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1 Hong Kong Baptist University HKBU Institutional Repository HKBU Staff Publication 014 Tail probability ratios of normal and student s t distributions Tiejun Tong Hong Kong Baptist University tongt@hkbu.edu.hk Heng Peng Hong Kong Baptist University hpeng@hkbu.edu.hk This document is the authors' final version of the published article. Link to published article: APA Citation Tong T. & Peng H. (014). Tail probability ratios of normal and student s t distributions. Communications in Statistics - Theory and Methods 43 (18) This Journal Article is brought to you for free and open access by HKBU Institutional Repository. It has been accepted for inclusion in HKBU Staff Publication by an authorized administrator of HKBU Institutional Repository. For more information please contact repository@hkbu.edu.hk.

2 Tail Probability Ratios of Normal and Student s t Distributions Tiejun Tong 1 and Heng Peng 1 1 Department of Mathematics Hong Kong Baptist University Hong Kong tongt@hkbu.edu.hk hpeng@hkbu.edu.hk Abstract The ratio of normal tail probabilities and the ratio of Student s t tail probabilities have gained an increased attention in statistics and related areas. However they are not well studied in the literature. In this paper we systematically study the functional behaviors of these two ratios. Meanwhile we eplore their difference as well as their relationship. It is surprising that the two ratios behave very different to each other. Finally we conclude the paper by conducting some lower and upper bounds for the two ratios. KEY WORDS: Lower bound; Normal distribution; Student s t distribution; Tail probability ratio; Upper bound 1

3 1 Introduction Let φ() = (π) 1/ e / be the probability density function and Φ() = φ(t)dt be the tail probability of the standard normal distribution. The ratio Φ()/φ() is known as Mills ratio which was well studied in the literature. See for eample Birnbaum (194) Sampford (1953) Boyd (1959) Ruben (196) Ruben (1964) Rabinowitz (1969) Kerridge & Cook (1976) Bryc (00) and Yun (009). It is also known that Mills ratio was etended to eplore the Student s t distribution (Soms 1976 Soms 1980 Shao 1999 Finner Dickhaus & Roters 008). Without loss of generality let a > 0 be a given constant. We define R(;a) = Φ(+a)/Φ() < < to be the ratio of two normal tail probabilities. The ratio R(;a) has gained an increased attention in statistics and related areas with typical eamples including: (1) Assuming that X N(01) the ratio R(;a) is the conditional probability of X > + a given that X >. That is R(;a) can be referred to the hazard rate of the standard normal distribution; () Tusnády s lemma plays an important role in the KMT/Hungarian coupling of the empirical distribution function with a Brownian bridge. In Carter & Pollard (004) the authors proposed some novel bounds for R(; a) and then successfully applied them to sharpen the Tusnády s inequality; (3) R(; a) was referred to as the epected adverse impact in Aguinis & Smith (007) for the purpose of comparing the minority to the majority group; And (4) R(;a) was also employed by Yuan Tong & Ng (011) to achieve the conditional type I error rate for superiority test conditioned on establishment of non-inferiority in clinical trials. In clinical trial studies it is often desirable to test superiority conditioned on establishment of non-inferiority based on the same primary endpoint. Ng (003) pointed out that switching between non-inferiority and superiority

4 without any adjustment has a logic flaw similar to a post-hoc specification for the null hypothesis even though the family-wise type I error rate is not inflated. To overcome this problem Yuan et al. (011) proposed to control the conditional type I error rate of the second-step superiority test at the nominal significance level which leads to a much lower significance level for the second-step superiority test. Nevertheless to derive the conditional type I error rate of the second-step superiority test we require R(; a) to be a monotonically decreasing function of on the entire line. In contrast to Mills ratio to the best of our knowledge little has been studied in the literature for the ratio R(; a). This motivates us to systematically study the functional behaviors of R(; a) in this paper including the curve behavior as well as the corresponding lower and upper bounds. Meanwhile we will study another closely related ratio namely the ratio of two Student s t tail probabilities. Specifically for any a > 0 and ν > 1 we define r ν (;a) = F ν (+a)/f ν () < < where f ν (t) = Γ( ) ν+1 ( νπγ ν ) ( 1+t /ν ) ν+1 and F ν () = f ν (t)dt are the probability density function and the tail probability of the Student s t distribution with ν degrees of freedom respectively. The remainder of the paper is organized as follows. In Section we eplore the curve behaviors of R(;a) and r ν (;a) as a function of. In Section 3 we discuss the relationship between the two ratios from an asymptotic point of view. We then conduct inequalities in Sections 4 and 5 that provide the lower and upper bounds for R(;a) and r ν (;a) respectively on the entire line of. 3

