REVISTA INVESTIGACION OPERACIONAL Vol. 21, No. 2, 2000

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1 REVISTA INVESTIGACION OPERACIONAL Vol., No., 000 A COMPARISON OF APPROXIMATIONS TO PERCENTILES OF THE NONCENTRAL t-distribution Hardeo Sahai, Department of Biostatistics and Epidemiology, University of Puerto Rico, San Juan, Puerto Rico Mario Miguel Ojeda, Facultad de Estadística e Informática, Universidad Veracruzana, Veracruz, México ABSTRACT In this paper we review the main proposed approximations to percentiles of the noncentral t-distribution. The approximations are examined for their acuracy over a wide range of values of the parameters of the distribution and for several percentil values. Tables summarizing the approximations are included. Key words: Cornish-Fishex expansions, Taylor Series, Percentiles RESUMEN En este artículo revisamos las principales aproximaciones propuestas para calcular percentiles de la distribución t nocentral. La precisión de las aproximaciones se examina para un amplio rango de valores de los parámetros de la distribución y para varios valores del percentil. Se incluyen tablas que resumen las aproximaciones. Palabras clave: Expansiones Cornish-Fishex, Series de Taylor, percentiles. INTRODUCTION AND SUMMARY It is widely recognized that the noncentral t-distribution is of considerable theoretical and practical importance in many mathematical and statistical applications. For instance, noncentral t-distribution is useful in evaluating power function of the Student s t-test (Owen, 968, 985; Jonhson et al., 995, pp ), calculating confidence interval of the coefficient of variation (Lehmann, 986, p. 5, Johnson et al., 995, pp. 50-5; Vangel, 996), approximating the distribution of sample coefficient of variation and calculation of its percentage points (McKay, 9; Iglewicz et al., 968; Vangel, 996), calculating confidence limits on the proportions in the tail of a normal distribution (Durrant, 978; Odeh and Owen, 980), constructing confidence limits on one-sided quantiles and tolerance limit for the normal distribution (Wolfowitz, 946; Johnson et al., 995, pp. 5-5), one-sided tolerance limits for the linear regression (Kabe, 976), and in the study of acceptance sampling plans involving proportion of defective items (Owe, 968; 985). Guenther (975) describes the use of noncentral t- distribution in testing hypotheses involving the quantiles of two normal populations. In many applications involving the noncentral t-distribution, one has to compute its percentiles involving the evaluation of the inverse probability functions (see, e.g. Bagui, 99, 996). However, the evaluation of such inverse functions is extremely tedious involving slow and expensive techniques of numerical iteration such as the Newton-Raphson procedure (see, e.g., Ralston and Wilf, 967; Carnahan et al.,969). There are a number of approximations for computing the percentage points of these distributions, at arbitrary probability levels, available in the literature. The applicability of several of these approximations is further enhanced by ease of their computational simplicity. The purpose of this paper is to compare these approximations to determine their accuracy. Some of these approximations were previously investigated by Johnson and Welch (940), (96), Kramer (97), Akahira (995) and Akahira et al. (995). A brief description of each procedure is given and appropriate tables comparing their accuracy, calculated for each procedure, are presented. A more comprenhensive set of tables is given in Sahai and Ojeda (998).. APPROXIMATIONS The noncentral t-distribution was first derived by Fisher (9) who also showed how the tables of the standard normal distribution could be used to approximate this distribution. There are several approximations

2 to percentiles of the noncentral t-distribution available in the literature. Some of the important ones are considered here. In this paper, t ( will be used to denote a noncentral t-variate with degrees of freedom and the noncentrality parameter δ. In addition, ( will denote its 00-th percentile defined by: [ t ( t ( δ ] = ) Pr,. Jennett and Welch (99), that t, normal distribution, independently) and K is a constant, gave the approximation X + K X is approximately normally distributed, where X has a standard X has a chi-square distribution with degrees of freedom (X and ( b ) ( δ z ) ( b ) X are distributed δb + z b + t, (, (.) b z where b = Γ{( + ) / } Γ( ) and z is determined by Pr z { Z z } = ( π ) exp z dz =. < Akira (995) obtained the approximation (.) as a special case of the Cornish-Fisher expansion by ignoring terms of higher order than o( - ). Johnson and Welch (940) simplified the approximation (.) leading to the approximation ( δ z ) δ + z + t δ, ( ) (.) z Masuyama (95) obtained values of this approximations using an improved binomial paper. Akahira (995) obtained the approximation (.) as a special case of (.) by letting b and - b /(). An approximation intermediate between (.) and (.) was given by (96) as ( δ z ) δb + z b + t, (. (.) b Z Approximations (.), (.) and (.), however, give real values for t, ( only for limited ranges of values of δ and z (see, e.g., Johnson et al., 995, p. 5).

