Efficient Estimator of Parameters of a Multivariate Geometric Distribution

U. J. Dixit; S. Annapurna

doi:10.2991/jsta.2018.17.4.6

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Volume 17, Issue 4, December 2018, Pages 636 - 646

Efficient Estimator of Parameters of a Multivariate Geometric Distribution

Authors

U. J. Dixit¹^{, *}, S. Annapurna²

¹Department of Statistics, University of Mumbai, Mumbai, India

²Department of Statistics, St. Xaviers College (Autonomous), Mumbai, India

^*

Corresponding author. Email: ullllhasdixit@yahoo.com.in

Received 10 March 2017, Accepted 27 December 2017, Available Online 31 December 2018.

DOI: 10.2991/jsta.2018.17.4.6 How to use a DOI?
Keywords: Multivariate geometric distribution; Maximum likelihood estimator; Uniformly minimum variance unbiased estimator; modified MLE
Abstract: The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) for the parameters of a multivariate geometric distribution (MGD) have been derived. A modification of the MLE estimator (modified MLE) has been derivedin which case the bias is reduced. The mean square error (MSE) of the modified MLE is less than the MSE of the MLE. Variances of the parameters and the corresponding generalized variance (GV) has been obtained. It has been shown that the MLE and modified MLE are consistent estimators. A comparison of the GVs of modified MLE and UMVUE has shown that the modified MLE is more efficient than the UMVUE. In the final section its application has been discussed with an example of actual data.
Copyright: © 2018 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. INTRODUCTION

It is appropriate and convenient to measure lifetime of devices such as on/off switches, bulbs, engines of an airplane on a discrete scale. Discrete random variables also help study lifetimes such as the incubation period of diseases like AIDS, the remission time of cancers as well as time to failure of engineering systems (see [1]). The discrete multivariate distributions are useful to measure lifetime data. The multivariate geometric distribution (MGD) has been vital in studying reliability analysis. Various models of the bivariate geometric distribution (BGD) have been proposed to study lifetime devices. Downtown [2] has described a model for developing a BGD. This arises in a shock model with two components. Downtown [2] describes this model asfollows. Suppose that the number of shocks suffered by each component before failure can be represented by a population in which proportions p₁ and p₂ affected the first and second components respectively, without failure and a proportion 1-p₁-p₂ of the shocks lead to failure of both the components. Hence X is number of shocks to component 1 prior to the first failure and Y is number of shocks to component 2 prior to the first failure. The joint probability function of (X, Y) is given by

(1)πX,Y(t1,t2)=(1−p1−p2)(1−p1t1−p2t2)−1

The corresponding joint probability mass function of (X,Y) is

(2)P (X=x, Y=y)=x+yx p1xp2yp3;x=0, 1, 2,… , y=0, 1, 2, …,0<p1<1; 0<p2<1; p3=1−p1−p20;otherwise

Hare Krishna and Pundir [3] have obtained the MLE and Bayeâ€™s estimators of the parameters for this BGD. Dixit and Annapurna [4] have further obtained the UMVUE estimators and have compared the MLE and UMVUE based on the mean square errors (MSEs). Phatak and Sreehari [5] introduced a version of bivariate geometric distribution as a stochastic model for giving the distribution of good and marginally good items that are produced by a production unit. Marshall and Olkin [6] constructed a BGD based on the sequence of Bernoulli random variables in which X was defined as the number of trials required for the rth occurence of an event A and Y was the number of trials required for the sth occurence of an event B. If we consider Bernoulli trials then (X,Y) will have 4 possible values (0,0), (0,1), (1,0) and (1,1).

We define PX=i,Y=j=pij;i,j=0,1.

Let p00+p01=p0+,p10+p11=p1+,p00+p10=p+0,p01+p11=p+1 Marginally X and Y have negative binomial distribution. X follows NB (r, p1+) and Y follows NB (s, p+1).

The marginal probability functions are of the usual form but the joint probability function of (X,Y) is quite complicated. Reference may be made to Marshall and Olkin ([6], Eq. 7.2).

