Zehir Kolli
email: kolli@inrets.fr

Ariane Dupont-Kieffer

Laurent Hivert
email: hivert@inrets.fr

INRETS Zéhir Kolli

Ariane Dupont
email: dupont@inrets.fr

Kolli

Car survival in a national fleet : a non-parametric approach based on French data

Keywords: Car survival, Kaplan-Meier estimator, car mileage, Beta and Weibull distributions

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INTRODUCTION

A longstanding question concerns household car survival in the national fleet. Planners concerned with energy, resource consumption and environmental quality will, more and more, take into account time for a national fleet to achieve green standards, time of diffusion of a new technology in a fleet, or expected time for disappearances of the 5, 10 or 15 percent of oldest and most polluting cars from a given fleet. With the aim of having elements of answer, related forecasts on car fleet are required. In the majority of cases, car fleet statistics are not accurate enough. In order to have more accurate estimations and forecasts we can proceed to a dynamic modeling including car registration statistics that are usually well known and estimations of cars that are scrapped through a survival data analysis. A serious examination of the longevity of cars will also permit having better understanding of car ownership behavior, better estimations and assessments of GHG emissions, as well as estimations and forecasts of car fleet composition in the absence of direct taxes3 . It can also help to simulate the reaction to a small, medium or massive introduction of a new technology like electric or hybrid cars. Much existing research related to car longevity or survival in the U.S. market (or linked to cash for clunkers programs) have used a logistic functional form to estimate car survival rates per year or per vintage. [START_REF] Greene | Scrappage and Survival Rates of Passenger Cars and Light Trucks in the U.S. 1966-77[END_REF] used a logistic equation separately for domestic U.S. cars and imported cars and for three vehicle types (trucks, light duty vehicles and automobiles). Their results show better survival expectancy for light trucks compared to light cars (median lifetimes respectively equal to 14.5 and 9.9 years) [START_REF] Berkovec | New Car Sales and Used Car Stocks: A Model of the Automobile Market[END_REF] estimated a scrappage model in which he regressed the log of the death rate on auto prices, prices squared, and dummy variables for vintage and class. [START_REF] Greenspan | Motor Vehicle Stocks, Scrappage, and Sales, Finance and Economics Discussion Series[END_REF] estimated car mortality as an input to the study of the demand for new cars. They captured engineering scrappage by estimating a pooled regression (where y denotes the fraction of vehicles of model year i remaining at age t). In order to be consistent, they assumed that scrappage is equal to zero the first three years. [START_REF] Gallez | Renouvellement du parc automobile et prospective transportenvironnement[END_REF] fitted longitudinal annual rates of survival with the assumption that they follow a Lognormal survival function with a maximum age (28 years) after which the number of survivors is equal to zero. Using data from 1968 to 1992, Gallez found that median lifetimes for diesel cars and gasoline cars are respectively equal to 10.82 and 11.23 and that mean lifetimes are equal to 11.74 and 12.22 years in France. Hamilton and Macaulay (1998) illustrated the nature and size of possible sources of improvements in cars' longevity, including improvements in intrinsic durability and other economic forces which induce consumers to maintain cars into older age. They followed the automobile-mortality literature in assuming that the pattern of car mortality could be characterized by a logistic function. More recent research showed that drivers' travel behavioral attitudes also affect car use. [START_REF] Choo | What type of vehicle do people drive? The role of attitude and lifestyle in influencing vehicle type choice[END_REF] found that travel attitudes, personality, lifestyle, and mobility factors are useful in predicting the most-used vehicle within a household. [START_REF] Chen | A mass point vehicle scrappage model[END_REF] applied a duration model of the Weibull form with two mass points to approximate the unobserved heterogeneity to estimate car survival rates on a sample of 678 cars. The main result of their vehicle scrappage model suggests that vehicle age and car mileage are important determinants of scrappage decision.

