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Tests for a Change Point in the Shape Parameter ofGamma Random VariablesAsoka Ramanayake a ba Department of Mathematics and Statistics , University of Wisconsin Oshkosh , Oshkosh,Wisconsinb Department of Mathematics and Statistics , University of Wisconsin Oshkosh , Oshkosh,WI, 54901Published online: 15 Feb 2007.

To cite this article: Asoka Ramanayake (2005) Tests for a Change Point in the Shape Parameter of Gamma RandomVariables, Communications in Statistics - Theory and Methods, 33:4, 821-833, DOI: 10.1081/STA-120028728

To link to this article: http://dx.doi.org/10.1081/STA-120028728

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Tests for a Change Point in the Shape Parameter ofGamma Random Variables

Asoka Ramanayake*

Department of Mathematics and Statistics, University of WisconsinOshkosh, Oshkosh, Wisconsin

ABSTRACT

Tests for detecting a change in the shape parameter of a sequence ofgamma random variables are introduced. The asymptotic propertiesof the tests under the null and the alternate hypothesis are studied.A Monte Carlo study is used to compare the critical values formoderate to large sample sizes. The tests are applied to two data sets,one on time intervals between coal mine explosions and the other oninter arrival times of aircrafts, to detect possible changes in theshape parameter.

Key Words: Change-point; Likelihood; Gamma distribution; Shapeparameter.

*Correspondence: Asoka Ramanayake, Department of Mathematics andStatistics, University of Wisconsin Oshkosh, Oshkosh, WI 54901; E-mail:ramanaya@uwosh.edu.

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DOI: 10.1081/STA-120028728 0361-0926 (Print); 1532-415X (Online)

Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com

COMMUNICATIONS IN STATISTICS

Theory and Methods

Vol. 33, No. 4, pp. 821–833, 2004

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1. INTRODUCTION

The Gamma distribution occurs frequently in a variety of applica-tions, especially in reliability, survival analysis and when modelingincome distributions. The family of distributions defined by the gammadistribution is given by, F ¼ f f ðx; z; yÞ : z > 0; y > 0g where

f ðx; z; yÞ ¼ 1

GðzÞyzxz�1 expf�x=yg; x > 0: ð1Þ

Here z is known as the shape parameter and y is known as the scaleparameter. The shape parameter z is especially of interest in reliabilitybecause the gamma distribution is DFR (decreasing failure rate),constant or IFR (increasing failure rate) according to whether z� 1 isnegative, zero or positive. The shape parameter also plays an importantrole in renewal theory, when modeling arrival times of events. Also noticethat when y ¼ 2, the gamma distribution is a chi-squared distributionwith parameter 2z. And chi-squared distribution has a wide range ofapplications.

Let X1;X2; . . . ;Xn be a sequence of independent gamma randomvariables as defined in (1). In this paper we will assume that the shapeparameter z is susceptible to a change at an unknown point in thesequence. The scale parameter y is not assumed to change. We will derivestatistics to test the hypotheses;

H0 : zi ¼ z0 i ¼ 1; . . . ; n

vs. the alternative:

HA : zi ¼ z0 i ¼ 1; . . . ; k

zi ¼ z0 þ d i ¼ k þ 1; . . . ; n; where k, and d unknown:

This is known in the literature as a change point problem. Changepoint problems have received considerable attention, mainly due towide variety of applications and recent developments of computationalmethods. Many authors have considered the case of the normaldistribution. Kander and Zacks (1966) proposed a test statistic fortesting a change in the one-parameter exponential family. Hsu (1979)considered change points for the scale parameter of gamma random vari-ables, assuming that the shape parameter was constant. Worsely (1986)discuss tests and confidence regions for the exponential distribution.

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Gupta and Ramanayake (2001) discuss linear trend change for theexponential distribution. Gombay and Hovarth (1990) discuss theasymptotic behavior of the likelihood ratio statistic. For related workin change point problems see Csorgo and Hovarth (1997).

