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PROBABILITY AND MATHEMATICAL STATISTICS Vol. 30, Fasc ), pp INEQUALITIES FOR QUANTILES OF THE CHI-SQUARE DISTRIBUTION BY TADEUSZ I N G L OT WROCŁAW) Abstract. We obtain a new sharp lower

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PROBABILITY AND MATHEMATICAL STATISTICS Vol. 30, Fasc ), pp INEQUALITIES FOR QUANTILES OF THE CHI-SQUARE DISTRIBUTION BY TADEUSZ I N G L OT WROCŁAW) Abstract. We obtain a new sharp lower estimate for tails of the central chi-sqare distribtion. Using it we prove qite accrate lower bonds for the chi-sqare qantiles covering the case of increasing nmber of degrees of freedom and simltaneosly tending to zero tail probabilities. In the case of small tail probabilities we also provide pper bonds for these qantiles which are close enogh to the lower ones. As a byprodct we propose a simple approximation formla which is easy to calclate for the chi-sqare qantiles. It is expressed explicitly in terms of tail probabilities and a nmber of degrees of freedom AMS Mathematics Sbject Classification: Primary: 62E17; Secondary: 60E15, 62E15, 62Q05, 65C60. Key words and phrases: Chi-sqare qantiles, normal qantiles, lower and pper bonds, tails of chi-sqare distribtion, Wilson Hilferty formla. 1. INTRODUCTION Approximation formlae for the chi-sqare distribtion qantiles have been investigated in nmeros papers beginning from Fisher [3] and Wilson and Hilferty [9]. Nowadays, very accrate approximation formlae are available see e.g. Zar [10] and Johnson et al. [6], [7] or Ittrich et al. [5] and references therein). On the contrary, ineqalities for the central chi-sqare qantiles rarely appear in the literatre althogh they play an important role in some statistical considerations. Larent and Massart [8] gave an exponential ineqality for tails of the noncentral chi-sqare distrbtion and sed it to determine ris bonds for penalized estimator of the sqared norm of a mean in a Gassian linear model. This ineqality is eqivalent to some global pper bond for qantiles covering all vales of parameters involved. Brain and Mi [2] proved some pper and lower bonds which are expressed solely in terms of a nmber of degrees of freedom and applied them to an interval estimation problem. Inglot and Ledwina [4] obtained lower bonds depending both on and tail probabilities α and, employing them, described an asymptotic behavior of the qantiles when increases and simltaneosly α tends 340 T. Inglot to zero. Sch considerations were necessary to stdy an asymptotic optimality of some adaptive test proposed by Barad et al. [1]. In the present note we prove a new sharp lower estimate for tails of the central chi-sqare distribtion. Next, sing it, reslts of Inglot and Ledwina [4] are essentially improved. In effect, we provide accrate pper and lower bonds for the chi-sqare qantiles for small α. Their form sggests to propose a new simple approximation formla for the chi-sqare qantiles which for typical between 3 and 100 and typical α between 0.1 and gives comparable relative errors as the celebrated Wilson Hilferty formla. Being expressed explicitly in terms of and α it is easy for hand calclations which may be regarded as its additional advantage. For the sae of completeness, we provide also some lower and pper bond for the normal qantiles. 2. BOUNDS FOR THE NORMAL QUANTILES In this section we give some bonds for qantiles of the standard normal distribtion. First, recall the well-nown ineqality for tails of this distribtion which we shall se to derive or reslt. Namely, let Φx) be the standard normal cdf. Then for every 0 we have 2.1) 2 h) 1 Φ) h), where h) = exp{ log + log 2π)}. Note that the pper bond eqals ϕ)/ and the lower bond eqals ϕ)/1 + 2 ), where ϕ) is the standard normal density. Now, denote by z α the 1 α qantile of the standard normal distribtion, i.e. defined by the eqation Φz α ) = 1 α. Or reslt is intended especially for small α, say, α 0.1. Then, applying 2.1), we obtain the following estimate for z α : THEOREM 2.1. For every 0 α 0.1 we have 2.2) 2 log1/α) log 4 log1/α) ) log1/α) z α 2 log1/α) log 2 log1/α) ) + 3/ log1/α) The proof of Theorem 2.1 is provided in Section 6. Chi-sqare qantiles TAILS OF THE CHI-SQUARE DISTRIBUTION Let χ 2 denote a random variable with central chi-sqare distribtion with degrees of freedom. For varios applications explicit lower and pper bonds for tails of χ 2 are important. Some estimates are well nown. In Lemma 1 of Inglot and Ledwina [4], the following version of sch an estimate was proved: 3.1) 1 2 E ) P χ 2 1 ) π + 2 E ), 2, 2, where E ) = exp { 1 2 2) log/) + log )}. It follows from the proof of 3.1) that the lower bond holds tre for all 0 and any 2. This observation will be sed in 6.7) below. The pper bond in 3.1) seems to be accrate enogh while the lower bond for not mch greater than is far from being precise and differs from the pper one significantly. In order to manage this problem we propose an essentially better lower bond than that given in 3.1). PROPOSITION 3.1. For all 2 and all 1 we have 3.2) P χ 2 ) 1 e 2 2 Proposition 3.1 is proved in Section E ). 4. UPPER BOUNDS FOR THE CHI-SQUARE QUANTILES From now on denote by α, ) the qantile of order 1 α, α 0, 1), of the central chi-sqare distribtion with degrees of freedom, i.e. satisfying the relation 4.1) P χ 2 α, )) = α. A global pper bond for qantiles of the noncentral chi-sqare distribtion follows from Lemma 1 of Larent and Massart [8]. In a special case of the central chi-sqare distribtion it may be stated as follows: THEOREM A Larent and Massart [8]). For every 1 and every α 0, 1) we have 4.2) α, ) + 2 log1/α) + 2 log1/α). The relation 4.2) is a simple conseqence of the pper bond in 3.1). That is why we reprove Theorem A in Section 6. As one can expect, 4.2) being tre for the whole range of and α is far from being precise. In particlar, the last term on the right-hand side taes mch too large vales for very small α. More precisely, the following fact taes place. 342 T. Inglot THEOREM 4.1. For any constant C 0, 2) there exists a = ac) 0 sch that for every 1 and every α e a we have 4.3) α, ) + 2 log1/α) + C log1/α). The proof of Theorem 4.1 is given in Section 6 and is a modification of or proof of Theorem A. To see to what range of α it applies in the assmption tae, for example, C = 1 and C = 1/4. Then from 6.11) it follows that a1) and a1/4) 900. Theorem 4.1 sggests that for very small α the third term in 4.3) being of the form log1/α) has too large order. An pper bond in which this term has more proper order is stated in the next theorem and proved in Section 6. we have THEOREM 4.2. For every 2, 0 d 1 and every α, α exp { exp [ 2)/2 ] 1/d)}, 4.4) α, ) + 2 log1/α) + 2 log 1+d log1/α). The above theorem shows that a proper order of the third term in expansion of the qantiles for very small α seems to be log log1/α). We shall see in the next section that, in fact, it is the case. 5. LOWER BOUNDS FOR THE CHI-SQUARE QUANTILES In some statistical considerations lower bonds for the chi-sqare qantiles seem to be as sefl as the pper ones. For example, both lower and pper bonds were sed by Brain and Mi [2] to prove some properties of confidence bonds for the maximm lielihood estimators. However, it trns ot that it is hard to find in the literatre sfficiently accrate lower bonds for α, ) in the case when α tends to zero and simltaneosly increases. As was said previosly, some reslts were obtained, among others, in Lemma 3 of Inglot and Ledwina [4]. A global lower bond which may be considered as a conterpart of the Larent and Massart pper bond was proposed in 4) of Lemma 3 in [4]. Using 3.2) it can be significantly improved. The corresponding reslt is stated in the next proposition. PROPOSITION 5.1. For every 2 and every α 0.17 we have 5.1) α, ) + 2 log1/α) 5/2. A simple proof of Proposition 5.1 is given in Section 6. A similar proof shows that at the cost of enlarging to 9 and diminishing α to α 1/17 one can remove the constant 5/2 in 5.1) to obtain α, ) + 2 