However, many researches show that the natural time scale of the survival process is not calendar or clock time. Mathematical research on different time scales has been carried out [START_REF] Oakes | Multiple time scale in survival analysis[END_REF] and Kordovsky and Gertsbakh, (1997) looked at multiple running time scales in survival data analysis. [START_REF] Duchesne | Alternative time scales and failure time models[END_REF] and Duchesne and Rosenthal, (2003) described various advances in running time models for survival data. Cox and Oakes, (1984, Section 1.2, p. 3-4) pointed out that "often the 'scale' for measuring time is clock time, although other possibilities certainly arise, such as the use of operating time of a system, 3 mileage of a car, or some measures or cumulative load encountered". The traditional variable of lifetime measurement is the age of the car. But in order to have an overview on car survival we choose to take into account car use intensity through the annual driven (e.g., declared) mileage variable or average driven (e.g., declared) mileage per month. This variable is treated in addition to the age of the car, the traditional variable of lifetime measure. For the non parametric approach, the main idea is to compare car use (which can depend on its motorization, status, horse power, cylinder capacity, etc...) measured by an average mileage per month with car life expectancy measured by age. This comparison show that car life duration is negatively correlated with use for each car category and consequently that drivers' behavior can explain their life duration. In order to illustrate the strong correlation between car age and car mileage and their ability for measuring car life duration some bivariate pdf for different categories of cars are shown. Then, we show with a parametric approach that mileage of cars (besides car age) is a major variable to explain car life duration. This parametric approach is also useful to show the best pdf that fit our data and hence the most appropriate function to model survival curves by age or by mileage.

DATA

We used `Parc-Auto' an annual car fleet database built from annual postal surveys conducted with a panel of 10 000 French households4 .'Parc Auto' database is merged to 2006 with car and household's characteristics concerned back to 2000. Hence a panel data set from 2000 to 2006 was built up including 6795 observed cars and 4794 household (e.g., Table1). From this database we use some car's specificity and characteristics like total mileage and age that are also well described. We study the impact of variables like motorization (e.g., Table 2) and car status in car life duration. Table 3 shows the proportion of cars by status. Three principal categories are considered by households for their cars and it can be a sole car (1), the main car (2) or a secondary car (3) with the respective proportions given as following: 63.88, 29.45 and 6.67 percents of our database (e.g., Table 3).

NON PARAMETRIC ANALYSIS: Kaplan-Meier estimator

Usually, a first step in the analysis of survival data is the estimation of the distribution of the survival times. In biometrics, an analysis of [START_REF] Kaplan | Nonparametric estimation from incomplete observations[END_REF] is practical to calculate the empirical life expectancy of a population of individuals. In that way, if we consider a car like an individual with a 'birthdate' considered as the date of construction and if each car is followed until death, the curve may be estimated simply by computing the fraction surviving at each time. In our case study, we have no information about car destructions and consequently all observations are considered as right censored. This is an important limitation but Kaplan-Meier analysis allows us to estimate survival trends over time, even when cars lifetime drop out or are studied for different lengths of time. For each interval, survival probability is calculated as the number of car still in the fleet divided by number of cars at risk. Cars that have been scrapped, abandoned, or that have not reached the time yet are not counted as 'at risk'. In a systematic way, cars that are scrapped or destructed are considered censored and are not counted in the denominator. Probability of surviving to any point is estimated from cumulative probability of surviving each of the preceding time intervals (calculated as the product of preceding probabilities). The survival distribution provides the proportion of cars surviving at the start of any interval and is equal to the probability of surviving each of the preceding intervals multiplied together. If we let S(t) be the probability that an item from a given population will have a lifetime exceeding t. For a sample from this population of size N let the observed times until death of N sample members be :

ݐ ଵ ൏ ݐ ଶ ൏ ݐ ଷ ൏ ڮ ൏ ݐ
Corresponding to each ݐ is ݉ ; the number at risk just prior to time ݐ (risk-set) and ݀ the number of deaths at time ݐ . Note that the intervals between each time typically will not be uniform. The Kaplan-Meier estimator is the nonparametric maximum likelihood estimate of

ܵ ௧ ൌ ෑሺ1 െ ߠ ሻ ௧ ೕ ழ௧
5

Is a product given by:

ܵ ௧ ൌ ෑሺ1 െ ߠ ሻ ௧ ೕ ழ௧ ൌ ෑሺ ݎ െ ݀ ݎ ሻ ௧ ೕ ழ௧
When there is no censoring, ݊ is just the number of survivors just prior to time ݐ . With censoring, ݊ is the number of survivors less the number of losses (censored cases). It is only those surviving cases that are still being observed (have not yet been censored) that are "at risk" of an (observed) death.

