Dumaria R. Tampubolon, Ph - usd.ac.id fileDumaria R. Tampubolon, Ph.D Statistics Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, Bandung,

Dumaria R. Tampubolon, Ph.DStatistics Research Division

Faculty of Mathematics and Natural Sciences

Institut Teknologi Bandung, Bandung, Indonesia

SEAMS School – Universitas Sanata Dharma – Yogyakarta August 2016

Outline What is “Actuarial Science”?

What is “General Insurance” and what is its nature?

What are the major problems in General Insurance?

What is a “runoff triangle”?

What is “outstanding claims liability”?

Using “Leverage” as a tool to measure the sensitivity of anestimate of outstanding claims liability due to small changes inthe incremental payments (paid claims).

Determining a probability model for the moment magnitudes ofearthquake mainshocks where the source of earthquake is in theregion of Megathrust Mid 2 Sumatera

Determining the average recurrence interval of certain momentmagnitudes.

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D. R. Tampubolon: SEAMS School at Universitas

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Actuarial Science A branch of knowledge which applies mathematics,

probability and statistics, economics, and finance inassessing risks of financial losses due to measurablenon-prevented events. Such events are those whichwill occur at a certain time in the future withprobability greater than zero but the time ofoccurrence of the event and the correspondingamount of the financial losses cannot be determinedwith certainty at the time of valuation.

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Definition Insurance:

A vehicle to transfer pure economic (or financial) risks

or

liability to compensate for loss or damage arisingfrom specified contingencies such as natural disaster,fire, theft, negligence, injury, death, etc

General Insurance:

Relates to the insurance of property and liability; mayalso relate to the insurance of the person which is notcovered by life insurance.

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Nature of General InsuranceSeverity (amount of claims) and Frequency (number of

claims )can not be determined with certainty at the time ofvaluation

Time of disastrous event which lead to financialcompensation can not be determined with certainty at thetime of valuation

Claims are not usually paid as soon as they occur. Hence, there is a delay: between occurrence of event and reporting of a claim; and between the time of reporting and settlement of the claim.

A closed claim might be reopened and additional money need to be paid

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Major Problems in General Insurance How to determine the premium?

Risk (Adjusted) Premium = Pure Premium + Loadings

Pure Premium is the Expected of Claims, that is the mean of the distribution of claims.

How to determine the loading factor?

How to determine the outstanding claims reserve?

OR

How to determine the outstanding claims liability?

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Outstanding Claims Liability The present values of expected future payments which include

Incurred But Not Reported (IBNR) claims, the evaluation of future payments on claims already notified and the management expenses of future claims payments of claims incurred as at the balance date.(Hart, D. G., Buchanan, R.A., and Howe, B. A. (1996). The Actuarial Practice of General Insurance, page 26)

The provision for the outstanding claims is usually the largest component of a general insurance company’s liabilities; hence, changes in the outstanding claims liability have a direct, and possibly large, impact on the company’s profits and tax liabilities.

(General) Insurance companies need to reserve enough of their premium income to cover future claim payments from past and current policies

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Sensitivity of the Estimate

Gain insights on the forecasting methodologyused:

→ very or moderately or not sensitive?

Gain insights on the data:

→ absolute and relative importance

Gain insights on the uncertainty of theestimate of the outstanding claims liability

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D. R. Tampubolon: SEAMS School at

Universitas Sanata Dharma 9

Measurement of Sensitivity

Measurement of the sensitivity of the estimate of the outstanding claims liability to smallperturbation in an incremental payment:

EXAMPLE Estimate of the outstanding claims

liability of AFG Data using Hertig’s model

The data used as an example is the AutomaticFacultative General (AFG) Liability, excluding Asbestosand Environmental, from the Historical LossDevelopment study, which was also considered byMack (1994b). Following Mack, the runoff triangle ofthe incurred payments of the AFG data is representedin thousands ($’000).

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EXAMPLE Estimate of the outstanding claims

liability of AFG Data using Hertig’s model

The estimate of the outstanding claims liability forthe AFG data, using Hertig’s Model (Hertig, 1985) is$86.889 million or approximately $87million.

What happen if there is an additional $1,000 in theincremental payment on a cell of the runoff triangle?Or, what happen if there is a delay of payment (of asmall amount) of $1,000?