On the same lines, Gultekin and Bairamov [7] constructed a trivariate geometric distribution and the corresponding multivariate extension. Srivastava and Bagchi [8] introduced the multivariate version of a geometric distribution and obtained certain characterizations. Vasudeva and Srilakshminarayana [9] established some properties of the MGD and also obtained a characterization assuming it to follow the power series distribution. Sreehari and Vasudeva [10] have given characterization of the MGD based on conditional distributions. Esary and Daniel [11] studied properties of MGDs that were generated by a cumulative damage process. In this paper we look at another form of the MGD and estimate its parameters by a new approach that reduces the bias.

Consider a system which comprises of k components namely C₁,C₂,…,C_k. The system is so designed that at any given time not more than one component can function. The system functions when any one component functions. The system initially functions because C₁ functions. When C₁ stops functioning C₂ starts functioning in the next trial keeping the system functioning. Thus system continues to function in this manner till C_k functions. Let probability that component C_i fails be p_i, i = 1,2,3, …, k. Let X_i denote the trial at which component C_i fails, i = 1,2, …, k.

The joint probability mass function of (X₁,X₂, …, X_k) is given as

(3)P(X1=x1, X2=x2, … , Xk=xk) =(1−p1)x1−1p1(1−p2)x2−x1−1p2⋯(1−pk)xk−xk−1−1pk ;1≤x1<x2<⋯<xk; 0<pi<1; i=1, …, k 0 ; otherwise

The probability generating function (pgf) is given as

(4)PX1X2⋯Xkt1,t2,…,tk=p1p2⋯pkt1t22⋯tkk1−1−p1t1t2⋯tk1−1−p2 t2t3⋯tk⋯1−1−pktk

In this paper we obtain UMVUE and MLE of the parameters in Eq. (1) and their functions.

2. UNIFORM MINIMUM VARIANCE UNBIASED ESTIMATOR (UMVUE)

Here we obtain the UMVUE of the parameters as well as of the functions of the parameters. We consider the trivariate case which has three parameters, p₁, p₂ and p₃. Here p_i denotes the corresponding probability of the ith component in the system failing.

Consider the case where k = 3. Eqs. (1) and (2) become

(5)P(X1=x1,X2=x2,X3=x3)=(1−p1)x1−1p1(1−p2)x2−x1−1p2(1−p3)x3−x2−1p3;1≤x1<x2<x3; 0<pi<1; i=1, 2, 3 0; otherwise

(6) PX1X2X3t1,t2,t3=p1p2p3t1t22t331−1−p1t1t2t31−1−p2t2t31−1−p3t3

The pgf of S1=∑i=1nX1i, S2=∑i=1nX2i and S3=∑i=1nX3i is

(7)PS1S2S3t1,t2,t3=p1p2p3t1t22t331−1−p1t1t2t31−1−p2t2t31−1−p3t3n

Hence the pmf of S₁, S₂ and S₃ is co-efficient of t1s1t2s2t3s3 in Eq. (7) and is given as

(8)Ps1,s2,s3=s3−s2−1s3−s2−n s2−s1−1s2−s1−n s1−1s1−np1np2np3n(1−p1)s1−n(1−p2)s2−s1−n(1−p3)s3−s2−n; n≤s1<s2<s3<∞;0<p1<1;0<p2<1;0<p3<1

Theorem 2.1.

The UMVUE of p1a1p2a2p3a3(1−p1)b1(1−p2)b2(1−p3)b3 is

(9)s1−b1−a1−1s1−b1−n s2−s1−b2−a2−1s2−s1−b2−n s3−s2−b3−a3−1s3−s2−b3−ns1−1s1−n s2−s1−1s2−s1−n s3−s2−1s3−s2−n

Proof.

The trivariate joint distribution of X11X21X31,X12X22X32, …, X1nX2nX3n belongs to the exponential family and (S₁, S₂ and S₃) is sufficient and complete for Eq. (1). Hence by using Rao-Blackwell theorem we can obtain the UMVUE of p1a1p2a2p3a3(1−p1)b1(1−p2)b2(1−p3)b3.