Results of the Non Parametric Estimation

Motorization

From 2000 to 2006 gasoline cars are, on average, 131 months old for an estimated average mileage of 115 112 kilometers. Diesel car drivers declare much more mileage for less old cars. They travel 148 286 kilometers and their cars have 108 months in average (e.g., Table 4). Gasoline cars travel monthly on average 1002 km, for diesel cars the value is much higher with a monthly value of 1681 km per month. This statistical analysis results show a substantial difference between use and age for gasoline and diesel cars. As a start, we try to compare car use for each type of motorization in order to identify their differences in terms of life expectancy (e.g., Figure 1). 72563) 120 ( 69)

Car status

Considering motorized households only, the considerate cars can be the sole car in the household, the main car (as freely defined by respondent, usually household head) out of several cars, or a secondary car (second or more) out of several cars. This variable (sole, main, secondary) designates the car status. Figure 4 depicts increasingly poorer survival rates with lower status category. As expected the subset of sole cars demonstrates extremely poor survival rates and the survival curve is the only one that seems to be convex. The half-population of sole cars disappear in only 95 months (nearly 8 years) (95 percents confidence interval=92-97). Survival curve decreasing is not so fast for main and secondary cars that spend respectively 141(almost 12 years) (95 percents confidence interval=137-144) and 156 months (13 years) (95 percents confidence interval=149-169) to see their half population disappear.

PARAMETRIC ANALYSIS

Preliminary

If we suppose that T is a continuous random variable. Then, we can define F(t) = Pr(T < t) as the probability that the random variable T is less than some value t. The corresponding density function is given by f(t) ݐ݀/ሻݐሺܨ݀= . It is also useful to define the survival function ܵሺݐሻ = 1 -ܨሺݐሻ = ܲሺݐ ܶሻ that correspond to the probability that the random variable ܶ will equal or exceed the value t. A particular useful function for duration analysis is the hazard function defined as ݄ሺݐሻ ൌ ݂ሺݐሻ/ܵሺݐሻ and gives the probability of exit from the state immediately after time t given that the state is still occupied at t. These relations shows that if either ݄ሺݐሻ, ܵሺݐሻ, ݂ሺݐሻ ݎ ܨሺݐሻ is given, the others can be derived.

As we are computing Beta, Gamma, Lognormal and Weibull three-parameters distributions that are usually written with two-parameters, a theoretical background of these distributions written with three parameters can be useful:

Beta distribution

The Beta probability distribution function is useful for modeling random probabilities and proportions, particularly in the context of Bayesian analysis. The Beta pdf has a left parameter α and a right parameter β. The Beta three-parameters distribution is given by:

݂ሺ,ݐ ߙ, ߚ, ߠሻ ൌ ൞ ሺݐ െ ߠሻ ఈିଵ ൫1 െ ሺݐ െ ߠሻ൯ ఉିଵ  ሺݑ െ ߠሻ ఈିଵ ଵ ൫1 െ ሺݑ െ ߠሻ൯ ఉିଵ ݑ݀ ݂݅ ݐ ߠ 0 ݂݅ ݐ ൏ ߠ
where ߙ and ߚ, respectively right and left parameters are positive.

Gamma distribution

The Gamma three-parameters pdf of a Gamma-distributed random variable t is given by :

݂ሺ,ݐ ߙ, ߚ, ߠሻ ൌ ൞ ൬ ݐ െ ߠ ߚ ൰ ఈିଵ ߚ ିఈ ݁ ሺି ௧ିఏ ఉ ሻ ߚ Γሺߙሻ ݂݅ ݐ ߠ 0 ݂݅ ݐ ൏ ߠ
With ߙ 0 and ߚ 0 and Γሺߙሻ ൌ  ݐ ఈିଵ ݁ ି௧ ן ݐ݀ ߙ: shape parameter ߚ: scale parameter and ߠ the location parameter (ߠ = 0 yields to the two-parameter Gamma distribution).

