Modeling uncertainty in periodic random environment: applications to environmental studies

MODELING UNCERTAINTY IN PERIODIC RANDOM

ENVIRONMENT: APPLICATIONS TO

ENVIRONMENTAL STUDIES

BOYAN DIMITROV,1* STEFANKA CHUKOVA1 AND MOHAMMED A. EL-SAIDI2

1Department of Sciences and Mathematics, Kettering University, Flint, MI 48504-4898, USA2Ferris State University, USA

SUMMARY

Recently an active research on the e�ects of a random environment of periodic nature on the properties ofuncertainty has been conducted by a number of authors. It re¯ects the resulting random variables andassociated random processes, and involves theoretical research, writing and implementing models andalgorithms. The research aspects are focused on the study of relevant uncertain quantities that result fromthe impact of a periodic random environment. This paper presents speci®c probability models relevant toenvironmental studies, with constant periodicity in additive models, and with periodicity with a drivingnon-stationary components. Some applications are brie¯y noticed. In particular, it is shown that varyinge�ects are modeled by extended in time Poisson trials with periodically changed components. Suchprocesses appear when modeling natural disasters like tornados, hurricane activities, etc. with a clearseasonal pattern. A representation of the waiting time up to the occurrence of such an event as a sum ofindependent components may be of considerable interest in the analysis of various environmentalcharacteristics of periodic nature. Illustrations are given. We expect these results to be useful for applicationin environmental studies, insurance and risk, as well as for developing suitable approaches for statisticaldata analysis. Copyright # 1999 John Wiley & Sons, Ltd.

KEY WORDS almost lack of memory; non-stationary Poisson process; periodic failure rate; processes ofaccumulation; risk process

1. INTRODUCTION

Periodicity is commonly observed in the environmental conditions of real life phenomena. This iswhat motivated our study. Few examples of periodic real life phenomena are listed below:

(i) Periodic alternations between cold and warm areas on Earth.(ii) Periodic phases in the solar activity and in the positions of the stars.(iii) Periodic tides, climatic and hydro-meteorological conditions.(iv) Production processes with periodic ¯uctuation in alternating day/night conditions.(v) Car accident occurrences during the year and their similar pattern from year to year.(vi) Risk-associated events, such as house ®res, diseases, fertility or mortality rates, pollution

intensity, currency exchange rates.

CCC 1180±4009/99/040467±19$17.50 Received 17 April 1998Copyright # 1999 John Wiley & Sons, Ltd. Revised 5 February 1999

ENVIRONMETRICS

Environmetrics, 10, 467±485 (1999)

*Correspondence to: B. Dimitrov, Department of Sciences and Mathematics, Kettering University, Flint, MI 48504-4898, USA.

We study the e�ects of a periodic random environment on random variables (r.v.) that representwaiting time until some event of interest occurs, and count the number of such events that occurwithin a given time interval. Probability properties, such as almost lack of memory, periodicfailure rates representation as sum of two independent r.v.s, and others characterize thesevariables. They appear similar to the exponential variables in non-periodic models. Then a non-stationary Poisson process with periodic failure rate re¯ects the e�ects of a random environmentof periodic nature. Such processes are useful in modeling claims generated by process ofpurchased items, by the subscribers of insurance, by car accidents, by emergency calls, etc.

2. PROBABILITY DISTRIBUTIONS WITH PERIODIC FAILURE RATE

2.1. Periodicity is naturally modeled

Consider the following notations: X, a non-negative r.v. that represents life time, or waiting timeuntil some event of interest occurs; FX�x� � PfX4 xg, its cumulative distribution function(c.d.f.); fX�x� � F0X�x�, its probability density function (p.d.f.).

The Hazard Distribution Function (HDF) of r.v. X, de®ned by

LX�x; t� � PfX ÿ x5 t jX5 xg � FX�x � t� ÿ FX�x�1 ÿ FX�x�

; x; t4 0;

is the conditional probability of the event occurrence within the interval �x; x � t�, given that itdid not occur before time x. It is called by Kotz and Shanbhag (1980) the hazard measure relativeto FX�x�.

The Failure Rate Function (FRF)

lX�x� �fX�x�

1 ÿ FX�x�represents the conditional probability of occurrence of the event at time x, given that this eventhas not occurred up to x.

The Cumulative Hazard Function

LX�x� �Z x

0

lX�u� du; x5 0;

is another useful measure related to a r.v. X.Our main assumption throughout this paper is the following: The impact of a random

environment on r.v.s is recorded either by the HDF LX�x; t�, or by the failure rate function lX�x�,if it exists. If a variable is embedded into an environment of periodic nature, these two measureswould re¯ect periodic properties by being periodic functions of same periodicity. This property isexpressed by the following de®nitions.

