Shenton, 1954 - University of WisconsinâMadison
[ 177 ] INEQUALITIES FOR THE NORMAL INTEGRAL INCLUDING A NEW CONTINUED FRACTION BY L. R. SHENTON
1. INTRODUCTION
In this note we consider the related normal integral ratios
sometimes called Mills's ratio, and
f\{x)dxjg{t),
Jo
where g(x) = ,
. e"***, and R(t) + R(t) = ll[2g(t)]. We give a new continued fractionf
for R(t) which is rapidly convergent for small values of t and which incidentally provides a new set of inequalities. The rapidity of convergence is compared with a series for R(t) and with the Laplace C.F. for R(t). This assessment is similar to recent work by Teichroew (1952) on the comparative rapidity of convergence of series ando.F.'s for the elementary function e?, hi (1 + x) and arc tan x. Lane (1944) has also considered the same sort of thing for interpolation, comparing Newton's series and Thiele's C.F. in the case of the function 2X. We also prove and generalize a conjecture of Birnbaumf (1950) that for (^0, it being shown that there are two sequences of similar inequalities, increasing and decreasing to the limiting value R(t). In the Appendix we set out a brief summary of results relating to the normal integral. 2. THE CONTINUED FRACTION FOB R(t)
The new C.F. is given by
^
fi
3*
Z*
(See, for example, Perron, 1913, pp. 311-14; Wall, 1948, pp. 349-55.) f Continued fraction will be abbreviated to C.F. throughout, j Birabaum actually conjectured (**— 1) i2» — 3tR + 2 > 0. Biometrika 41
Downloaded from biomet.oxfordjournals.org at University of Wisconsin Madison - General Library System on January 14, 2011
College of Technology, Manchester
178
Inequalities for the normal integral
The expression (1) now follows since R(t) = 1^(1; f; \t*). Denoting the *th convergent of (1) by rs = pjq,, it follows that pt and qt satisfy the recurrence relations \ w i t h ^ 0 = 0,Pl = t, q0 =l,q1=
(s= 1,2,...),
(5)
1.
ro-t>)'eipi(t*-x')dx. JO
(d) Formulae for computational checks:
Squ-pu
2st*
n terms leading to the approximation ru+, for R{t). (e) Partial sums of series for R(t): «-i r_01.3
ttr+i
_ * (2 +l)~T-t*
«' 30).
where
(This result is mentioned by Wishart (1927).) In particular
tRlt) = 1
(6)
1
1
1
5
h
ff(z)dx
1. INTRODUCTION
In this note we consider the related normal integral ratios
sometimes called Mills's ratio, and
f\{x)dxjg{t),
Jo
where g(x) = ,
. e"***, and R(t) + R(t) = ll[2g(t)]. We give a new continued fractionf
for R(t) which is rapidly convergent for small values of t and which incidentally provides a new set of inequalities. The rapidity of convergence is compared with a series for R(t) and with the Laplace C.F. for R(t). This assessment is similar to recent work by Teichroew (1952) on the comparative rapidity of convergence of series ando.F.'s for the elementary function e?, hi (1 + x) and arc tan x. Lane (1944) has also considered the same sort of thing for interpolation, comparing Newton's series and Thiele's C.F. in the case of the function 2X. We also prove and generalize a conjecture of Birnbaumf (1950) that for (^0, it being shown that there are two sequences of similar inequalities, increasing and decreasing to the limiting value R(t). In the Appendix we set out a brief summary of results relating to the normal integral. 2. THE CONTINUED FRACTION FOB R(t)
The new C.F. is given by
^
fi
3*
Z*
(See, for example, Perron, 1913, pp. 311-14; Wall, 1948, pp. 349-55.) f Continued fraction will be abbreviated to C.F. throughout, j Birabaum actually conjectured (**— 1) i2» — 3tR + 2 > 0. Biometrika 41
Downloaded from biomet.oxfordjournals.org at University of Wisconsin Madison - General Library System on January 14, 2011
College of Technology, Manchester
178
Inequalities for the normal integral
The expression (1) now follows since R(t) = 1^(1; f; \t*). Denoting the *th convergent of (1) by rs = pjq,, it follows that pt and qt satisfy the recurrence relations \ w i t h ^ 0 = 0,Pl = t, q0 =l,q1=
(s= 1,2,...),
(5)
1.
ro-t>)'eipi(t*-x')dx. JO
(d) Formulae for computational checks:
Squ-pu
2st*
n terms leading to the approximation ru+, for R{t). (e) Partial sums of series for R(t): «-i r_01.3
ttr+i
_ * (2 +l)~T-t*
«' 30).
where
(This result is mentioned by Wishart (1927).) In particular
tRlt) = 1
(6)
1
1
1
5
h
ff(z)dx