2. A BAYES TYPE STATISTIC

Under HA, the likelihood function for fixed k ðk ¼ 1; . . . ; n� 1Þ canbe written as,

f ðx; z; d; y;kÞ

¼ exp �Xni¼1

xiyþXki¼1

½ðz� 1Þ lnxi � lnGðzÞ � z ln y�(

þXni¼kþ1

½ðzþ d� 1Þ lnxi � lnGðzþ dÞ � ðzþ dÞ ln y�):

Next define Zðxi; z; yÞ ¼ ðz� 1Þ ln xi � lnGðzÞ � z ln y, for i¼1; . . . ;n.Now as in Kander and Zacks, if we assume the change-point is equallylikely to fall within any point in the sequence of observations, then themarginal likelihood under HA is,

f ðx; z; d; yÞ ¼Xn�1

k¼1

ðn� 1Þ�1f ðx; z; d; y; kÞ

¼ ðn� 1Þ�1 exp �Xni¼1

xiy

( )

�Xn�1

k¼1

expXki¼1

Zðxi; z; yÞ þXni¼kþ1

Zðxi; zþ d; yÞ( )

:

But notice that in a close neighborhood of z, we can write,

Zðxi; zþ d; yÞ ¼ Zðxi; z; yÞ þ dZ0ðxi; z; yÞ þ oðdÞ¼ Zðxi; z; yÞ þ d½ln xi � ln y�CðzÞ� þ oðdÞ;

for i ¼ 1; . . . ; n:

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Where CðzÞ is defined to be the first derivative of lnGðzÞ. Thus themarginal likelihood under HA can be expressed as,

f ðx; z;d;yÞ ¼ ðn� 1Þ�1 exp �Xni¼1

xiyþ Zðxi; z;yÞ

( )

�Xn�1

k¼1

exp dXni¼kþ1

½lnxi � lny�CðzÞþ oðdÞ�( )

¼ ðn� 1Þ�1 exp �Xni¼1

xiyþ Zðxi; z;yÞ

( )

�Xn�1

k¼1

1þ dXni¼kþ1

½lnxi � lny�CðzÞþ oðdÞ�( )

; as d! 0:

Therefore, the likelihood ratio for testingH0 vs.HA can be expressed,as d ! 0, by

Lðx; z; d; yÞ ¼ 1þ dn� 1

Xn�1

k¼1

Xni¼kþ1

½ln xi � ln y�CðzÞ�

¼ 1þ dn� 1

Xn�1

i¼1

i ln xiþ1 � dn2fCðzÞ þ ln yg þ oðdÞ: ð2Þ

But CðzÞ þ ln y is unknown. Therefore we will use the maximumlikelihood estimator of CðzÞ þ ln y, under H0 which is 1

n

Pni¼1 ln xi.

Thus using this relation in (2) we have

Lðx; z; d; yÞ ¼ 1þ dn� 1

Xn�1

i¼1

i ln xiþ1 � d2

Xni¼1

ln xi þ oðdÞ

¼ 1þ d2ðn� 1Þ

Xni¼1

ð2i � n� 1Þ ln xi þ oðdÞ:

Hence for testing null hypothesis against the one-sided alternative wecan use the statistic,

T ¼Xni¼1

ð2i � n� 1Þ lnXi:

In the case of the two sided alternative we suggest the statisticT� ¼ jPn

i¼1½ð2i � n� 1Þ lnXij.

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2.1. Moments of Statistic T Under H0

Define Yi ¼ lnðXi=yÞ, for i ¼ 1; . . . ; n. Next notice that T ¼Pni¼1ð2i � n� 1ÞðYi þ ln yÞ ¼ Pn

i¼1ð2i � n� 1ÞYi and hence the distribu-tion of Yi is independent of y. Thus the distribution of T will also befree of y. Also notice that Xi=y are iid gamma random variables withparameters ðz; 1Þ. Thus Yi, for i ¼ 1; . . . ; n are lngammaðzÞ, with pdf

f ðyÞ ¼ 1

GðzÞ expfzy� eyg; �1 < y < 1: ð3Þ

It can be shown that the cumulant generating function of Yi iskrðYiÞ ¼ Cðr�1ÞðzÞ, where CðrÞðzÞ is the rth derivative of CðzÞ. Hencethe first four central moments of Yi can be calculated as, mðYiÞ ¼ CðzÞ;m2ðYiÞ ¼Cð1ÞðzÞ; m3ðYiÞ ¼Cð2ÞðzÞ and m4ðYiÞ ¼Cð3ÞðzÞþ3½Cð1ÞðzÞ�2. Nowthe moments of T can be calculated as,

m¼EðTÞ ¼ 0; s2 ¼VarðTÞ ¼ 1

3nðn2� 1ÞCð1ÞðzÞ;

m3 ¼EðT3Þ ¼ 0; m4 ¼EðT4Þ ¼ 1

9nðn2� 1Þ

�3

5ð3n2� 7ÞCð3ÞðzÞ

þ 3nðn2� 1Þ½Cð1ÞðzÞ�2�

b1 ¼m23

ðs2Þ3 ¼ 0; b2 ¼m4

ðs2Þ2 ¼ 3þ 3

5

ð3n2� 7ÞCð3ÞðzÞnðn2� 1ÞCð1ÞðzÞ2 :

