A new estimator for stationary distribution of the inventory model of type (s, S)
Esra Gokpinar1,
Tahir Khaniyev2 , Hamza Gamgam1, Fikri Gokpinar1
1
Gazi University, Department of Statistics, 06500, Teknikokullar, Ankara, Turkey.
2
TOBB University of Economics and Technology, Department of Industrial Engineering, Sogutozu,
06500, Ankara, Turkey; Institute of Cybernetics of Azerbaijan National Academy of Sciences, Az
1141, Baku, Azerbaijan.
Abstract
We consider inventory model of type (s, S) which is used mostly in stock control policy. It is
very important to know characteristics of an inventory model of type (s, S), such as stationary
distribution. Using the straight line approach of Frees (1986a), we establish estimator for
ergodic distribution of inventory model of type (s, S) and investigate asymptotic properties of
this estimator such as consistency, asymptotic unbiasedness and asymptotic normality.
Key Words: Inventory model of type (s, S), Ergodic distribution, Estimation, Consistency, Asymptotic
unbiasedness, Asymptotic normality.
1. Introduction
Renewal process, renewal-reward process, random walk and various modifications of these
processes have a wide range of applications in reliability theory, queueing theory, insurance
applications and etc. There are many valuable studies on these processes in the literature
(Brown and Solomon, 1975; Gihman and Skorohod, 1975; Borovkov, 1976; Alsmeyer, 1991;
Aras and Woodroofe, 1993; Khaniyev et al., 2008;). Many interested problems of stock
control are also expressed by means of these processes. For example, in the analysis of most
inventory processes, it is customary to assume that the pattern of demands forms a renewal
process. Most of the standard inventory policies induce renewal sequences, e.g., the times of
replenishment of stock. The stock control policy is one of the important topics related to
production policy of managements, since running a right stock policy for managements will
increase company's profit. In the literature, one of the most used inventory models is the
inventory model of type (s, S). In recent years, the inventory model of type (s, S) have been
extensively considered (Prabhu, 1981; Sahin, 1983; Zheng and Federgruen, 1991; Chen and
Zheng, 1992, 1997; Sethi and Cheng, 1997; Janssen et al., 1998; Heisig, 2001; Gavirneni,
2001; Khaniyev and Mammadova, 2006; Khaniyev and Atalay, 2010; Khaniyev et al., 2012;
etc.). In real world application, it is very important to know characteristics of an inventory
model of type (s, S), such as stationary distribution, moments, etc. For this reason, the ergodic
distribution of an inventory model of type (s, S) is considered in this study. The ergodic
distribution is obtained by using renewal function of demand quantity which has distribution
function F. However, the functional form of the distribution function F or the parameters of
the distribution or both of them are unknown in many cases. Thus, it is desirable to have an
estimator of the renewal function. Frees (1986a) proposed estimation of a straight line
1
approximation of the renewal function instead of direct estimation of the renewal function.
This approximation is based on a limit expression for large values of time parameter t. It is
easy to apply this estimator in practice, especially for large value of t. To estimate of renewal
function, especially in the cases of fixed values of t, there are many studies on this topic in the
literature (Vardi, 1982; Frees, 1986b; Grübel and Pitts, 1993; Zhao and Rao, 1997; Guedon
and Cocozza-Thivent, 2003; Markovich and Krieger, 2006; Bebbington et al., 2007). To
calculate these estimators in most of these studies, it is needed to the considerable amount of
computation, especially in the cases of large values of t. Contrary to Frees's estimator, it is a
disadvantage for these estimators. Because of simple form of Frees's estimator, it can be used
as estimator for applying in stochastic models including the complicated function associated
with renewal function for large value of t, for example, estimation of ergodic distribution of
inventory model of type (s, S).
The estimator for ergodic distribution makes it possible to estimate many characteristics
arising in inventory model of type (s, S). Although the literature of inventory model of type (s,
S) include many articles, none of them investigate the estimation problem for this model.
Because of that, in this study, we deal with estimator for the ergodic distribution of inventory
model of type (s, S). So this article is organized as follows. In Section 2-3, the inventory
model of type (s, S) and the ergodic distribution of this process are defined. In section 4, we
give estimator for ergodic distribution of inventory model of type (s, S), by using Frees’s
estimator. Besides, we investigate some important statistical properties of this estimator, such
as consistency, asymptotic unbiasedness and asymptotic normality. In section 5, the estimator
is applied to a simulated dataset. Concluding remarks are summarized in section 6.
The model
In this section, we define the inventory model of type (s, S) and the ergodic distribution of
the process expressed by this model. Before giving the mathematical definition, let us give the
essential notations as follows:
s
: Stock control level,
S
: The maximum stock level,
Y(t) : Stock level in a depot at time t,
Xn
: Demand quantity,
ξn
: Interarrival time,
2
τ1
: The first crossing time of the control level s by the process Y(t),
N1 : The necessary number of demands up to the first crossing time τ1
It is assumed that stock level (Y(t)) in a depot at the initial time (t=0) is Y(0)=Y0=S.
Furthermore, it is assumed that in a depot at random times T1, T2,...,Tn,... the stock level (Y(t))
in the depot decreases according to X1, X2,..., Xn,... until the stock level falls below the
predetermined control level s as follows:
Y(T1)=Y1=S-X1;
Y(T2)=Y2=S-(X1+X2); ... ; Y(Tn)=Yn=S-(  i 1 X i ), ...
n
where Xn represents the quantity of the nth demand. This variation of the system continues
until the certain random time τ1 which is the first crossing time of the control level s by the
process Y(t). At time τ1, stock level in a depot is immediately replenished to the maximum
stock level S. Thus, the first period has been completed. Afterwards, the system continuous
similar to the first period.
2. Mathematical construction of the process Y(t)
Let
  , X 