log1/α). However, both Chi-sqare qantiles ) as well as 5) and 6) of Lemma 3 in [4] leave a wide gap to the pper bonds 4.4) and 4.2). So, it seems desirable to loo for better bonds by an application of 3.2). Note that for + 2 log1/α) the estimate 3.2) can be better than 3.1) only if is not too small. Below we give for sch s a qite accrate lower bond mch better than that in 5) of Lemma 3 in [4] and which corresponds to the logarithmic order term in expansion in 4.4). Its proof needs somewhat delicate nmerical considerations and is given in Section 6. THEOREM 5.1. For every 32 and every α e /8 or any 18 and every α e /3 ) we have 5.2) α, ) + 2 log1/α) + log log1/α). If α is not very close to 0, then the order of the third term in 5.2) can be enlarged to that appearing in the pper bonds 4.2) and 4.3). A reslt of sch a type is given in the next theorem. It improves 6) of Lemma 3 in [4]. THEOREM 5.2. For every 17 and every α [e 560, 1/17] we have 5.3) α, ) + 2 log1/α) log1/α). It is worth to mention that, in fact, 5.2), 5.3) as well as 4.4) remain tre for mch wider ranges of α and than we are able to prove. For example, nmerical calclations show that 5.2) holds tre for all 18 and α 0.05 or for all 28 and α PROOFS P r o o f o f T h e o r e m 2.1. Let s pt t = log1/α). Then t log 10. Let s write z = 2t log 4t + 2)/2 2t and z = 2t log 2t + 3/2)/2 2t. To prove the right-hand side of the estimate 2.2) it is enogh, de to 2.1), to show that hz ) α or, eqivalently, 6.1) z ) log z + log 2π 2t. Inserting the form of z one can redce 6.1) to the ineqality log 2t + 3/2 v + 2 log1 v) + log 2π 3 2 2, where we have defined v = vt) = log 2t + 3/2)/4t. Since v is decreasing with respect to t, it taes vales in 0, 1/3). As log 2t 3/2, it is enogh to show 6.2) 3v + 4 log1 v) + 2 log 2π 3, v 0, 1 ). 3 344 T. Inglot The left-hand side of 6.2) is decreasing with respect to v and its minimal vale is attained at v = 1/3, so it is greater than log8π/9) which, in trn, is greater than 3. To prove the left-hand side of 2.2) it is enogh, again by 2.1), to chec hz )z ) 2 / 1 + z ) 2) α. Ptting = t) = log 4t + 2)/4t and repeating similar calclations as above we redce this ineqality to an eqivalent form ) 6.3) 2t log log π 2. 2t1 ) Applying the ineqality log1 + y) y to the second term in 6.3) it is enogh to prove 6.4) log 2 4t 8t /1 ) 2t + log π 2 for t log 10. Since is decreasing with respect to t in log 10, ), the left-hand side of 6.4) is also decreasing with respect to t and its largest vale is attained at t = log 10, being obviosly less than 2. This completes the proof. P r o o f o f P r o p o s i t i o n 3.1. For 4 and 0 integration by parts gives 6.5) x 2)/2 e x/2 dx = 2 2)/2 e /2 + 2) x 4)/2 e x/2 dx. From the proof of Lemma 1 in [4] we obtain c)=2/2e) /2 / Γ/2) ) 1/2 for all 2, which together with 6.5) imply that for 4 and 0 the tail probabilities of the chi-sqare distribtion satisfy a recrrence formla 6.6) P χ 2 ) 1 2 E ) + P χ 2 2 ). Let s pt r =, where a is the integer part of a nmber a. If 6, then iteration of 6.6) r times yields 6.7) P χ 2 ) 1 2 E ) E 2) E 2r +2) + P χ 2 2r ) 1 2 E ) E 2r ) for 0, where the last ineqality follows from 3.1). Chi-sqare qantiles 345 Since [/ 2)] 1)/2 e for all 3 insert x = 2/ 2) into the ineqality x + 2) logx + 1) 2x which holds for x 0), we infer that E 2 ) [ 2)/]E ) for 3 and 0. Conseqently, for 6 and 0 P χ 2 ) 1 [ 2 E ) )... 2r ] ) 6.8) r 1 [ 2 E ) ) 2 r ] = 1 [ ) 2 E 2 r +1] ) It is easy to see that for 2 we have 2 ) r +1 ) 2 r +1 e 2, 2 which together with 6.8) proves 3.2) for 6. For = 2, 3, 4 and 5 the righthand side of 3.2) is for 1 smaller than E )/2 and or reslt follows immediately from 3.1). REMARK 6.1. More carefl considerations in 6.8) lead to a slightly stronger estimate than 3.2), which has the form P χ 2 ) 1 e E ) and which holds tre for all 6 and. Using this estimate, we can frther improve 5.2) and 5.3). We omit details. In all proofs below we shall still se the notation t = log1/α) for α 0, 1). Obviosly, the relation α, ) holds if and only if P χ 2 ) α. Hence, to get this ineqality for 2 it is sfficient to show, de to the pper bond in 3.1), that 6.9) + 2 E ) πα. P r o o f o f T h e o r e m A. For = 1, 4.2) follows immediately from 3.7) of Lemma 3 in [4] or from or Theorem 2.1 after some easy calclations. For 2 we need to chec 6.9) with = + 2t + 2 t. Inserting this form of into 6.9), taing logarithms of both sides and rearranging we get an eqivalent form of 6.9): t 2 t log 1 + 2t + 2 ) log + 2t + 2 ) t + log π 0. 