an asymptotic approximation,

In the context of an  process, this can be writen

Thus, we get the following well-known relationship

In the case where  is an i.i.d. sequence, i.e. , then we get the relationship mentioned initially. And in the case of a random walk… unfortunately, we cannot use that relationship. But observe that


which can be written

If we compare the true value and the approximation, we get the following graph,

> V=function(phi,s2=1,n=100){
+ g0=s2/(1-phi^2)
+ if(phi<1){
+ if(phi==0){v1=g0/n}
+ if(phi>0){v1=g0/n^2*(n+2*((n-1)*
+ phi^(-1)-n+phi^(n-1))/(phi^(-1)-1)^2)}
+ v2=g0/n*(1+phi)/(1-phi)
+ }
+ if(phi==1){
+ v1=(2*n+1)*(n+1)*s2/(6*n)
+ v2=NA
+ }
+ return(c(v1,v2))}
> Vphi=function(phi) V(phi,1,100)
> x=seq(.01,1,by=.02)
> M=matrix(unlist(lapply(x,V)),nrow=2)
> plot(x,M[1,],type="l",col="red",log="y",
+ ylab="Variance of the average (log scale)",
+ xlab="Autoregressive coefficient")
> lines(x,M[2,],col="blue")

Visualizing overdispersion (with trees)

This week, we started to discuss overdispersion when modeling claims frequency. In my previous post, I discussed computations of empirical variances with different exposure. But I did use only one factor to compute classes. Of course, it is possible to use much more factors. For instance, using cartesian products of factors,

> X=as.factor(paste(sinistres$carburant,sinistres$zone,
+ cut(sinistres$ageconducteur,breaks=c(17,24,40,65,101))))
> E=sinistres$exposition
> Y=sinistres$nbre
> vm=vv=ve=rep(NA,length(levels(X)))
>   for(i in 1:length(levels(X))){
+  	   ve[i]=Ei=E[X==levels(X)[i]]
+  	   Yi=Y[X==levels(X)[i]]
+   vm[i]=meani=weighted.mean(Yi/Ei,Ei)    # moyenne 
+   vv[i]=variancei=sum((Yi-meani*Ei)^2)/sum(Ei)    # variance
+  cat("Class ",levels(X)[i],"average =",meani," variance =",variancei,"\n")
+ }
Class D A (17,24]  average = 0.06274415  variance = 0.06174966 
Class D A (24,40]  average = 0.07271905  variance = 0.07675049 
Class D A (40,65]  average = 0.05432262  variance = 0.06556844 
Class D A (65,101] average = 0.03026999  variance = 0.02960885 
Class D B (17,24]  average = 0.2383109   variance = 0.2442396 
Class D B (24,40]  average = 0.06662015  variance = 0.07121064 
Class D B (40,65]  average = 0.05551854  variance = 0.05543831 
Class D B (65,101] average = 0.0556386   variance = 0.0540786 
Class D C (17,24]  average = 0.1524552   variance = 0.1592623 
Class D C (24,40]  average = 0.0795852   variance = 0.09091435 
Class D C (40,65]  average = 0.07554481  variance = 0.08263404 
Class D C (65,101] average = 0.06936605  variance = 0.06684982 
Class D D (17,24]  average = 0.1584052   variance = 0.1552583 
Class D D (24,40]  average = 0.1079038   variance = 0.121747 
Class D D (40,65]  average = 0.06989518  variance = 0.07780811 
Class D D (65,101] average = 0.0470501   variance = 0.04575461 
Class D E (17,24]  average = 0.2007164   variance = 0.2647663 
Class D E (24,40]  average = 0.1121569   variance = 0.1172205 
Class D E (40,65]  average = 0.106563    variance = 0.1068348 
Class D E (65,101] average = 0.1572701   variance = 0.2126338 
Class D F (17,24]  average = 0.2314815   variance = 0.1616788 
Class D F (24,40]  average = 0.1690485   variance = 0.1443094 
Class D F (40,65]  average = 0.08496827  variance = 0.07914423 
Class D F (65,101] average = 0.1547769   variance = 0.1442915 
Class E A (17,24]  average = 0.1275345   variance = 0.1171678 
Class E A (24,40]  average = 0.04523504  variance = 0.04741449 
Class E A (40,65]  average = 0.05402834  variance = 0.05427582 
Class E A (65,101] average = 0.04176129  variance = 0.04539265 
Class E B (17,24]  average = 0.1114712   variance = 0.1059153 
Class E B (24,40]  average = 0.04211314  variance = 0.04068724 
Class E B (40,65]  average = 0.04987117  variance = 0.05096601 
Class E B (65,101] average = 0.03123003  variance = 0.03041192 
Class E C (17,24]  average = 0.1256302   variance = 0.1310862 
Class E C (24,40]  average = 0.05118006  variance = 0.05122782 
Class E C (40,65]  average = 0.05394576  variance = 0.05594004 
Class E C (65,101] average = 0.04570239  variance = 0.04422991 
Class E D (17,24]  average = 0.1777142   variance = 0.1917696 
Class E D (24,40]  average = 0.06293331  variance = 0.06738658 
Class E D (40,65]  average = 0.08532688  variance = 0.2378571 
Class E D (65,101] average = 0.05442916  variance = 0.05724951 
Class E E (17,24]  average = 0.1826558   variance = 0.2085505 
Class E E (24,40]  average = 0.07804062  variance = 0.09637156 
Class E E (40,65]  average = 0.08191469  variance = 0.08791804 
Class E E (65,101] average = 0.1017367   variance = 0.1141004 
Class E F (17,24]  average = 0           variance = 0 
Class E F (24,40]  average = 0.07731177  variance = 0.07415932 
Class E F (40,65]  average = 0.1081142   variance = 0.1074324 
Class E F (65,101] average = 0.09071118  variance = 0.1170159