5 Functional Behaviors of R(;a) and r ν (;a) Theorem 1. For the ratio R(;a) with a > 0 we have (1) lim R(;a) = 0; () lim R(;a) = 1; (3) R(;a) is a monotonically decreasing function of on R. Proof. (1) Noting that for any a > 0 both lim Φ( + a) = 0 and lim Φ() = 0. Then by L Hôpital s rule (Abramowitz & Stegun 197) we have Φ (+a) lim R(;a) = lim Φ () where Φ () denotes the first derivative of Φ() on. = lim e (+a)/ e / = 0 () Note that lim Φ() = 1 > 0 and lim Φ(+a) = 1. We have (3) The first derivative of R(;a) is R (;a) = lim Φ(+a) lim R(;a) = lim Φ() = 1. ( ) ( / dt +a e t e t / dt ) ( ( e t / dt ) +a e t / dt )( ) / dt e t / / dt+e = e (+a) e t / / dt +a ( e t e t / dt ). (1) To show that R(;a) is a monotonically decreasing function of it suffices to show that R (;a) 0 for any R. Or equivalently by (1) it suffices to show that e (+a) / +a e t / dt e / 4 e t / dt < <. ()

6 For ease of notation we define g() = e / / dt. By the Newton-Leibnitz formula e t g () = e / = e / e t / dt+e / e t / dt 1. ( ) e / When 0 we have g () < 0. When > 0 by the result of Gordon (1941) e t / dt 1 e / we still have g () 0. This implies that g() is a monotonically decrease function of on R. Therefore g(+a) g() i.e. () holds. This completes the proof. Theorem. For the ratio r ν (;a) with a > 0 and ν > 0 we have (1) lim r ν (;a) = lim r ν(;a) = 1; () r ν (;a) is a monotonically increasing function of on [ ν ). Proof. (1) For ease of notation we define k ν = Γ ( ) ν+1 ( /( νπγ ν ) ). Given that lim F ν(+a) = lim F ν () = 0 for any a > 0 by the L Hôpital s rule lim r F ν ν(;a) = lim (+a) F ν () k ν (1+(+a) /ν) ν+1 = lim k ν (1+ /ν) ν+1 ( ) +a+a ν+1 +ν = lim +ν = 1. On the other side given that lim F ν(+a) = lim F ν() = 1 we have lim r ν(;a) = lim F ν(+a) lim F ν() 5 = 1.

7 () The first derivative of r ν (;a) is given as r ν () = g() [ ] (1+t /ν) ν+1 dt where g() = ) ν+1 (1+ ν +a ) ν+1 (1+ t dt ν (1+ (+a) ν ) ν+1 ) ν+1 (1+ t dt. ν Let C() = (1+ /ν) ν+1 (1+(+a) /ν) ν+1. With some algebra we have { [1+(+a) ]ν+1 /ν g() = C() = C() 1+(t+a) /ν { [h(a)] ν+1 [h(0)] ν+1 } dt [ 1+ /ν 1+t /ν } ]ν+1 dt where h( ) is defined in Lemma 1. When ν by Lemma 1 we know that h(y) is a monotonically increasing function of y on [0 ). This implies that h(a) > h(0) and thus g() > 0. Therefore r ν (;a) is a monotonically increasing function of on [ ν ). Lemma 1. Assume that ν > 0 and t. Define h(y) = 1+(+y) /ν 1+(t+y) /ν. (3) Then h(y) is a monotonically increasing function of y on [ ν ). Proof. The first derivative of h(y) is h (y) = (+y)[ν +(t+y) ] (t+y)[ν +(+y) ] [ν +(t+y) ] = (t )[(+y)(t+y) ν] [ν +(t+y) ]. When y > ν we have ( + y)(t + y) ( + y) ν by the fact that t. Further h (y) 0. This shows that h(y) is a monotonically increasing function of y on [ ν ). 6