346 T. Inglot Ptting t = v we obtain finally ) hv) + 2 log + 2 v + 2 ) t + log π 0, where hv) = 2 v log1 + 2v + 2 v). Since hv) is increasing on 0, ) and h0) = 0, the first term in 6.10) is positive. The sm of the two last terms in 6.10) is greater than log4πt) which, in trn, is positive for t 1/4π), i.e. for α e 1/4π) Bt, obviosly, for α 0.5 qantiles α, ) are smaller than, ths satisfy 4.2). This completes the proof. P r o o f o f T h e o r e m 4.1. For y 0 let s define the fnction gy) = e y 1 y)/y 2. Then g is increasing on 0, ) and taes vales in 1/2, ). Pt 6.11) a = ac) = [g 1 2/C 2 )/C] 2. Inserting = + 2t + C t into 6.9) and repeating the same calclations as in the preceding proof we see that it is enogh to show that 6.12) C v log1 + 2v + C v) ) + 2 log 2 + 2t + C ) t + log π 0 for t a or, eqivalently, v a. The relation v a is eqivalent to gc v) 2/C 2, which in trn means that the first term in 6.12) is nonnegative. Obviosly, g 2) 1. Conseqently, for C 2 we have a 1 and for t a the second term in 6.12) is positive, which proves 6.12) in this case. If C 2 and t 1/2, then the second term in 6.12) is positive. Finally, for C 2 and t 1/2, i.e. for α e 1/2 1/2, the ineqality 4.3) is trivially satisfied, as was seen in the previos proof. This concldes the proof. P r o o f o f T h e o r e m 4.2. For = 2 the relation 4.3) is obvios. So, assme 3. Applying 6.9) to = + 2t + 2 log 1+d t, repeating the same calclations as in the previos proofs and setting log t = we see that it is enogh to chec that 6.13) ψ ) + 2 log1 + e + 1+d e ) + log 4π 0 for 3, 0 d 1 and 0, d) = 2)/2 ) 1/d, where ψ ) = 2 1+d + 2 log e + 2 ) 1+d. A standard calclation shows that for every 3 and 0 d 1 the fnction ψ ) is increasing on 0 /2, ). Chi-sqare qantiles 347 For 8 we omit the second term in 6.13) and it remains to show that 6.14) ψ 0 ) + log 4π 2 = log + e ) 0e 0 + log 4π is nonnegative. Since the expression nder the logarithm in 6.14) is decreasing with respect to 0 and d ranges from 0 to 1, it attains its largest vale for the minimal vale of 0, i.e. 3/ 8. So, the expression nder the logarithm in 6.14) can be bonded by 0.97, which redces this expression to /33 + log4π/) and which is obviosly greater than zero for all. For 3 7 and 0 d 1 we have 0 0, 1). So, ψ ) is increasing on 1, ). Hence, to prove 6.13) in 1, ) we omit the second term in 6.13) and it remains to chec that ψ 1) + log 4π = log 1 + 2e + 2 ) + log 4π 0, which we do directly for each vale of nder consideration. For 0, 1) we shall consider = 5, 6, 7 and = 3, 4, separately. For = 5, 6 and 7 we see by a rotine calclation that the expression θ = e + 1+d e is either increasing on 0, 1) or increasing on some interval 0, 1 d, ) ) and decreasing on 1 d, ), 1 ). Hence it attains the minimal vale at = 1 or at = 0. As 0 0, 1), a simple calclation shows that θ is at least 5e 1 /2. This means that the second term in 6.13) can be bonded from below by 2 log1 + 5e 1 /2) 1.3. Again by the monotonicity of ψ on 0, 1) we infer that it is enogh to chec that ) log + e ) 0e log 4π log ) log 4π 0. A straightforward calclation shows 6.15) for = 5, 6 and 7. For = 3 and 4 the second term on the left-hand side of 6.13) in 0, 1) is larger than 2 log1 + e 1 ) 0.6. Arging as above we see that it is enogh to chec that log1 + 2/) log4π/) 0, which obviosly holds for = 3 and 4. This concldes the proof. Similarly to the reasoning for the pper bond and de to 3.2) in order to show that α, ) for some it is enogh to prove that 6.16) 1 e E ) α. For the ftre se we pt κ = 2 log 