加拿大留学生作业-减少摩托车保险费率：Modeling Motorcycle Insurance Rate Reduction
几项研究已经表明,与摩托车相关的司机和乘客死亡的风险事故远远超过汽车事故。这个惊人的统计数据已经引起了国家汽车部门的注意,国家卫生部门和保险行业开始致力于降低摩托车事故的发生率。例如,康涅狄格引入强制性安全摩托车驾驶培训课程。本文的目的是开发一个统一的框架来量化这种强制性的项目，并且有效的降低保险费率。我们使用保险索赔数据来指定强制性的项目,通常有两种类型:一个是基本的(BRC)，这是先进的摩托车安全课程，另一个是Integer-Valued自回归(INAR)模型，它是用来评估摩托车事故的索赔频率（已经考虑不同类型的培训项目）。该模型允许我们估算调整激励因素来反映由于强制性的计划可能降低的保险费率。
Modeling Motorcycle Insurance Rate Reduction
due to Mandatory Safety Courses
Shujuan Huang*, Vadiveloo Jeyaraj*, Valdez Emiliano*, Garry D. Lapidus**
*Math Department, University of Connecticut
** Connecticut Institute for Clinical and Translational Science, UConn Health Center
Abstract
Several studies have indicated that the risk of driver and passenger fatality associated with motorcycle accidents far exceeds that of automobile accidents. This alarming statistic has caused increased attention from state motor vehicle departments, state health departments and the insurance industry to pursue efforts to introduce programs to dramatically reduce motorcycle accidents. For example, Connecticut introduced mandatory training courses for safety motorcycle driving. It is the purpose of this paper to develop a unifying framework to quantify the effectiveness of such mandatory programs and to translate this in terms of a possible insurance rate reduction. To calibrate the proposed framework, we used insurance claims data drawn from states where such mandatory programs have been introduced and where typically there are two types: a Basic (BRC) or an Advanced Motorcycle Safety Course (ARC). An Integer-Valued Autoregressive (INAR) model is employed to assess the claims frequency of motorcycle accidents taking into account the differences in the type of training program. A heterogeneity factor is injected into the model to assess the impact of the training programs. Finally, the model allows us to estimate incentive adjustment factors to reflect possible insurance rate reduction as a result of the mandatory program.
1 Introduction
1.1 Research Purpose
. Intense call on Insurance Industry Involvement from Public Transportation and Health Department
In 2008, 5,290 U.S. motorcyclists were killed and 96,000 injured. Since 1998, there has been a significant increase in deaths and non-fatal injuries. Per vehicle mile traveled, motorcyclists are about 37 times more likely than passenger car occupants to die in a motor vehicle crash and 9 times more likely to be injured. Motorcycle-related deaths account for 14 percent of total traffic fatalities in 2008, although motorcycles only made up about 3 percent of registered vehicles. . This alarming statistic has caused increased attention from state motor vehicle departments, state health departments and the insurance industry to pursue efforts to introduce programs to dramatically reduce motorcycle accidents.#p#分页标题#e#
In the blueprint for project "National Agenda for Motorcycle Safety" which is supported by Motorcycle Safety Foundation (MSF) and National Highway Traffic Safety Administration (NHTSA), it is strongly recommended that the insurance industry should collect, organize, analyze, and distribute motorcycle-specific loss data to better understand safety issues, and develop guidelines to tie approved training, licensing, and safe-riding practices to premium reductions. However, at present practice, insurers employ limited avenues to enhance and encourage motorcycle safety; Motorcycle insurers are not currently required to provide motorcycle-specific loss data for analysis or use in a safety-related database as they are, for example, with automobiles. It also should be noted that most research or projects which focus on evaluating the effectiveness of motorcycle safety course systematically and comprehensively are before 2000, and have shown mixed results on effectiveness, due to different kinds of Methodological Pitfalls and unavailability of training indicator in our current national traffic accident database. While some insurers have both claim records and policyholder information about whether they have taken the training or not, it would be extremely helpful to evaluate these public transportation projects if we could employ more resources in insurance industry.
. Reduce Insurers’ Own Losses by Supporting Certain Responsible Riding Practices with Incentives.
