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A new method for hypotheses testing using spacings
ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1345–1347 www.elsevier.com/locate/stapro
A new method for hypotheses testing using spacings Hamzeh Torabi Department of Statistics, Yadz University, Yadz, Iran Received 26 July 2005; received in revised form 16 January 2006 Available online 9 March 2006
Abstract A method for hypotheses testing which makes use of the spacings is proposed. Finally, we give an example. r 2006 Elsevier B.V. All rights reserved. MSC: 62F03; 62F10 Keywords: Spacing; Hypotheses testing; Maximum likelihood estimator; UMVUE; Likelihood ratio test
1. Introduction The maximum spacings estimator (MSPE) is introduced by Cheng and Amin (1979, 1983) and independently is discussed by Ranneby (1984). Cheng and Amin (1983) show that in such situations as a three-parameter Gamma, Lognormal or Weibull distribution where the ML method breaks down due to unboundedness of the likelihood, the MSPE methods produces consistent and asymptotically efficient estimators. See Ghosh and Jammalamadaka (2001) for more details. In this paper, we propose a method for hypotheses testing about y based on the spacings, i.e., the gaps between successive order statistics. In Section 2, we review the definition of the maximum spacings estimator and give some examples. The spacings ratio test for hypotheses testing is given in Section 3. 2. Maximum spacings estimator Let X 1 ; . . . ; X n1 be a random sample from continuous distribution function F y , y 2 Y with support on R. Here, the unknown parameter y may be a vector. Let the order statistics be denoted by Y 1 ; . . . ; Y n1 . Define Di ðyÞ ¼ F y ðY i Þ F y ðY i1 Þ;
i ¼ 1; . . . ; n,
where F y ðY 0 Þ ¼ 0 and F y ðY n Þ ¼ 1. The maximum spacing estimator (MSPE) is discussed by Cheng and Amin (1979, 1983) and Ranneby (1984) corresponds to estimating y by maximizing the product n Y ½F y ðY i Þ F y ðY i1 Þ. i¼1
E-mail address: [email protected]. 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.02.003
ARTICLE IN PRESS H. Torabi / Statistics & Probability Letters 76 (2006) 1345–1347
1346
As the following three examples given in Ghosh and Jammalamadaka (2001) illustrate, MSPE gives superior estimators in many cases. Example 2.1. Suppose that X 1 ; . . . ; X n1 be a random sample from Uðy 12; y þ 12Þ, y 2 R. It can be shown that the MLE for y is a point in the interval ðY n1 12; Y 1 þ 12Þ, whereas the MSPE for y is the midrange, y^ ¼ ðY 1 þ Y n1 Þ=2, which is also the UMVUE for y. Example 2.2. Let X 1 ; . . . ; X n1 be a random sample from the double exponential distribution given by the following density: f y ðxÞ ¼ 12 expfjx yjg;
x 2 R;
y 2 R.
For odd sample size (say, n 1 ¼ 2k þ 1), the MSPE gives the same estimator as the MLE: Y kþ1 . For n 1 ¼ 2k, the MLE is not unique; it is any value in the interval ðY k ; Y kþ1 Þ whereas in this case the MSPE is the midrange, i.e., ðY 1 þ Y n1 Þ=2 which again corresponds to the UMVUE. Example 2.3. Suppose that X 1 ; . . . ; X n1 is a random sample from Uð0; yÞ, y40. It can be shown that the MLE for y is a point in the interval Y n1 , whereas the MSPE for y is ðnÞ=ðn 1ÞY n1 , which is again the UMVUE for y. 3. Spacings ratio test Now, consider the following testing problem: ( H0 : y 2 Y0 ; H1 : y 2 Y1 ; where Y0 \ Y1 ¼ ; and Y0 [ Y1 ¼ Y. The spacing ratio test (SRT) reject H0 against H1 if Q supy2Y0 ni¼1 Di ðyÞ Q ok, lðxÞ ¼ supy2Y ni¼1 Di ðyÞ for intuitively is justifiable because if H1 is true, the Qn some 0pkp1, where x ¼ ðx1 ; . . . ; xn1 Þ. This test Q n D ðyÞ is maximised for some y in Y , hence sup i 1 y2Y0 i¼1 i¼1 Di ðyÞ and therefore lðxÞ will be small and so on. Example 3.1. Remember Example 2.3. Suppose that we want to test ( H 0 : y ¼ y0 ; H1 :
yay0 :
The SRT reject H0 against H1 if Qn Qn Di ðy0 Þ Di ðy0 Þ i¼1Q ¼ Qi¼1 ok, lðxÞ ¼ n n ^ supy40 i¼1 Di ðyÞ i¼1 Di ðyÞ where y^ ¼ ðn=ðn 1ÞÞyn1 . But D1 ðyÞ ¼ F y ðy1 Þ 0 ¼
y1 , y
Di ðyÞ ¼ F y ðyi Þ F y ðyi1 Þ ¼
yi yi1 ; y
Dn ðyÞ ¼ 1 F y ðyn1 Þ ¼ 1
yn1 . y
i ¼ 2; . . . ; n 1,
ARTICLE IN PRESS H. Torabi / Statistics & Probability Letters 76 (2006) 1345–1347
1347
