ON THE NORMAL MOMENT DISTRIBUTIONS

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1 ON THE NORMAL MOMENT DISTRIBUTIONS Akin Olosunde Department of Statistics, P.M.B. 40, University of Agriculture, Abeokuta. 00, NIGERIA. Abstract Normal moment distribution is a particular case of well known Kotz-type elliptical distributions, the density which depends on a shape parameters α, such that α = 0 corresponds to the normal probability density function; several important properties of this class of density function and relationship with existing distribution are established. Keywords and phrases: Normal Moment Distribution, Cumulative Distribution Function,Probability Density Function, Generalized Birnbaum-Saunder Distribution, Reflected Gamma Distribution. AMS(000) Mathematics Subject Classification:Primary 60E05. Introduction Some univariate distributions that have been of great importance lately is the family of elliptical contour densities or simply elliptical distributions. This family includes distributions with lesser or greater kurtosis than the normal distribution. Furthermore, the elliptical family has the normal distributions as a special case. The elliptical laws have been studied by numerous authors and most important results were obtained by Fang,Kotz, and Ng [990]. The use of elliptical distribution as a generalization of normal distribution is not based on empirical arguments or on physical laws. In general, its reasoning is purely statistical and /or mathematical, in the sense that (a) the theory developed under normal distribution is a particular case of the theory derived within elliptical distributions; (b) many of the properties of the normal distribution can be generalized to the case of the elliptical distributions; (c) some important statistics in the theory of normal inference are invariant within the elliptical family. for these reasons, currently, a large part of normal theory is being reconstructed using elliptical distributions, allowing that any statistical analysis in which a normal distribution All correspondence to be addressed to this author - akinolosunde@yahoo.ca

2 can be assumed, can be generalized to this whole family. For a random variable (one dimensional case), the elliptical distributions are symmetrical distributions in R. A r.v. X with elliptical distribution is characterized by the parameters µ and σ and is characterized by a function generator of densities,g, for which the notation X is EC(µ, σ;g) is used. In general, µ and σ are position and scale parameters respectively. µ=e(x) when the first moment of the distribution exists. Var(X)=c 0 σ when the first two moments exists, where c 0 = φ (0) and φ is the derivative of the function φ associated with characteristic function of X. in this article we introduce parameter α which is the shape parameter such that α= 0 corresponds to normal distribution and we called the distribution normal moment distribution. The normal moment distribution can be derived from Kotz-type distribution see Fang et al [990]. Let X be continuous r.v. such that X is (0,;g). Thus, the characteristics function of X is given by ψ(x) = φ(x ); x R, () with φ:r + R, and the p.d.f. of X is given by f(x) = cg(x ); x R, () where g is the kernel of the pdf of X and c the normalization constant. X follows the Kotz-type symmetric distribution with parameters r, s > 0 and q > / and the pdf is f(x) = xr(q )/s Γ((q )/s) x(q ) exp( rx s ), x R. (3) Where normal distribution is a particular case of the Kotz type when q=s= and r=/ in this work we found the pdf of normal moment by introducing a shape parameter α which is define below starting from Kotz-type distribution.. Definition A r.v. X from standardized pdf in (4)above is said to be normal moment distribution if r=/, s= and q=α+. Therefore, X is a normal moment r.v. if it has pdf given by f(x) = e x α+ Γ(α + < x <, α 0 (4) )xα Where α is the shape parameter and Γ(.) is gamma function. Then we say that X is a normal moment random variable with parameter α; for brevity we shall say that X is NM(α), (4) is a proper density function because;if f:r R > 0 denote a positive real valued-function. We say f is very rapidly decreasing (VRD) if every smooth compactly supported function g is a uniform limit (on R) of functions of the form f(x)p(x), where P(x) is a polynomial, thus we have this theorem; Theorem. If f(x) is VRD, then the moment integrals; f(x)xα dx converges for all α. 3

