We present a function ρ (F1, F2, t) which contains Matusita's affinity and expresses the affinity between moment generating functions. An interesting results is expressed through decomposition of this affinity ρ (F1, F2, t) when the functions considered are k-dimensional normal distributions. The same decomposition remains true for other families of distribution functions. Generalizations of these results are also presented

Dorival Campos, A.

UPCommons. Portal del coneixement obert de la UPC

QU¨ESTIIO´, vol. 23, 2, p. 225-237, 1999SOME RESULTS ENVOLVING THE CONCEPTS OFMOMENT GENERATING FUNCTION ANDAFFINITY BETWEEN DISTRIBUTION FUNCTIONS.EXTENSION FOR r k−DIMENSIONAL NORMALDISTRIBUTION FUNCTIONSA. DORIVAL CAMPOSUniversidade de Sa˜o PauloWe present a function ρ (F1, F2, t) which contains Matusita’s affinity andexpress the «affinity» between moment generating functions. An interestingresult is expressed through decomposition of this «affinity» ρ (F1, F2, t)when the functions considered are k−dimensional normal distributions.The same decomposition remains true for others families of distributionfunctions. Generalizations of these results are also presented.Keywords: Affinity, moment generating functions, distance, inner pro-duct, multivariate normal distributions, probability densityfunctions, absolutely convergent.AMS Classification (MSC 2000): primary 62E99, secondary 62H20.* Dpto. de Matema´tica Aplicada a` Biologia. Faculdade de Medicina de Riberira˜o Preto. Universidade de Sa˜oPaulo. Brasil.– Received June 1998.– Acepted March 1999.2251. INTRODUCTION AND PRELIMINARIESLet F1 and F2 be two distribution functions defined on R and let us denote by fi(x) theprobability density function of Fi with respect to a measure m in R, for i = 1, 2.We find in the literature several forms of defining distance between distributions of a sa-me class. Matusita (1955) by making use of the distance function denoted by d (F1, F2)and expressed byd (F1, F2) ={(f1/21(x) − f1/22(x), f1/21(x) − f1/22(x))}1/2,introduced in the statistical literature the concept of affinity between the distributionsF1 and F2 denoted by ρ2 (F1, F2) and defined byρ2 (F1, F2) =(f1/21(x), f1/22(x))which is related to d (F1, F2) through the expressiond2 (F1, F2) = 2 {1 − ρ2 (F1, F2)}where (f, g) denotes the inner product of f(x) and g(x) defined by:(f, g) =∫Rf(x) g(x) dmThe importance and usefulness of the notions of distance and affinity between distribu-tions, in statistics, were stressed in a series of papers by Matusita (1954, 1955, 1956,1961, 1964, 1967b, 1973), Matusita & Motoo (1955), Matusita & Akaike (1956), Khan& Ali (1971) and others.Concrete expressions for the affinity between two multivariate normal distributions we-re established by Matusita (1966). As a following step, Matusita (1967) extended thenotion of affinity to cover the case where there are r distributions involved and esta-blished concrete expressions when the r distributions are k−dimensional normal.Our work is characterized by the introduction of the concept of a function, denoted byP (t), functionally expressed through the moment generating functions relative to thedistributions considered and another expression denoted by ρ (F1, F2, t) that containsas a particular case the affinity between the distribution functions F1 and F2, or in otherwords, express the «affinity» between the moment generating functions relative to F1and F2. We also present a result that express the decomposition of ρ(F1, F2, t˜) in aproduct of two factors identified as the affinity and the moment generating functionwhen F1 and F2 are k−dimensional normal distributions. This result is extended tocover the case where there are r k−dimensional normal distributions.226In this same way, the concept of a more general function Dr(s1, . . . , sr; t˜j) is introdu-ced, and the results obtained through it contains those developed in this paper as thoseones established by Matusita (1966, 1967a) and Campos (1978).2. RESULTSDefinition 1. Let F1 and F2 be two distribution