De®nition 1: We say that a r.v. X has a Periodic Hazard Distribution Function (PHDF) of periodc if for any t5 0 it is ful®lled

LX�x � c; t� � LX�x; t� for all x5 0:

Note from the de®nition of the HDF that this function is periodic only with respect to its ®rstargument (the time elapsed without event occurrence).

Copyright # 1999 John Wiley & Sons, Ltd. Environmetrics, 10, 467±485 (1999)

468 B. DIMITROV ET AL.

De®nition 2: A r.v. X has a Periodic Failure Rate Function (PFRF) of period c if and only if itholds

lX�x � c� � lX�x� for all x5 0:

It means that the failure rate, x time units after c, is the same as x time units after the processonset.

2.2. Important analytical results

The following statement is true:

Theorem 1: A r.v. X has a PHDF of period c4 0 if and only if:(a) Its c.d.f. FX(x) has the form

FX�x� � 1 ÿ a�x=c� 1 ÿ �1 ÿ a�FY x ÿ x

c

h ic

� �n o; x5 0;

where a 2 (0, 1) and FY(y) is a c.d.f. of a r.v. Y with support on the interval [0, c), and [x/c] is theinteger part of the number x/c.

(b) Its p.d.f., if it exists, is

fX�x� � �1 ÿ a�a�x=c�fY x ÿ x

c

h ic

� �; x5 0: h

The proof can be found in Dimitrov et al. (1997).

2.3. Probability properties and equivalent representations

Chukova and Dimitrov (1992) introduced the additive almost lack of memory (ALM) propertyof r.v.s in the following way

De®nition 3: A r.v. X is said to have ALM if and only if the equation

PfX ÿ c5 t jX5 cg � PfX5 tg �4�holds for any t5 0 and for in®nitely many distinct values of c.

Analogously:

De®nition 4: A r.v. X is said to have the usual lack of memory (LM) property at a ®xed point c if(1) holds for all t5 0 at that value c.

The next theorem summarizes important probability properties of random variables producedby a periodic environment.

Theorem 2: A r.v. X has LM property at time c4 0 if and only if:(a) X has a PHDF of period c. Moreover,

PfX ÿ �nc � y�5 t jX5 nc � yg � PfX ÿ y5 t jX5 yg �5�holds for all t5 0, for arbitrary y 2 [0, c), and for arbitrary integer n � 0; 1; 2; . . .


PERIODIC RANDOM ENVIRONMENT 469

(b) The following representation

X � Y � cZ

of the r.v. X as a sum of two independent random components Y and Z holds. Here:

the r.v. Y with probability 1 is located on the interval [0, c);the r.v. Z has a geometric distribution PfZ � kg � ak�1 ÿ a�, k � 0; 1; 2; . . . ; with some

a 2 [0, 1]. h

The proof of this statement can be found in Dimitrov (1998). Moreover, the quantity a is relatedto the other components by the equations

a � eÿR c

0l�x� dx

; or a � P�X5 c�:

Theorems 1 and 2 justify the notation ALM�a; c;FY� for the considered family of probabilitydistributions. Theorem 2(a) shows that the ALM property is invariant to the time shift less than cwith respect to the initial time. This fact is of main importance for the applications of theseprobability distributions. In other words it means, that the start of counting a period (which ismeasured by amount y) does not change the global properties of random variable X.

4.2. Other properties

Dimitrov and Khalil (1992) mention that generalized Bernoulli trials also lead to the class ofALM�a; c;FY� distributions. These are de®ned by assuming that to perform a Bernoulli trial, atime of duration c4 0 is needed, and success is immediately registered. Introduce

a � Pfevent of interest will not be observed during a single trialg;

and

Y � ftime from the beginning of the trial for the event to occurg:

Then the r.v.

X � fthe total elapsed time until the event has occurred after the trials have begung;

has an ALM�a; c;FY� distribution.Also Lin (1994) gives the following interpretation of random variables from the ALM�a; c;FY�

class of distributions: `X with an ALM property is the total service time of a customer on a non-reliable server, where the server has ®xed (constant) life times of duration c4 0, a is theprobability that a service will not be completed within an attempt, Y is the conditional serviceduration if it is completed within an attempt, and every interrupted service is immediatelyrestarted like a new request.'



3. TWO COUNTING PROCESSES

The following two counting processes can be associated with any random variable X, which is aresult from random environment, below:(1) A Renewal Process generated by a sequence of i.i.d. r.v.s with the same distribution as ther.v. X.

De®nition 5: Given the sequence fXngn5 1 of i.i.d. r.v.s, for any t5 0 de®ne

N�1�t � maxfn;X1 � X2 � � � � � Xn 5 tg

the renewal process associated with r.v. X.

(2) A non-stationary Poisson process (NPP) with an intensity rate equal to the failure rate of ther.v. X.