3 Proof. The space of smooth compactly supported functions is of infinite dimension, and all such spaces are uniform limits of sequences f(x)p α (x). Therefore, f(x)p(x) L (R) for some polynomial P of arbitrary large degree. Therefore, f(x)= (x α ) for all α, and the lemma follows.thus, x α e x < x <, α 0 (5) exists and the integrand is a positive continuous function which is bounded by an integrable function. Also, the normal moment distribution can be attributed as a special case of the Sobolev (q-r)family of life distribution which is widely used in fatigue life modeling Sobolev (994). The importance of the statistical theory and application of the normal distribution is well known. Since the proposed distribution is more flexible than the normal distribution, it is anticipated that this class of the distribution will give a better fit for some sets of data for a selected values of the shape parameter α, in this paper we shall consider only standardized type that is where X is NM(α,0,). The first section have introduced the topic and some literature review. In second section some discussions about the shape of the distribution are presented. The third section; the even moment, properties and the relationship with well known distributions are established.the last section shows the estimation of the parameters of the distribution and finally some remarks on this distribution. SHAPE One of the unique properties that distinguished normal moment distribution is the shape of the pdf, it can easily be seen that dlogf(x)/dx= α, see (figure.0. Appendix A).We notes that as α increases the degree of peakedness also increases, f(0)=0 and 0 f(x)= 0 f(x), also, the shape is similar to reflected gamma distribution introduced by Borghi(965) which includes Laplace distribution as a special case when the shape parameter α=. The cumulative distribution function is not in close form the tables of approximate values for some selected α has been produced by Olosunde (007) using numerical computation method. 3 THE MOMENTS AND RELATED DISTRIBU- TIONS In what follows we derived the rth moment of the distribution associated even rth moment is given as If X is NM(α) then the E(x r ) = xr f(x)dx = r Γ(α + r + ) Γ(α + ) (6) 4

4 it can thus be seen that E(X) = 0 (7) var(x) = α + (8) γ = 0 (9) γ = α + 3 (0) The expressions γ and γ are the skewness and kurtosis respectively for the standard normal moment distribution. We establish some properties and prove theorems that shows relationships between the normal moment distribution and some well known distributions. Theorem 3. Let X be a continuous random variable having the even normal moment distribution and α 0. Then the following properties hold; (a) NM(0)=N(0,), (b) if X is NM(α) then -X is NM(α), (c) X =Y is χ (α+). Proof: (a)and(b) are obvious, (c) can easily be proved that; P[Y y]= P[X y]=p[- y X y] F (y) = y 0 r/ Γ(r/) zr/ e z/ dz, for y > 0, () and when r=α+,the result follows Theorem 3. Let X and X be two independent continuously distributed random variable each having the even normal moment distribution symmetric about 0 with parameters α and α respectively, then the random variable F = X /(α + ) X/(α + 0 < f < () ), has an f-distribution with r = α + and r = α + degree of freedom. Proof: and h(x ) = k(α )x α e x / (3) h(x ) = k(α )x α e x / (4) 5

5 from equation () above since x is it can be shown that f( α + α + )p, x = p (5) h F,P (f, p) = J h X,X [g (f, p), g (f, p)], (6) where J is the jacobian of the transformation which is nonzero and from theorem above we have define a new random variable and propose finding the marginal p.d.f. g (f) of F,given the equation for the joint p.d.f. g(f,p)of the random variables f and x =p that is; g(f, p) = f α + / α +α Γ(α + )Γ(α + )(α α + / )α +/ p α +α + e p [+f( α +/ α +/ )], (7) then the marginal pdf g of F is f r / ( r r ) r / Γ( r +r ) Γ( r )Γ( r )[ + f( r r )] (r +r )/ 0 < f < (8) This is the pdf of f-distribution with degree of freedom r =α + and r =α +. Theorem 3.3 Let X be a random variable of continuous type having the normal moment distribution symmetric about 0 and α=. then Y = ln x has Gumbel pdf having the shape parameter λ= Proof: it can be shown that f Y (y) = d dy g (y) f X (g (y)) e y e e y, forλ =, < y <, (9) The corresponding moment generating function is t Γ( t) (0) This is the moment generating function of Gumbel distribution the shape parameter λ=. Theorem 3.4 Let X and X be two stochastically independent random variables which are N(0, ) and normal moment distribution respectively, then Y = x x has Cauchy-type distribution. 6