Let ϕs1,s2,s3 be the UMVUE of p1a1p2a2p3a3(1−p1)b1(1−p2)b2(1−p3)b3

(10)E(ϕ(s1,s2,s3))=∑s1=n∞∑s1=s2+n∞∑s3=s2+n∞ϕ(s1,s2,s3)(s3−s2−1s3−s2−n) (s2−s1−1s2−s1−n) (s1−1s1−n)× p1np2np3n(1−p1)s1−n(1−p2)s2−s1−n(1−p3)s3−s2−n;=p1a1p2a2p3a3(1−p1)b1(1−p2)b2(1−p3)b3

Hence

(11)∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞ϕ(s1, s2, s3) (s3−s2−1s3−s2−n) (s2−s1−1s2−s1−n) (s1−1s1−n) × p1n−a1p2n−a2p3n−a3(1−p1)s1−n−b1(1−p2)s2−s1−n−b2(1−p3)s3−s2−n−b3=1;

Therefore

(12)ϕs1,s2,s3=s1−b1−a1−1s1−b1−n s2−s1−b2−a2−1s2−s1−b2−n s3−s2−b3−a3−1s3−s2−b3−ns1−1s1−n s2−s1−1s2−s1−n s3−s2−1s3−s2−n

Particular Cases

a1=a2=a3=1 and b1=b2=b3=0
(13)p1p2p3^=(s1−2s1−n) (s2−s1−2s2−s1−n) (s3−s2−2s3−s2−n)(s1−1s1−n) (s2−s1−1s2−s1−n) (s3−s2−1s3−s2−n)=(n−1)3(s1−1)(s2−s1−1)(s3−s2−1)
a1=1 and a2=a3=b1=b2=b3=0
(14)p1^=s1−2s1−ns1−1s1−n=n−1s1−1
a2=1 and a1=a3=b1=b2=b3=0
(15)p2^=s2−s1−2s2−s1−ns2−s1−1s2−s1−n=n−1s2−s1−1
a3=1 and a1=a2=b1=b2=b3=0
(16)p3^=s3−s2−2s3−s2−ns3−s2−1s3−s2−n=n−1s3−s2−1
a1=b1=1 and a2=a3=b2=b3=0
(17)p1(1−p1)^=(s1−3s1−n−1)(s1−1s1−n)=(n−1)(s1−n)(s1−1)(s1−2)

Similarly it is possible to obtain UMVUE for various combinations of p1a1p2a2p3a3(1−p1)b1(1−p2)b2(1−p3)b3 for different values of a₁,a₂,a₃,b₁,b₂ and b₃.

Theorem 2.2.

The UMVUE in the multivariate case of

(18)∏i=1kpiai(1−pi)bi=∏i=1ksi−si−1−bi−ai−1si−si−1−bi−nsi−si−1−1si−si−1−n; si=∑j=1nXij , s0=0

Proof is similar to proof of Theorem 2.1.

3. MAXIMUM LIKELIHOOD ESTIMATOR (MLE)

In the earlier section we have obtained an estimator based on the criteria of unbiasedness and minimum variance. We now look at another very popular principle used namely method of maximum likelihood to obtain the estimators of the functions of the parameters. These shall be compared with the corresponding estimators obtained by UMVUE in order to study their efficiency. The likelihood function based on n systems put on test strictly under the same conditions will be

(19)L =∏i=1n (1−p1)x1i−1p1(1−p2)x2i−x1i−1p2⋯(1−pk)xki−xki−1−1pk= (1−p1)s1−np1n(1−p2)s2−s1−np2n⋯(1−pk)sk−sk−1−npkn

Taking log and differentiating w.r.t p_i, i = 1,2, …, k, the MLEs are obtained as