Let us ®rst recall the de®nition of a NPP. Consider a NPP with intensity rate l(t). Then

L�t� ÿZ t

0

l�u� du; t4 0

is called the hazard function of the NPP.It is known, that the number of events (points) N�0;t� of this NPP observed over the interval

[0, t) is a Poisson distributed r.v. with parameter L(t). Also, N�t;t�t�, the number of events of thisNPP that occur within the �t; t � t�, (we call t the initial age of the process), is a Poisson r.v. withparameter L�t � t� ÿ L�t�, and intensity rate l(u), u 2 �t; t � t�.

Beichelt (1991), Block et al. (1993) exploit the concept of minimal repair in reliabilitymaintenance policies in the following sense: Let a new technical item have a random life time ofduration X. After its failure it is immediately renewed (transferred from non-working to workingstate) without changes in its current failure rate (e.g. by replacing it with a working item of thesame age). Assume that the same procedure is applied to the item after any other failure in®nitelymany times. This is called a minimal repair policy. Then, the above authors prove that the pointprocess formed by the item's failures is a NPP with intensity rate lX�t�.

We propose the above NPP as a second type of counting process related to the r.v. X. Ourmotivation is to use it in the environmental studies and in risk theory modeling, where theintensity rate of risk events does not change with the event occurrence. Wewill refer to this secondprocess as that associated with the r.v. X, when its hazard function is LX�t�. We drop thesuperscript whenever it is clear which process is being considered.

Theorem 3: The number of repairs of an item maintained by a minimal repair policy is a NPP withperiodic failure rate of period c4 0, if and only if the item's life time X belongs to the ALM(a, c, FY)family of distributions. h

The proof of this statement can be found in Dimitrov (1998). It also explains that the ¯ow ofevents that occur under the in¯uence of random environment is a NPP with an appropriateperiodic rate function which re¯ects the chance of an event to occur.

The following assertion gives a useful characterization of the counting processes that resultfrom a random environment of periodic nature. Its proof can be found in Chukova et al. (1993).



Theorem 4: A NPP has periodic failure rate l(t) of period c4 0, if and only if the following twoproperties hold:(i) For a ®xed constant c and for an arbitrary t5 0 it is true that

PfN�c;c�t� � 0g � PfN�0;t� � 0g and

(ii) The r.v.s N�c;c�t� and N�0;c� are independent, for any t5 0. h

The following property of NPP is of main importance for applications in environmentalmodeling. The proof can be found in Chukova et al. (1993).

Theorem 5: A NPP counting process {Nt ; t5 0} is generated by a periodic random environment ofperiod c4 0 if and only if for any t4 0 it can be represented in the form

Nt � M1 �M2 � � � � �M�t=c� � Ntÿ�t=c�c;

where {Mn}n51 are i.i.d. Poisson r.v.s with parameter L�c� � R c0 l�u� du, independent of thecomponent Ntÿ�t=c�c, itself a Poisson process with a hazard function L�y� � R y0 l�u� du, fory � t ÿ �t=c�c 2 �0; c�. h

4. ACCUMULATING FAILURE RATES IN PERIODIC RANDOM ENVIRONMENT

Here we discuss the waiting time up to the occurrence of a random event governed by a randomenvironment with driving periodic (or double periodic) structure. In the double periodic case,after given number of periods, say m, the initial conditions start to repeat, same as from the verybeginning. As an example, we consider month-by-month changes of the environmental condi-tions in a short-term periodic environment and the 12-months period of a year as a long-termperiodicity. Such processes appear when considering, say, car accidents, or tornado occurrence.Following the approach in Dimitrov et al. (1996), one could use the extended in time Poissontrials to model short-term periodicity, and get results for the random variables generated by thedriving periodic conditions. We brie¯y give the view of these results, using an example of tornadowatch.

4.1. The model and main results

Let the tornado watch starts on 1 January (at time t0 � 0). The time that follows is naturallydivided in months (we assume here months are periods of equal duration, say c4 0). Thus theinterval ��k ÿ 1�c; kc� is the kth month (kth period of time) of this uniform partition, k �1; 2; 3; . . . : Then we watch for tornado occurrence continuously during each month (i.e. weperform the trial extended in time of duration c, and watch for the occurrence of an event duringthe trial). Let the probability to have a tornado (named `success' in the trial process model)during the kth month be ak , where 05 ak 5 1. The sequence fakg1k�1represents the drivingperiodic properties of the environmental conditions.