Nowadays, in some states, most motorcycle insurance companies offer up to a 10% discount per year for the next 3 years, for the successful completion of the Basic or Experienced Motorcycle Safety Course. Actually, the courses are uniformly designed and guided by MSF, the content and effectiveness for each one should be almost the same. However, the discount rate varies a lot from company to company which is shown in table 1. From the perspective of the insurers’ own interest, we do need to develop a quantitative methodology to evaluate the effectiveness of the Motorcycle Safety Courses and the insurance discount rate，to enhance motorcycle safety and optimize own risk management as well.Several studies have indicated that the risk of driver and passenger fatality associated with motorcycle accidents far exceeds that of automobile accidents. This alarming statistic has caused increased attention from state motor vehicle departments, state health departments and the insurance industry to pursue efforts to introduce programs to dramatically reduce motorcycle accidents. For example, Connecticut introduced mandatory training courses for safety motorcycle driving. It is the purpose of this paper to develop a unifying framework to quantify the effectiveness of such mandatory programs and to translate this in terms of a possible insurance rate reduction. To calibrate the proposed framework, we used insurance claims data drawn from states where such mandatory programs have been introduced and where typically there are two types: a Basic (BRC) or an Advanced Motorcycle Safety Course (ARC). An Integer-Valued Autoregressive (INAR) model is employed to assess the claims frequency of motorcycle accidents taking into account the differences in the type of training program. A heterogeneity factor is injected into the model to assess the impact of the training programs. Finally, the model allows us to estimate incentive adjustment factors to reflect possible insurance rate reduction as a result of the mandatory program.#p#分页标题#e#
Table.1 Facts about Current Motorcycle Insurance Discount Rate for Safety Course
Insurance CompanyDiscount rate Details
PROGRESSIVENASafety Course – Completing an approved safety course could earn you a discount.
GEICO10% discount for completing a Motorcycle Safety Foundation or Military Safety Course
Allstate5%Save 5% if you’ve voluntarily passed a Motorcycle Safe Driving in the past 36 months.
USAA5%approved safety course within the last three years
FOREMOSTNAMotorcycle safety course discount
Nationwideup to 5%Save up to 5 percent on your motorcycle insurance when you complete an approved safety course.
MARKELNA Safety Course Discount
Dairyland CycleNAMotorcycle safety course completion
RiderNo discount
1.2 Facts about Motorcycle Safety Course
Nearly all the approved safety courses by different insurance companies refer to the Motorcycle Safety Foundation (MSF) RiderCourses, which are adopted by most states DMV. The Motorcycle Safety Foundation is the internationally recognized developer of the comprehensive, research-based, Rider Education and Training System (MSF RETS). It is a national, not-for-profit organization sponsored by BMW, BRP, Ducati etc.
There are different levels of Rider Courses, for example: Basic RiderCourse(BRC), Basic RiderCourse 2( License waiver, skill practice), Street RiderCourse 1, Returning Rider Basic Rider Course, 3-Wheel Basic RiderCourse (3WBRC), Scooter Basic RiderCourse (SBRC), Street RiderCourse 2 (SRC2), Advanced RiderCourse (ARC). Some of them are just introduced or in the process of developing or designing. Different states and different driving school approved by MSF may offer different levels of instruction. For example, in Waterbury, CT, the courses offered Rider Education Program include Basic RiderCourse, Intermediate RiderCourse, and Experienced RiderCourse; while in New York State, they call the intermediate Rider Course as Basic RiderCourse 2. But the content of the Basic RiderCourse 1 and 2(sometimes called Intermediate RiderCourse) are similar, both are for those who don’t have a license yet. While the advanced RiderCourse are for those who have been riding for some time. Therefore, for the discussion of convenience, in this paper we will only consider two categories: BRC (Basic RiderCourse) and ARC (Advanced RiderCourse).
