Example 2 Probability Density Function]:Suppose the PDF g(r)of Xt is a smooth function with unbounded support.We can expand g(x)=(e)B,H(e), j=0 where the function 1 =V2元p(-2) is the N(0,1)density function,and [H()}is the sequence of Hermite polynomials, defined as （-1yΦ(@)=-耳-=e)p()forj>0 where (is the N(0,1)CDF.For example, H(x)=1, H1(x)=x, H2(x)=(x2-1) H3(x)=x(x2-3), H4(x）=x4-6x2+3. See,for example,Magnus,Oberhettinger and Soni (1966,Section 5.6)and Abramowitz and Stegun (1972,Ch.22). Here,the Fourier coefficient g(x)Hj(x)o(x)dz. Again,,月，一0asj一ogiven∑go号<oo. The N(0,1)PDF o(r)is the leading term to approximate the unknown density g(x), and the Hermite polynomial series will capture departures from normality(e.g.,skewness and heavy tails). To estimate g(r),we can consider the sequence of truncated probability densities gn(c)=Cp(x)月，H(c, i=0 where the constant Hj(x)o(z)drExample 2 [Probability Density Function]: Suppose the PDF g(x) of Xt is a smooth function with unbounded support. We can expand g(x) = (x) X1 j=0 jHj (x); where the function (x) = 1 p 2 exp(￾ 1 2 x 2 ) is the N(0; 1) density function, and fHj (x)g is the sequence of Hermite polynomials, deÖned as (￾1)j d j dxj (x) = ￾Hj￾1(x)(x) for j > 0; where (
) is the N(0; 1) CDF. For example, H0(x) = 1; H1(x) = x; H2(x) = (x 2 ￾ 1) H3(x) = x(x 2 ￾ 3); H4(x) = x 4 ￾ 6x 2 + 3: See, for example, Magnus, Oberhettinger and Soni (1966, Section 5.6) and Abramowitz and Stegun (1972, Ch.22). Here, the Fourier coe¢ cient j = Z 1 ￾1 g(x)Hj (x)(x)dx: Again, j ! 0 as j ! 1 given P1 j=0  2 j < 1: The N(0; 1) PDF (x) is the leading term to approximate the unknown density g(x), and the Hermite polynomial series will capture departures from normality (e.g., skewness and heavy tails). To estimate g(x); we can consider the sequence of truncated probability densities gp(x) = C ￾1 p (x) X p j=0 jHj (x); where the constant Cp = X p j=0 j Z Hj (x)(x)dx 8