Let X be the total time until a tornado occurs, measured from the start t0 � 0 of the tornadowatch. Also let Yk be the local time from the beginning of the kth month ( from the beginning ofthe kth trial of duration c) until the tornado occurs (until a success occurs during the kth trial).The support of the r.v.s Yk is on the time interval [0, c) with probability 1. The time Yk is part ofthe time X only if there were no tornados up to the kth month (no successes up to the kth trial,



and ®rst success is obtained during the kth trial). In terms of probability these facts are expressedby the equation

PfX4 kc � y j kc4X5 �k � 1�cg � FYk�1�y�; y 2 �0; c�; k � 1; . . . �1�

Here FYk�y� � PfYk 4 yg is the c.d.f. of the r.v. Yk , k � 1; 2; . . . ; which re¯ects the `short-term'

periodicity in the corresponding trial process. Thus, we have

FYk�0� � 0; FYk

�c� � 1; k � 1; 2; . . . �2�

From equation (1) and from the assumption of independence of what happens during separatesubintervals, we state and prove the following theorem (notation [z] is for the integer part of z andan empty product is assumed to be equal to 1).

Theorem 6: The total time X to occurrence of success in a sequence of extended Poisson trials indriving environmental conditions, described by the sequence of probabilities fakg1k�1 and thesequence of random variables fYkg1k�1, is determined by the c.d.f.

FX�x� �Xkj�1

ajYjÿ1i�1�1 ÿ ai� � ak�1

Yki�1�1 ÿ ai�

( )FYk�1

x ÿ x

c

h ic

� �: �3� h

Proof: For any x4 0 there is an integer k and a number y such that x � kc � y, where k � �x=c�and y � x ÿ �x=c�c 2 �0; c�. By making use of total and conditional probability rules, whileexpressing the c.d.f. of the r.v. X, we obtain

FX�x� � PfX4 kc � yg

�Xkj�1

PfX4 kc � y j �j ÿ 1�c4X5 jcgPf�j ÿ 1�c4X5 jcg

� PfX4 kc � y j kc4X5 �k � 1�cgPfkc4X5 �k � 1�cg

�Xkj�1

ajYjÿ1i�1�1 ÿ ai� � ak�1

Yki�1�1 ÿ ai�

( )FYk�1

�y�:

Here the relation (1), the equation

Pfkc4X5 �k � 1�cg � ak�1Yki�1�1 ÿ ai�;

and

PfX4 kc � y j �j ÿ 1�c4X5 jcg � 1; for j4 k

are used. e

Another useful characteristic is the survival function of the r.v. X

GX�x� � PfX4 xg � 1 ÿ FX�x�;



which represents the chances to have no success (no tornado in the observed area) for at least xunits of time. Using Theorem 6 it is easy to obtain the following results:

Corollary 1: The survival function of the time X until an event occurs in a scheme of extendedPoisson trials de®ned by Theorem 6 is given by the equation

GX�x� �Y�x=c�j�1�1 ÿ aj�

!1 ÿ a�x=c��1 � a�x=c��1GY�x=c��1

x ÿ x

c

h ic

� �� : �6�

where GYk�y� is the survival function of the r.v. Yk , k � 1; 2; . . . : e

From (6) one may conclude that if some probability ak � 1, then the waiting time X until ®rstsuccess will be ®nite with probability 1, because one will get the success for sure on this kth trial.Moreover, mathematical calculations show that if the product P1j�1�1 ÿ aj� � b is convergent(i.e. not equal to 0), then there exists a positive probability b that the success (tornado for ourexample) will never occur. It is a mathematical fact that a necessary and su�cient condition tohave b5 0 is the series S � S1j�1 ln�1 ÿ aj� to be convergent. Then the relation b � eS holds. Justa qualitative criterion for ®nite X is that the series S1j�1aj diverges.

Theorem 6 provides helpful conclusions about the probability density function and the failurerate of the r.v. X. It holds

Corollary 2: (a) If each of the r.v.s Yk �k � 1; 2; . . .� has a p.d.f. fk�y�, then the probability densityfunction of the r.v. X in the scheme of extended Poisson trials is given by the expression

fX�x� � a�x=c��1Y�x=c�j�1�1 ÿ aj�

!fY�x=c��1

x ÿ x

c

h ic

� �: �7�

(b) The failure rate function of the r.v.X exists under the conditions of (a), and it is determined bythe expression

lX�x� � fY�x=c��1x ÿ x

c

h ic

� � 1 ÿ a�x=c��1a�x=c��1

� GY�x=c��1x ÿ x

c

h ic

� �( )ÿ1; �8�

where GYk�y� is the survival function of the r.v. Yk , k � 1; 2; . . . :

Proof: (a) The form (7) follows directly from the relation fX�x� � dFx�x�=dx; and afterdi�erentiation in (3). The points x � jc are points where the derivative may fail to exist, and weassume the existence of the right derivatives in both sides of (3) at these points. (b) By making useof the de®nition of FRF given in Section 2.1, the form (6) of the survival function, and equation(7), after some manipulations we get (8). e