i
random
i
i 1
be a sequence of independent and identically distributed pairs of positive
variables
probability space {,F,P} We denote
(t )  P  1  t  , F  x   P  X1  x  , x   s, S  , t  0. Define the renewal sequences {Tn} and {Zn}
as follows:
n
n
Tn   i ,
Zn   X i ,
i 1
i 1
defined
on
some
n  1, T0  Z 0  0,
and a sequence of integer valued random variables {Nn}, n0 as
N0  0;
N1  min k  1: S  Zk  s;


 
Nn1  min k  Nn  1: S  Zk  Z Nn  s .
nth crossing time of the control level s by the process Y(t) can be written as
Nn
n  TNn   i , n  1, 0  0.
i 1
Then the desired stochastic process (Y(t)) expressed by inventory level at time t can be written
as
3
Y (t )  S  Zv (t )  Z Nn , n  t  n1 ,
n  0,1,2,...,
where v(t )  max n  0 : Tn  t , t  0.
This model is known as an inventory model of type (s, S). A sample path of the process Y(t) is
given in Figure 1.
Figure 1. A sample path of the process Y(t)
The main purposes of this study is to estimate ergodic distribution of the inventory model of
type (s, S). In following section, initially we give definition of ergodic distribution of this
model.
3. Ergodic distribution for the inventory model of type (s, S)
The ergodic distribution of the process Y(t) is given as Q  x   lim P  Y(t )  x  , x [s, S ).
t 
Nasirova et al. (1998) obtained the ergodic distribution as following Lemma 3.1.
Lemma 3.1. (Nasirova et al.,1998) Suppose that E(  n)<∞, then, the ergodic distribution of
the process Y(t) is given as:
4
Q  x  1 
U X  S  x
UX S  s
s  x< S
,
(3.1)
Here UX (z) is the renewal function generated by the sequence of {Xn, n=1,2,…}. Note that
UX (S-x) and UX (S-s) can be written as follows for sufficiently large values of (S-s) and (S-x)
(Feller, 1971):
U X S  s 
S  s 
2
 o(1)
212
S  x 
2
 o(1),
212
1
and
U X S  x 
1
where k  E  X nk  is kth moment of the demands. Then the ergodic distribution Q (x) can be
rewritten as follows:
Q  x   Q1 ( x; 1 , 2 )  o(1),
where Q1 ( x; 1 ,  2 ) 
2  x  s  1
21   2
(3.2)
and β=(S-s). Note that if ξ1 and X1 are mutually independent
and 0  E  1   , then the ergodic distribution of the process Y(t) is independent from the
moments of the random variable ξ1.
4. Estimator for ergodic distribution of inventory model of type (s, S)
In this section, we give estimator for ergodic distribution in Eq. 4.1 of inventory model of
type (s, S) by using Frees’s estimator (1986a) of renewal function. We first investigate some
important statistical properties of the proposed estimator Qˆ1n ( x) for Q1 ( x; 1 ,  2 ) . After that
we will show that the proposed estimator Qˆ1n ( x) converges in probability to Q ( x) when the
value of parameter  =S-s is sufficiently large and n→∞.
5
We propose an estimator for ergodic distribution as follows:
Uˆ  S  x 
Qˆ1n  x   1  X
,
Uˆ  S  s 
sxS
(4.1)
X
where
 S  s   ˆ 2
Uˆ X  S  s  
ˆ 1
2ˆ 12
and
 S  x   ˆ 2 .
Uˆ X  S  x  
ˆ 1
2ˆ 12
Here ˆ 1  X   i 1 X i n and ˆ 2  X 2   i 1 X i2 n are estimators of µ1 and µ2 based on the
n
n
random sample X 1 , X 2 ,..., X n . Then this estimator can be rewritten as follows:
2( x  s ) X
Qˆ1n  x  
.
2X  X 2
(4.2)
Now, we first investigate some important statistical properties of estimator Qˆ1n  x  for
Q1 ( x; 1 , 2 )  2  x  s  1  21  2  .
Let us investigate the asymptotic unbiasedness of this estimator. To show unbiasedness
properties of this estimator, we first need to prove following Proposition 4.1 about the
covariance between X and X 2 .
Proposition 4.1. Let X 1 , X 2 ,..., X n be independent and identically distributed random variables
with distribution function F and µi (i=1,2,...) are ith moment of the distribution.
X k  i 1 X ik n (k=1,2,...) and X m  i 1 X im n (m=1,2,…) are kth and mth sample
n
n
moment, respectively. The covariance between