6 proof: Given the joint density function of X, X and y as stated above, for y = x, the transformation such that the joint pdf of y and y is, f Y,Y (y, y ) = J f X,X (g (y, y ), g (y, y )), < y <, < y < = f(y, y ) = k(α) π y α e y and marginal pdf of y say g (y ) and solving, we have (+y ) () k(α) π 0 y α+ e y (+y ) dy = Γ(α + ) Γ(α + ) π ( + y ) α+ () This is Cauchy type distribution, note that when α = 0 the distribution reduces to Cauchy distribution. and g (y ) does not converge for α=0, where the µ r moment is Γ(r + )Γ(α r + ) Γ(α + ) (3) 4 ESTIMATION We consider estimation by the method of maximum likelihood. from (4) introducing a new variable of integration, say y, by writing then (4) becomes f(y; α, µ, σ) = α+/ Γ(α + /) x = y µ, σ > 0, (4) σ (y µ) α σ α+ The log-likelihood for a random sample x,...x n n exp[ / ( y µ i= σ ) ], < µ <, σ > 0, < y <. (5) logl(α, µ, σ) = n(α+ n )ln nlnγ(α+ )+α ln(x i µ) n(α+)lnσ n (x i= σ µ). i= (6) The first order derivatives of (6) with respect to the three parameters are: lnl = n n(α+) i= + (xi µ), where σ σ σ 3 ni= (x ˆσ i µ) =, (7) n(α + ) 7

7 lnl α = n i= ln(x µ) nlnσ nln Γ (α+ ) Γ(α+ ) and lnl µ = σ ni= (x i µ) α n i= (x i µ). Setting the expressions to zero give the equations as stated above, the case of µ and α can be solve using numerical computation.γ (.) represent digamma function. 5 Remarks In this work, we have discussed a wider extension of the normal distribution starting with Kotz-type distribution, obtaining its density and some of its graphics to see how the shape parameter influences its behavior. We have also outlined some important properties of this new model, some of its anticipated usefulnes is in fatigue life modeling following the recent work of Diaz-Garcia and Leiva-Sanchez(005) where they presented the generalization of Birnbaum-Saunder distribution(gbs),which is denoted by T GBS(δ, β; g), then we have this theorem that: Theorem 5. Let T GBS(δ, β; g). Then, the pdf of T is given by f T (t, δ) = Γ(α + )( δ )α+ t ( β + β t )α exp[ δ ( t β + β )][t 3/ t (t + β) ]. (8) δβ / proof: This result is simply proof following the same procedure used in Diaz-Garcia and Leiva-Sanchez(005) References [] Allan, J.Mathematical Formulars and Integrals. Academic Press, Inc. [] Allan,S., and Keith, J. (985)Kendall s Advanced Theory of Statistics. Charles Griffin and Co. Ltd. London. [3] Balakrishnan,N., and Kocherlakota, S. (985). On the double weibull distribution: Order Statistics and estimation, Sankhya, Series B, 48, [4] Borghi,O.(965).Sobre una distribucion de frecuencias, Trabajos de Estadistica,6, 7-9. [5] Diaz-Garcia, J.A. and Leiva-Sanchez,V.(005). A new family of life distributions based on the contoured elliptical distributions. Journal of Statistical Planning and Inference, 8(),

8 [6] Fang,K.T., Kotz,S. and Ng, K.W.(990). Symmetric multivariate and related distribution. Chapman and Hall, London. [7] Olosunde, A.A., and Ojo, M.O. (007). Some properties of normal moment distribution, Ife Journal of Science, (To appear) [8] Sobolev, I.G.(994). Interrelations between probabilty distributions belonging to the q-r family, Izmeritel naya Tekhnika,0,-4. 9

9 Appendix A. Figure.0. The Pdf of Normal Moment Distributions for selected values of α=,,3 respectively and µ=0, σ =.