Lognormal distribution

The random variable t follows the Lognormal distribution if and only if ܶ ൌ ݊ܮሺݐሻ follows a standard normal distribution N(ߙ; ߚ). Thus, the probability density function for the threeparameters Lognormal distribution is:

9 ݂ሺ,ݐ ߙ, ߚ, ߠሻ ൌ ቐ 1 .ݐ ߙ√2ߨ ݁ ି ሾሺ௧ିఏሻିఉሿ మ ଶఈ మ 0 ݂݅ ݐ ൏ ߠ ݂݅ ݐ ߠ

Weibull and exponential distributions

The Weibull distribution has been found to provide a reasonable model for lifetimes of several types of units. The three-parameters Weibull pdf is defined as :

݂ሺ,ݐ ߙ, ߚ, ߠሻ ൌ ቐ ߙ ߚ ሺ ݐ െ ߠ ߚ ሻ ఈିଵ ݁ ିሺ ௧ିఏ ఉ ሻ ഀ 0 ݂݅ ݐ ൏ ߠ ݂݅ ݐ ߠ
where ߙ and ߚ respectively shape and scale parameters are positive and ߠ the location parameter (ߠ = 0 yields to the two-parameter Weibull distribution). Note that If ߙ = 1 we fall into an exponential model. The pdf given by age in month is particularly well fitted5 by a Beta distribution (e.g., Appendix A for qq-plot tests; Appendix B Table 8 for quantile distributions given for Beta and Weibull distributions) although the Weibull distribution give similar adequacy results (e.g., Appendix B Table 8). The Lognormal distribution doesn't provide a good fit of our sample. It has been pointed out [START_REF] Lawless | Statistical Models and Methods of life time data[END_REF] that the hazard function for the Lognormal model decreases for longer values of cycles. This does not agree with our knowledge concerning cars' progressive deterioration resulting from fatigue process, nevertheless the Lognormal pdf is often used satisfactorily to match shorter values of time-to-failure [START_REF] Lawless | Statistical Models and Methods of life time data[END_REF].

MODELISATION

Modelisation by age

The Lognormal pdf cannot represent cars' lifetime exactly. The Lognormal is quite limited to represent lifetime data compared to the Weibull. For example, the Weibull can be either positively or negatively skewed but the Lognormal can only be positively skewed.

The Gamma distribution gives better adequacy parameters than the Lognormal but still far behind the Weibull or the Beta (e.g., Appendix A). It is known that, the Weibull distribution is based on more physically convincing arguments than the Gamma distribution. Moreover, the Weibull pdf is well suited for certain procedures of statistical extrapolation to large systems and has the advantage of dealing easily with our censored data. More importantly, from the view of statistics, as the data are well fitted by the Beta's and Weibull's pdf, these two distributions might be applicable to model our data hazard function. Therefore cars' longevity (in age) might be matched accurately by Beta or Weibull survival functions. The pdf given by mileage is also particularly well fitted by the Beta distribution (e.g., Appendix A for qq-plot tests and Appendix B Table 8). The superiority of the Beta distribution is obvious against the Lognormal's and the Gamma's. But like previously, the Weibull distribution is not far of these results and from the empirical quantile distribution (Appendix B Table8). Like for car age , as the data are well fitted by the Beta's and Weibull's pdf then our cars' longevity given in mileage might be matched accurately by Beta or Weibull survival functions.

Modelisation by mileage

These results from parametric estimation shows that both the Beta and Weibull pdf often seems to provide a good fit of cycles to failure data for our marker process Y(r) which is car mileage and for our parent process X(r) which is car age. More importantly this distribution given by mileage shows the interest of a comparison by total driven mileage versus the classic comparison by age. We can also use mileage as a determinant of car life duration. More precisely, if we consider that car age is a parent process, car mileage can be called marker process defined by E.A. [START_REF] Peña | Dynamic Modeling and Statistical Analysis of Event Times[END_REF]: "A marker process refers to an external process that covaries with the parent process and hence, It assists in tracking progress of the parent process if the parent process is latent or only infrequently observed. Marker processes may also be in scientific interest in their own right.

As markers of the parent process, they offer potential insights into the causal forces that are generating the movements of the parent process".

The basic analytical framework for a marker process conceives of a bivariate stochastic process [X(r),Y(r)] where the parent process X(r) would be car age and the marker process Y(r) the mileage (Both are one dimensional and defined in the running time scale r). In the next section we work on these bivariate distributions to catch the link between our marker and parent process.