Xk
and
Xm
can be given as:

Cov X k , X m    k  m   k  m  n .
Proof. The covariance between X k and X m is obtained as follows:
6



Cov X k , X m  E  X k   k


 X
m



  m   E X k X m   k  m .

(4.3)

Here E X k X m term is obtained as


 

1 n
1 n
 1  n
E X k X m  E   X ik  X im   2 E   X ik  m  2  X ik X mj   k  m   k  m  k m .
n
n
n
n
n
i 1
1 i  j  n
 i 1

 i 1



Then, Cov X k , X m    k  m   k  m  n .

This completes the proof of Proposition 4.1.
Corollary

4.1.
The
covariance
between
X
and
X2
can
be
given
as

Cov X , X 2   3  1 2  n .


Remark. As it is seen from Corollary 4.1, lim Cov X , X 2  0 when 3  E  X13   . This
n 
result shows that X and X 2 are asymptotically uncorrelated.
By using Corollary 4.1, we can prove that the estimator Qˆ1n ( x) is an asymptotic unbiased
estimator of Q1 ( x) .
Theorem 4.1. Suppose that 4  E  X14    , then
lim E  Qˆ1n ( x)  
n 
2( x  s)1
 Q1 ( x; 1 , 2 )
21  2
holds, that is, Qˆ1n ( x) is asymptotic unbiased estimator of Q1 ( x; 1 ,  2 ) . Here after, for
simplicity, we take Q1 (1 ,  2 ) instead of Q1 ( x; 1 ,  2 ) .
Proof. Let us first find the second order Taylor expansion of Qˆ1 ( x) at µ1 and µ2
7
 Qˆ1n  x  
 X  1 

Qˆ1n  x   Q1  1 ,  2    X  1  

   X   X 
2

 2 1
2
X  2
X

2
 2

2
2
 2

  Qˆ1n  x  
  X  1  X 2   2

2 
2
  X
 X 

 2 1

 
  2 Qˆ  x  
1n


 X 2  2
   X 2 

 X 1


X 2  2


ˆ
 Q1n  x 

  X  X 2

 
X  2
 ˆ

 Q1n  x  


  X 2  X 

 2 1
 
X  2


 Rˆ 21

 X 
 2 1
(4.4)
X  2
where R̂21 is the remainder term. The expected value of Qˆ1n ( x) could be given as follows.
 Qˆ  x  
E  X  1 

E Qˆ1n  x   E  Q1  1 ,  2    E  X  1   1n

   X   X 
2

 2 1


X  2


E X 2  2
2

2
 2

  Qˆ1n  x  
 E  X  1  X 2   2

2 
2
  X
 X 

 2 1

 