Remark 1: If we multiply the numerator and denominator in (8) by GY�x=c�and separate terms, we

obtain

lX�x� � lY�x=c��1 x ÿ x

c

h ic

� � GY�x=c��1x ÿ x

c

h ic

� �1 ÿ a�x=c��1a�x=c��1

� GY�x=c��1x ÿ x

c

h ic

� � :



By introducing the representation x � kc � y and notations

ck�y� �GYk�y�

1 ÿ akak� GYk

�y�; �9�

we obtain the relations

lX�kc � y� � lYk�1�y�ck�1�y�; y 2 �0; c�; k � 0; 1; 2; . . . �10�

Therefore, the FRF of the time to occurrence of success in a series of extended Poisson trials isexpressed by the Cox proportional failure rate model (10) on each interval of the form�kc; �k � 1�c�, with the functions ck�y� from (9), and ak are the probabilities for success within thekth trial. Thus, such an approach could be used to model accelerated environmental processes infull analogy of what is used for reliability studies (e.g. Cox and Oakes 1984).

The following representation is of great importance for simulation purposes of processesresulting from environmental periodic driving conditions.

Theorem 7: Under the conditions of Theorem 6, the total time to occurrence of the ®rst successequals in distribution to the sum of two random components of the form

X � YZ�1 � cZ: �11�

Here the integer valued r.v. Z is de®ned by the distribution

PfZ � kg � ak�1Ykj�1�1 ÿ aj�; k � 0; 1; 2; . . . ; �12�

and is independent from the sequence fYkg1k�1. h

Proof: To prove (11) we derive the L.S.T. of the distribution of X. Taking into account (1) weobtain

jX�s� �Z 10

eÿsx

dFX�x� �X1k�0

Z �k�1�ckc

eÿsx

dFX�x�

�X1k�0

eÿkcsak�1

Ykj�1�1 ÿ aj�

Z �k�1�ckc

eÿs�xÿkc�

dFYk�1�x ÿ kc�

�X1k�0

eÿkcsak�1

Ykj�1�1 ÿ aj�jYk�1

�s�:

The same last expression would be obtained if one directly calculates the L.S.T. of the right-handside of (11) by the use of total expectation rule (we omit these details). e



Corollary 3: The expected value and the variance of the r.v. X from Theorem 6, resulting from thedriving periodic environment, are given by the expressions

E�X� �X1k�1�E�Yk� � �k ÿ 1�c�ak

Ykÿ1j�1�1 ÿ aj�;

Var�X� �X1k�1�Var�Yk� � �k ÿ E�Z��2c2�ak

Ykÿ1j�1�1 ÿ aj�

� 2cX1k�1�k ÿ E�Z��E�Yk�ak

Ykÿ1j�1�1 ÿ aj�;

where

E�Yk� �Z c

0

y dFYk�y�; Var Yk �

Z c

0

y2dFYk�y� ÿ �E�Yk��2;

E�Z� �X1k�1�k ÿ 1�ak

Ykÿ1j�1�1 ÿ aj�:

Proof: The expressions are derived from equation (11) by applying the rules of mathematicalexpectation and variance, and by taking into account the independence between the componentsin this representation. e

Remark 2: The extended Bernoulli trials modeled by Dimitrov et al. (1996) can be obtained as aparticular case of the above one when all Yk , k � 1; 2; . . . are i.i.d. same as a r.v. Y, i.e.Yk �d Y.Then in (11) we obtain the representation X �d Y � cZ.

The r.v. X discussed in Section 2 can be obtained from the extended Poisson model, if Yk �d Yand ak � a for all k � 1; 2; . . . :

4.3. The double-periodic model

We ®nish the present considerations with a particular case which corresponds to the tornado-watch model, i.e. when the sequence {ak} is periodic.

Chances to have a tornado are not the same during di�erent months of the year but thesechances are mathematically the same for the speci®c month of any year. This means that thesequence fakg1k�1 is periodic of period m, where m is a positive integer, and the equation

ak�m � ak �13�

holds for k � 1; 2; 3; . . . : The periodic sequence is a tool to model long term periodicity (secondlevel of periodicity). For our tornado watch example the period for the sequence of probabilities{an} is m � 12 (a year in months). If a sequence is periodic of period m, then it is also periodic ofperiod rm and this will correspond to r years (in long run) for the tornado watch process. In thisway the global environmental conditions are considered repeatedly the same over the timeintervals of the form �rmc; �r � 1�mc�, r � 0; 1; 2; . . . : Thus, double periodicity implies



(r)-periodicity. It is also natural to assume that the components of sequence of r.v.s {Yk} areformed under periodically repeated conditions of the same period m, i.e. it should be true

Yk�m �d Yk; k � 1; 2; . . . :

Here the equation `�d ' means equality in distribution. Then it is true that:

Theorem 8: (Double-Periodic Model) If the sequences of probabilities {ak} and the times untilsuccess {Yk} on separate periods are periodic of same period m, then the waiting time X until the ®rstsuccess is represented by either of the following ways:

(a) As the sum of three components

X � YZ0�1� cZ0 � �mc�Zm; �14�

where the random variables Z0 , and Zm and of the sequence fYkgmÿ1k�1 , are mutually independent. Thedistribution of integer-valued r.v. Z0 is

PfZ0 � kg �ak�1

Ykj�1�1 ÿ aj�

1 ÿYmj�1�1 ÿ aj�

; k � 0; 1; 2; . . . ;m ÿ 1; �15�

and the r.v. Zm has a geometric distribution

PfZm � ng � dn�1 ÿ d�; n � 0; 1; 2; . . .

with parameter

d �Ymj�1�1 ÿ aj�:

(b) The c.d.f. of X is of the form

FX�x� � 1 ÿ d�x=mc�1 ÿ �1 ÿ d�FY* x ÿ x

mc

h imc

� �� ; �16�

where the r.v. Y* is determined by the c.d.f.

FY*�y� �1

1 ÿ d

X�y=c�ÿ1j�0

aj�1Yji�1�1 ÿ ai� � a�y=c��1FY�y=c��1

y ÿ y

c

h ic

� � !Y�y=c�j�1�1 ÿ aj��; y 2 �0;mc�:

�17�



Proof: The proof of part (a) is just like an exercise on suitable algebraic manipulations with theexpression of the L.S.T. of X in order to show that it can be written in the form

jX�s� �

Xmÿ1k�0

eÿskcak�1jYk�1

�s�Ykj�1�1 ÿ aj�


eÿsmcYmj�1�1 ÿ aj�

1 ÿ 1 ÿYmj�1�1 ÿ aj�

!eÿsmc

:

This is a product of the L.S.T.s of the random variables on the right-hand side of (14). Since theproduct of L.S.T.s holds only for independent sum of random components, the statement (a) isproved.

To derive statement (b) a little bit more complicated algebraic manipulations must beimplemented on the expression (3). By regrouping the terms in the products and by introducingan auxiliary notation for the r.v. Y* we get the expression (16). An alternative proof could beconducted by analyzing the double-periodic process structure: there are long periods of lengthmcwhere everything resembles the model discussed in Section 2. Within each of these periods thereare acting some driving periodic conditions on the interval [0, mc) dispersed on the periods�0; c�; �c; 2c�; . . . ; ��m ÿ 1�c;mc�, and then they are repeated. The quantity d is the probability thaton such a long interval there will be no success, i.e. d � PfX4mcg, and Y* is the conditionalexpectation of X given that X4mc (i.e. given that a success occurs within the time interval[0, mc)). By combining the results of Theorem 1 and Theorem 6, we obtain the result ofTheorem 8. e

Other combinations of the above results provide other speci®c representations for the expectedduration until ®rst occurrence of the success in double-periodic extended Poisson model, and itsvariance or higher moments.

Corollary 4: The expected value and the variance of the waiting time X until the ®rst successoccurs in a scheme of extended Poisson trials according to conditions of Theorem 8, are given bythe equations

E�X� � 1


Xmk�1�E�Yk� � kc�ak

Ykÿ1j�1�1 ÿ ak� � mc

1

1 ÿ d;

Var�X� �Xmk�1�Var�Yk� � �k ÿ E�Z0��2c2�ak


�2cXmk�1�k ÿ E�Z0��E�Yk�ak


!1


� m2c2d1 ÿ d

;



where E(Yk), Var(Yk) are de®ned as in Corollary 3, and

E�Z0� �

Xmk�1

kakYkÿ1j�1�1 ÿ aj�


:

Proof: It follows after an accurate explicitly written expression for the expected value andthe variance of the sum in the right-hand side of equation (14). It takes into account thedependence between the terms YZ0�1 and Z0 , and their independence of the component Zm , usestotal expectation rule, and some other well known properties of expected values. We omit thedetails. e

Remark 3: After a thorough analysis of the structure of the components in the representation(14), one could obtain the following equivalent representations of the time X until the ®rst successoccurs in a series of extended Poisson trials and double-periodic environment:

X � Ym � �mc�Zm; where Ym � YZ0�1 � cZ0;

X � ~Ym � cZ0; where ~Ym � YZ0�1 � �mc�Zm;

X � YZ0�1 � cZ; where Z � Z0 � mZm:

Each of these representations re¯ects the speci®c structure of the r.v. X. They could be useful fordi�erent purposes, either in simulations, or in statistical analysis of data, related to these types ofphenomena. The bene®t is that only variables with ®nite support are used: Y1;Y2; . . . ;Ym withsupport on the interval [0, c), and the integer-valued r.v. Z0 whose support is on the set ofintegers 0; 1; . . . ;m ÿ 1. The r.v. Zm has only one parameter, d, which is easily constructed.