8 From Theorems 1 and we observe that the two ratios behave very different to each other. For visualization we set a = 1 and plot the curve of R(;1) in Figure 1 as well as the curves of r ν (;1) with various ν values. 3 Relationship between R(;a) and r ν (;a) In this section we investigate the relationship between the two ratios from the asymptotic pointofview. Specifically wehavethefollowingasymptoticresultforr ν (;a)withrespect to the degrees of freedom ν. Theorem 3. For any fied ( ) we have lim r ν(;a) = R(;a). ν Proof. Let m ν (t) = (1+t /ν) ν+1. For any fied ν 1 by Lemma we have m ν (t) ) ν+1 (1+ t ν +1 ( 1+ t ) 1 = n(t) < t < where n(t) = (1+t /) 1. Note that n(t) is integrable on the interval [ ) i.e. for any value n(t) dt <. Thus by the dominated convergence theorem the integration is interchangeable with the limit operation for the sequence m ν (t). That is ( lim 1+t /ν ) ν+1 dt = ν = ( lim 1+t /ν ) ν+1 dt ν e t / dt. 7

9 Further since the limit of the denominator e t / dt is non-zero for any fied ( ) we have This completes the proof. lim r ν(;a) = lim ν +a (1+t /ν) ν+1 dt ν lim ν (1+t /ν) ν+1 dt / dt +a = e t e t / dt = R(;a). Lemma. Let s(y) = (1+t /y) y where t is a constant. Then s(y) is a strictly decreasing function of y on (0 ). Proof. The first derivative of s(y) is s (y) = 1 (1+ t y ) y [ log ( ) ] 1+ t t /y. y 1+t /y Let ( ) r(y) = log 1+ t t /y y 1+t /y. Note that lim y r(y) = 0 and r (y) = t /y 1+t /y + t /y (1+t /y) = t4 /y 3 0 y > 0. (1+t /y) We have r(y) 0 for any y > 0. This implies that s (y) 0 for any y > 0. Therefore s(y) is a strictly decreasing function of y on (0 ). 4 Bounds of R(;a) For the bounds of Φ() a seminal inequality was given by Mills (196) ( 1 1 ) φ() < Φ() < 1 φ() > 1. (4) 3 8

10 It is known that (4) has been improved in the literature e.g. in Birnbaum (194) and Sampford (1953). Let p(γ) = (γ + 1)/{[ + (/π)(γ + 1) ] 1/ + γ}. Boyd (1959) proposed the following inequality for Φ() p(γ min )φ() < Φ() < p(γ ma )φ() > 0 (5) where γ ma = /(π ) and γ min = π 1. It can be shown that p(γ min ) 1/ 1/ 3 and p(γ ma ) 1/. Therefore (5) gives more accurate bounds for Φ() compared to (4). By (5) we have the following bounds for R(;a) p(+aγ min )φ(+a) p(γ ma )φ() < R(;a) < p(+aγ ma)φ(+a) > 0. (6) p(γ min )φ() Recently Carter & Pollard (004) proposed another two important inequalities for R(; a) that were used to sharpen the Tusnády inequality: ep[ aρ(+a)] R(;a) ep[ aρ()] (7) ep[ aρ() a /] R(;a) ep[ aρ(+a)+a /] (8) where ρ() = φ()/φ(). Note that both the lower and upper bounds in (7) and (8) depend on the term Φ( ) itself through ρ(). To have inequality bounds of R(;a) independent of Φ( ) by the fact that 1 p(γ ma ) < ρ() < 1 p(γ min ) > 0 we propose the following two new inequalities [ ] [ ] a a ep R(;a) ep > 0 (9) p(+aγ min ) p(γ ma ) [ ] [ ] a ep p(γ min ) a a R(;a) ep p(+aγ ma ) + a > 0. (10) 9