2
  2Qˆ  x  
1n


 E X 2  2
   X 2 

 X 1

X 2  2

ˆ
 Q1n  x 

  X  X 2

 
X  2








 
X  2


 E Rˆ 21

 X 
 2 1
 
(4.5)
X  2

It is seen that E  X  1   0, E X 2   2  0, E  X  1   2  12  n , E X 2  2
2

 ˆ

 Q1n  x  


  X 2  X 

 2 1

2
  4  22  n

and E  X  1  X 2  2  Cov X , X 2   3  12  n from Corollary 4.1. It can be proved that

 

E Rˆ21  D n3 (see, Appendix). So E Qˆ1n  x  is obtained as below:
2( x  s )1  4  x  s   2
E Qˆ1n  x  

21   2   21   2 3


     2   2  x  s  1      2 
1
2
 2
 4
  

3 
 n

n
   21   2   


 2  x  s  21   2        
1
2 1
 3

 o 

3


n
n

 21   2 


2( x  s) A1
1
E Qˆ1n  x   Q1 ( x) 
 o ,
n
n


(4.6)
8
where
 2( 
A 
3 1
1
  22 )   41  3 2 
 21  2 
3
. If µ4<, A1< for all 0<β<. We have the bias of
Qˆ1n ( x) as follows:
2( x  s) A1
1
Bias Qˆ1n ( x) 
 o ,
n
n



x  (s, S ].
(4.7)

n 
ˆ ( x) is asymptotic unbiased
Thus, Bias Qˆ1n ( x) 
 0 holds. That is, it is seen that Q
1n

estimator of Q1 ( x) .
We need to obtain the variance of Qˆ1n ( x) before show that Qˆ1n ( x) is a consistent for Q1 ( x) .
Thus, Lemma 4.1 gives the variance of Qˆ ( x).
1n
Lemma 4.1. Suppose that 4  E  X14   . Then, the variance of Qˆ n ( x) can be represented as
follows as n→∞
A
1 
Var Qˆ1  x   4( x  s ) 2  2  o    ,
n 
 n


 2  312  2 213  3 221   5 412  3 3 21  4 2212  2 32 
.
where A2  
4
 21   2 
Proof. It is known tha

Var Qˆ1  x 

2
 2( x  s) X    2( x  s ) X  
 E
 E
 .
 2X  X 2    2X  X 2  

  

2
(4.8)
For calculating the first term in Eq. (4.8), we need to obtain the second order Taylor

expansion of Qˆ12n ( x)  2( x  s) X
 2X  X 
2
2
at µ1 and µ2 as given Eq. (4.9). We have the
following result:
9
 Qˆ 2 ( x) 
 X  1 

Qˆ12n ( x)  Q12  1 ,  2    X  1   1n

   X   X 
2

 2 1
X  2
X

 2
2

2
 2 2

  Qˆ1n ( x) 
  X  1  X 2   2

2 
2
 X
 X 

 2 1

 
2

2
  2 Qˆ 2 ( x) 
1n



   X 2 
X


 1
X 2  2

2
 Qˆ1n ( x)

  X  X 2

 
X  2

 ˆ2

Q1n ( x) 

X  2 

  X 2  X 

 2 1
2

 
X  2


 Rˆ 22 .

 X 
 2 1
(4.9)
X  2
The expected value of Qˆ12n ( x) could be given as shown below:
 2( x  s ) X
E
 2X  X 2

2
2
2 2
3

4( x  s ) 2 12 8( x  s )  3 41  23 21  2 21   2 

 
2
4
n  21   2 
 21   2 

2
(4.10)
16( x  s )    2  2 1   
2

3
1
2
2
n  21   2 
2
3 1
4
  o  1 .
 
n
The second term given in Eq. (4.8) is the square of the expression given in Eq. (4.6), i.e.,
  2( x  s) X
E 
2

  2X  X
where B1 
B2 
2

4( x  s)2 12
B B 
 1 
 4( x  s)2  1  22   o  2  ,
  
2
n 
 n n 
 21  2 
 
 2( 
2
3 1
  221 )   412  3 21 
 21  2 
4
(4.11)
and
42  3212  232221   24   4    4312   4 221  32 21  3 32    4212  2 43 21  32 22
 21  2 
6
.
Consequently, the variance of Qˆ1n ( x) , by using the results shown in Eq. (4.10) and Eq. (4.11),
is given by
A
1 
Var Qˆ1n ( x)  4( x  s ) 2  2  o    ,
n 
 n


10
where A2 
2  312  2 213  3 221   5 412  33 21  4 2212  2 32
 21  2 
4
.
□
This completes the proof.
Corollary 4.2. It can be shown that


nVar Qˆ1n  x   4( x  s)2 A2 , as n.
(4.12)
Now we can give the consistent estimator for Q1 ( x) as given in Lemma 4.2.
Lemma 4.2. Suppose that E  X i4   4  , then, Qˆ1n ( x) is a consistent estimator of Q1 ( x) .