Again, one can plot the variance against the average,

> plot(vm,vv,cex=sqrt(ve),col="grey",pch=19,
+ xlab="Empirical average",ylab="Empirical variance")
> points(vm,vv,cex=sqrt(ve))
> abline(a=0,b=1,lty=2)


An alternative is to use a tree. The tree can be obtained from another variable (the insured had, or had not, a claim, during the period considered) but it should be rather close to the one we would like to model (the number of claims over the period considered). Here, I did use the whole database (with more that 600,000 lines)

> library(tree)
> T=tree((nombre>0)~as.factor(zone)+as.factor(puissance)+
+ as.factor(marque)+as.factor(carburant)+as.factor(region)+
+ agevehicule+ageconducteur,data=baseFREQ,
+ split =  "gini",minsize =25000)

The tree is the following

> plot(T)
> text(T)


Now, each knot defines a class, and it is possible to use it to define a class. Which is supposed to be homogeneous.

> X=as.factor(T$where)
> E=sinistres$exposition
> Y=sinistres$nbre
> vm=vv=ve=rep(NA,length(levels(X)))
>   for(i in 1:length(levels(X))){
+  	   ve[i]=Ei=E[X==levels(X)[i]]
+  	   Yi=Y[X==levels(X)[i]]
+   vm[i]=meani=weighted.mean(Yi/Ei,Ei)    # moyenne 
+   vv[i]=variancei=sum((Yi-meani*Ei)^2)/sum(Ei)    # variance
+  cat("Class ",levels(X)[i],"average =",meani," variance =",variancei,"\n")
+  }
Class  6 average =   0.04010406  variance = 0.04424163 
Class  8 average =   0.05191127  variance = 0.05948133 
Class  9 average =   0.07442635  variance = 0.08694552 
Class  10 average =  0.4143646   variance = 0.4494002 
Class  11 average =  0.1917445   variance = 0.1744355 
Class  15 average =  0.04754595  variance = 0.05389675 
Class  20 average =  0.08129577  variance = 0.0906322 
Class  22 average =  0.05813419  variance = 0.07089811 
Class  23 average =  0.06123807  variance = 0.07010473 
Class  24 average =  0.06707301  variance = 0.07270995 
Class  25 average =  0.3164557   variance = 0.2026906 
Class  26 average =  0.08705041  variance = 0.108456 
Class  27 average =  0.06705214  variance = 0.07174673 
Class  30 average =  0.05292652  variance = 0.06127301 
Class  31 average =  0.07195285  variance = 0.08620593 
Class  32 average =  0.08133722  variance = 0.08960552 
Class  34 average =  0.1831559   variance = 0.2010849 
Class  39 average =  0.06173885  variance = 0.06573939 
Class  41 average =  0.07089419  variance = 0.07102932 
Class  44 average =  0.09426152  variance = 0.1032255 
Class  47 average =  0.03641669  variance = 0.03869702 
Class  49 average =  0.0506601   variance = 0.05089276 
Class  50 average =  0.06373107  variance = 0.06536792 
Class  51 average =  0.06762947  variance = 0.06926191 
Class  56 average =  0.06771764  variance = 0.07122379 
Class  57 average =  0.04949142  variance = 0.05086885 
Class  58 average =  0.2459016   variance = 0.2451116 
Class  59 average =  0.05996851  variance = 0.0615773 
Class  61 average =  0.07458053  variance = 0.0818608 
Class  63 average =  0.06203737  variance = 0.06249892 
Class  64 average =  0.07321618  variance = 0.07603106 
Class  66 average =  0.07332127  variance = 0.07262425 
Class  68 average =  0.07478147  variance = 0.07884597 
Class  70 average =  0.06566728  variance = 0.06749411 
Class  71 average =  0.09159605  variance = 0.09434413 
Class  75 average =  0.03228927  variance = 0.03403198 
Class  76 average =  0.04630848  variance = 0.04861813 
Class  78 average =  0.05342351  variance = 0.05626653 
Class  79 average =  0.05778622  variance = 0.05987139 
Class  80 average =  0.0374993   variance = 0.0385351 
Class  83 average =  0.06721729  variance = 0.07295168 
Class  86 average =  0.09888492  variance = 0.1131409 
Class  87 average =  0.1019186   variance = 0.2051122 
Class  88 average =  0.05281703  variance = 0.0635244 
Class  91 average =  0.08332136  variance = 0.09067632 
Class  96 average =  0.07682093  variance = 0.08144446 
Class  97 average =  0.0792268   variance = 0.08092019 
Class  99 average =  0.1019089   variance = 0.1072126 
Class  100 average = 0.1018262   variance = 0.1081117 
Class  101 average = 0.1106647   variance = 0.1151819 
Class  103 average = 0.08147644  variance = 0.08411685 
Class  104 average = 0.06456508  variance = 0.06801061 
Class  107 average = 0.1197225   variance = 0.1250056 
Class  108 average = 0.0924619   variance = 0.09845582 
Class  109 average = 0.1198932   variance = 0.1209162