1.3 Introduction to Automobile Insurance and Priori Rating System
In an insurance portfolio, the potential risks exposed by policyholders vary; specifically for automobile insurance, the likelihood of having accidents varies among the insured drivers. One of the main tasks of actuaries is to fairly allocate the burden of baring the potential losses among policyholders, which is materialized by quantitative analysis to specify individual risks and thus to determine the premiums. This procedure is called pricing or rate-making. There are two main phases involved. A base premium is determined when the policy is issued, and then the premium will be adjusted by discounts or surcharges as the policy is carried out. Discount for the motorcyclists who have taken the safety course designed by MSF is such kind of discount. We should try to make everyone pay a premium corresponding to his own risk, and evaluate this discount rate according to the effectiveness on risk reduction of this program.#p#分页标题#e#
1.4 Why research studies about the effectiveness have shown mixed results
It should not be surprising that the results of research studies looking at the effectiveness of rider training have shown mixed results. Most of the studies reviewed a training program that essentially consisted of a single course. Most government and insurance company involvement in the U.S. is through the licensing function, and therefore, limited primarily to a basic novice course. In addition, MSF Rider Education and Training System (RETS) is expanding in breadth and depth to meet the growing needs of current and prospective riders all the while. For example, the Street RiderCourse was just introduced in last year (2010), there may be a lag to show on the effectiveness of the program. Some courses are still under the process of developing and have not been introduced yet. These may lead to a fallacy of a single training course serving as an in-total countermeasure.
2 Effectiveness of Different Levels of Motorcycle Safety Courses
2.1 Definition of effectiveness
Effectiveness means the capability of producing an effect. In other words, it means doing "right" things, i.e. setting right targets to achieve an overall goal (the effect). Then we need to define our overall goal first.
To put it simply, if the goal is to assure the minimum riding skills for initial entry into the motorcycling environment, then we can say MSF safety course has achieved at a 85-90% success rate in basic courses according to the records of training schools.
In fact, in most cases, we need to consider a more comprehensive goal of safety courses which is to assess and address what riders actually need, and furthermore reducing risk prior to any crash involvement. It includes: quality education and training, knowledge, skills, attitude, habits, values, risk management skills, self-awareness and self-assessment.
Since most motorcycle insurance companies offer up to a 10% discount for the successful completion of the Basic or Experienced Motorcycle Safety Course, the effectiveness of the safety course would directly affect the costs of insurance companies. From the point of view of Motorcycle insurance pricing, we could define actuarial-effectiveness as follows:
Definition:
Actuarial-Effectivenss for Motorcycle Safety Training: The percent reduction in average incurred loss per unit exposure(or average claim frequency) to insurance company that claimed by a population of untrained motorcyclists if all were to take the safety training, with nothing else changing(ex: Weather, Motorcycle Physical Condition, Transportation Environment etc.) This is essentially the same as the potential risk reduction a random motorcyclist obtains by changing from non-attending to safety training. For example: Actuarial-Effectiveness of 10 percent means that an insurance company can reduce their incurred loss for one policy holder (or frequency of claims) by 10% simply by the behavior of such policy holder attending a safety course.#p#分页标题#e#
2.2 Formulation of Actuarial-Effectiveness
It should be noted that for the novice training course, before training, they don’t have licenses, then there should be no previous claim record for them, only for those Experienced Motorcycle Safety Course learners may have previous records. Therefore, there should be different formula for different courses’ effectiveness. As we mentioned in section 1.2, different states and different driving schools approved by MSF may offer different levels of instruction. But since almost all of them are designed by MSF, the content of the Basic RiderCourse 1 and 2(sometimes called Intermediate RiderCourse) are similar, both are for those who don’t have a license yet. While the advanced RiderCourse are for those who have been riding for some time. Therefore, for the discussion of convenience, in this paper we will only consider two categories: BRC (Basic RiderCourse) and ARC (Advanced RiderCourse).
Firstly, let’s look at what kind of data we may obtain from insurance company..... I will begin by addressing the situation where no censoring or truncation is present in the data.
2.2.1 Effectiveness for Basic Motorcycle Safety Course
For the Basic Motorcycle Safety Course, we need to compare the reduction on incurred loss per unit of exposure ”after training“ with those ”without training“ since before training the BRC learners don’t have a license yet. In particular, only when the chosen samples have similar characteristics like age, gender, motorcycle models, years riding, miles ridden per year and primary purpose of riding (commuting, recreation, etc.); the estimated effectiveness makes sense. Hence matched-pair approach could be employed here to calculate the effectiveness for basic motorcycle safety course. Usually, for insurers, we have the number of incurred claims, unites of exposures, dollars of incurred losses per year, as well as individual level data about the policyholder about whether they have taken the training or not, if yes, what kind of course they have taken. The corresponding claim data about both claim frequency and severity of the policyholder should be also available. For the matched-pair samples, suppose we have obtained the following summary data about exposures, claim count and incurred losses.