Proof. Recall that lim E Qˆ1n ( x)  Q1 ( x) from Theorem 4.1. To show that Qˆ1n ( x) is a
n


consistent estimator of Q1 ( x) , we need to prove that lim P Qˆ1n ( x)  Q1 ( x)    0 . It is well
n 


2
known that P Qˆ1n ( x)  Q1 ( x)    E  Qˆ1n ( x)  Q1 ( x)  2 from Markov inequality. Then, we need
to show that lim E  Qˆ1n ( x)  Q1 ( x)   0. It is seen that
n 
2
lim
E  Qˆ1n ( x)  Q( x)   lim
n 
n 
2
Var Qˆ (x)   Bias(Qˆ (x))   lim 4(x  s) A
2
1n
1n
2
n 
2

n   2( x  s) A1  n 2  0,
2
that is, Qˆ1n ( x) is consistent estimator for Q1 ( x) .
Now we can show that Qˆ1n ( x) converges in probability to Q ( x) when the value of parameter
β is sufficiently large.
P
 Q( x) when the value of
Theorem 4.2. Suppose that E  X i4   4  , then, Qˆ1n ( x) 
n 
parameter β is sufficiently large.
Proof. It is known that

 



P Qˆ1n ( x)  Q( x)    P Qˆ1n ( x)  Q1 ( x)  Q1 ( x)  Q( x)   
 P Qˆ1n ( x)  Q1 ( x)   2 and


Q1 ( x)  Q( x)    2 .
11



We have P Qˆ1n ( x)  Q1 ( x)    1 from Lemma 4.2. It is possible to find a number of β0>0
for each   0 , so that  Q1 ( x)  Q( x)    and P   Q1 ( x)  Q( x)      1. If En and Fn are two
sequences of events, then P  En  
1, P  Fn  
1
n 
n
1
implies P  En  Fn  
n 
(Lemma 2.1.2 in Lehmann, 1998).






Thus, P Qˆ1n ( x)  Q1 ( x)   2 and  Q1 ( x)  Q( x)    2  1, implies P Qˆ1n ( x)  Q( x)    1 ,
P
 Q( x) when the value of parameter β is sufficiently large.
that is, Qˆ1n ( x) 
n 

P
ˆ ( x) is a consistent and asymptotic unbiased
 Q( x) then Q
Remark. Since Qˆ1n ( x) 
1n
n 
estimator
for
Q( x)
when
the
value
of
parameter
β
is
sufficiently
large.
To show asymptotic normality of the estimator Qˆ1n ( x) , we first need to following Lemma
4.3.
Lemma 4.3. (Multivariate Delta Method, Casella and Berger, 2002) Define the random vector
Z=(Z1,...,Zr) with mean µ=(µ1,...,µr) and covariances Cov  Zi , Z j   ij . Let Z1),...,Z(n) be a
n
random samples of the population Z and Zˆi   k 1 Zi( k ) , i=1,...,r be the sum of observation for
each variables. For a given function g(Z) with continues first partial derivatives and a specific
value of µ=(µ1,...,µr) for which 2  i  j ij gi  g j     0. Then


d
n g (Zˆ (1) ,..., Zˆ ( n ) )  g (1 ,...,  p ) 
 N  0, 2  . Here
n 
gi     g  1 ,..., r  i
(i=1,…r).
Now we can give asymptotic normality of the estimator Qˆ1n ( x) .
Theorem 4.3. Suppose that E  X i4   4  , then Qˆ1n ( x) is asymptotically normal, that is,


d
n Qˆ1n ( x)  Q1 ( x) 
 N  0, 2  ,
n
where 2 
4( x  s)2
 21  2 
4
 
2
4 1
 2321  32  .
Proof. By the Multivariate Delta Method (Casella and Berger, 2002), we obtain that
 Qˆ   ,   
 Qˆ   ,   
Qˆ   ,   Qˆ1n  1 ,  2 
  Var ( X1 )  1n 1 2   Var ( X12 )  1n 1 2   2Cov  X1 , X12  1n 1 2
,




  1 
  2 
  1 
  2 




2
2
2
12
where Var ( X1 )  2  12 , Var ( X12 )  4  22 and Cov  X1 , X12   3  21. Then,
2 
4( x  s)2
 21  2 
4
 
Thus, we have that
4
2
1
 2321  32  .


d
n Qˆ1n  x   Q1 ( x) 
 N  0, 2 .
n
This completes the proof.