Here, when ploting the empirical variance (per knot) against the empirial average of claims, we get


Here, we can identify classes where remaining heterogeneity.

It is “simply” the average value

For some obscure reasons, simple things are usually supposed to be simple. Recently, on the internet, I saw a lot of posts on the “average time in which you hold a stock“, and two rather different values are mentioned

  • Take any stock in the United States. The average time in which you hold a stock is – it’s gone up from 20 seconds to 22 seconds in the last year” (Michael Hudson on http://www.telegraph.co.uk/) or “The founder of Tradebot, in Kansas City, Mo., told students in 2008 that his firm typically held stocks for 11 seconds” (on http://www.nytimes.com/) among many others
  • Based on the NYSE index data, the mean duration of holding period by US investors was around 7 years in 1940. This stayed the same for the next 35 years.  The average holding period had fallen to under 2 years by the time of the 1987 crash. By the turn of the century it had fallen to below one year. It was around 7 months by 2007” (on http://topforeignstocks.com/ see also the graph below) or “Two-thirds [of the managers of more than 800 institutional funds interviewed in a study] had higher turnover than they predicted […] Even though most are judged by performance over three-year horizons, their average holding period was about 17 months, and 19% of the managers held the typical stock for one year or less” (mentioned on http://online.wsj.com/) again among many others

How comes that on the one hand, some people talk about less than 20 sec. for the “average time in which you hold a stock“, and on the other, around a year. How can we have such a difference ? We are talking about an average time here, not a rare event probability…

To understand what might be wrong, consider the following case, with a market, and two stocks: one is kept over a year (52 weaks) while the other other is traded – and exchanged – every week (52 times per year). What is the “average time in which you hold a stock” ? Is it

  • 26.5 weeks ? the average time for the first stock is 52 weeks, while it is 1 for the second one, i.e. 53 over 2
  • 1.96 weeks ? over a year the first stock has been traded once, while it was exchanged 52 times for the second one, i.e. 104 over 53 (total time over the total number of transactions)

Obviously, there is a selection bias in that study (see here for an illustration of that concept, in French). In order to get a better understanding, consider the following simple model, with a large number of simulated stocks. At each transaction, they can be hold by 3 types of investors,

  • with probability 70%, hold – on average – for 20 sec.
  • with probability 20%, hold – on average – for 15 days
  • with probability 10%, hold – on average – for 10 years

As claimed by Warren Buffett, “my favorite time frame for holding a stock is forever“, so it might not be absurd to consider investors who keep a stock for a long period of time. Assume further that the time frame for holding a stock is exponentially distributed (the rate depending on the kind of investor). Assume that those stocks are observed during a period of time of 20 years (which might sound reasonable). Several techniques can be used to estimate the “average time in which you hold a stock

  • The first one is to calculate the mean, per stock, of the holding time, and to consider the average over all the stocks. Maybe it would be a good idea to exclude the last observation (since data were censored),
  • The second one is to divide the (total) period of time by the (total) number of investors that hold the stock during that time frame (or number of transactions)
  • A third idea might be to use the first method, but instead of removing the last one, to use an estimator of the mean based on Kaplan-Meier estimate
  • A fourth idea is to look at what happened at a specific date (say after 10 years), i.e. which investor had the stock, and how long he kept it.