An important note is that if at least one of the probabilities a1; a2; . . . ; am is not equal to 0, thenthe waiting time X to the occurrence of the ®rst success is ®nite with probability 1. Moreover, inlong run while the same double-periodic environmental conditions hold (assume they holdin®nitely long), the success will occur in®nitely many times with probability 1.

5. APPLICATIONS TO ENVIRONMENTAL STUDIES

Theorem 5 is of overall, principle importance for the applications of the additive ALM propertyto environmental modeling, risk theory and elsewhere (see e.g. Gerber 1979; Beard et al. 1984;Asmussen and Rolski, 1994). We discuss here its consequences on the properties of thecorresponding compound counting processes. We also propose a class of intensity functions tomodel the periodic NPP in practical situations.

5.1. Modeling processes of accumulation in a periodic random environment

Let x be a r.v. that represents the size (or amount) of a single severity and fxngn5 1 be the sequenceof recorded amount of severities. Assume that the occurrence times of each of these severities



form a NPP fNt; t5 0g with periodic intensity l(t) of same period c4 0 as that of the environ-mental conditions.

For convenience we re-scale time to c � 1. The aggregate severity process over the time interval[0, t) is the compound sum

Zt �XNt

n�0xn with x0 � 0:

Theorem 9: The aggregate severity process fZt; t5 0g driven by a periodic NPP can be decomposedinto the form

Zt � S1 � S2 � � � � � S�t� � Ztÿ�t�: �18�

Here fSngn5 1 are i.i.d. r.v.s distributed as the compound Poisson sum

Z1 �XN1

n�0xn;

and N1 is a Poisson r.v. with parameter L�1� � R 10 l�s� ds. The term Ztÿ�t� in (18) is also acompound Poisson sum, independent from the previous terms with parameter L�t� � R t0 l�s� ds, fort4 1. h

The proof of this theorem is given in Dimitrov (1997b).The Laplace transform of the aggregate severity process is

jZt�s� � E�eÿsZt � � expfL�1��jx�s� ÿ 1�g;

and it is easily obtained from (18). Hence the expected total severities within an arbitrary timeperiod [0, t), and the corresponding variance are given by:

E�Zt� � �t�L�1�E�x� � L�t ÿ �t��E�x�;V�Zt� � �t�L�1�V�x� � L�t ÿ �t��V�x� � fE�x�g2f�t�L�1� � L�t ÿ �t��g:

This last expression shows that, as in the classical compound Poisson model, the variance ofthe aggregate severity process is proportional to the expected number of severities and to thesecond moment E(x2) of the size of a single severity.

Likewise, recursive evaluations of the distribution of Zt , for ®xed t, extend the classicalcompound Poisson case (see Beard et al. 1984, p. 100) to this periodic NPP. Note that

pn � PfNt � ng � eÿL�t�L�t�nn!

� L�t�n

pnÿ1;

for n5 1, with p0 � eÿL�t�.



5.2. Modeling the severity intensity function during a separate period

The above considerations show that the main characteristic of a risk process in a periodic randomenvironment is the severity's intensity. The natural seasonal environment leads to a period of oneyear to be used in the applications of our results. The ®rst year pattern is periodically repeated atany future period of the same duration. We suggest here the family of Beta intensity functions asthe most ¯exible, suitable to model ®nancial counting processes in a wide range of practicalsituations.

De®nition 6: The b-family of intensity functions is de®ned by

l�t� � gtp�1 ÿ t�q for t 2 �0; 1�

where p4 ÿ 1, q4 ÿ 1 and g4 0. It serves to model the intensity of NPPs on a unit period oftime, with periodic intensity recurring as l�t� � l�t ÿ �t�� for t5 1.

This parametric family allows for the estimation of rates with a wide range of statistical data.Some examples of the rates shaped as the proposed b-family, are the following:

Car's accident rate with a natural origin on January 1; a bathtub-shape can be expected innorthern countries, where the winter intensity rate is higher than the one for the dry summermonths. This means that it is expected to have a b-family with p5 1, q5 1, and possibly p � q.

Hurricane activity in many coastal U.S. states has to be modeled with a bell-shape intensitywith a mode towards the second half of the year. Thus, they will have representatives from a b-family with p4 1, q4 1, and p4 q. The same type of shape should be expected to model the caraccident activity for the southern countries, in contrast to that at north.

Wood-®re intensity is expected to be a symmetric bell-shaped function within a period.House-trade intensity is expected to be a bell-shaped curve with a maximum in the ®rst half of

the calendar year (April±May), i.e. it would have p5 q, and both p, q greater than 1.An open question of mathematical interest here is how the choice of the start to count of the

calendar time may a�ect the best model that will ®t a given set of statistical data for the intensityfunction.