functions belonging to the same classand let f1(x) and f2(x) their respective probability density functions with respect toa measure m defined on R. Let us suppose that there is a scalar t, −h 6 t 6 h(h > 0) such that the integral below, defined through the inner product, is absolutelyconvergent.Now, we define:(2.1)P (t) =(exp {tx/2}{f1/21(x) − f1/22(x)}, exp {tx/2}{f1/21(x) − f1/22(x)})where (f, g) denotes the inner product of f(x) and g(x), defined by:(f, g) =∫Rf(x) g(x) dmFrom (2.1), we obtain:P (t) = M1(t) + M2(t) − 2 ρ (F1, F2, t)where:Mi(t) represent the moment generating function for the distribution Fi whose probabi-lity density function is fi(x), i = 1, 2;and(2.2) ρ (F1, F2, t) =(exp {tx/2} f1/21(x), exp {tx/2} f1/22(x))From (2.1) and (2.2) we verify that:i) P (0) = d2 (F1, F2),ii) F1 = F2 implies P (t) = 0 for all −h 6 t 6 hiii) ρ (F1, F2, 0) = ρ2 (F1, F2)iv) F1 = F2 = F implies ρ(F, t) = M(t)where:d2 (F1, F2) =(f1/21(x) − f1/22(x), f1/21(x) − f1/22(x))227and ρ2 (F1, F2) is the affinity between the distributions F1 and F2 as defined by Matu-sita (1966).Teorema 1. Let F1 and F2 be k−dimensional nonsingular normal distributions, whoseprobability density functions are given by:(2pi)−k/2|A˜|1/2 exp{−1/2(A˜−1(x˜− a˜), x˜− a˜)}and(2pi)−k/2|B˜|−1/2 exp{−1/2(B˜−1(x˜− b˜), x˜− b˜)},respectively, where:x˜is a k−dimensional (column vector);A˜and B˜are covariance matrices de degree K anda˜, b˜are k−dimensional mean (column) vectors.In these conditions, we have:ρ(F1, F2, t˜) = ρ2 (F1, F2) · MG(t˜)where:t˜is k−dimensional (column) vector andMG(t˜) is the moment generating function of a k−dimensional normal distribution withmean vector C˜−1(A˜−1a˜+ B˜−1 b˜) and covariance matrix 2C˜−1, given by(2.3) MG(t˜) = exp{(C˜−1(A˜−1 a˜+ B˜−1 b˜), t˜) + (C˜−1 t˜, t˜)}with C˜= A˜−1 + B˜−1.Proof: From (2.2), we have:(2.4) ρ(F1, F2, t˜) = (2pi)−k/2|A˜B˜|−1/4∫Rkexp {−1/4Q} d x1, . . . , d xkwhere:Q = (A˜−1(x˜− a˜), x˜− a˜) + (B˜−1(x˜− b˜), x˜− b˜) − 4(x˜, , t˜)228By working with this algebraic sum of inner products, we obtain:(2.5) Q = ((A˜−1 +B˜−1)x˜, x˜) −2(A˜−1a˜+B˜−1b˜+2 t˜, x˜)+(A˜−1a˜, a˜)+(B˜−1 b˜, b˜)If we define the transformation:y˜= C˜1/2 x˜with C˜= A˜−1 + B˜−1 and Jacobian equal to mod |C˜|−1/2, (2.5) may be written asfollows:Q = (y˜, y˜) − 2(C˜−1/2(A˜−1a˜+ B˜−1b˜+ 2t˜), y˜) + (A˜−1a˜, a˜) + (B˜−1b˜, b˜)That is,(2.6) Q = Q1−(C˜−1(A˜−1a˜+B˜−1b˜+2t˜), A˜−1a˜+B˜−1b˜+2t˜)+(A˜−1a˜, a˜)+(B−1b˜, b˜)where:Q1 = (y˜− C˜−1/2(A˜−1a˜+ B˜−1b˜+ 2t˜), y˜− C˜−1/2(A˜−1a˜+ B˜−1b˜+ 2t˜))We have also:(A˜−1 a˜, a˜) = (A˜−1a˜, C˜−1 A˜−1 a˜) + (A˜−1 a˜, C˜−1 B˜−1 a˜)and(B˜−1 b˜, b˜) = (B˜−1b˜, C˜−1 A˜−1 b˜) + (B˜−1 b˜, C˜−1 B˜−1 b˜)By using these results in (2.6), we obtain, after same algebraic manipulations:(2.7)Q = Q1 − (C˜−1 B˜−1 b˜, A˜−1 a˜) − (C˜−1 A˜−1 a˜, B˜−1 b˜) + (A˜−1 a˜, C˜−1 B˜−1 a˜)+(B˜−1b˜, C˜−1A˜−1b˜) + (C˜−1(A˜−1a + B˜−1b˜+ 2t˜), 2t˜) − (2C˜−1t˜, A˜−1a˜+ B˜−1b˜)From (2.7), it follows that:Q = Q1 + ((B˜C˜A˜)−1b˜, b˜− a˜) − ((A˜C˜B˜)−1a˜, b˜− a˜)−− 4(C˜−1(A˜−1a˜+ B˜−1 b˜), t˜) − 4(C˜−1 t˜, t˜)229Since C˜= A˜−1 + B˜−1, we have:A˜C˜B˜= B˜C˜A˜= A˜+ B˜Or,(2.8) Q = Q1+((A˜+B˜)−1(b˜−a˜), b˜−a˜)−4(C˜−1(A˜−1a˜+B˜−1 b˜), t˜)−4(C˜−1 t˜, t˜)Using (2.8) in (2.4), we obtain:(2.9)ρ(F1, F2, t˜) = (2pi)−k/2|A˜B˜|−1/4 exp{−1/4((A˜+ B˜)−1 (b˜− a˜), b˜− a˜)}exp{(C˜−1(A˜−1 a˜+ B˜−1 b˜), t˜) + (C˜−1 t˜, t˜)}·∫Rkexp{−14Q1}|C˜|−1/2 d y1, . . . , d ykWe easily verify that:∫Rkexp{−14Q1}d y1, . . . , d yk = 2k/2 (2pi)k/2Or,(2.10)ρ(F1, F2, t˜) = |A˜B˜|1/4∣∣∣∣12(A˜ + B˜ )∣∣∣∣−1/2exp{−1/4((A˜+ B˜)−1 (b˜− a˜), b˜− a˜)}·· exp{(C˜−1(A˜−1 a˜+ B˜−1 b˜), t˜) + (C˜−1 t˜, t˜)}since|C˜|−1/2 = |A˜|1/2 |A˜+ B˜|−1/2 |B˜|1/2It follows from theorem demonstrated by Matusita and (2.3) that (2.10) may be writtenas:ρ(F1, F2, t˜) = ρ2(F1, F2)MG(t˜)Corolary 1. When A˜= B˜it follows thatρ(F1, F2, t˜) = exp{−1/8(A˜−1(b˜− a˜), b˜− a˜)}MG(t˜)230where:MG(t˜) = exp{(1/2(a˜+ b˜), t˜) + 1/2(A˜t˜, t˜)}Corolary 2. If a˜= b˜it follows that:ρ(F1, F2, t˜) =∣∣∣A˜B˜∣∣∣1/4∣∣∣∣12(A˜+ B˜)∣∣∣∣−1/2exp{(a˜, t˜) +(C˜−1 t˜, t˜)}Corolary 