As a result, it is shown that Qˆ1n ( x) is consistent, asymptotic unbiased and asymptotically
normal estimator for Q ( x) when the value of parameter β is sufficiently large.
5. An Example
A company operating in the energy sector produces, stores, fills, and distributes liquefied
petroleum gas (LPG). Domestic LPG distribution is carried out through pipelines and land
transport. Where there is no pipeline installation, gas is distributed through land transport.
LPG is carried from the LPG production center (a city in Turkey) to the 30 dealers by tankers.
After delivering the needed amount of gas to the dealer, the tanker waits in its position until
the nest order of any dealer. Each dealer has a storage capacity of S=30m3. Random amounts
of LPG (Xn) are sold from these storage tanks at random times (n) . The level of LPG in the
tank of the dealer falls below the control level s=S/5, a demand signal is automatically sent
online to the production center. As a response to this demand, the nearest tanker to the dealer
is directed to the demanding dealer. If there is no tanker near to the dealer, a full tanker is sent
from the production center. The dealers fill the full capacity S of their tanks. Therefore, the
process that expresses the working principle of the storage tank can be considered as the
renewal process with inventory model of type (s, S). A dealer need to know ergodic
distribution of the process Y(t) to obtain some important characteristics of the process. To do
this we should know the distribution of demands and its parameters at least their first two
moments. Knowing the distribution or its parameters are almost impossible in most of the
times. Nevertheless these basic characteristics, e.g. first two moments of the demands can be
estimated from the calculated data. To obtain these moments dealer collected data until full
storage tank emptied twice. The collected data is given at Table 5.1 and generated from
Gamma (2,1).
Table 5.1.The demand quantity collected from dealers
13
1,14
1,91
2,56
1,50
1,90
2,75
0,28
0,40
3,01
1,99
1,59
0,29
2,89
2,62
1,36
1,62
1,88
0,80
3,05
1,56
0,77
1,87
10,42
2,08
3,32
0,61
1,58
Figure 2. A figure of the process according to example
The first and second sample moments are X  1.923 and X 2  7.147. By using the
moments, estimator for Q ( x) given in Eq. (4.2) is obtained as
2( x  s ) X
Qˆ1n  x  
 0.035 x  0.104.
2X  X 2