Here the timeline should be based on the motorcyclists’ training year, we denote the year they took training as , one year after they took the training as , two years after they took training as . While for those who have not taken any training, we don’t need to consider such time constraint.
Table 2 Matched-Pair Sample Summary for BRC
Types of policyholders when loss occursExposures for year Total Claim CountTotal Incurred Loss
Have taken BRC[1]
No training course
If we define the effectiveness by the measure of reduction in incurred loss, the effectiveness for year of BRC would be: (When the value of is negative, we take it as 0, mean it is not effective at all)#p#分页标题#e#
In we define the effectiveness by the measure of reduction in claim frequency, the effectiveness for year of BRC would be: = Then we could estimate the overall effectiveness, let’s simply estimate the average value as where
2.2.2 Effectiveness for Advanced Motorcycle Safety Course
For the Advanced Motorcycle Safety Course, we can compare the claim data for the same policyholder before and after they took the course. The timeline should be also based on the motorcyclists’ training year, but for different policyholders, their training year may be different. Suppose for policyholder , before training there are years’ records, after training there are years’ records.
Table3 Claim History Statistics for Motorcyclists who Took the ARC
3.For those who has taken ARCExposuresTotal Claim CountTotal Incurred Loss
Before taken ARC
After taken ARC
If we define the effectiveness by the measure of reduction in loss, the effectiveness of ARC would be: If we define the effectiveness by the measure of reduction in claim frequency, the effectiveness for year of BRC would be: 3 Determination of Discount Rate
In the following sections, we basically will use the effectiveness by the measure of reduction in claim frequency, that is the (frequency part of the) pure premium[2]. Firstly, we will estimate the overall discount rate using insurance claim data drawn from states where such mandatory programs have been introduced. Secondly, similar to the bonus-malus scheme for each policy holder, when we need to determine the specific discount rate for each person, we would consider both the policy holder’s claim history and overall discount rate.
3.1 Reflection on Current Discount Rate Policy
As shown in table 1 of section 1.1, some insurance companies use unified discount rate, such as 5% in Allstate and 10% in GEICO, while other insurance companies use flexible policies for different policy holders, such as up to 5% in Nationwide or determined case by case in PROGRESSIVE. It should also be noted that some insurers require the safety course should be taken within 3 years like Allstate and USAA, while others not.
Based on those differences, from the point of view of risk management and profit maximization of insurers, we will first evaluate whether we need to add the constraints that the safety course should be taken within 3 years, then determine what specific value or upper bound we should use.
3.2 Evaluation about the ”3-year“ constraints
This could be easily carried out by check the effectiveness of the BRC and ARC.
. For BRC, as we discussed before = , where we denote the year they took training as , one year after they took the training as , two years after they took training as , and so on. Then we can compare with , and by a simple statistical test, if there are significant difference between them, we could add the ”3-year“ constraint, otherwise, it is unnecessary. Meanwhile, we could evaluate other possible ”k-year“ policy using similar methods.#p#分页标题#e#
. For the ARC, as we discussed before . In order to evaluate the ”3-year“ policy, we could fix as 1, 2, 3, 4 or more. Then compare whether there is any significant statistical difference when is larger than 3. If it is, we suggest keep the ”3-year“ policy, otherwise not.
3.3 Overall Discount Rate Estimation Using Past Insurance Claim Data
Firstly, we could use past insurance claim data to estimate the overall discount rate, taken into consideration both the effect on pure premium and the demand. Then based on this estimated overall discount rate, discount for motorcyclist could be adjusted according to their own claim history.