3. For F1 = F2 = F , we have:ρ(F, t˜) = exp{(a˜, t˜) + 1/2(A˜t˜, t˜)}= MX(t˜)where MX(t˜) is the moment generating function of F .The conclusion (result) of theorem 1 is naturally generalized for r k−dimensionalnormal distributions. To accomplish this objective, we first generalize the concept ofρ(F1, F2, t˜), by considering r distributions F1, . . . , Fr, defined over the same spaceR, with probability density functions f1(x), . . . , fr(x) with respect to a measure on R,and let us suppose that the integral below be absolutely convergent. Then we define:Definition 2ρ(F1, . . . , Fr, t) =∫Rr∏j=1exp(tx) fj(x)1/rdmIf F1, . . . , Fr denote r k− dimensional non singular normal distributions whose pro-bability density functions are given for j = 1, . . . , r by(2.11) (2pi)−k/2 |A˜ j|−1/2 exp{−1/2(A˜−1j(x˜− a˜j), x˜− a˜j)}where A˜ jis the covariance matrix and a˜jthe mean vector of Fj , respectively, we havethe following result whose proof we omit:Theorem 2ρ(F1, . . . , Fr, t˜) = ρr(F1, . . . , Fr)MG(t˜)231whereρr(F1, . . . , Fr) =r∏j=1|A˜ j|−1/2r∣∣∣∣∣∣1rr∑j=1A˜−1j∣∣∣∣∣∣−1/2· exp−1/2rr∑j = 2l 6 i < j((A˜ jD˜A˜ i)−1 a˜j, a˜j− a˜i−−r−1∑i = 1i < j 6 r((A˜ iD˜A˜ j)−1 a˜i, a˜j− a˜i)is the affinity between r k−dimensional normal (non singular) distributions obtainedby Matusita (1967a) and expressed in another form by Campos (1978); andMG(t˜) = expD˜−1 r∑j=1A˜−1ja˜j , t˜+ 12(r D˜−1 t˜, t˜)is the moment generating functions of a k−dimensional normal distribution with meanvector D˜−1 r∑j=1A˜−1ja˜j and covariance matrix r D˜−1 with D˜=r∑j=1A˜−1jand t˜ak−dimensional (column) vector.With the objective of generalizing these results, as those obtained by Matusita (1966,1967a) or Campos (1978) we introduce the following definition.Definition 3. Let F1, . . . , Fr be multivariate distributions defined on the same spaceR and let f1(x), . . . , fr(x) be their respective probability density functions. Let ussuppose that there are r scalars s1, . . . , sr such that:r∑j=1sj = 1 and 0 6 sj 6 1 for j = 1, . . . , r.In these conditions, and if the integral below is absolutely convergent, we define:Dr(s1, . . . , sr, t˜j) =∫Rkr∏j=1exp{sj(x˜, t˜j)}fsjj (x) d x1 . . . d xk232where t˜jis a k−dimensional (column) vector with components tj1 , . . . , tjk , j = 1, . . . , r.If F1, . . . , Fr denote k−dimensional normal (non singular) distributions defined as(2.11) we establish the following result:Theorem 3(2.12) Dr(s1, . . . , sr; t˜j) = Dr(s1, . . . , sr) MG r∑j=1sj t˜jwhere:(2.13)Dr(s1, . . . , sr) =r∏j=1|A˜ j|−sj/2∣∣∣∣∣∣r∑j=1sj A˜−1j∣∣∣∣∣∣−1/2· exp−1/2r∑j = 2l 6 i < j(si sj(A˜ jC˜A˜ i)−1 a˜j, a˜j− a˜i)−−r∑i = 1i < j 6 r(si sj(A˜ iC˜A˜ j)−1 a˜i, a˜j− a˜i)and MG r∑j=1sj t˜j is the moment generating function for a k−dimensional normaldistribution with mean vector C˜−1 r∑j=1sj A˜−1ja˜j and covariance matrix C˜−1, ex-pressed by:(2.14)MG r∑j=1sj t˜j = expC˜−1 r∑j=1sj A˜−1ja˜j ,r∑j=1sj t˜j++1/2C˜−1 r∑j=1sj t˜j ,r∑j=1sj t˜j233with C˜=r∑j=1sj A˜−1j.Proof: By the definition 3, we have:(2.15)Dr(s1, . . . , sr; t˜j) = (2pi)−k/2r∏j=1|A˜ j|−sj/2∫Rkexp{−12Q}d x1 . . . d xkwhereQ =r∑j=1sj(A˜−1j(x˜− a˜j), x˜− a˜j)− 2r∑j=1sj(x˜, t˜j)That is,(2.16) Q = (C˜x˜, x˜) − 2(b˜, x˜) − 2(t˜, x˜) +r∑j=1(sj A˜−1ja˜j, a˜j)withb˜=r∑j=1sj A˜−1ja˜jandt˜=r∑j=1sj t˜jAfter the transformationy˜= C˜1/2 x˜whose Jacobian is mod |C˜|−1/2 and same intermediate steps, (2.16) may be written(2.17)Q =(y˜− C˜−1/2(b˜+ t˜), y˜− C˜−1/2(b˜+ t˜))−(C˜−1 b˜, t˜)−−(C˜−1 t˜, b˜)−(C˜−1 t˜, t˜)+r∑j=1(sj A˜−1ja˜j, a˜j)−(C˜−1 b˜, b˜)234One may also prove that:r∑j=1(sj A˜−1ja˜j, a˜j)−(C˜−1 b˜, b˜)=r∑j = 2l 6 i < j(si sj(A˜ jC˜A˜ i)−1 a˜j, a˜j− a˜i)−−r−1∑i = 1i < j 6 r(si sj(A˜ iC˜A˜ j)−1 a˜i, a˜j− a˜i)On applying this result, (2.17) is expressed as:(2.18) Q = Q3 + Q1 − Q2 − 2(C˜−1 b˜, t˜)−(C˜−1 t˜, t˜)where:Q3 =(y˜− C˜−1/2(b˜+ t˜), y˜− C˜−1/2(b˜+ t˜))Q1 =r∑j = 2l 6 i < j(si sj(A˜ jC˜A˜ i)−1 a˜j, a˜j− a˜i)andQ2 =r−1∑i = 1i < j 6 r(si sj(A˜ iC˜A˜ j)−1 a˜i, a˜j− a˜i)By substitution of (2.18) in (2.15), we obtain:(2.19)Dr(s1, . . . , sr; t˜j) = (2pi)−k/2r∏j=1|A˜ j|−sj/2·· exp{−12(Q1 − Q2)}exp{(C˜−1 b˜, t˜)+(12C˜−1 t˜, t˜)}··∫Rkexp{−12Q3}|C˜|−1/2 d y1 . . . d ykThe integral of (2.19) after the transformationz˜= y˜− C˜−1/2(b˜+ t˜)becomes equal|C˜|−1/2(2pi)k/2235By using this result in (2.19), we obtain, in accord with (2.13) and (2.14) the result thatwe have established through theorem 3, that is,Dr(s1, . . . , sr; t˜j) = Dr(s1, . . . , sr) · MG r∑j=1sj t˜jThis result is a generalization in the sense of summarize the results established throughtheorems 1 and 2 as those obtained by Matusita (1966, 1967a) or Campos (1978).REFERENCESAhmad, I.A. (1980). «Nonparametric estimation of an affinity measure between twoabsolutely continuous distributions with hypotheses testing applications». Annalsof the Institute of Statistical Mathematics, 32, 223-240.Ahmad, I.A. (1980). «Nonparametric estimation of Matusita’s measure of affinity bet-ween absolutely continuous distributions». Annals of the Institute of Statistical Ma-thematics, 32, 241-246.Campos, A.D. (1978). «Afinidade entre distribuic¸o˜es normais multivariadas». Atas do3.o Simpo´sio Nacional de Probabilidade e Estatı´stica (ed. IME-USP), 75-79.Cuadras, C.M. (1988). «Statistical distances». Estadı´stica Espan˜ola, 119, 295-378.Cuadras, C.M. (1989). «Distance analysis in discrimination and classification usingboth continous and categorical variables». In Statistical Data Analysis and Infe-rence, (Y. Dodge, ed.), Elsevier, Amsterdam, 459-473.Cuadras, C.M. and Fortiana, J. (1993). «Applying distances in statistics». Qu¨estiio´, 17,39-74.Khan, A.H. & Ali, S.M. (1971). «A new coefficient of association». Annals of the Ins-titute of Statistical Mathematics, 23, 41-50.Krzanowski, W.J. (1983). «Distance between populations using mixed continuous andcategorical variables». Biometrika, 70, 235-243.Matusita, K. (1954). «On the estimation by the minimum distance method». Annals ofthe Institute of Statistical Mathematics, 5, 59-65.Matusita, K. (1955). «Decision rules based on the distance for problems of fit, twosamples and estimation». Annals of the Institute of Statistical Mathematics, 26, 631-640.Matusita, K. (1956). «Decision rule based on the distance for the classification pro-blem». Annals of the Institute of Statistical Mathematics, 8, 67-77.236Matusita, K. (1961). «Interval estimation based on the notion of affinity». Bulletin ofthe International Statistical Institute, 38, 4, 241-244.Matusita, K. (1964). «Distance and decision rules». Annals of the Institute of StatisticalMathematics, 16, 305-315.Matusita, K. (1966). «A distance and related statistics in multivariate analysis». Multi-variate Analysis - Proceedings of an International Symposium (ed. P.R. Krishnaiah).Academic Press New York, 187-200.Matusita, K. (1967a). «On the notion of affinity of several distributions and some of itsapplications». Annals of the Institute of Statistical Mathematics, 19, 181-192.Matusita, K. (1967b). «Classification based on distance in multivariate Gaussian ca-ses». Proceedings of the Fifth Berkeley Simposium on Mathematical Statistics andProbability. University California Press, 1, 299-304.Matusita, K. (1973). «Correlation and affinity in Gaussian cases». Multivariate Analy-sis - III - Proceedings of the Third International - Symposium (ed. P.R. Krishnaiah),Academic Press, New Yor, 345-349.Matusita, K. (1973). «Discrimination and the affinity of distributions». In: Discrimi-nant Analysis and Applications. (T. Cacoullos, ed.), Academic Press, N.Y., 213-223.Matusita, K. (1977). «Cluster analysis and affinity of distributions. Recent Develop-ments in Statistics». Proceedings of the 1976 European Meeting of Statisticians,537-544.Matusita, K. & Motoo, M. (1955). «On the fundamental theorem for the decision rulebased distance II.»Annals of the Institute of Statistical Mathematics, 7, 137-142.Matusita, K. & Akaike, H. (1956). «Decision rules based on the distance for the pro-blems of independence, invariance and two samples». Annals of the Institute ofStatistical Mathematics, 7, 67-80.237