The MLE of p_i; i = 1,2,. …, k is

(20)pi^=nsi−si−1 ; s0=0

By invariance property, MLE of ∏i=1kpiai(1−pi)bi is

(21)∏i=1kpiai(1−pi)bi^=∏i=1knsi−si−1aisi−si−1−nsi−si−1bi

4. MODIFIED MAXIMUM LIKELIHOOD ESTIMATOR (MODIFIED MLE)

In the earlier two sections we have applied two procedures to obtain the estimators and either of them could be good. We now try to improve on the MLE by reducing the bias and thus derive a modified estimator namely modified MLE. We have further shown thatthis modified MLE is better than the UMVUE. Hence to derive a modification that reduces the bias and the MSE of the MLE of p_i, i = 1,2,3, …, k, we apply the Taylor Series two-parameter expansion to

(22)pi^=ϕsi−1,si=nsi−si−1 ; s0=0, i=1,2,⋅⋅⋅,k.

where j = 1,2, …, k and

(24)μsi=ESi=∑i=1nnpi ,i=1,2,…,k.

On substition of Eqs. (21) in (22) we obtain

(25)pi^=pi+(si−1−∑j=1i−1npj)pi2n+(si−∑j=1inpj)−pi2n+(si−1−∑j=1i−1npj)22!2pi3n2+(si−∑j=1inpj)22!2pi3n2+(si−∑j=1inpj)(si−1−∑j=1i−1npj)2!−2pi3n2+O1n

On taking expectaion of Eq. (24) we obtain

(26)Epi^=pi1+1n−pi2n+O1n

We observe that

pi^ is an asymptotically unbiased estimator of p_i

Consider a linear function of the MLE of p_i given as

pi˜=α pi^+β, α and β are constants

Hence E(pi˜)=α.E(pi^)+β

(27)=αpi1+1n−pi2n+O1n+β

If the coefficient of p_i is set equal to 1 and the constant term set equal to zero, we obtain aproximate equality between Epi˜ and p_i.

This gives us α=nn+1 and β=0.

Therefore we obtain a modified MLE of p_i

(28)pi˜=n2n+1si−si−1; i=1,2,3,…,k, s0=0.

Since

(29)pi˜=nn+1pi^

E(pi˜−pi)2=nn+12E(pi^−pi)2+pi2n2−2piEpi^−pin+O1n

E(pi˜−pi)2=nn+12E(pi^−pi)2−pi2n2

Thus

(30)E(pi˜−pi)2<E(pi^−pi)2

Thus the MSE of modified MLE of pi namely pi˜ is less than the mean square error of the corresponding MLE of p_i i.e. pi^, i = 1,2, …, k.

5. CONSISTENCY OF MODIFIED MLE

If we collect a large number of observations, then we can obtain a lot of information about the unknown parameter. We can thus construct an estimator T(X) with a small MSE and we can call it a consistent estimator if limn→∞MSETX=0.

Theorem 5.1.

p1^, p2^, …, pk^ are consistent estimators where pi^ is the MLE of p_i, i = 1,2, …, k. where

(31)pi^=nsi−si−1 ; s0=0

Proof.

From Eq. (25) it is clear that Epi^ tends to p_i as n tends to ∞, i = 1,2, …, k. We shall prove by method of induction that Vpi^ tends to 0 as n tends to ∞.

Let k = 1

(32)p1^=ns1

Applying the Taylor series expansion

(33)p1^=p1+s1−np1−p12n+s1−np122!2p13n2+O1n

On taking expectaion of (p1^−p1)2 we obtain

(34)E(p1^−p1)2=Es1−np12p14n2+Es1−np14p16n4+O1n

(35)E(p1^−p1)2=n1−p1p12p14n2+n1−p1p12−3p1n+2+3n+2p14p16n4+O1n

Hence Vp1^=E(p1^−p1)2 tends to 0 as n tends to ∞.

Thus p1^ is a consistent estimator of p₁.

Consider k = 2

Here

(36)p2^=ns2−s1

Applying Taylor Series two parameter expansion we obtain

(37)E(p2^−p2)2=Es1−np12p24n2+Es2−np1−np22p24n2+Es1−np14p26n4+Es2−np1−np24p26n4+Es1−np12s2−np1−np224p26n4+O1n

(38)E(p2^−p2)2=np12−np1p24n2+np12+np22−np1−np2p24n2+n1−p1p12−3p1n+2+3n+2p14p16n4+O1n

Hence Vp2^=E(p2^−p2)2 tends to 0 as n tends to ∞. Thus p2^ is a consistent estimator of p₂.