1 e 2 )/2 ) 348 T. Inglot P r o o f o f P r o p o s i t i o n 5.1. First consider 6. Inserting = + 2t 5/2 into 6.16) and taing logarithms of both sides we see that it is enogh to prove 6.17) log 1 + ) 2t 5/2 2 log 1 + t 5/4 ) κ + log 4 5/2 for 6 and t log Since for each 6 the left-hand side of 6.17) is increasing with respect to t and for t = log 0.17 is increasing with respect to, we only chec that 6 log 1 2 log /2 6 ) ) log /4 2 log 1 κ + log 4 5/2 6 by a direct calclation. For = 2 the ineqality 5.1) is trivially satisfied. For = 3, 4, and 5 we se 3.1) rather than 3.2) and instead of 6.16) and 6.17) we need to chec that 6.18) 2) log 1 + 2t 5/2 ) log 4 5/2 for t log Since again the left-hand side of 6.18) is increasing with respect to t, we insert t = log 0.17 and verify 6.18) by a straightforward calclation. This completes the proof. P r o o f o f T h e o r e m 5.1. Let s pt = + 2t + log t. Inserting into 6.16), taing logarithms of both sides, setting v = t/ and rearranging we redce 6.16) to 6.19) log ) log v 1 + 2v + log v 2 log2 + 2 v + log v) κ 0. A rotine althogh laborios) calclation shows that at least for every 12 the left-hand side of 6.19) is increasing with respect to v in the interval [1/8, ). So, for the first case, it is enogh to chec that 6.20) log log/8) ) log log log ) κ 0. 8 Using the facts that the fnction ζy) = log y)/y is decreasing on the interval e, ), that the expression log y)/ y attains maximal vale 2e 1 and 32, we can write 2 log log ) 2ζ2 + [ log 4) log/8) ] ζ log 4) e Chi-sqare qantiles 349 Hence and from the ineqality log1 + y) y y 2 /2 the relation 6.20) can be redced to log log log κ [ log e 32 log 5 4 [ e ] log ) κ ] log κ 0, which clearly holds tre. This completes the proof of the first case. The second case, i.e. 18 and v 1/3, can be proved in a similar way. P r o o f o f T h e o r e m 5.2. Proceeding in a similar way to that in 6.17) bt for = + 2t + t/4 we redce 5.3) to the following ineqality: 6.21) ξ v) = log 1 + 2v + v 4 ) v 2 log 2 log + 2v + 4 ) v κ 4 for 17 and v [log 17)/, 560]. A rotine calclation shows that for every 17, ξ v) is increasing on some interval v 1 ), v 2 ) ), 0 v 1 ) 9 v 2 ), and decreasing on the complementary intervals in 0, ). Moreover, for v = 0 as well as for v the relation 6.21) is not satisfied. So, if for some v1 ) v 2 ) we shall have ξ v i ) ) κ, i = 1, 2, then 6.21) holds also in [v1 ), v 2 )]. Now, we chec that for all 17 we can tae v2 ) = 560. Indeed, 6.21) for v = 560 has the form [log ) 35] 2 log2 + [ ] ) κ. Since the left-hand side of the above ineqality is increasing with respect to and for = 17 the ineqality holds tre, or claim is proved. Finally, we show that we can tae v1 ) = log 17)/. In other words, we need to prove that for 17 [ log ) log 17 log ) log 17 1 ] log 17 log log 17 log 17 2 log ) log 17 κ. 4 Clearly, for 17 we have log 17)/ log 17)/ Conseqently, the last term on the left-hand side of 6.22) is greater than Moreover, the 350 T. Inglot first term is increasing with respect to. Indeed, this follows since the fnction [log1 + 2v + v/4) v/4]/v is decreasing on 0, 1). So, inserting = 17 to the first term of 6.22) we see that the left-hand side is greater than 1.741, which concldes the proof. 7. APPENDIX In this section, as a byprodct of or stdy we propose sing the reslts of previos sections simple, explicit approximation formlae, which are easy to calclate for the normal and chi-sqare qantiles. Normal qantiles. Ptting w = 2t = 2 log1/α) one can give an approximation ẑ α for the normal qantiles z α by fitting constants A and B in the formla cf. 2.2)) A.1) ẑ α = w A log w + B. w We propose to tae A = and B = With these constants A.1) gives for α [ , 0.1] a qite precise approximation as is shown in Table 1. For larger or smaller vales of α this approximation also wors althogh is less accrate. Table 1. Approximation of the normal qantiles α z α ẑ α We have fitted the constants A and B to obtain a minimal vale of th

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