Some results envolving the concepts of moment generating function and affinity between distribution functions. Extension for r k-dimensional normal distribution functions

Revistes Catalanes amb Accés Obert

QU¨ESTIIO´, vol. 23, 2, p. 225-237, 1999
SOME RESULTS ENVOLVING THE CONCEPTS OF
MOMENT GENERATING FUNCTION AND
AFFINITY BETWEEN DISTRIBUTION FUNCTIONS.
EXTENSION FOR r k−DIMENSIONAL NORMAL
DISTRIBUTION FUNCTIONS
A. DORIVAL CAMPOS
Universidade de Sa˜o Paulo
We present a function ρ (F1, F2, t) which contains Matusita’s affinity and
express the «affinity» between moment generating functions. An interesting
result is expressed through decomposition of this «affinity» ρ (F1, F2, t)
when the functions considered are k−dimensional normal distributions.
The same decomposition remains true for others families of distribution
functions. Generalizations of these results are also presented.
Keywords: Affinity, moment generating functions, distance, inner pro-
duct, multivariate normal distributions, probability density
functions, absolutely convergent.
AMS Classification (MSC 2000): primary 62E99, secondary 62H20.
* Dpto. de Matema´tica Aplicada a` Biologia. Faculdade de Medicina de Riberira˜o Preto. Universidade de Sa˜o
Paulo. Brasil.
– Received June 1998.
– Acepted March 1999.
225
1. INTRODUCTION AND PRELIMINARIES
Let F1 and F2 be two distribution functions defined on R and let us denote by fi(x) the
probability density function of Fi with respect to a measure m in R, for i = 1, 2.
We find in the literature several forms of defining distance between distributions of a sa-
me class. Matusita (1955) by making use of the distance function denoted by d (F1, F2)
and expressed by
d (F1, F2) =
{(
f
1/2
1
(x) − f
1/2
2
(x), f
1/2
1
(x) − f
1/2
2
(x)
)}1/2
,
introduced in the statistical literature the concept of affinity between the distributions
F1 and F2 denoted by ρ2 (F1, F2) and defined by
ρ2 (F1, F2) =
(
f
1/2
1
(x), f
1/2
2
(x)
)
which is related to d (F1, F2) through the expression
d2 (F1, F2) = 2 {1 − ρ2 (F1, F2)}
where (f, g) denotes the inner product of f(x) and g(x) defined by:
(f, g) =
∫
R
f(x) g(x) dm
The importance and usefulness of the notions of distance and affinity between distribu-
tions, in statistics, were stressed in a series of papers by Matusita (1954, 1955, 1956,
1961, 1964, 1967b, 1973), Matusita & Motoo (1955), Matusita & Akaike (1956), Khan
& Ali (1971) and others.
Concrete expressions for the affinity between two multivariate normal distributions we-
re established by Matusita (1966). As a following step, Matusita (1967) extended the
notion of affinity to cover the case where there are r distributions involved and esta-
blished concrete expressions when the r distributions are k−dimensional normal.
Our work is characterized by the introduction of the concept of a function, denoted by
P (t), functionally expressed through the moment generating functions relative to the
distributions considered and another expression denoted by ρ (F1, F2, t) that contains
as a particular case the affinity between the distribution functions F1 and F2, or in other
words, express the «affinity» between the moment generating functions relative to F1
and F2. We also present a result that express the decomposition of ρ(F1, F2, t
˜
) in a
product of two factors identified as the affinity and the moment generating function
when F1 and F2 are k−dimensional normal distributions. This result is extended to
cover the case where there are r k−dimensional normal distributions.
226
In this same way, the concept of a more general function Dr(s1, . . . , sr; t
˜j
) is introdu-
ced, and the results obtained through it contains those developed in this paper as those
ones established by Matusita (1966, 1967a) and Campos (1978).
2. RESULTS
Definition 1. Let F1 and F2 be two distribution functions belonging to the same class
and let f1(x) and f2(x) their respective probability density functions with respect to
a measure m defined on R. Let us suppose that there is a scalar t, −h 6 t 6 h
(h > 0) such that the integral below, defined through the inner product, is absolutely
convergent.
Now, we define:
(2.1)
P (t) =
(
exp {tx/2}
{
f
1/2
1
(x) − f
1/2
2
(x)
}
, exp {tx/2}
{
f
1/2
1
(x) − f
1/2
2
(x)
})
where (f, g) denotes the inner product of f(x) and g(x), defined by:
(f, g) =
∫
R
f(x) g(x) dm
From (2.1), we obtain:
P (t) = M1(t) + M2(t) − 2 ρ (F1, F2, t)
where:
Mi(t) represent the moment generating function for the distribution Fi whose probabi-
lity density function is fi(x), i = 1, 2;
and
(2.2) ρ (F1, F2, t) =
(
exp {tx/2} f
1/2
1
(x), exp {tx/2} f
1/2
2
(x)
)
From (2.1) and (2.2) we verify that:
i) P (0) = d2 (F1, F2),
ii) F1 = F2 implies P (t) = 0 for all −h 6 t 6 h
iii) ρ (F1, F2, 0) = ρ2 (F1, F2)
iv) F1 = F2 = F implies ρ(F, t) = M(t)
where:
d2 (F1, F2) =
(
f
1/2
1
(x) − f
1/2
2
(x), f
1/2
1
(x) − f
1/2
2
(x)
)
227
and ρ2 (F1, F2) is the affinity between the distributions F1 and F2 as defined by Matu-
sita (1966).
Teorema 1. Let F1 and F2 be k−dimensional nonsingular normal distributions, whose
probability density functions are given by:
(2pi)
−k/2
|A
˜
|1/2 exp
{
−1/2(A
˜
−1(x
˜
− a
˜
), x
˜
− a
˜
)
}
and
(2pi)
−k/2
|B
˜
|−1/2 exp
{
−1/2(B
˜
−1(x
˜
− b
˜
), x
˜
− b
˜
)
}
,
respectively, where:
x
˜
is a k−dimensional (column vector);
A
˜
and B
˜
are covariance matrices de degree K and
a
˜
, b
˜
are k−dimensional mean (column) vectors.
In these conditions, we have:
ρ(F1, F2, t
˜
) = ρ2 (F1, F2) · MG(t
˜
)
where:
t
˜
is k−dimensional (column) vector and
MG(t
˜
) is the moment generating function of a k−dimensional normal distribution with
mean vector C
˜
−1(A
˜
−1a
˜
+ B
˜
−1 b
˜
) and covariance matrix 2C
˜
−1
, given by
(2.3) MG(t
˜
) = exp
{
(C
˜
−1(A
˜
−1 a
˜
+ B
˜
−1 b
˜
), t
˜
) + (C
˜
−1 t
˜
, t
˜
)
}
with C
˜
= A
˜
−1 + B
˜
−1
.
Proof: From (2.2), we have:
(2.4) ρ(F1, F2, t
˜
) = (2pi)−k/2|A
˜
B
˜
|−1/4
∫
Rk
exp {−1/4Q} d x1, . . . , d xk
where:
Q = (A
˜
−1(x
˜
− a
˜
), x
˜
− a
˜
) + (B
˜
−1(x
˜
− b
˜
), x
˜
− b
˜
) − 4(x
˜
, , t
˜
)
228
By working with this algebraic sum of inner products, we obtain:
(2.5) Q = ((A
˜
−1 +B
˜
−1)x
˜
, x
˜
) −2(A
˜
−1a
˜
+B
˜
−1b
˜
+2 t
˜
, x
˜
)+(A
˜
−1a
˜
, a
˜
)+(B
˜
−1 b
˜
, b
˜
)
If we define the transformation:
y
˜
= C
˜
1/2 x
˜
with C
˜
= A
˜
−1 + B
˜
−1 and Jacobian equal to mod |C
˜
|−1/2, (2.5) may be written as
follows:
Q = (y
˜
, y
˜
) − 2(C
˜
−1/2(A
˜
−1a
˜
+ B
˜
−1b
˜
+ 2t
˜
), y
˜
) + (A
˜
−1a
˜
, a
˜
) + (B
˜
−1b
˜
, b
˜
)
That is,
(2.6) Q = Q1−(C
˜
−1(A
˜
−1a
˜
+B
˜
−1b
˜
+2t
˜
), A
˜
−1a
˜
+B
˜
−1b
˜
+2t
˜
)+(A
˜
−1a
˜
, a
˜
)+(B−1b
˜
, b
˜
)
where:
Q1 = (y
˜
− C
˜
−1/2(A
˜
−1a
˜
+ B
˜
−1b
˜
+ 2t
˜
), y
˜
− C
˜
−1/2(A
˜
−1a
˜
+ B
˜
−1b
˜
+ 2t
˜
))
We have also:
(A
˜
−1 a
˜
, a
˜
) = (A
˜
−1a
˜
, C
˜
−1 A
˜
−1 a
˜
) + (A
˜
−1 a
˜
, C
˜
−1 B
˜
−1 a
˜
)
and
(B
˜
−1 b
˜
, b
˜
) = (B
˜
−1b
˜
, C
˜
−1 A
˜
−1 b
˜
) + (B
˜
−1 b
˜
, C
˜
−1 B
˜
−1 b
˜
)
By using these results in (2.6), we obtain, after same algebraic manipulations:
(2.7)
Q = Q1 − (C
˜
−1 B
˜
−1 b
˜
, A
˜
−1 a
˜
) − (C
˜
−1 A
˜
−1 a
˜
, B
˜
−1 b
˜
) + (A
˜
−1 a
˜
, C
˜
−1 B
˜
−1 a
˜
)+
(B
˜
−1b
˜
, C
˜
−1A
˜
−1b
˜
) + (C
˜
−1(A
˜
−1a + B
˜
−1b
˜
+ 2t
˜
), 2t
˜
) − (2C
˜
−1t
˜
, A
˜
−1a
˜
+ B
˜
−1b
˜
)
From (2.7), it follows that:
Q = Q1 + ((B
˜
C
˜
A
˜
)−1b
˜
, b
˜
− a
˜
) − ((A
˜
C
˜
B
˜
)−1a
˜
, b
˜
− a
˜
)−
− 4(C
˜
−1(A
˜
−1a
˜
+ B
˜
−1 b
˜
), t
˜
) − 4(C
˜
−1 t
˜
, t
˜
)
229
Since C
˜
= A
˜
−1 + B
˜
−1
, we have:
A
˜
C
˜
B
˜
= B
˜
C
˜
A
˜
= A
˜
+ B
˜
Or,
(2.8) Q = Q1+((A
˜
+B
˜
)−1(b
˜
−a
˜
), b
˜
−a
˜
)−4(C
˜
−1(A
˜
−1a
˜
+B
˜
−1 b
˜
), t
˜
)−4(C
˜
−1 t
˜
, t
˜
)
Using (2.8) in (2.4), we obtain:
(2.9)
ρ(F1, F2, t
˜
) = (2pi)−k/2|A
˜
B
˜
|−1/4 exp
{
−1/4((A
˜
+ B
˜
)−1 (b
˜
− a
˜
), b
˜
− a
˜
)
}
exp
{
(C
˜
−1(A
˜
−1 a
˜
+ B
˜
−1 b
˜
), t
˜
) + (C
˜
−1 t
˜
, t
˜
)
}
·
∫
Rk
exp
{
−
1
4
Q1
}
|C
˜
|−1/2 d y1, . . . , d yk
We easily verify that:
∫
Rk
exp
{
−
1
4
Q1
}
d y1, . . . , d yk = 2
k/2 (2pi)k/2
Or,
(2.10)
ρ(F1, F2, t
˜
) = |A
˜
B
˜
|1/4
∣∣∣∣12(A˜ + B˜ )
∣∣∣∣
−1/2
exp
{
−1/4((A
˜
+ B
˜
)−1 (b
˜
− a
˜
), b
˜
− a
˜
)
}
·
· exp
{
(C
˜
−1(A
˜
−1 a
˜
+ B
˜
−1 b
˜
), t
˜
) + (C
˜
−1 t
˜
, t
˜
)
}
since
|C
˜
|−1/2 = |A
˜
|1/2 |A
˜
+ B
˜
|−1/2 |B
˜
|1/2
It follows from theorem demonstrated by Matusita and (2.3) that (2.10) may be written
as:
ρ(F1, F2, t
˜
) = ρ2(F1, F2)MG(t
˜
)
Corolary 1. When A
˜
= B
˜
it follows that
ρ(F1, F2, t
˜
) = exp
{
−1/8(A
˜
−1(b
˜
− a
˜
), b
˜
− a
˜
)
}
MG(t
˜
)
230
where:
MG(t
˜
) = exp
{
(1/2(a
˜
+ b
˜
), t
˜
) + 1/2(A
˜
t
˜
, t
˜
)
}
Corolary 2. If a
˜
= b
˜
it follows that:
ρ(F1, F2, t
˜
) =
∣∣∣A
˜
B
˜
∣∣∣1/4
∣∣∣∣12
(
A
˜
+ B
˜
)∣∣∣∣
−1/2
exp
{
(a
˜
, t
˜
) +
(
C
˜
−1 t
˜
, t
˜
)}
Corolary 3. For F1 = F2 = F , we have:
ρ(F, t
˜
) = exp
{
(a
˜
, t
˜
) + 1/2(A
˜
t
˜
, t
˜
)
}
= MX(t
˜
)
where MX(t
˜
) is the moment generating function of F .
The conclusion (result) of theorem 1 is naturally generalized for r k−dimensional
normal distributions. To accomplish this objective, we first generalize the concept of
ρ(F1, F2, t
˜
), by considering r distributions F1, . . . , Fr, defined over the same space
R, with probability density functions f1(x), . . . , fr(x) with respect to a measure on R,
and let us suppose that the integral below be absolutely convergent. Then we define:
Definition 2
ρ(F1, . . . , Fr, t) =
∫
R