3.3.1 Incentive Adjustment on Discount Rate
Since from January 1st, 2011, Connecticut require all the ages motorcyclists who want to get the license to take the BRC, insurance company don’t need to use discount rate to attract the customs but should offer basic discount based on the reduced risk compared before. While the ARC is voluntary, then it is possible that someone who took the course just for the discount, and the riders’ view on effectiveness of the ARC. Therefore, the number of policyholder who has taken ARC (within three year) should be a function of discount rate , the effectiveness of the course and the total demand , let’s say . For most states where the BRC is not required, insurer could also consider the incentive adjustment on the discount rate to increase the number of policyholder to reduce the total risk and increase profit. There are three factors which can affect total profit and risk management of insurance company as follows:
Table 4: Three Factors that affect profits of insurer
TypevariableImpact on insurance company profit
Reduction in individual premium because of the discount offered to customsDiscount rate ( and )-
Possible Reduction in Claim Cost because the training effectEffectiveness and +
Possible increasing number of policy holders because of the discount incentives or state government requirement on trainingExposures +/-
3.3.2 Discount Rate for ARC and BRC learners
As we discussed before, for the ARC learners, we can compare their claim history before and after the training. If we have derived (the frequency part of) pure premium before training and after training separately, we could calculate the discount rate as . Meanwhile, comprehensively considering the pure premium and demand, we will also consider introducing the adjustment factor . Suppose among all the policyholders who has taken the safety course for year , the proportion for BRC is and the proportion for ARC is . Once we have estimated and based on the changes on pure premium, the unified discount rate could be
Step 1: Estimate and based on the changes on pure premium
#p#分页标题#e#
Actually, here we could directly use the value of and to initially estimate and because the formula for and exactly reflects the change on (severity and frequency part of) pure premium using the insurance claim data.
Step 2: Estimate the total in-force exposure at a certain time . For current year’s exposure, we could use the basic methods in ratemaking to estimate the total in-force exposures at a certain time, let’s denote it as . Then we could have . For future year’s total exposure, we could use time series forecasting methods to estimate them. It should be noted that we need to consider the government mandatory policy’s affect on the number of motorcyclists who have taken the course in the course of analysis.
Step 3: Formulate and . should depend on the policies of different State DMV. Some states require all-age motorcyclists should take the BRC if they want to get the license, like Connecticut, Texas etc; which some states require the motorcyclists under 18 (or 16, 21) should take course. In this paper, we will look at the situation in Connecticut. Since we just introduced the policy in 2011, we could use estimate by all the new license issuing pattern in the past years by the method of time series estimation.
. As we discussed before, the number of policyholder who has taken ARC (within three years) should be a function of discount rate , the effectiveness of the course and the total demand , that is . Objectively speaking, the function f should be monotone non-decreasing function of both and when is given. We assume to be a constant in a certain period. could be estimated by the data in section 2.2. Here we treat (or adjustment factor ) as a decision variable in our programming.
In fact, the function f could be treated as the utility function on the effectiveness of the training and the insurance discount for customs. Similar to the commonly used Exponential Utility[3] in insurance industry, we assume
(3.1)
Where the factor can be determined subjectively or objectively. To put it simply, we could let both and be 0.5.
Step 3 Final Programming
Objective: Max (3.2)
St. (3.3)
(3.4)
Where need to be estimated or calculated by past data. are based the previous years’ data file. are estimated in section 2.2, the same for and .
This is a nonlinear programming problem. We could use MATLAB to solve it. We could get the optimized value for firstly, then the value for is easily followed.
3.4 Individual Discount Rate Adjustment by Personal Claim History
Since we have estimated the overall discount rate in section 3.3, now we aim at determining the specific discount rate for each person similarly to the bonus-malus scheme for each policy holder.
3.4.1Discount Rate for ARC learners#p#分页标题#e#
In this paper, the Integer-Valued Autoregressive (INAR) method will be used to model individual annual claim count in consecutive policy years. Al-Osh and Alzaid (1987) proposed what they have called an integer-valued first order autoregressive (INAR(1)) model. Later, Gourieroux and Jasiak (2004) applied it to car insurance in bonus-malus system design. Here we will use the similar INAR(1) model, but interpret the heterogeneity as training impact factor.
Let’s consider the policy holder and denote the number of claims per year submitted by this individual. We assume are independent conditional on an unobservable heterogeneity factor . We assume that the heterogeneity is a time independent random variable and follows a Gamma distribution . Here the parameters are design to ensure the expectation of to be ; where is the overall discount rate estimated in section 3.3.
(3.5)
(3.6)
[4] (3.7)
(3.8)
Where is a sequence of random variables taking nonnegative integer values; is the so-called Binomial thinning factor, which is independent of the error term and defined by , is a sequence of i.i.d. Bernoulli random variables with autoregressive parameter .The mixture operation ‘ ’ is called binomial thinning.
Note:
. Intuitively, is introduced into two parts, one is the lagged claim counts from previous years , the second part is the newly arrived claim counts . The first part lagged claim counts are introduced because during the loss settlement period, which can be many years in duration, additional facts regarding individual claims and trends often will become known (including unpaid, and often unreported, losses to their ultimate settlement values)
. For expository purpose we focus on the autoregressive process of order 1, but the approach is easy to extend to higher autoregressive orders.