To compare Q  x  and Qˆ1n  x  we take different values of x from s to S. Then values of Q  x 
and Qˆ1n  x  are obtained for 13 value of x. Also we compare these values according to their
accurate percentage (AP) values, that is AP(%)=100- Qˆ1n  x   Q  x  Q  x  . These values are
given in Table 5.2.
14
Table 5.2. A comparison of the values of Q  x  and Qˆ1n  x 
x
4
6
8
10
12
14
16
18
20
22
24
26
28
Qˆ1n  x 
0,0351
0,1053
0,1754
0,2456
0,3158
0,3860
0,4561
0,5263
0,5965
0,6667
0,7368
0,8070
0,8769
Q  x
AP
0,0350
0,1051
0,1751
0,2452
0,3153
0,3853
0,4554
0,5254
0,5955
0,6656
0,7356
0,8057
0,8757
99,99
99,98
99,97
99,96
99,95
99,94
99,92
99,91
99,90
99,89
99,88
99,87
99,87
Here U ( z)  z 2  3 4  1 4 exp(2 z) for Gamma (2,1).
As seen from Table 5.2 the all AP values are greater than 99%. These results show that the
proposed estimator Qˆ1n  x  for Q  x  gives very accurate results.
6. Conclusion and Remarks
In real world application, it is very important to know characteristics of an inventory model of
type (s, S), such as ergodic distribution, moments, etc. The ergodic distribution is obtained by
using renewal function of demand quantity which has distribution function F. However, the
functional form of the distribution function F or the parameters of the distribution or both of
them are unknown in many cases. Thus, it is desirable to have a estimator of the renewal
function. For this reason, in this article, we give an estimator for ergodic distribution of
inventory model of type (s, S) by using the straight line approach for Frees (1986a). We show
that the proposed estimator is consistent, asymptotically unbiased and asymptotically normal
for large values of β. The proposed estimator is also applied to a dataset. The results indicate
that the accurate percentage (AP) values are all above 99%. It is meant that the estimator
gives the results very close to Q  x  . However, it is observed that the AP values get smaller as
15
the x values get closer to S. The estimator given in this study can be applied to other complex
stochastic models including renewal function. Therefore, in future studies, it can be
considered to apply this estimator in different inventory models of type (s, S).
References
[1] Alsmeyer, G., "Some relations between harmonic renewal measure and certain first
passage times", Statistics & Probability Letters, 12(1):19–27, (1991).
[2] Aras, G. and Woodroofe, M., "Asymptotic expansions for the moments of a randomly
stopped average", Annals of Statistics, 21:503–519, (1993).
[3] Bebbington, M., Davydov, Y. and Zitikis, R., "Estimating the renewal function when the
second moment is infinite", Stochastic Models, 23:27-48, (2007).
[4] Borovkov, A.A., Stochastic Processes in Queuing Theory, Spinger-Verlag, New York,
(1976).
[5] Brown, M. and Solomon, H.A., "Second-order approximation for the variance of a
renewal-reward process", Stochastic Processes and their Applications, 3:301–314, (1975).
[6] Casella, G. and Berger, R.L., Statistical Inference. 2nd edn. Pacific Grove, CA,
Duxbury/Thomson Learning, (2002).
[7] Chen, F. and Zheng, Y., "Waiting time distribution in (T,S) inventory systems",
Operations Research Letters, 12:145–151, (1992).
[8] Chen, F. and Zheng, Y.S., "Sensitivity analysis of an (s,S) inventory model", Operations
Research Letters, 21:19–23, (1997).
[9] Feller, W. An introduction to probability theory and its applications, Vol. 2. Second
edition, New York: Wiley, (1971).
[10] Frees, E.W., "Warranty analysis and renewal function estimation", Naval Research
Logistics Quart, 33:361-372, (1986a).
[11] Frees, E.W., "Nonparametric renewal function estimation", Annals of Statistics, 14(4),
1366-1378, (1986b).
[12] Gavirneni, S., "An efficient heuristic for inventory control when the customer is using a
(s; S) policy", Operations Research Letters, 28:187–192, (2001).
[13] Gihman, I.I. and Skorohod, A.V., Theory of Stochastic Processes, II. Springer: Berlin,
(1975).
[14] Grübel, R. and Pitts, S., "Nonparametric estimation in renewal theory 1: the empirical
renewal function", Annals of Statistics, 21(3):1431-145, (1993).
16
[15] Guedon, Y. and Cocozza-Thivent, C., "Nonparametric estimation of renewal processes
from count data", The Canadian Journal of Statistics, 3(12), (2003).
[16] Heisig, G., "Comparison of (s,S) and (s,nQ) inventory control rules with respect to
planning stability", International Journal of Production, 73:59–82, (2001).
[17] Janssen, F., Heuts, R. and Kok, T., "On the (R, s, Q) inventory model when demand is
modeled as a compound Bernoulli process", European Journal of Operational Research, 104,
423–436, (1998).
[18] Khaniev, T. and Mammadova, Z., "On the stationary characteristics of the extended
model of type (s,S) with Gaussian distribution of summands", Journal of Statistical
Computation and Simulation, 76(10):861–874, (2006).
[19] Khaniyev, T., Kokangül, A. and Aliyev, R., "An asymptotic approach for a semimarkovian inventory model of type (s, S)", Applied Stochastic Models in Business and
Industry, 29:439-453, (2013).
[20] Khaniyev, T. and Atalay, K. "On the weak convergence of the ergodic distribution for an
inventory model of type (s, S)", Hacettepe Journal of Mathematics and Statistics, 39(4):599611, (2010).
[21] Khaniyev, T., Kesemen, T., Aliyev, R. and Kokangül, A., "Asymptotic expansions for
the moments of a semi-Markovian random walk with exponentional distributed interference
of chance", Statistics & Probability Letters, 78(6):785–793, (2008).
[22] Markovich, M.N. and Krieger, R.U., "Nonparametric estimation of the renewal function
by empirical data", Stochastic Models, 22:175-199, (2006).
[23] Nasirova, T.I.,Yapar, D., Khaniev, T., "Probability Characteristics of the Inventory Level
in (s,S) Model", Cybemetics and Systems Analysis, 34(5):689-695, (1998).
[24] Prabhu ,N.U., Stochastic Storage Processes, Springer-Verlag, New York, (1981).
[25] Sahin, I., "On the continuous-review (s, S) inventory model under compound renewal
demand and random lead times", Journal of Applied Probability, 20:213–219, (1983).
[26] Sethi, S.P. and Cheng, F., "Optimality of (s, S) policies in inventory models with
Markovian demand", Operations Research, 45(6):931–939, (1997).
[27] Vardi, Y., "Nonparametric estimation in renewal processes", Annals of Statistics, 10:
772-785, (1982).
[28] Zheng, Y.S. and Federgruen, A., "Computing an optimal (s,S) policy is as easy as a
single evaluation of the cost function", Operations Research, 39:654–665, (1991).
[29] Zhao, Q. and Rao, S.S., "Nonparametric renewal function estimation based on estimated
densities", Asia-Pacific Journal of Operational Research, 14:115-12, (1997).
17
Appendix A.
The expected value of remainder term R̂2 goes to zero as n. To show this, initially, we
need to obtain Lagrange form of the remainder term R̂2 as follows:



2
16( x  s )3 1   X  1
3  8( x  s )
ˆ
R21   X  1  

C13
C14


X  
1
2
X
 X    X
2
1
X
2
 2
2
 2
 2

2






2

 8( x  s ) 24( x  s ) 1   X  1


3
C14
 C1




 2( x  s ) 12( x  s ) 1   X  1


3
C14
 C1



   
   






 2( x  s ) 1    X  1  


4
C1



 
3
   
where 0<<1 and C1  2  1    X  1    2    X  2  . According to law of large
P
P
 1 and X 2 
numbers, X 
 2 , then n can be chosen so large that X  1 
X 2  2 
2
2n
1
2n
and
where 0<1;2<1. Hence we have inequality as shown below:
1    X  1   1 
1
2n


and 2   X 2  2  2 
 2
2n
.
Then, we obtain the upper bound of R̂ 21 as follows:
18


 
Rˆ 21 X , X 2   1 
 2n 
3
3
2

 8( x  s ) 16( x  s )  1   1 2n   




3
4
C2
C2




2
2
 8( x  s ) 24( x  s )  1   1 2n   

 1   2  
    


3
4
C2
 2n   2n  
 C2


    
  1  2 
 2n  2n 
 
 2 
 2n 
3
2

 2( x  s ) 12( x  s )  1   1 2n   




3
4
C2


 C2


 2( x  s )  1   1 2n   



4
C
2




(A.1)
where C2  2 1  1 2n   2   2 2n   . 1 (0<1<1) and 2 (0<2<1) are chosen so
small that let 1  1 2  1 2 and 2   2 2  2 2. In this case, Eq. (A.1) similarly
holds by using C3   2 1 2  2 2 instead of C2.
1 (0<1<1) and 2 (0<2<1) are chosen so small that let 1 2  1 and  2 2  2 . Then,


3
2
2
    8 ( x  s ) 32 ( x  s )1   1    2   8( x  s ) 48( x  s) 1 
Rˆ 21 X , X 2   1  



    
3
4
3
4
C3
C3
C3
 2n  
  2n   2n   C3

3
2
      2( x  s ) 24( x  s )1    2   4( x  s )1 
  1  2  

  
.
3
C34
C34
 2n  2n   C3
  2n  

2
3
Let max(1; 2)= , then
2
3
2
3  ( x  s)     1/ 4  ( x  s)1  4  6  3  1/ 2  
2
ˆ
R21 X , X  3 


n 
C33
C34




Then,


D
Rˆ21 X , X 2  3 ,
n
19
2
3
2

 ( x  s)     1/ 4  ( x  s)1  4  6  3  1/ 2  

where D   

 .
3
4
C3
C3




3
It is known that if X1 X2 in probability 1, then E  X1   E  X 2  (Roussas, 1997). So,
 
E Rˆ 21 X , X 2
  E  Rˆ  X , X    E  D n   D n
2
21
3
3
 

. Therefore, E Rˆ21 X , X 2  D n3 .
20