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Hertig’s Model Leverage(1 unit increase)

0 1 2 3 4 5 6 7 8 9

0 -1.292 -1.311 -0.513 -0.11 0.48 1.201 2.116 3.237 5.489 12.161

1 -161.585 1.03 -1.596 0.762 0.877 1.323 2.073 3.707 6.455

2 -1.352 -0.629 -0.034 0.257 0.643 1.142 1.677 2.678

3 -0.659 -0.469 0.025 0.47 0.626 0.996 1.528

4 -7.935 0.318 0.254 0.626 0.804 0.996

5 -3.322 0.037 0.671 0.842 1.454

6 -13.908 0.367 1.51 1.405

7 -3.344 1.177 1.664

8 2.265 2.309

9 22.815

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Hertig’s Model Leverage(1 unit increase)

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Sensitivity of the Estimate

Given the Hertig’s model, the leverage iscalculated as follows. Let us say that there is a$1000 increase in cell (1,0) of the runofftriangle of the incremental payments (anincrease of $1000 is small enough sinceincreases of $500 and $1 in the incrementalpayments also result in the same leveragevalues).

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Sensitivity of the Estimate The resulting leverage is -161.585. This means

that there is a decrease in the estimate of theoutstanding claims liability of almost 162 timesthe increase in the cell. In other words, had theclaims paid in accident year 1 and developmentyear 0 been $107,000 instead of $106,000, thenthe resulting Hertig’s model estimate of theoutstanding claims liability will be approximately$162,000 lower than the original estimate of$86,889,000.

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Sensitivity of the Estimate In another example, for accident year 0, let us say

that there is an increase of $1000 in the paidclaims at the final development year (at the tail).Then the change in estimated total outstanding isapproximately 12 times as much. This meansthat, had the $1000 claims been paid later, theresulting Hertig’s model estimate of theoutstanding claims liability will be approximately$12,000 higher than the original estimate.

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Hertig’s Model Leverage What happens if claim payments are delayed?

For a particular accident year:

Pay early → a “decrease” in outstanding claims liability estimate

Pay later → an “increase” in outstanding claims liability estimate

What happens where there are very few observations available to do forecasting?

Large leverage in the last accident year and at the tails

There is an extremely large leverage in cell (1,0). It turned out that the incremental payment in that cell is “unusual” compare to the other incremental payments in development year 0. The leverage values of the Hertig’s model indicate “unusual observations” in the incremental payment data.

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Earthquake Insurance

How to determine the premium to cover loss due toearthquake hazard?

How to determine the outstanding claims liability?

There is an expression used in seismology andgeophysics ; for example:

“10% PE in 50 years given the return period of

475 years “.

What does that mean?

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Determining Premium using Catastrophe Model for Earthquake

Let say someone asks an insurance company to provideprotections against earthquakes; that is a coverage ondamages to a property or on business’ interruptioncaused by an earthquake. What is the premium needto be charged by the insurance company?

To determine the premium, one need to estimate thefinancial loss caused by the earthquake; or one need todetermine the “probable maximum loss”; or one needto determine the probability of the loss to exceed acertain amount.

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Peta Tim Revisi Gempa 2010

2010 Indonesia Hazard Map

(Open a Different File)

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Significant earthquakes in Indonesia 2004 – 2011 (Source: USGS website)

26 December 2004 Sumatera-Andaman Islands (Aceh): Mw 9.1 (tsunami); 227,898 fatalities

28 March 2005 Northern Sumatera (Nias region):

Mw 8.6; 1,313 fatalities

27 May 2006 Java (Yogyakarta): Mw 6.3;

5,749 fatalities

12 September 2007 Southern Sumatera (Bengkulu):

Mw 8.5; 25 fatalities

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Significant earthquakes in Indonesia 2004 – 2011 (Source: USGS website)

16 November 2008 North Sulawesi (Gorontalo):

Mw 7.4; 6 fatalities

30 September 2009 Southern Sumatera (Padang):

Mw 7.6; 1,117 fatalities

25 October 2010 Kepulauan Mentawai :

Mw 7.7 (tsunami); 670 fatalities

4 April 2011 South Java: Mw 6.7

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Catastrophe Model

23

-Earthquake (Moment) Magnitude

-Distance between the Site and the

Source of Earthquake

-Soil Condition

HAZARD

VULNERABILITY

INVENTORY

LOSS

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Case Study Megathrust Mid 2 Sumatera