Assume p^i−1 is a consistent estimator of pi−1.

To prove that pi^ is a consistent estimator of p_i.

We need to prove that Vpi^ tends to 0 as n tends to ∞, i = 2,3, …, k.

(39)pi^=p^i−1+n2si−1−si−si−2si−si−1si−1−si−2

To obtain E(pi^−p^i−1)2 we apply the Taylor series expansion to n2si−1−si−si−2si−si−1si−1−si−22 and then take the expectation. Hence

En2si−1−si−si−2si−si−1si−1−si−22=(pi−pi−1)2+O1n

E(pi^−p^i−1)2=(pi−pi−1)2+O1n

Thus as n tends to ∞, E((pi^−p^i−1)2) tends to (pi−pi−1)2

We now consider

Vpi^−p^i−1=E(pi^−p^i−1)2−(Epi^−p^i−1)2

Vpi^−p^i−1=E(pi^−p^i−1)2−(Epi^−Ep^i−1)2

Thus from Eq. (39) and since pi^ is asymptotically unbiased from Eq. (26) we can conclude that Vpi^−p^i−1 tends to 0 as n tends to ∞.

But

(40)Vpi^−p^i−1=Vpi^+Vp^i−1−2Covp^i−1,p^i

The Covp^i−1,p^i can also be shown as tending to 0 as n tends to ∞ by applying the Taylor series expansion.

Thus, since Vpi^−p^i−1,Covp^i−1,p^i and Vp^i−1 all tend to 0 as n tends to ∞, we conclude that Vpi^ also tends to 0 as n tends to ∞. Hence pi^, the MLE of p_i, is a consistent estimator of pi,i=1,2,⋯,k.

Note: The MSE of modified MLE is less than MSE of MLE from Eq. (29).

From Eq. (28) we can conclude that the modified MLE, pi˜ is a consistent estimator of pi,i=1,2,⋅⋅⋅,k.

6. CONCLUSION AND COMPARISION OF ESTIMATORS

We have observed in the earlier section that an improvement over the MLE is the modified MLE. We now have two estimators the UMVUE and the modified MLE. Our objective is to compare both the estimators with respect to efficiency. We make a comparative study of the two based on the determinant of the variance covariance matrix also called as the generalised variance (GV). We consider the trivariate case namely k = 3. The variances and covariances of the UMVUE and modified MLE of the parameters can be obtained as below. Consider the case when k = 3.

Variance of UMVUE of p_i, i = 1, 2, 3 where s0=0 is

(41)∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞si−si−1−2si−nsi−si−1−1si−si−1−n2Ps1s2s3−pi2

Covariance of the UMVUEs of p_i and p_j, i, j = 1, 2, 3 and s0=0 is

(42)∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞si−si−1−2si−nsi−si−1−1si−si−1−nsj−sj−1−2sj−nsj−sj−1−1sj−sj−1−nPs1s2s3−pi.pj;

Variance of modified MLE of p_i when k = 3 is

(43)∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞(n2(n+1)(si−si−1))2P(s1s2s3) −(∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞(n2(n+1)(si−si−1))2P(s1s2s3))2

where s0=0 ; i=1,2,3

Covariance of modified MLE of p_i and p_j, i, j = 1, 2, 3 and s0=0 is

∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞(n2(n+1)(si−si−1))(n2(n+1)(sj−sj−1))P(s1s2s3) −(∑s1=n∞∑s2=s1+n∞∑s3=s2+n∞(n2(n+1)(si−si−1))(n2(n+1)(sj−sj−1))P(s1s2s3))2

The determinant of the variance covariance matrix is calculated and are compared for the two estimators in the graphs below for a range of values of the parameters p₁, p₂ and p₃.

It can be observed from the graphs in Figs. 1 and 2 that the generalised variance of the modified MLE is less than the corresponding GV of the UMVUE for numerical values of p₁, p₂ and p₃ ranging from 0.1 to 0.9. Thus we obtain a new and better estimator called modified MLE which is consistent. It is an improvement over the MLE and is also more efficient than the UMVUE.