BIVARIATE DISTRIBUTIONS

In the following, figure (a) to (e) represents bivariate distributions given by age and by mileage. If we let ܺ ൌ ሺݔ , ݕ ሻ, ݅ א ۤ1, ۥ݊ be a sample of size n drawn from this distribution. The kernel density estimate of fሺx, yሻ based on this sample will be given by:

݂ መ ሺ,ݔ ݕሻ ൌ 1 ݊ ߮ሺݔ െ ܺ , ݕ െ ୀଵ ܻ ሻ ൌ 1 ݄݊ ௫ ݄ ௬ ߮ ሺ ݔ െ ܺ ݄ ௫ , ݕ െ ܻ ݄ ௬ ሻ ୀଵ
where ሺ,ݔ ݕሻ א Ը ଶ , ݄ ௫ 0, ݄ ௬ 0 are the bandwidth and ߮ ሺ,ݔ ݕሻ the rescaled normal density:

߮ ሺ,ݔ ݕሻ ൌ 1 ݄ ௫ ݄ ௬ ߮ሺ ݔ ݄ ௫ , ݕ ݄ ௬ ሻ
Where ߮ሺ,ݔ ݕሻ is the standard normal density function : show that for the gasoline car population the bivariate pdf is centered compared to diesel car one. For practial reasons, our bivariate distributions are given in a grid with an upper bound corresponding to 320 000 kilometers and to 320 months (26 years and 8 months). Gasoline car population reach its maximum for 81 months and 76520 km as for diesel cars it is around 54 months and 87625 km. The spread is also more important for diesel than for gasoline car population.

߮ሺ,ݔ ݕሻ ൌ 1 2ߨ ݁ ൬ି ௫ మ ା௬ మ ଶ ൰

Bivariate distributions by motorization

In addition to that a substantial proportion of the diesel car population still in the fleet after 240 000 kilometers (some after 320000 km) while for gasoline cars the bivariate pdf ends around 260 000 kilometers.

Bivariate distributions by car status

The bivariate pdf is given by category of status in Figures 8,(c) (d) and (e) is enough to see that the spreading of the density increase in a radical way with the status. Moreover, age and mileage of maximums and dispersion of points increases with status. We can say that secondary cars stay longer than main cars, and that main cars stay longer than secondary car. But we also know that a quite important proportion of cars are intended to change of status one, two or more times during their lifetime in the same or in an other household.

CONCLUSION

Car status (e.g., sole, main or secondary), an indicator of the intensity of use, appears to be a major determinant of car longevity in our sample of the French car fleet. We noticed better survival rates for cars declared as secondary cars. With the assumption that the status remains the same during the whole life of each car, secondary cars are maintained longer than main cars in the French fleet. And respectively, main cars are maintained longer in the fleet than sole cars. But we know that a car status is not fixed during a whole life duration and that status can move from one to another of our three categories. Motorization has a big impact too. Gasoline cars are maintained longer in the fleet than diesel cars and here too this is explained by an heterogeneity of car use behavior.

The reason of relatively good survival rates observed for some categories are mainly found in car use behavior. In that sense, this study underlined the negative correlation between car use and longevity. However, car survival duration could also be influenced by other variables: Bhat and Sen (2006), [START_REF] Bhat | The Impact of Demographics, Built Environment Attributes, Vehicle Characteristics, and Gasoline Prices on Household Vehicle Holdings and Use[END_REF] There is no doubt that car survival time can be explained both by its mileage which could be defined as a marker process or by car age defined as a classic calendar time which would be the parent process. The literature did not provide models integrating car mileage in the survival functions (as well as car status or rank, cylinder capacity or horsepower etc…). This research rests on the extensive existing literature, by examining the link between car use, car age and car longevity. But also by giving answers concerning functional forms of pdf, survival functions and main determinants (motorization and car status) that should be used to model and forecast car fleet. Economic survival rates among these different populations should also take into consideration the complex relationship among motorization and cars status of the car fleet population.

Both at the EU and US levels critical forecasts are drawn for the next years concerning greenhouse gas emissions. The need for estimating vehicle population for assessing air quality has become increasingly important (Zachariadis et al. , 1995;[START_REF] Bhat | The Impact of Demographics, Built Environment Attributes, Vehicle Characteristics, and Gasoline Prices on Household Vehicle Holdings and Use[END_REF].