97.298°E – 101.947°E and -5.418°S – 0.128°N

West Sumatera Province, Indonesia

(Irsyam et al., 2010)

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Modeling Hazard: Moment Magnitude Gutenberg-Richter Law

Gutenberg-Richter (1941; 1944) describe therelationship between the frequency and earthquakemagnitude through the equation

where N(m) is the number of earthquakes withmagnitudes greater than or equal to m; a and b areparameters which indicate the characteristics ofseismic activities at a particular site.

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log10 𝑁 𝑚 = 𝑎 − 𝑏𝑚

Modeling Hazard: Moment Magnitude In practice, it is of interest to examine earthquake

magnitudes which are greater than or equal to aparticular value mt . Hence, the Gutenberg-Richterequation becomes

or

where m is greater than or equal to mt ; and at is thelogarithm of the number of earthquakes withmagnitude greater than or equal to mt .

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log10 𝑁 𝑚 =𝑎𝑡 − 𝑏 𝑚 −𝑚𝑡

𝑁 𝑚 = 10𝑎𝑡−𝑏 𝑚−𝑚 𝑡

Modeling Hazard: Moment Magnitude

In this research, we will use the moment magnitudescale instead of the Richter scale to measure theearthquake magnitude. Let Mw be a random variablewhich denote the moment magnitude of anearthquake.

Let Z be a random variable which denote the scalarseismic moment in Newton-meter or Nm. Therelationship between earthquake moment magnitudeand the scalar seismic moment is

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𝑀𝑤 =2

3log𝑍 − 6

Modeling Hazard: Moment Magnitude Hence, the number of earthquakes with Mw greater

than or equal to M is given by the equation

or

Since at = log10 N(Mt) then it can be shown that theequation above is equivalent to:

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𝑁 𝑀 = 10𝑎𝑡−𝑏 𝑀−𝑀𝑡

𝑁 𝑀 = 10𝑎𝑡−𝑏 23

log 𝑍−23

log 𝑍𝑡


where

Hence,

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𝑁 𝑀 = 𝑁 𝑀𝑡 𝑍𝑡𝑍 𝛽

𝛽 =2

3𝑏

Pr 𝑀𝑤 > 𝑀 𝑀𝑤 ≥ 𝑀𝑡 = 𝑍𝑡𝑍 𝛽


So, given Mw greater than or equal to Mt , theequation above is the survival function of a Paretodistribution with parameters β and Zt . That is, giventhe moment magnitude is greater than or equal to athreshold Mt , the seismic moment Z follows a Paretodistribution with parameters β and Zt .

This result leads to a hypothesis that the momentmagnitude of earthquake mainshocks might follow aGeneralized Pareto distribution.

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Modeling Hazard: Moment Magnitude The data used in this research is the earthquake data

from the National Earthquake Information Centre –United States Geological Survey (USGS, 2012)earthquake catalog, from January 1973 to December2011.

The selected earthquake data are those of whichcentre are in the area of Megathrust Mid 2 Sumatera.We use the report by “Tim Revisi Peta GempaIndonesia tahun 2010” (Irsyam et al, 2010) in definingthe area of Megathrust Mid 2 Sumatera.

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The earthquake mainshocks are separated from theearthquake foreshocks and aftershocks. The processis called seismicity declustering. The Gardner –Knopoff algorithm (1974) and the program written byStiphout et al (2012) are used to decluster theearthquakes data.

After declustering, 6.82% or 137 earthquakes arecategorized as earthquake mainshocks. Thedescriptive statistics of the earthquake mainschoksdata are as follows:

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N 137 Mean 5.6814

Variance 0.2006 Standard Deviation 0.4479

Skewness 3.0609 Kurtosis 14.4983

Lower Quartile 5.4146 Median 5.5727

Upper Quartile 5.8155 Range 3.4066


We fit a Generalized Pareto distribution to themoment magnitudes of the earthquake mainshocksdata.