11 In Figure we compare the performance of the proposed three inequalities (6) (9) and (10) under various settings. We observe that the bounds in (6) performs better when a is large. In other cases the bounds in (9) or (10) performs better. We also observe that the bounds in (9) and (10) are comparable in general. Combining (6) (9) and(10) we propose the following compounded bounds for R(; a) L 1 (a) R(;a) U 1 (a) > 0 (11) where { [ p(+aγmin )φ(+a) L 1 (a) = ma ep p(γ ma )φ() { [ p(+aγma )φ(+a) U 1 (a) = min p(γ min )φ() ep ] a p(+aγ min ) a p(γ ma ) ] ep [ ep [ a p(γ min ) a a p(+aγ ma ) + a ]} ]}. Unlike most of the results on Mills ratio that hold only for > 0 for the ratio R(;a) we can derive the lower and upper bounds on the entire line of. Note that Φ() = 1 Φ( ) and φ( ) = φ(). For 0 we have ρ() = φ()/(1 Φ( )) and by (5) p( γ min )φ() < Φ( ) < p( γ ma )φ() 0. (1) This leads to φ() 1 p( γ min )φ() ρ() φ() 0. (13) 1 p( γ ma )φ() With (1) and (13) in what follows we present the bounds of R(;a) for 0. (i) When a < 0 noting that R(;a) = Φ(+a)/[1 Φ( )] we have L (a) R(;a) U (a) a < 0 10

12 where { p(+aγmin )φ(+a) L (a) = ma 1 p( γ min )φ() ep [ aφ() ep 1 p( γ ma )φ() a { [ p(+aγma )φ(+a) U (a) = min 1 p( γ ma )φ() ep [ ]} a ep p(+aγ ma ) + a. [ a p(+aγ min ) ]} ] aφ() 1 p( γ min )φ() ] (ii) When a noting that R(;a) = [1 Φ( a)]/[1 Φ( )] we have L 3 (a) R(;a) U 3 (a) a where { 1 p( aγma )φ(+a) L 3 (a) = ma 1 p( γ min )φ() [ aφ() ep 1 p( γ ma )φ() a [ ep ]} aφ(+a) 1 p( aγ ma )φ(+a) { [ 1 p( aγmin )φ(+a) aφ() U 3 (a) = min ep 1 p( γ ma )φ() 1 p( γ min )φ() [ ]} aφ(+a) ep 1 p( aγ min )φ(+a) + a. As anillustration we present infigure 3 thetrue ratio R(;a) andits lower andupper bounds on the entire line of for a = 0.5. The visualized plot shows that the proposed bounds are relatively accurate on the whole range. ] ] 5 Bounds of r ν (;a) For the bounds of F ν () an inequality analogous to (4) was first given by Soms (1976) ( ) 1 ν )(1+ f (ν +) 3 ν () < F ν () < 1 ) (1+ f ν () > 0. (14) ν ν 11