In that way, we need a better understanding of car renewal and identify which are the economic and regulatory instruments packages that can best promote technological car renewal and technology deployment. For all these reasons, we need to have a better understanding of the forces that are driving car longevity through age, mileage accumulated, car status or motorization, etc… 14

There are several areas that remain open for further exploration. Car scrapping data are not as readily available as car registration data but it would be valuable to collect such data in upcoming studies. Their contribution to further study could help to bypass the problem of right censored data and helpful to compute robust Kaplan-Meier survival curves. Recognition of vehicle use patterns along with unexpected vehicle loss (due to crashes and theft, for example) would enhance the information content and reliability of results.

In order to give robust survival parametric model we also discussed the problem of comparative distributions. Parametric models are not always robust and are sometimes unsubstantiated mainly because their adjustment is performed on a small amount of annuals survival rates. But we showed that Beta and Weibull distributions fit the data very well mostly because they are flexible distributions.

In the near future, It would be also of great importance to focus on determinants of multimotorization and diesel cars' diffusion on car life expectancy in France, E.U. and U.S. for example.

Finally, information about travel behavior, driving habits, frequency affect the use, fuel economy and emissions from household vehicle fleets may be important determinants of car longevity. An integration of declared mileage in the hazard specification using Weibull or Beta form would be of great interest to assess car life duration and to model car fleet with a dynamic survival curve depending on driving habits.

ACKNOWLEDGMENTS:

The authors are grateful to two anonymous referees for valuable suggestions. Thanks are also due to Jean-Luc Wingert for careful reading of a first version of the manuscript.

Figure 1 -

 1 Figure 1 -Kaplan-Meier curves of survival rate by motorization and per monthly average kilometers

Figure 2 -

 2 Figure 2 -Kaplan-Meier curves of survival rate by motorization and per months

Figure 3 -

 3 Figure 3 -Kaplan-Meier curves of survival rate by car status and per monthly average kilometers Sole cars can be considered as the most extensively used category in the fleet comparing to other categories. Households with a unique car travel in average 124741 km in 104 months (e.g., Table5). They also do in average 141162 km in 143 months with their main car and 153168 km in 168 months with their secondary car (e.g., Table5). A comparison of Kaplan-Meier curves for each category shows that sole cars is far away the most extensively used category. Fifty percents of households which have a unique car travel more than 1243 km per months (95 percents confidence interval=1220-1246). For main car category and secondary car categories, it is almost the same level of use: fifty percents of the main car "population" drive more than 1018 kilometers per months (95 percents confidence interval=991-1044). It is 1000 km per months (95 percents confidence interval=946-1071) for secondary car "population".

Figure 4 -

 4 Figure 4 -Kaplan-Meier curves of survival rate by car status and per months

Figure 5 -

 5 Figure 5 -Functions adequacy by age

Figure 6 -

 6 Figure 6 -Functions adequacy by mileage

Figure 7 -

 7 Figure 7 -(a) Bivariate distribution for gasoline cars -(b) Bivariate distribution for diesel cars

  Figures 7 (a) and (b) show that for the gasoline car population the bivariate pdf is centered compared to diesel car one. For practial reasons, our bivariate distributions are given in a grid with an upper bound corresponding to 320 000 kilometers and to 320 months (26 years and 8 months). Gasoline car population reach its maximum for 81 months and 76520 km as for diesel cars it is around 54 months and 87625 km. The spread is also more important for diesel than for gasoline car population. In addition to that a substantial proportion of the diesel car population still in the fleet after 240 000 kilometers (some after 320000 km) while for gasoline cars the bivariate pdf ends around 260 000 kilometers. Figures7 (a) and (b) enlighten us on a heterogeneousness of the behavior in automobile usage depending on motorization: diesel cars population travel more than gasoline cars population. Bivariate distributions looks equilibrated both by marker process (car mileage: Y(r)) and parent process (car age: X(r)).