Under the above assumptions explicit formulas for the distribution of the number, and of totalaccumulated severity (damages in other cases), both within a period and within an arbitrary timeinterval, can be obtained. Then these results can be used to compute various characteristics of theassociated risk process. Illustrations of this estimation will be considered in detail in futurearticles.

5.3. A tornado watch example

As an illustration of the double periodic model of Section 4.2 we consider a numerical tornado-watch example. We assume the duration of months equals 1 time unit, so that a year has 12 timeunits. Then the sequences {ak} and {Yk} are periodic with period of 12 units. We assume thatprobabilities to have tornado in the area for the months of the year are:

a1 � 0�01; a2 � 0�015; a3 � 0�02; a4 � 0�01; a5 � 0�02; a6 � 0�025;a7 � 0�03; a8 � 0�04; a9 � 0�03; a10 � 0�03; a11 � 0�02; a12 � 0�01:



These numbers are close to some statistical estimations based on observations on the severehurricane activity in the USA for the last 100 years. Further, let us assume that for the ®rst 6months corresponding times Yk , k � 1; 2; . . . ; 6 of tornado occurrence have a uniform distribu-tion. For next three months we assume higher chances to observe the tornado at the end of themonths then at the beginning. This would lead us to the triangular distribution for the r.v.s Y7 ,Y8 , and Y9 on the interval [0, 1), given by the p.d.f.

fY7�y� � fY8

�y� � fY9�y� � y if y 2 �0; 1�;

0 otherwise.

nFinally, for the last three months we assume also a triangular distribution on the interval[0, 1), i.e.

fY10�y� � fY11

�y� � fY12�y� � 1 ÿ y if y 2 �0; 1�;

0 otherwise

nwhich expresses higher chances the tornado to happen at the beginning of the month then at theend.

One may write the explicit expression of the c.d.f. FX(x) in terms of the given data. The graphof the respective p.d.f. for time period of 5 years is given by Figure 1 and its corresponding c.d.f.on Figure 2. Numerical calculations show that the average waiting time until a tornado occursand its variance are

E�X� � 59�1754 months; Var�X� � 486�567; and sX � 22�06 months:

The chances to have one year without tornado is given by the probability d � 1 ÿ FX�12� �0�768442. The chances to survive 2.5 years without tornado are estimated by the value of thesurvival function at x � 30 time units, i.e.

P�at least 2�5 years without tornado� � GX�30� � 0�533809:

Other purposeful quantities also can be obtained from the model. For instance, the number oftornados during the active season, say, from May to September will have, according to results ofTheorem 3, a Poisson distribution of parameter L�4; 9� � ln�1 ÿ FX�4�� ÿ ln�1 ÿ FX�9�� 0�147261, which also gives the expected number of tornados during that period. The averagenumber of tornadoes per year is L�12� � ÿln�1 ÿ FX�12�� 0�26339. The solution of theequation ÿln�1 ÿ FX�T�� 1 gives the average length of time between tornado occurrence. Forour example we do have T � 45.1268 months. These numbers could be of use in possible costanalysis of damages, insurance premiums, and other related assessments.

We believe this illustration could serve as a hint for many other meaningful applications ofsimilar models.

6. CONCLUSIONS

Periodicity in surrounding environmental conditions may have a considerable impact on randomevents, variables and processes they generate, and a�ects the properties of these uncertainties.



Figure 1. The probability density function of the ®rst tornado occurrence time for a period of 60 months

Figure 2. The cumulative distribution function of the ®rst tornado occurrence (i.e. chances to have tornado within a givennumber of months)



A natural description of these e�ects produces simple analytical form of the respectiveprobability distributions. Their mathematical models belong to three-parameter family of ALMdistributions, that are ¯exible enough to describe a wide range of practical situations.

The NPP with periodic failure rate provides good models for the counting processes in aperiodic random environment.

The periodic nature of intensity functions which are the main result of acting periodicenvironment, and used in ®nancial mathematics modeling, and risk processes could berepresented by b-type of periodic functions. This is a ¯exible and powerful tool in the study ofmany problems in environmental modeling. Statistical procedures are sought for the search of thebest ®t b-model in evaluating the parameters of a periodic NPP, proposed in section 5.2.

Non-stationary compound Poisson process with periodic intensity is an appropriate tool formodeling phenomena associated with process of accumulation in a periodic random environ-ment. Double periodic models o�er a suitable approach to extend the periodic ones and to get adeeper insight to non-stationarity within a prolonged periodicity. The same also suits to describea number of natural phenomena with apparent seasonal subordination.

ACKNOWLEDGEMENT

This work was partially supported by a grant from Kettering University NFPDG 3-58501.

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Modeling uncertainty in periodic random environment: applications to environmental studies

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