r∏
j=1
exp(tx) fj(x)


1/r
dm
If F1, . . . , Fr denote r k− dimensional non singular normal distributions whose pro-
bability density functions are given for j = 1, . . . , r by
(2.11) (2pi)−k/2 |A
˜ j
|−1/2 exp
{
−1/2(A
˜
−1
j
(x
˜
− a
˜j
), x
˜
− a
˜j
)
}
where A
˜ j
is the covariance matrix and a
˜j
the mean vector of Fj , respectively, we have
the following result whose proof we omit:
Theorem 2
ρ(F1, . . . , Fr, t
˜
) = ρr(F1, . . . , Fr)MG(t
˜
)
231
where
ρr(F1, . . . , Fr) =


r∏
j=1
|A
˜ j
|−1/2r


∣∣∣∣∣∣
1
r
r∑
j=1
A
˜
−1
j
∣∣∣∣∣∣
−1/2
· exp


−1/2r


r∑
j = 2
l 6 i < j
((A
˜ j
D
˜
A
˜ i
)−1 a
˜j
, a
˜j
− a
˜i
−
−
r−1∑
i = 1
i < j 6 r
((A
˜ i
D
˜
A
˜ j
)−1 a
˜i
, a
˜j
− a
˜i
)




is the affinity between r k−dimensional normal (non singular) distributions obtained
by Matusita (1967a) and expressed in another form by Campos (1978); and
MG(t
˜
) = exp



D
˜
−1

 r∑
j=1
A
˜
−1
j
a
˜j

 , t
˜

+ 1
2
(
r D
˜
−1 t
˜
, t
˜
)

is the moment generating functions of a k−dimensional normal distribution with mean
vector D
˜
−1

 r∑
j=1
A
˜
−1
j
a
˜j

 and covariance matrix r D
˜
−1 with D
˜
=
r∑
j=1
A
˜
−1
j
and t
˜
a
k−dimensional (column) vector.
With the objective of generalizing these results, as those obtained by Matusita (1966,
1967a) or Campos (1978) we introduce the following definition.
Definition 3. Let F1, . . . , Fr be multivariate distributions defined on the same space
R and let f1(x), . . . , fr(x) be their respective probability density functions. Let us
suppose that there are r scalars s1, . . . , sr such that:
r∑
j=1
sj = 1 and 0 6 sj 6 1 for j = 1, . . . , r.
In these conditions, and if the integral below is absolutely convergent, we define:
Dr(s1, . . . , sr, t
˜j
) =
∫
Rk
r∏
j=1
exp
{
sj(x
˜
, t
˜j
)
}
f
sj
j (x) d x1 . . . d xk
232
where t
˜j
is a k−dimensional (column) vector with components tj1 , . . . , tjk , j = 1, . . . , r.
If F1, . . . , Fr denote k−dimensional normal (non singular) distributions defined as
(2.11) we establish the following result:
Theorem 3
(2.12) Dr(s1, . . . , sr; t
˜j
) = Dr(s1, . . . , sr) MG