Figure 8 -

 8 Figure 8 -(c) Bivariate distribution for sole cars (d) Bivariate distribution for main cars (e)Bivariate distribution for secondary cars

Table 1 -

 1 Parc Auto annual waves

	Year	Number of cars	Number of households
	2000	8115	6368
	2001	8177	6438
	2002	8249	6466
	2003	8256	6523
	2004	8170	6291
	2005	8186	7162
	2006	6257	5600
	Total	56315	43943
	Observations used	6795	4794

Table 4 -

 4

	Gasoline cars	115112 (62517)	131 (70)
	Diesel cars	148286 (78174)	108 (66)
	Total sample	131472 (

Car use by motorization

Motorization

Average mileage (std-error) Average age (std-error)

Table 5 -

 5 Car use by status

	Status	Average mileage (std-error) Average age (std-error)
	Sole cars	124741 (70875)	104 (60)
	Main cars	141162 (74056)	143 (71)
	Secondary cars	153168 (72715)	168 (90)
	Total	131472 (72563)	120 (69)

Table 6 -

 6 Estimated parameters for distribution functions given by car age

	Status	Location (ߠ) Scale (ߚሻ	Shape (ߙሻ	Mean	Variance
	Beta(,ݐ ߠ, ߙ, ߚ, ߪሺ݈݁ܽܿݏሻ) Gamma(,ݐ ߠ, ߙ, ߚ) Lognormal(,ݐ ߠ, ߙ, ߚ) Weibull(,ݐ ߠ, ߙ, ߚ)	-12.5 -12.5 -12.5 -12.5	ߪ=700 37.826 4.731 149.731	ߙ =2.829 ߚ=12.074 3.502 0.579 2.011	120.39 119.93 122.37 120.18	68.84 70.78 85.17 69.01

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10

Table 7 -

 7 Estimated parameters for distribution functions given by car mileage

	Status	Location (ߠ) Scale (ߚሻ	Shape (ߙሻ	Mean	Variance
	Beta(,ݐ ߠ, ߙ, ߚ, ߪሺ݈݁ܽܿݏሻ) Gamma(,ݐ ߠ, ߙ, ߚ) Lognormal(,ݐ ߠ, ߙ, ߚ) Weibull(,ݐ ߠ, ߙ, ߚ)	-10000 -10000 -10000 -10000	ߪ=700000 39896 11.711 160053	ߙ =2.876 ߚ=11.345 3.546 0.578 2.064	131565 131472 137790 131780	72068 75127 105881 72048

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  conclude that vehicle-holdings and miles of travel vary with demographic characteristics, vehicle attributes, fuel costs etc... In our case study, car use and miles of travel depends mostly of car status and car motorization determinants. As car life duration depends mostly of car use, it also depends on status and motorization. The correlation between age and mileage vary by status and by motorization .precisely because there is an heterogeneity in use behavior and miles of travel in each of these subpopulationsThe idea of integrating car mileage in survivor function is not new but was not widely explored mainly because of practical reasons. And especially because odometer data are not always available or well described as explained by Zachariadis et al

6 , (1995)

: "Thinking about vehicle populations, one could reasonably assume that the most suitable unit to express their age is the mileage accumulated per vehicle since the beginning of its useful life. Expressing age in terms of years since entering the market is nevertheless equally appropriate and more practical (…) Statistics always group vehicles according to their model year or year of entering the market, not according to their accumulated mileage". Moreover, contrary to what is advanced by

Zachariadis et al., (1995)

cars' age and cars' mileage enlighten us differently on car use behavior mostly because their correlation is different for each subpopulation of car status or/and of motorization.

In France a direct tax called `vignette' was aborted in 2000

About one hundred questions were asked in the survey about car ownership, car characteristics, main and secondary users, previous car characteristics, car use behavior, attitudes toward automobiles, etc...

th WCTR, July 11-15, 2010 -Lisbon, Portugal

The maximum likelihood method of estimation was used to fit all four models. The Newton-Raphson algorithm, having a quadratic rate of convergence, was used to optimize the likelihood equations solving for the unknown parameters with maximum likelihood estimates and for each fixed location parameter.

The study gives a median lifetime of 11.5 years for France and Spain, 9.6 years for the former West Germany, 12.9 years for Italy, 20.6 years for Portugal and 9.7 years for the United Kingdom.

APPENDIX A-QQ-plots QQ-plots are quite useful to compare data distribution with each family of distribution that vary in location and scale. They are here computed for each distribution family to compare the quantiles of the data distribution with the quantiles of the theoretical distribution family. In the following the reference line (in red) represent the particular theoretical distribution that depends on the location and scale parameters. This theoric distribution have an intercept and a slope that are equal to the location and scale parameters of that distribution.