Hence hQ i ðy1 =y0 Þ n1 ðy y Þ=y ð1 yn1 =y0 Þ 0 i1 i¼2 i hQ i lðxÞ ¼ n1 ^ ^ ^ ðy1 =yÞ i¼2 ðyi yi1 Þ=y ð1 yn1 =yÞ ¼
nn ðyn1 =y0 Þn1 ð1 yn1 =y0 Þ. ðn 1Þn1
Let t ¼ ðyn1 Þ=y0 , 0oto1 and gðtÞ ¼ ðn 1Þ ln t þ ln ð1 tÞ. It is clear that lðxÞok is equivalent to gðtÞok0 , k0 40. But g0 ðtÞ ¼ ððn 1Þ=tÞ 1=ð1 tÞ ¼ ððn 1Þ ntÞ=tð1 tÞ. Hence g0 ððn 1Þ=nÞ ¼ 0, g0 ðtÞ40 for toðn 1Þ=n and g0 ðtÞo0 for t4ðn 1Þ=n. Therefore, gðtÞok0 is equivalent to tok1 or t4k2 , 0ok1 ok2 o1. Thus, the SRT gives the same critical region as the likelihood ratio test, i.e., C ¼ fxjyn1 oc1
or yn1 4c2 ; c1 oc2 g,
see Shao (2003). Acknowledgments Author is grateful to the referees of the journal for their suggestions and for their help in writing the paper in acceptable form. References Cheng, R.C.H., Amin, N.A.K., 1979. Maximum product of spacings estimation with application to the lognormal distribution. Mathematical Report 79-1, Department of Mathematics, UWIST, Cardiff, 1979. Cheng, R.C.H., Amin, N.A.K., 1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403. Ghosh, K., Jammalamadaka, S.R., 2001. A general estimation method using spacings. J. Statist. Plann. Inference 93, 71–82. Ranneby, B., 1984. The maximum spacing method: an estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112. Shao, J., 2003. Mathematical Statistics, second ed. Springer, New York.
Further reading Casella, G., Berger, R.L., 2002. Statistical Inference, second ed. Duxbury Press, North Scituate, MA. Lehmann, E.L., 1994. Testing Statistical Hypotheses. Chapman & Hall, New York. Lehmann, E.L., Casella, G., 1998. Theory of Point Estimation. Springer, Berlin.
Statistics & Probability Letters 76 (2006) 1345–1347 www.elsevier.com/locate/stapro
A new method for hypotheses testing using spacings Hamzeh Torabi Department of Statistics, Yadz University, Yadz, Iran Received 26 July 2005; received in revised form 16 January 2006 Available online 9 March 2006
Abstract A method for hypotheses testing which makes use of the spacings is proposed. Finally, we give an example. r 2006 Elsevier B.V. All rights reserved. MSC: 62F03; 62F10 Keywords: Spacing; Hypotheses testing; Maximum likelihood estimator; UMVUE; Likelihood ratio test
1. Introduction The maximum spacings estimator (MSPE) is introduced by Cheng and Amin (1979, 1983) and independently is discussed by Ranneby (1984). Cheng and Amin (1983) show that in such situations as a three-parameter Gamma, Lognormal or Weibull distribution where the ML method breaks down due to unboundedness of the likelihood, the MSPE methods produces consistent and asymptotically efficient estimators. See Ghosh and Jammalamadaka (2001) for more details. In this paper, we propose a method for hypotheses testing about y based on the spacings, i.e., the gaps between successive order statistics. In Section 2, we review the definition of the maximum spacings estimator and give some examples. The spacings ratio test for hypotheses testing is given in Section 3. 2. Maximum spacings estimator Let X 1 ; . . . ; X n1 be a random sample from continuous distribution function F y , y 2 Y with support on R. Here, the unknown parameter y may be a vector. Let the order statistics be denoted by Y 1 ; . . . ; Y n1 . Define Di ðyÞ ¼ F y ðY i Þ F y ðY i1 Þ;
i ¼ 1; . . . ; n,
where F y ðY 0 Þ ¼ 0 and F y ðY n Þ ¼ 1. The maximum spacing estimator (MSPE) is discussed by Cheng and Amin (1979, 1983) and Ranneby (1984) corresponds to estimating y by maximizing the product n Y ½F y ðY i Þ F y ðY i1 Þ. i¼1
E-mail address: [email protected]. 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.02.003
ARTICLE IN PRESS H. Torabi / Statistics & Probability Letters 76 (2006) 1345–1347
1346
As the following three examples given in Ghosh and Jammalamadaka (2001) illustrate, MSPE gives superior estimators in many cases. Example 2.1. Suppose that X 1 ; . . . ; X n1 be a random sample from Uðy 12; y þ 12Þ, y 2 R. It can be shown that the MLE for y is a point in the interval ðY n1 12; Y 1 þ 12Þ, whereas the MSPE for y is the midrange, y^ ¼ ðY 1 þ Y n1 Þ=2, which is also the UMVUE for y. Example 2.2. Let X 1 ; . . . ; X n1 be a random sample from the double exponential distribution given by the following density: f y ðxÞ ¼ 12 expfjx yjg;
x 2 R;
y 2 R.