Notice that the distribution of T is centered around 0 with positivekurtosis and 0 symmetry. Moreover the kurtosis tends to zero asn ! 1. Let T1 be the standardized statistic of T . That is

T1 ¼ Tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðn2 � 1ÞCð1ÞðzÞ=3

q :

It can be shown from the Lyapunov’s central limit theorem that T1

converges to a standard normal distribution as n ! 1. Also note that

ðYi �CðzÞÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCð1ÞðzÞ

qconverges in distribution to a standard normal

distribution when z ! 1. See Lawless (1982) for details. Thus T1 con-verges to a standard normal distribution as z ! 1 as well. Since themoments of T1 are known, we can use an Edgeworth expansion to obtainapproximate critical values for T1 under H0 for moderate sample sizes.

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In the case where z is unknown, replace Cð1ÞðzÞ, by the variance of Yis.If y is unknown replace it by the mle of y under H0, which is �XX . Thenby Slutsky’s theorem we can show that this statistic too will convergeto a standard normal distribution as n ! 1. It can also be shown thatthe statistic T� converges in distribution to

R 10 B

2ðtÞdt, where BðtÞ denotesa Brownian bridge process.

Table 1 gives the critical values of the distribution T1 for someselected values of z and n. The simulated values are based on MonteCarlo simulations based on 5000 replications. It is clear that the criticalvalues are close to the standard normal critical values for large n andlarge z. The edgeworth expansion seems to give a better approximationthan the standard normal critical values for small sample sizes andsmall z.

Table 1. Critical values of the statistic T1 for selected values of z and n.

Simulated Edgeworth

n z 90 95 99 90 95 99

15 3 1.2803 1.6432 2.3881 1.2750 1.6430 2.349015 5 1.2997 1.6972 2.4706 1.2780 1.6440 2.339015 7 1.3338 1.6960 2.4178 1.2790 1.6450 2.335015 9 1.3027 1.6906 2.4163 1.2800 1.6450 2.330015 11 1.3169 1.6998 2.4197 1.2800 1.6450 2.330015 13 1.3250 1.6865 2.4194 1.2810 1.6450 2.331015 15 1.2871 1.6988 2.3561 1.2810 1.6450 2.3310

25 3 1.2693 1.6323 2.2760 1.2780 1.6440 2.340025 5 1.2784 1.7728 2.5577 1.2800 1.6450 2.334025 7 1.2573 1.5702 2.4234 1.2800 1.6450 2.330025 9 1.2678 1.6744 2.3988 1.2800 1.6450 2.330025 11 1.2158 1.5412 2.1899 1.2800 1.6450 2.330025 13 1.3564 1.7259 2.4721 1.2800 1.6450 2.330025 15 1.2869 1.7062 2.3397 1.2800 1.6450 2.3300

50 3 1.2777 1.7043 2.2635 1.2800 1.6450 2.330050 5 1.2526 1.5985 2.5045 1.2810 1.6450 2.331050 7 1.3043 1.6140 2.3083 1.2800 1.6450 2.330050 9 1.2552 1.5808 2.5393 1.2800 1.6450 2.330050 11 1.3124 1.6603 2.2746 1.2800 1.6450 2.330050 13 1.3163 1.6701 2.1425 1.2800 1.6450 2.330050 15 1.2008 1.6430 2.2599 1.2800 1.6450 2.3300

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2.2. Power Function for T1

Assume HA is true. That is the first k observations X1; . . . ;Xk followa gamma distribution with parameters z and y and the next (n–k)observations Xkþ1; . . . ;Xn follow a gamma distribution with parameterszþ d and y. Then the moments of T1 can be calculated as,

mA ¼ EðT1Þ ¼ffiffiffi3

pkðn� kÞg0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

nCð1ÞðzÞðn2 � 1Þq

sA2 ¼ VarðT1Þ ¼ ð�nþ n3 þ 6 nk2 � 3 n2k� 4k3 þ kÞg1nCð1ÞðzÞðn2 � 1Þ þ 1

ðnþ 1Þ ;

ð4Þwhere gi ¼ CðiÞðzþ dÞ �CðiÞðzÞ.

Now if we assume that kn ! l as n ! 1, then we have that,

ffiffiffin

pmA!

ffiffiffi3

plð1�lÞg0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCð1ÞðzÞ

q

s2A!ð�4l3þ6l2�3lþ1Þg1

Cð1ÞðzÞ þ1

ffiffiffin

pm3A!