The code to generate that process is the following

> set.seed(1)
> invest=sample(size=ns,c("A","B","C"),
+ prob=c(.7,.2,.1),replace=TRUE)
> lambda=(invest=="A")*20/(365*24*60*60)+
+        (invest=="B")*15/365+
+        (invest=="C")*10
> E=rexp(ns,rate=1/lambda)
> T=cumsum(E)
> T=T[T<20]
> plot(c(T,50),0:length(T),type="s",xlim=c(0,20),col="blue")

with the following trajectory for the number of investor that did hold that specific stock between time 0 and time 20.

Then, the different techniques are the following,

# method 1
> E1=diff(T)
> m1=mean(E)
> M1[s]=m1

for the first one (means of time length, per stock),

# method 2
> if(length(T)>1){
+ n2=length(T)-1
+ d2=T[length(T)-l]-T[1]
+ N2[s]=n2; D2[s]=d2
+ }

for the second one (time length and number of transactions),

+ # method 3
+ T3=c(T,20)
+ C3=c(rep(0,length(T)-1),1)
+ km=survfit(Surv(diff(T3), 1-C3)~1)
+ m3=summary(km,rmean='individual')$table[5]
+ M3[s]=m3

for the third one (based on a prediction of the expected mean, from Kaplan-Meier estimate) and

# method 4
> T0=c(0,T,20)
> m4=min(T0[T0>10])-max(T0[T0<10])
> M4[s]=m4

for the fourth one (based on what happened at time 10). Using monte carlo simulations, we get very different quantities, that can all be interpreted as the “average time in which you hold a stock”

> sum(D2,na.rm=TRUE)/sum(N2,na.rm=TRUE)
[1] 0.3692335
> mean(M1,na.rm=TRUE)
[1] 0.5469591
> mean(M3,na.rm=TRUE)
[1] 1.702908
> mean(M4,na.rm=TRUE)
[1] 12.40229

If we change to probabilities (and assume that high frequency investors are much more important than long-term ones), e.g.

> invest=sample(size=ns,c("A","B","C"),
+ prob=c(.9,.09,.01),replace=TRUE)

then the first two estimates are rather different. But not the last two.

> sum(D2,na.rm=TRUE)/sum(N2,na.rm=TRUE)
[1] 0.04072227
> mean(M1,na.rm=TRUE)
[1] 0.06393767
> mean(M3,na.rm=TRUE)
[1] 0.2504322
> mean(M4,na.rm=TRUE)
[1] 12.05508

So I have to confess that the “average time in which you hold a stock” can be almost anything from 10 sec. to 10 years, it clearly depends on the way the average is calculated. The second point is that if the proportion of high frequency trading is extremely high, I should not affect the last one (which is, from my point of view, the most interesting one, an might also be improved by here also integrate a censored variate). So I guess people should be careful when discussing such quantities… And if anyone is willing to share data on that topic, I’d be glad to look at them…

Warming in Paris: minimas versus maximas ?

Recently, I received comments (here and on Twitter) about my previous graphs on the temperature in Paris. I mentioned in a comment (there) that studying extremas (and more generally quantiles or interquantile evolution) is not the same as studying the variance. Since I am not a big fan of the variance, let us talk a little bit about extrema behaviour.

In order to study the average temperature it is natural to look at the linear (assuming that it is linear, but I proved that it could reasonably be assumed as linear in the paper) regression, i.e. least square regression, which gives the expected value. But if we care about extremes, or almost extremes, it is natural to look at quantile regression.

For instance, below, the green line is the least square regression, the red one is 97.5% quantile, and the blue on the 2.5% quantile regression.

It looks like the slope is the same, i.e. extremas are increasing as fast as the average…


(here I plot randomly some points to avoid a too heavy figure, since I have too many observations, but I keep all the observations in the regression !).

Now, if we look at the slope for different quantile level (Fig 6 in the paper, here, but on minimum daily temperature, here I look at average daily temperature), the interpretation is different.

for(i in seq(.005,.995,by=.005)){

with the following graph below,

for(i in seq(.005,.995,by=.005)){

Here, for minimas (quantiles associated to low probabilities, on the left), the trend has a higher slope than the average, so in some sense, warming of minimas is stronger than average temperature, and on other hand, for maximas (high probabilities on the right), the slope is smaller – but positive – so summer are warmer, but not as much as winters.
Note also that the story is different for minimal temperature (mentioned in the paper) compared with that study, made here on average daily temperature (see comments)… This is not a major breakthrough in climate research, but this is all I got…