The distribution function of a Generalized Paretodistribution with parameters ξ and θ, and threshold uis:

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𝐹𝑋 𝑥 =

1 − 1 + 𝜉 𝑥 − 𝑢

𝜃

−1𝜉

, if 𝜉 ≠ 0 , 𝜃 > 0

1 − 𝑒− 𝑥−𝑢 𝜃 , if 𝜉 = 0 , 𝜃 > 0


To estimate the parameters, the Maximum LikelihoodEstimation method is applied.

At 5% significance level, the Cramér-von Mises teststatistics showed that, given a threshold of momentmagnitude Mt = 5.4, the moment magnitudes of theearthquake mainshocks (with the source of earthquakesin the area of Megathrust Mid 2 Sumatera) follows aGeneralized Pareto distribution with parameters

ξ = 0.14447 and θ = 0.30891.

With the parameters obtained, the distribution of themoment magnitudes has mean 5.76107 and standarddeviation 0.42820.

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Modeling Hazard: Average Recurrence Interval

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𝑀𝑤 be a random variable which denote the moment magnitudes of earthquakes (mainshocks)

𝑉 be a random variable which denote the number of years needed until an earthquake with

moment magnitude at least a certain value, 𝑀𝑤 ≥ 𝑀𝑡 , occurs for the first time.

Modeling Hazard: Average Recurrence Interval

V follows a Geometric distribution with parameter

The expected number of years needed until an earthquake with Mw at least Mt occurs for the first time is

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𝑝 = Pr 𝑀𝑤 ≥ 𝑀𝑡

𝐸 𝑉 =1

Pr 𝑀𝑤 ≥ 𝑀𝑡

Modeling Hazard:Average Recurrence Interval

The “average recurrence interval“ (some literatureused the term “return period”) is defined as theexpected number of years until an earthquake withMw at least M, given a moment magnitude thresholdMt , occurs for the first time in a region.

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Average Recurrence Interval = 𝜏 =1

Pr 𝑀𝑤 >𝑀 𝑀𝑤>𝑀𝑡


Let N be the random variable which denote the number of earthquakes with Mw at least M, given a moment magnitude threshold Mt , occurring in t years in a region.

It is assumed that the earthquake (mainshock) is independent of time and independent of past earthquakes (mainshocks).

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The value 1

𝜏 is called the “average recurrence rate”.


The random variable N may be modeled by a Poisson distribution, that is

for n = 0,1,2,…

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Pr 𝑁 = 𝑛 =𝑒−

𝑡𝜏

𝑡𝜏 𝑛

𝑛!


The probability of at least one earthquake with Mw

at least M, given a moment magnitude threshold Mt ,occurring in t years in a region is:

The above equation can be used to calculate theseismic risk expressed as: “x% PE in t years” (x%Probability of Exceedance in t years) for a givenrecurrence interval of earthquakes with a certainmoment magnitude or greater.

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Pr 𝑁 ≥ 1 = 1 − 𝑒𝑥𝑝 𝑡

𝜏


Example:

Let t = 50 years and let the probability of exceedance0.1

Then

or the average recurrence interval is approximately

475 years

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0.1 = 1 − 𝑒𝑥𝑝 −50

𝜏

Modeling Hazard:Averange Recurrence Interval

For a threshold Mt = 5.4, let Mw follows a Generalized Pareto distribution with parameters

and

Let

Then M is approximately 8.475

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𝜉 = 0.14447

𝜃 = 0.30891

Pr 𝑀𝑤 > 𝑀 = 𝑆𝑀𝑤 𝑀 =

1

475


This means that “the probability of at least oneearthquake with moment magnitude at least 8.475occurring in 50 years in the region, given the averagerecurrence interval of 475 years, is 10%”.

Using the expression usually used by seismologists:

“10% PE in 50 years given the average recurrenceinterval of 475 years with moment magnitude at least8.5 “.

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Another example:

The probability of at least one earthquake withmoment magnitude at least 7.935 occurring in 50years in the region, given the average recurrenceinterval of 224 years, is 20%.