3ffiffiffi3

plð1�lÞð2l2�2lþ1Þg2

½Cð1ÞðzÞ�3=2

nb1A!�27ð1�lÞ2ð2l2�2lþ1Þ2l2g22

ð�g1�Cð1ÞðzÞþ3g1l�6g1l2þ4g1l

3Þ3

nb2A!�9ð40l3g3�Cð3ÞðzÞ�g3þ5lg3�40l4g3þ16l5g3�20l2g3Þ

5ð�g1�Cð1ÞðzÞþ3g1l�6g1l2þ4g1l

3Þ2

as n ! 1. Here mA; s2A and m3A are the first three central moments and

b1A; b2A are the skewness and the kurtosis of T1 under HA. Notice thatfor fixed z; mA is maximized when l ¼ 1=2 . The variance s2A is a decreas-ing function of l. The maximum of s2A ¼ Cð1Þðzþ dÞ=Cð1ÞðzÞ, is achievedwhen l ¼ 0 and the minimum of s2A ¼ 1, is achieved when l ¼ 1. It can beshown by the Lyapunov’s CLT that ð ffiffiffi

np ðT1 � mAÞÞ=sA converges in dis-

tribution to a standard normal distribution as n ! 1. Next if we definethe power of the test statistic T1 to be b,

b ¼ P(reject H0/HA true) ¼ 1� F�za � mA

ffiffiffin

psA

�;

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where za is defined by a ¼ R1za

fðxÞdx. By using (4) in the aboveexpression the powers of statistic T1 can be calculated.

The power curves of statistic T1 for certain values of z; l; n andd are displayed in Fig. 1. The first graph of Fig. 1 shows the powercurves for statistic T1 for values of zð¼1; . . . ; 4Þ; l ¼ 0; . . . ; 1 with nfixed at 50 and d fixed at 1. It can be seen from this graph that thepower is maximized when l is 0.5 and the power is higher when z issmaller. The second graph shows the power curves for statistic T1 forvalues of dð¼0:5; 1; 2; 3Þ; n ¼ 1; . . . ; 200 with z fixed at 1 and l fixedat 0.5. It can be seen from this graph that the power is an increasingfunction of both d and n. The third graph shows the power curvesfor statistic T1 for values of dð¼0:5; 1; 2; 3Þ; n ¼ 1; . . . ; 200 with z fixedat 3 and l fixed at 0.5. This figure also shows that the power is anincreasing function of both z and n, but notice that the overall powersare lower than the corresponding powers in the second graph. Thefourth graph shows the power curves for statistic T1 for values ofzð¼1; 2; 3; 4Þ; n ¼ 1; . . . ; 200 with d fixed at 1 and l fixed at 0.5. Itcan be seen from this graph that the power is an increasing functionof n and a decreasing function in z.

It is clear from these graphs in Fig. 1 that the power of statisticincreases when d and n increases. The power is higher when the changeoccurs in the middle of the sequence. The power is higher when z issmaller for fixed d. This is not unexpected because small changesin d will be more distinguishable when z is small as opposed to large z.

3. THE LIKELIHOOD RATIO STATISTIC

Next we will derive the likelihood ratio statistic. Assume thatthe observations X1; . . . ;Xn are independent gamma random variableswith y known. Define Lk to be the likelihood ratio for testing H0 vs.

HA for fixed k. Set Yk ¼ 1k

Pki¼1 lnðXi=yÞ, Yk� ¼ 1

ðn�kÞPn

i¼kþ1 lnðXi=yÞand Yn ¼ 1

n

Pni¼1 lnðXi=yÞ. Then

�2 lnLk ¼ k½z1z1Yk � lnGðz1z1Þ� þ ðn� kÞ½z2z2Y �k � lnGðz2z2Þ�

� n½z0z0Yk � lnGðz0z0Þ�

where z0z0 is the maximum likelihood estimator (mle) of z under H0; z1z1 andz2z2 are the mles of z and zþ d under HA. But the location of change k is

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Figure 1. The power curves of the statistic T1.

