Distribution of N = (N0, N1)
As the estimators defined in (4) are expressed as functions of N0 and N1 we first study their distribution. Using a Gaussian approximation, we have
ℒ ( [ N 0     N 1       ]   ︸  N   )  ≃ N ( [ E 0     E 1       ]   ︸  E  , [ C 0 , 0     C 0 , 1      C 1 , 0     C 1 , 1        ]   ︸  C   )        ( 5 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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aaGaayjkaiaawMcaaiaaxMaacaWLjaGaeiikaGIaeGynauJaeiykaKcaaa@6A7A@
where, for i, j ∈ {0, 1}, Ei ∈ ℝdi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIDesOdaahaaWcbeqaaiabdsgaKnaaBaaameaacqWGPbqAaeqaaaaaaaa@3122@, and Ci,j ∈ ℝdi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIDesOdaahaaWcbeqaaiabdsgaKnaaBaaameaacqWGPbqAaeqaaaaaaaa@3122@ × ℝdj MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIDesOdaahaaWcbeqaaiabdsgaKnaaBaaameaacqWGQbGAaeqaaaaaaaa@3124@ with di = km+i. One can note that C0,0 and C1,1 are symmetric, and t (C1,0) = C0,1 (where t is the matrix transpose operator).
In the stationary case, exact expression of E and C can be computed according to [5].
Expectation is simply given ∀w ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@m by
E0(w) = (n - m + 1) μ(w)     E1 (wa) = (n - m) μ(w)Π(w, a)     ∀(w, a) ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@m × A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@     (6)
In order to give more fluidity to this paper, the expression of the covariance matrix C have been moved in appendix A. Let us remark, before going forward that substituting N by E in (4) immediately gives
μ E  = μ   and   π E  = ( 1 − 1 n − m + 1    )  π       ( 7 )        MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeqadaaabaacciGae8hVd02aaSbaaSqaaGqabiab+veafbqabaGccqGH9aqpcqWF8oqBaeaacqqGHbqycqqGUbGBcqqGKbazaeaacqWFapaCdaWgaaWcbaGae4xraueabeaakiabg2da9maabmaabaGaeGymaeJaeyOeI0YaaSaaaeaacqaIXaqmaeaacqWGUbGBcqGHsislcqWGTbqBcqGHRaWkcqaIXaqmaaaacaGLOaGaayzkaaGae8hWdaNaaCzcaiaaxMaadaqadaqaaiabiEda3aGaayjkaiaawMcaaaaaaaa@49E8@