13 Denote k ν = Γ( ν+1)/( νπγ( ν )) as in Section. Let γ ma = 4k ν and γ 1 4kν min = ν (ν +)k ν 1. Soms (1980) showed that for any ν ( 1+ /ν ) f ν ()p ν (γ min ) F ν () ( 1+ /ν ) f ν ()p ν (γ ma ) > 0 (15) where p ν (γ) = (1+γ)/ { [ +4kν (1+γ) ] 1/ +γ }. For the special case when ν = γ ma = γ min = 1 so that (15) is actually an equality. When ν = 1 the inequality (15) still holds with the definition of γ ma and γ min interchanged. It can be shown that (15) gives more accurate bounds for F ν () compared to (14). By (15) we have the following bounds for r ν (;a) c 1 (a) r ν (;a) d 1 (a) > 0 (16) where c 1 (a) = [1+(+a) /ν]f ν (+a)p ν (+aγ min ) (1+ /ν)f ν ()p ν (γ ma ) d 1 (a) = [1+(+a) /ν]f ν (+a)p ν (+aγ ma ). (1+ /ν)f ν ()p ν (γ min ) In what follows we propose a new inequality for r ν (;a). Let Ψ ν () = logf ν () and its derivative ψ ν () = (d/d)ψ ν () = f ν ()/F ν (). By Lagrange mean value theorem we have Ψ ν (+a) Ψ ν () = aψ ν (ξ) = aψ ν(ξ) where < ξ < +a. This equivalents to r ν (;a) = ep{ aψ ν (ξ)}. By (15) we have 1 (1+ /ν)p ν (γ ma ) ψ ν() 1 (1+ /ν)p ν (γ min ). By Lemma A from Soms (1980) we have γ ma > 0 for any ν 1. By Lemmas A3 and A5 from Soms (1980) we have γ min > 0 for any ν 1. Now given that both γ ma and 1

14 γ min are positive it is easy to verify that p ν (γ min ) and p ν (γ ma ) are both decreasing functions of on (0 ). This gives Further we have 1 [1+(+a) /ν]p ν (γ ma ) ψ ν(ξ) 1 (1+ /ν)p ν (+aγ min ). c (a) r ν (;a) d (a) > 0 (17) where { a c (a) = ep (1+ /ν)p ν (+aγ min ) { d (a) = ep } a [1+(+a) /ν]p ν (γ ma ) }. To compare the performance of the proposed inequalities (16) and (17) we present the bounds under various settings in Figure 4 for ν = 1 and in Figure 5 for ν = 10. For both ν values we observe that the bounds in (16) performs better when a is large whereas the bounds in (17) performs better when a is small. By combining (16) and (17) we have l 1 (a) r ν (;a) u 1 (a) > 0 (18) where l 1 (a) = ma{c 1 (a)c (a)} and u 1 (a) = min{d 1 (a)d (a)}. The same as in Section 4 we now derive the lower and upper bounds of r ν (;a) for 0. Note that F ν () = 1 F ν ( ) and f ν ( ) = f ν (). Also by (15) we have ( 1+ /ν ) f ν ()p ν ( γ min ) F ν ( ) ( 1+ /ν ) f ν ()p ν ( γ ma ) 0. (i) When a < 0 noting that r ν (;a) = F ν (+a)/[1 F ν ( )] we have l (a) r ν (;a) u (a) a < 0 13

15 where l (a) = [1+(+a) /ν]f ν (+a)p ν (+aγ min ) 1 (1+ /ν)f ν ()p ν ( γ min ) u (a) = [1+(+a) /ν]f ν (+a)p ν (+aγ ma ). 1 (1+ /ν)f ν ()p ν ( γ ma ) (ii) When a noting that r ν (;a) = [1 F ν ( a)]/[1 F ν ( )] we have l 3 (a) r ν (;a) u 3 (a) a where l 3 (a) = 1 [1+(+a) /ν]f ν (+a)p ν ( aγ ma ) 1 (1+ /ν)f ν ()p ν ( γ min ) u 3 (a) = 1 [1+(+a) /ν]f ν (+a)p ν ( aγ min ). 1 (1+ /ν)f ν ()p ν ( γ ma ) Finally we present in Figure 6 the true ratio r ν (;a) and its lower and upper bounds on the entire line of for ν = 5 and a = 0.1. The visualized plot shows that the proposed bounds perform quite well especially when is negative or when is large. References Abramowitz M. and Stegun I. (197). Handbook of Mathematical Functions Dover New York. Aguinis H. and Smith M. A. (007). Understanding the impact of test validity and bias on selection errors and adverse impact in human resource selection Personnel Psychology 60: Birnbaum Z. W. (194). An inequality for Mill s ratio Annals of Mathematical Statistics 13:

16 Boyd A. V. (1959). Inequalities for Mill s ratio Reports of Statistical Applications Research Japanese Union of Scientists and Engineers 6: Bryc W. (00). A uniform approimation to the right normal tail integral Applied Mathematics and Computation 17: Carter A. and Pollard D. (004). Tusnády inequality revisited Annals of Statistics 3: Finner H. Dickhaus T. and Roters M. (008). Asymptotic tail properties of Student s t-distribution Communications in Statistics: Theory and Methods 37: Gordon R. D. (1941). Values of Mill s ratio of area to bounding ordinate of the normal probability integral for large values of the argument Annals of Mathematical Statistics 1: Kerridge D. F. and Cook G. W. (1976). Yet another series for the normal integral Biometrika 63: Mills J. P. (196). Table of the ratio: Area to bounding ordinate for any portion of normal curve Biometrika 18: Ng T. H. (003). Issues of simultaneous tests for non-inferiority and superiority Journal of Biopharmaceutical Statistics 13: Rabinowitz P.(1969). New Chebyshev polynomial approimations to Mills ratio Journal of the American Statistical Association 64: Ruben H. (196). A new asymptotic epansion for the normal probability integral and Mill s ratio Journal of the Royal Statistical Society Series B 4:

17 Ruben H. (1964). Irrational fraction approimations to Mills ratio Biometrika 51: Sampford M. R. (1953). Some inequalities on Mill s ratio and related functions Annals of Mathematical Statistics 4: Shao Q. (1999). A Cramér type large deviation result for Student s t-statistic Journal of Theoretical Probability 1: Soms A. P. (1976). An asymptotic epansion for the tail area of the t-distribution Journal of the American Statistical Association 71: Soms A. P.(1980). Rational bounds for the t-tail area Journal of the American Statistical Association 75: Yuan J. Tong T. and Ng T. H. (011). Conditional type I error rate for superiority test conditioned on establishment of non-inferiority in clinical trials Drug Information Journal 45: Yun B.(009). Cumulative normal distribution using hyperbolic tangent based functions Journal of the Korean Mathematical Society 46:

18 Ratios of Tail Probabilities ratio r(;1) with d.f.= r(;1) with d.f.=5 r(;1) with d.f.=10 r(;1) with d.f.=0 R(;1) Figure 1: The curve of R(;1) and the curves of r ν (;1) with various degrees of freedom ν. 17

19 R(;0.1) Bounds for Normal Ratios R(;0.) Bounds for Normal Ratios R(;0.5) R(;1) Figure: BoundsfortheratioR(;a)withvariousavalues wherethesolidlinesrepresent the true ratio R(; a) the dashed lines represent the bounds in (6) the dotted lines represent the bounds in (9) and the dash-dotted lines represent the bounds in (10). 18

20 Lower and Upper Bounds for R(0.5) R(; 0.5) Figure 3: Lower and upper bounds for R(;0.5) on the entire line of where the solid line represents the true ratio R(; 0.5) and the dashed lines represent its lower and upper bounds. 19

21 r(; 0.05) with nu= Bounds for t Ratios r(; 0.075) with nu= Bounds for t Ratios r(; 0.1) with nu= r(; 0.15) with nu= Figure4: Boundsfortheratior 1 (;a)withvariousavalues wherethesolidlinesrepresent the true ratio r 1 (;a) the dashed lines represent the bounds in (16) and the dotted lines represent the bounds in (17). 0

22 Bounds for t Ratios Bounds for t Ratios r(; 0.05) with nu= r(; 0.1) with nu= r(; 0.15) with nu= r(; 0.) with nu= Figure 5: Bounds for the ratio r 10 (;a) with various a values where the solid lines represent the true ratio r 10 (;a) the dashed lines represent the bounds in (16) and the dotted lines represent the bounds in (17). 1

23 Lower and Upper Bounds for r(; 0.1) with nu=5 r(; 0.1) with nu= Figure 6: Lower and upper bounds for r 5 (;0.1) on the entire line of where the solid line represents the true ratio r 5 (;0.1) and the dashed lines represent its lower and upper bounds.