7. AN EXAMPLE FOR k = 3

A game of cricket has been considered. When a batsman is out he is replaced by another batsman in the next ball and the game continues. When the replaced batsman is declared out another replacement is sent forth and the game continues. Consider the 2016 season of Cricket's Indian Premium League ‘IPL 2016’. A total of 17 matches were played by the winning team. Sunrisers Hyderabad, of which 15 were suitable for our study. We have recorded the following details.

Let

X_i denote the ball at which the first player becomes out in the ith match.

Y_i denote the ball at which the player who replaces the first batsman becomes out in the ith match.

Z_i denote the ball at which the player who replaces the second batsman becomes out in the ith match, i = 1, 2, …, 15.

Match No i	X_i	Y_i	Z_i
1	25	29	35
2	7	16	27
3	4	26	38
4	31	32	52
5	4	18	20
6	11	49	52
7	17	26	42
8	33	38	61
9	14	51	56
10	30	54	56
11	8	9	20
12	16	41	50
13	10	31	61
14	4	10	23
15	25	30	53

Table 1

IPL 2016: Team Hyderabad Sunrisers

Hence p1=PFirst player is out, p2=PSecond player is out and p3=PThird player is out.

Thus for n = 15, we obtain s₁ = 239, s₂ = 460 and s₃ = 646. The MLE, UMVUE and modified MLE for the following parameters are as shown in Table 2.

	P₁	P₂	P₃
UMVUE	0.05882	0.063636	0.075676
MLE	0.06276	0.06787	0.0806
Modified MLE	0.05884	0.063631	0.075605

Table 2

Values of MLE, UMVUE and modified MLE for parameters

From the example of IPL2016, it is observed that the UMVUE, MLE and modified MLE estimates calculated for p₁, p₂ and p₃ are very close to each other.

ACKNOWLEDGEMENT

We are thankful to the referees and the editor for their valuable comments and constructive suggestions which has improved this manuscript.

REFERENCES

1.J. Li, Modeling with Bivariate Geometric Distribution, The State University of New Jersey-Newark. Department of Mathematical Sciences(NJIT). Department of Mathematics and Computer Science, Rutgers, 2010. A dissertation submitted to the Faculty of New Jersey Institute of Technology(NJIT) and Rutgers

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 17 - 4
Pages: 636 - 646
Publication Date: 2018/12/31
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.2018.17.4.6 How to use a DOI?
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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TY  - JOUR
AU  - U. J. Dixit
AU  - S. Annapurna
PY  - 2018
DA  - 2018/12/31
TI  - Efficient Estimator of Parameters of a Multivariate Geometric Distribution
JO  - Journal of Statistical Theory and Applications
SP  - 636
EP  - 646
VL  - 17
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.2018.17.4.6
DO  - 10.2991/jsta.2018.17.4.6
ID  - Dixit2018
ER  -

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Match No i	X_i	Y_i	Z_i
1	25	29	35
2	7	16	27
3	4	26	38
4	31	32	52
5	4	18	20
6	11	49	52
7	17	26	42
8	33	38	61
9	14	51	56
10	30	54	56
11	8	9	20
12	16	41	50
13	10	31	61
14	4	10	23
15	25	30	53

Match No i	X_i	Y_i	Z_i
1	25	29	35
2	7	16	27
3	4	26	38
4	31	32	52
5	4	18	20
6	11	49	52
7	17	26	42
8	33	38	61
9	14	51	56
10	30	54	56
11	8	9	20
12	16	41	50
13	10	31	61
14	4	10	23
15	25	30	53

Match No i	X_i	Y_i	Z_i
1	25	29	35
2	7	16	27
3	4	26	38
4	31	32	52
5	4	18	20
6	11	49	52
7	17	26	42
8	33	38	61
9	14	51	56
10	30	54	56
11	8	9	20
12	16	41	50
13	10	31	61
14	4	10	23
15	25	30	53