. is the average annual claim frequency for policy holder during the year (We denote the year before they took ARC as , the year they took training as , one year after they took the training as , two years after they took training as , etc.) Therefore, here the index is not indicating the exact year, but a relative time compare to the time when the motorcyclists took the training. Then is the corresponding statistics. Here we only consider one year before the training because there should be no training impact for consecutive years before the training, hence mismatching with formula (3.5) and (3.6).Several studies have indicated that the risk of driver and passenger fatality associated with motorcycle accidents far exceeds that of automobile accidents. This alarming statistic has caused increased attention from state motor vehicle departments, state health departments and the insurance industry to pursue efforts to introduce programs to dramatically reduce motorcycle accidents. For example, Connecticut introduced mandatory training courses for safety motorcycle driving. It is the purpose of this paper to develop a unifying framework to quantify the effectiveness of such mandatory programs and to translate this in terms of a possible insurance rate reduction. To calibrate the proposed framework, we used insurance claims data drawn from states where such mandatory programs have been introduced and where typically there are two types: a Basic (BRC) or an Advanced Motorcycle Safety Course (ARC). An Integer-Valued Autoregressive (INAR) model is employed to assess the claims frequency of motorcycle accidents taking into account the differences in the type of training program. A heterogeneity factor is injected into the model to assess the impact of the training programs. Finally, the model allows us to estimate incentive adjustment factors to reflect possible insurance rate reduction as a result of the mandatory program.#p#分页标题#e#
At year , the (frequency part of ) pure premium is
Where is predetermined by the basic ratemaking category of such policy holder.
The risk on the count variable can be measured by
As suggested by many investigations, the period that the safety courses play a key role to reduce risk is within 3 years after the course taken time (Of course it is easy to generalize 3 to any other number k if necessary), here we only need to consider the short claim history , that pertains to a customer with a seniority for up to 3 years or new customers with similar history.
Proposition 3.1 Conditional Distribution of (i) For , the conditional distribution of is (ii) For , the conditional distribution of given is
(iii) For , the conditional distribution of given , is
Where
(iv) For , the conditional distribution of given , , is
Where
Proof:
Case The joint distribution of and is
Where INAR(1) defines a count process which has a marginal Poisson distribution with a modified parameter Therefore, the conditional distribution of given is
The proof for other cases are similar to this and easy to generalized.
Proposition 3.2 Prediction of the Course Impact Factor and (Frequency Part of) Pure Premium
(i) For , , (ii) For , (iii) For , (iv) For ,
Proposition 3.3 Individual Discount Rate
(i) For , (ii) For , (iii) For , Where is the discount rate for policy holder in the year ( is a relative time to the training). , , are estimated through Proposition 3.2.
3.4.2 Individual Discount Rate for BRC learners
For the BRC learners, we don’t have the BRC learners’ previous claim history record. One straightforward solution is to use the overall discount rate estimated in section 3.3, which already comprehensively considered the effectiveness of BRC program and other factors.
4 Numerical Illustrations
5 Conclusion
6 References
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[5] Highway Loss Data Institute Bulletin. December 2009. Motorcycle collision coverage claims in states with required motorcycle rider training. Vol 26(12).#p#分页标题#e#
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[11] Pacific Institute for Research and Evaluation, (2010 updated) Injury Prevention: What Works? A summary of Cost-Outcome Analysis for Motor vehicle
[12] Ting Zhang, (2009) Integer-Valued Autoregressive Processes with Dynamic Heterogeneity and their Applications in Automobile Insurance, Master Degree Thesis,
[13] Tim Buche, (2010),Giving Motorcyclists The Best in Training: Designing Principle-Based, Safety-Oriented Education And Training Programs, Motorcycle Safety Foundation, 2010 Conference
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[1] For those have taken both BRC and ARC in history, only consider the most recent one, in other words, the ARC.
[2] The complete pure premium includes also the cost of the claim. It is equal to the frequency part times the expected cost per claim, when cost per claim and claim occurrence are independent.
[3] ”Utility-Theoretic Underwriting,“ http://www.casact.org/sections/care/0905/handouts/halliwell.pdf
[4] Recall: For gamma distribution, , where is the shape parameter and is the scale parameter. Mean