 r∑
j=1
sj t
˜j


where:
(2.13)
Dr(s1, . . . , sr) =


r∏
j=1
|A
˜ j
|−sj/2


∣∣∣∣∣∣
r∑
j=1
sj A
˜
−1
j
∣∣∣∣∣∣
−1/2
· exp


−1/2


r∑
j = 2
l 6 i < j
(si sj(A
˜ j
C
˜
A
˜ i
)−1 a
˜j
, a
˜j
− a
˜i
)−
−
r∑
i = 1
i < j 6 r
(si sj(A
˜ i
C
˜
A
˜ j
)−1 a
˜i
, a
˜j
− a
˜i
)




and MG

 r∑
j=1
sj t
˜j

 is the moment generating function for a k−dimensional normal
distribution with mean vector C
˜
−1

 r∑
j=1
sj A
˜
−1
j
a
˜j

 and covariance matrix C
˜
−1
, ex-
pressed by:
(2.14)
MG

 r∑
j=1
sj t
˜j

 = exp



C
˜
−1

 r∑
j=1
sj A
˜
−1
j
a
˜j

 ,
r∑
j=1
sj t
˜j

+
+

1/2C
˜
−1

 r∑
j=1
sj t
˜j

 ,
r∑
j=1
sj t
˜j




233
with C
˜
=
r∑
j=1
sj A
˜
−1
j
.
Proof: By the definition 3, we have:
(2.15)
Dr(s1, . . . , sr; t
˜j
) = (2pi)−k/2
r∏
j=1
|A
˜ j
|−sj/2
∫
Rk
exp
{
−
1
2
Q
}
d x1 . . . d xk
where
Q =
r∑
j=1
sj
(
A
˜
−1
j
(x
˜
− a
˜j
), x
˜
− a
˜j
)
− 2
r∑
j=1
sj(x
˜
, t
˜j
)
That is,
(2.16) Q = (C
˜
x
˜
, x
˜
) − 2(b
˜
, x
˜
) − 2(t
˜
, x
˜
) +
r∑
j=1
(
sj A
˜
−1
j
a
˜j
, a
˜j
)
with
b
˜
=
r∑
j=1
sj A
˜
−1
j
a
˜j
and
t
˜
=
r∑
j=1
sj t
˜j
After the transformation
y
˜
= C
˜
1/2 x
˜
whose Jacobian is mod |C
˜
|−1/2 and same intermediate steps, (2.16) may be written
(2.17)
Q =
(
y
˜
− C
˜
−1/2(b
˜
+ t
˜
), y
˜
− C
˜
−1/2(b
˜
+ t
˜
)
)
−
(
C
˜
−1 b
˜
, t
˜
)
−
−
(
C
˜
−1 t
˜
, b
˜
)
−
(
C
˜
−1 t
˜
, t
˜
)
+
r∑
j=1
(
sj A
˜
−1
j
a
˜j
, a
˜j
)
−
(
C
˜
−1 b
˜
, b
˜
)
234
One may also prove that:
r∑
j=1
(
sj A
˜
−1
j
a
˜j
, a
˜j
)
−
(
C
˜
−1 b
˜
, b
˜
)
=
r∑
j = 2
l 6 i < j
(
si sj(A
˜ j
C
˜
A
˜ i
)−1 a
˜j
, a
˜j
− a
˜i
)
−
−
r−1∑
i = 1
i < j 6 r
(
si sj(A
˜ i
C
˜
A
˜ j
)−1 a
˜i
, a
˜j
− a
˜i
)
On applying this result, (2.17) is expressed as:
(2.18) Q = Q3 + Q1 − Q2 − 2
(
C
˜
−1 b
˜
, t
˜
)
−
(
C
˜
−1 t
˜
, t
˜
)
where:
Q3 =
(
y
˜
− C
˜
−1/2(b
˜
+ t
˜
), y
˜
− C
˜
−1/2(b
˜
+ t
˜
)
)
Q1 =
r∑
j = 2
l 6 i < j
(
si sj(A
˜ j
C
˜
A
˜ i
)−1 a
˜j
, a
˜j
− a
˜i
)
and
Q2 =
r−1∑
i = 1
i < j 6 r
(
si sj(A
˜ i
C
˜
A
˜ j
)−1 a
˜i
, a
˜j
− a
˜i
)
By substitution of (2.18) in (2.15), we obtain:
(2.19)
Dr(s1, . . . , sr; t
˜j
) = (2pi)−k/2
r∏
j=1
|A
˜ j
|−sj/2·
· exp
{
−
1
2
(Q1 − Q2)
}
exp
{(
C
˜
−1 b
˜
, t
˜
)
+
(
1
2
C
˜
−1 t
˜
, t
˜
)}
·
·
∫
Rk
exp
{
−
1
2
Q3
}
|C
˜
|−1/2 d y1 . . . d yk
The integral of (2.19) after the transformation
z
˜
= y
˜
− C
˜
−1/2(b
˜
+ t
˜
)
becomes equal
|C
˜
|−1/2(2pi)k/2
235
By using this result in (2.19), we obtain, in accord with (2.13) and (2.14) the result that
we have established through theorem 3, that is,
Dr(s1, . . . , sr; t
˜j
) = Dr(s1, . . . , sr) · MG

 r∑
j=1
sj t
˜j


This result is a generalization in the sense of summarize the results established through
theorems 1 and 2 as those obtained by Matusita (1966, 1967a) or Campos (1978).
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237


Some results envolving the concepts of moment generating function and affinity between distribution functions. Extension for r k-dimensional normal distribution functions

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