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unknown. Thus the maximum likelihood ratio test statistic for testing H0

vs. HA can be written by,

T2 ¼ max1�k<n

�2 lnLk: ð5Þ

Gombay and Hovarth (1990) discuss the asymptotic behavior of T2 andshow that under H0

limn!1P

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ln lnn

pT

1=22 � tþDðlnnÞ

n o¼ exp ð�2 expf�tgÞ; for all t;

ð6Þ

where DðxÞ ¼ 2 ln xþ 12 ln ln x� ln

ffiffiffip

p. Csorgo and Hovarth note that

the convergence in (6) gives a conservative rejection region and they pointout that very large sample sizes will be needed if we want to use this limittheorem. Thus for small to moderate sample sizes they suggest using thefollowing convergence of T2 under H0.

����T1=22 � sup

1=n�t�1�1=n

BnðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitð1� tÞp ���� ¼ Opðexpð� ln nÞ1�EÞ;

for all 0 < E < 1;

where fBnðtÞ; 0 � t � 1g is a sequence of a stochastic process thatconverges in distribution to fBðtÞ; 0 � t < 1g for each n and BðtÞ isdefined to be a Brownian bridge process.

3.1. Asymptotic Distribution of T2 Under HA

Define HðxÞ ¼ �C�1½CðxÞ�CðxÞ þ ln GðC�1½CðxÞ�; t1 ¼ Cðz1Þand t2 ¼ Cðz2Þ. Then we have that Hðt1Þ ¼ �z1Cðz1Þ þ lnGðz1Þand Hðt2Þ ¼ z� z2Cðz2Þ þ lnGðz2Þ. Now if we assume that underHA; limn!1 k=n ¼ l, where 0 < l < 1, then it can be shown as in Csorgoand Hovarth (1997) that,

ðT2 � 2z�Þffiffiffiffiffiffiffiffiffiffi4ns2

p !D Nð0; 1Þ ð7Þ

where z� ¼kHðt1Þþðn�kÞHðt2Þ�nHðknt1þ n�kn t2Þ and s2 ¼ l fH 0ðt1Þ�

Hðlt1 þ ð1 � lÞðt2Þg2Cð1Þðz1Þ þ ð1 � lÞfH 0ðt2Þ � Hðlt1 þ ð1 � lÞðt2Þg2Cð1Þðz2Þ.

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4. ANALYSIS OF BRITISH COAL-MINING

DISASTERS

Maguire et al. (1952) examined the time interval between explosionsin coal-mines in Great Britain involving the loss of ten lives or more dur-ing the period from December 6, 1875 to May 29, 1951. They concludedthat the data were exponentially distributed with a constant mean overtime. Worsely (1986) tested the data for a change point in mean assumingthat the data was exponentially distributed. He used the likelihood ratiomethod, and concluded that the change-point may occur in the year 1890.The tests described in this paper were applied to the data to detect anincrease in shape parameter and T1 was 2.48. Since the sample size is109, the normal approximation was used to obtain the p-value, whichturned out to be 0.007.

The likelihood ratio Lk were calculated for values of k ¼ 2; . . . ; 108,the maximum of Lk was at 45, which turns out to be the year 1890 and isconsistent with Worsely’s result as shown in Fig. 2. A chi-square good-ness of fit tests were performed on the first 45 observations, and the last64 observation to check the model assumption and there was no signifi-cance evidence that the gamma model was invalid for the data. The mle’sof z1 and z2 were 0.629 and 1.036, respectively.

Figure 2. The likelihood ratios for coal-mining data as a function of changepoint k.

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5. ANALYSIS OF INTER ARRIVAL TIMES

Hsu (1979) examined aircraft inter arrival times collected from alow-altitude transitional control sector for the period from noon through8 PM on April 30, 1969. His analysis assumes that the 212 observationsare independent exponential random variables.

The tests described in this paper were applied to this dataset. Thevalue of T is 2199.75. The standardized statistic T1 is 1.039. Since thesample size is large here, the distribution of T1 will be a standard normalif H0 is true. Hence the p-value of the test is 0.85. Thus we conclude thatthere was no change in the shape parameter. This result is consistent withHsu’s results.

The likelihood ratio statistic T2 was computed for the data and it wasequal to 8.17. It is clear that a significant change in z has occurred at thispoint k ¼ 211. Figure 3 shows the frequency graph of the observationswhen max1�k<211ð�2 lnLkÞ is reached. The mles for y is 117.279 and zis 1.13739 for the data. The statistic T1 was unable to detect this changebecause the change occurs at the very end of the sequence and power ofT1 is small for such cases.

ACKNOWLEDGMENT

Research supported in part by a research grant from the FacultyDevelopment Board–University of Wisconsin Oshkosh.

Figure 3. The figure on the left gives the inter arrival times and the figure on theright gives the likelihood ratios for aircraft inter arrival data as a function ofchange point k.

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REFERENCES

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