For odd sample size (say, n 1 ¼ 2k þ 1), the MSPE gives the same estimator as the MLE: Y kþ1 . For n 1 ¼ 2k, the MLE is not unique; it is any value in the interval ðY k ; Y kþ1 Þ whereas in this case the MSPE is the midrange, i.e., ðY 1 þ Y n1 Þ=2 which again corresponds to the UMVUE. Example 2.3. Suppose that X 1 ; . . . ; X n1 is a random sample from Uð0; yÞ, y40. It can be shown that the MLE for y is a point in the interval Y n1 , whereas the MSPE for y is ðnÞ=ðn 1ÞY n1 , which is again the UMVUE for y. 3. Spacings ratio test Now, consider the following testing problem: ( H0 : y 2 Y0 ; H1 : y 2 Y1 ; where Y0 \ Y1 ¼ ; and Y0 [ Y1 ¼ Y. The spacing ratio test (SRT) reject H0 against H1 if Q supy2Y0 ni¼1 Di ðyÞ Q ok, lðxÞ ¼ supy2Y ni¼1 Di ðyÞ for intuitively is justifiable because if H1 is true, the Qn some 0pkp1, where x ¼ ðx1 ; . . . ; xn1 Þ. This test Q n D ðyÞ is maximised for some y in Y , hence sup i 1 y2Y0 i¼1 i¼1 Di ðyÞ and therefore lðxÞ will be small and so on. Example 3.1. Remember Example 2.3. Suppose that we want to test ( H 0 : y ¼ y0 ; H1 :
yay0 :
The SRT reject H0 against H1 if Qn Qn Di ðy0 Þ Di ðy0 Þ i¼1Q ¼ Qi¼1 ok, lðxÞ ¼ n n ^ supy40 i¼1 Di ðyÞ i¼1 Di ðyÞ where y^ ¼ ðn=ðn 1ÞÞyn1 . But D1 ðyÞ ¼ F y ðy1 Þ 0 ¼
y1 , y
Di ðyÞ ¼ F y ðyi Þ F y ðyi1 Þ ¼
yi yi1 ; y
Dn ðyÞ ¼ 1 F y ðyn1 Þ ¼ 1
yn1 . y
i ¼ 2; . . . ; n 1,
ARTICLE IN PRESS H. Torabi / Statistics & Probability Letters 76 (2006) 1345–1347
1347
Hence hQ i ðy1 =y0 Þ n1 ðy y Þ=y ð1 yn1 =y0 Þ 0 i1 i¼2 i hQ i lðxÞ ¼ n1 ^ ^ ^ ðy1 =yÞ i¼2 ðyi yi1 Þ=y ð1 yn1 =yÞ ¼
nn ðyn1 =y0 Þn1 ð1 yn1 =y0 Þ. ðn 1Þn1
Let t ¼ ðyn1 Þ=y0 , 0oto1 and gðtÞ ¼ ðn 1Þ ln t þ ln ð1 tÞ. It is clear that lðxÞok is equivalent to gðtÞok0 , k0 40. But g0 ðtÞ ¼ ððn 1Þ=tÞ 1=ð1 tÞ ¼ ððn 1Þ ntÞ=tð1 tÞ. Hence g0 ððn 1Þ=nÞ ¼ 0, g0 ðtÞ40 for toðn 1Þ=n and g0 ðtÞo0 for t4ðn 1Þ=n. Therefore, gðtÞok0 is equivalent to tok1 or t4k2 , 0ok1 ok2 o1. Thus, the SRT gives the same critical region as the likelihood ratio test, i.e., C ¼ fxjyn1 oc1
or yn1 4c2 ; c1 oc2 g,
see Shao (2003). Acknowledgments Author is grateful to the referees of the journal for their suggestions and for their help in writing the paper in acceptable form. References Cheng, R.C.H., Amin, N.A.K., 1979. Maximum product of spacings estimation with application to the lognormal distribution. Mathematical Report 79-1, Department of Mathematics, UWIST, Cardiff, 1979. Cheng, R.C.H., Amin, N.A.K., 1983. Estimating parameters in continuous univariate distributions with a shifted origin. J. Roy. Statist. Soc. Ser. B 45, 394–403. Ghosh, K., Jammalamadaka, S.R., 2001. A general estimation method using spacings. J. Statist. Plann. Inference 93, 71–82. Ranneby, B., 1984. The maximum spacing method: an estimation method related to the maximum likelihood method. Scand. J. Statist. 11, 93–112. Shao, J., 2003. Mathematical Statistics, second ed. Springer, New York.
Further reading Casella, G., Berger, R.L., 2002. Statistical Inference, second ed. Duxbury Press, North Scituate, MA. Lehmann, E.L., 1994. Testing Statistical Hypotheses. Chapman & Hall, New York. Lehmann, E.L., Casella, G., 1998. Theory of Point Estimation. Springer, Berlin.