Using the expression usually used by seismologists:

“20% PE in 50 years given the average recurrenceinterval of 224 years with moment magnitude at least7.9 “

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Still Need to be Done!!! Modeling the distribution of Peak Ground Acceleration

(PGA) utilizing the probability distribution of momentmagnitudes (Hazard Module)

Determining the Modified Mercalli Intensity (MMI)

Determining the Damage Curve (Vulnerability Module)

Determining the “Probable Maximum Loss”; ordetermining the Distribution of Loss (Loss Module).

Determining the Premium

Determining the Outstanding Claims Liability

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References [1] Andaria, R. (2013). Penentuan Average Recurrence

Interval, Peak Ground Acceleration dan ModifiedMercalli Intensity: Sumber Gempa Wilayah Megathrust Mid 2Sumatera, Tesis Program Studi Magister Matematika(Supervisor: Tampubolon, D. R.), FMIPA, Institut TeknologiBandung.

[2} Chen, W. and Scawthorn, C. (2003). Earthquake EngineeringHandbook. CRC Press.

[3] Choulakian, V. and Stephens, M. (2001). “Goodness-of-FitTests for the Generalized Pareto Distribution”.Technometrics, American Statistical Association andAmerican Society for Quality Control, 43, 4, 478-484

[4] Hertig, J. (1985), "A Statistical Approach to IBNR-Reserves

in Marine Reinsurance”, ASTIN Bulletin, 15, 2, 171-183.

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Reference [5] Irsyam, M., Sengara, I. W., Aldiamar, F., Widiyantoro, S.,

Triyoso, W., Natawidjaja, D. H., Kertapati, E., Meilano, I.,Suhardjono, Asrurifak, M., and Ridwan, M. (2010).Ringkasan Hasil Studi Tim Revisi Peta Gempa Indonesia2010, Technical Report, Departemen Pekerjaan Umum,Indonesia.

[6] Kagan, Y. (2002). “Seisimic Moment DistributionRevisited: I. Statistical Results”, Geophysical JournalInternational, 148, 520-541

[7] Klugman, S., Panjer, H., Willmot, G. (2004). LossModels: From Data to Decisions, 2nd edition, New York:Wiley.

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Reference [8] Mack, T. (1994b), "Measuring the Variability of Chain Ladder

Reserve Estimates”, Casualty Actuarial Society Forum, Spring,101-182.

[9] Pisarenko, V., Sornette, A., Sorenette, D., and Rodkin, M.(2008). “Characterization of the Tail of the Distributions ofEarthquake Magnitudes by Combining the GEV and GPDDescriptions of Extreme Value Theory”,

http://arxiv.org/ftp/arxiv/papers/0805/0805.1635.pdf

[10] Pradana, A. A. (2013). Pemodelan Magnitudo Gempa BumiMenggunakan Distribusi Peluang Generalized Pareto: StudiKasus Megathrust Mid 2 Sumatera dan Megathrust Jawa,Laporan Tugas Akhir Program Studi Sarjana Matematika(Supervisor: Tampubolon, D. R.), FMIPA, Institut TeknologiBandung.

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http://arxiv.org/ftp/arxiv/papers/0805/0805.1635.pdf

Reference [11] Putra, R. R., Kiyono, J., Ono, Y., dan Parajuli, H. R. (2012).

“Seismic Hazard Analysis for Indonesia”, Journal of Natural Disaster Science, 33, 2, 59-70.

[12] Stiphout, T., Zhuang, J., and Marsan, D. (2012). “Seismicity Declustering”, Community Online Resource for Statistical Seismicity Analysis.

[13] Tampubolon, D. R. (2008). Uncertainties in theEstimation of Outstanding Claims Liability in GeneralInsurance, PhD Thesis, Macquarie University, Australia

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Reference [14] Tse, Y. K. (2009). Non-life Actuarial Models: Theory,

Methods and Evaluation, New York: Cambridge University Press.

[15] United States Geological Survey (USGS) website. Earthquake search. http://earthquake.usgs.gov/earthquakes/eqarchives

[16] Wang, Z. (2007). “Seismic hazard and risk assessment in the intraplate environment: The New Madrid seismic zone of the central United States”, The Geological Society of America. Special Paper 425, 363-373.

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Thank You

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Documents

Dumaria R. Tampubolon, Ph - usd.ac.id fileDumaria R. Tampubolon, Ph.D Statistics Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung, Bandung,