3 For small values of δ and large values of (> 0), the simple approximation of the standardized ( variate by a standard normal variate yields the result (Johnson et al., 995, p. 5): ( + δ ) δ t, ( δb + z b (.4) ( ) The normal approximation (.4) is, of course, applicable for very small values of δ and large values of and is included here for the sake of completeness. Cornish and Fisher (97) (see also Fisher and Cornish, 960) expansion applied to the distribution of t ( (expansion up to and including terms in - ) yields the following approximation (, 96; Johnson et al.,. 995, p. 54): 5 [ z + z + ( z + ) δ + zδ ] + [ 5z + 6z t, ( z + δ + + z ( 4z + z + ) δ + 6 ( z + 4z ) δ 4 ( z ) δ z δ ]. + (.5) Shibata (98) derived approximation (.5) from the taylor series expansion of the characteristic function of t ( δ with a chi-square variate having degrees of freedom. Akahira (995) derived it by applying the Cornish-Fisher expansion and using the characteristic function of a chi-square variate with degrees of fredom. The corresponding Cornish-Fisher expansion applied to the central t-distirbution (δ = 0) gives the result (see, e.g., Sahai and Thompson, 974): t t, z zz + 4 5z z 96 + z. If these terms in (.5) are replaced by t,, (.5) becomes (, 96; Johnson et al.,. 995, p. 54). [ δ 4 ( t + δ + ( + z + δ z ) + δ ( 4z + z ) + 6 ( z + 4z ) δ 4 ( z ) δ zδ ] t,, (.6) We will call the approximations (.5) and (.6) as Cornish and Fisher s st and nd approximations respectively: Azorin (95), starting from the relationship var ( t ( ) = a + b [ E ( t ( δ ) )], with a =, ( ) b = Γ { [( )] } ( ) ( ) and (.7) E ( t ( ) = ( ) Γ ( ( ) ) Γ ( ) δ

4 obtained the transformation sinh b b b ( t ( ) sinh ( E ( t ( )) a a b, (.8) which is to be approximated as a standard normal variate. In addition, Azorin (95) suggested two similar transformations of simpler forms: ( t ( ) sinh (.9) and sinh ( t ( ). (.0) The standarized versions of (.9) and (.0), correcting for mean and standard deviation of the transformed variable to terms of order -, are ( t ( ) δ δ + ( δ ) sinh 4 δ (.) and sinh ( t ( ) δ ( δ ) + ( δ ) (.) We will call approximations (.8), (.) and (.) as Azorin s st, nd and rd approximations respectively. There is a considerable degree of skewness in the noncentral t-distribution for large values of δ and small values of (Johnson and Welch, 940). Thus, approximations (.8), (.) and (.) are expected to be rather very poor for simultaneously small values of and large values of δ. The results of numerical computations show that these approximations are not at all satisfactory; however, they have been included here for the sake completeness. Also, for small values of, the quality of approximations deteriorates rather rapidly as δ increases. For very large values of, the approximations improve somewhat, bull still are not to be recommended. Only Azorin s st approximation is included in our comparative study. Laubscher (960) also considered the transformation (.8). Furthermore, he proposed two modifications of (.8) of the form: L and L b b ( ) = sinh t ( sinh μ + b μ μ (.) b a b a 4 = L b μ 6 ( ) 5 where a, b and = E( t ( ) t ( given by μ [ μ ( a / b )], (.4) μ are given as in (.7) and μ and μ are the second and thirs central moments of 4

5 ( + δ ) ( δ + ) μ = μ and μ = μ μ. ( ) ( )( ) The approximations (.) and (.4) are expected to eliminate more bias than (.8). In addition, following Laubcher s (960) conjecture, we consider the approximations ( ) and [ ] (/ ) t ( [ ( + δ ) ( + δ ) ( + δ ) ] [( )t ( + ( + δ ) ( + δ )] (.5) ( / 9 ) (+ δ ) [ t ( ) ( ) ] δ + δ ( + δ ) 9( + δ ) / 9 (+ δ ) + ( / 9) [ t ( (+ δ )] /, (.6) where each is to be approximated as a standard normal variate. We will call approximations (.), (.4), (.5) and (.6) as Laubscher s st, nd, rd, and 4 th approximations respectively. Harley (957) suggested an approximation of t ( in terms of a function of the sample correlation coefficient (r), in a random sample of size n = + from a bivariate population with correlation coefficient ρ, by the relationship r ( + ) t ( = (.7) where ( r ) ( + + δ ) ρ = δ ( + + δ ) Approximation (.7) is of course valid for δ ( + ) ½ and is thus appplicable for only small values of δ. The percentiles of t ( can be approximated from the percentiles of r by using relationship (.7). One can obtain the percentiles of r by using the Fisher s Z-transformation + r Z = loge (.8) r + ρ which is considered as approximately normally distributed with mean loge and variance /(n-). ρ However, following the recommendation of David (98), we approximate (.8) by a normal random variable with mean μ and variance σ given by 5

6 μ = and σ + ρ loge + ρ ρ 5 + ρ + (n ) 4(n ) 4 ρ 6ρ ρ = + + n (n ) 6(n ) 4. This approximation is considered to be the most accurate one of all the existing normal approximations of r (see, e.g., Kraemer, 97). Another kind of approximation of r was considered by Ruben (966) who showed that distributed as r r is Z + χn ρ ( ρ ) χn, (.9) where Z is a unit normal variate, χ is a chi-variate with degrees of freedom, and Z, χ n- and χ n- are mutually independent. For large values of n, χ n- and χ n- may be approximated by normal variates using Fisher s approximation that χ is approximately distributed as a unit normal variate. Using these results, it can be shown that the transformed variate r ( ) r n ρ ( ρ ) + r ( r ) + ρ ( ρ ) 5 n (.0) has a standard normal distribution. We will call the approximations of the type (.7), based on percentiles of r via (.8), its improved version due to David, and (.0), as Harley s, st, nd, and rd approximations respectively. Merrington and Pearson (958) gave an approximation to the percentiles of t ( based on an approximations by a Pearson Type IV distribution. Let β and β be the moment rations, i.e., μ β = and μ μ β =, μ 4 where μ, μ and μ 4 denote the second, third and fourth central moments respectively of the noncentral t- distribution. Then, we have ( β, β ) t, ( μ + σ U,, (.) where ( β, β, ) U is the 00-th percentiles of the standardized Pearson Type IV distribution and μ and σ are the mean and standard deviation respectively of the noncentral t-distribution. The approximation (.) is of course applicable for > 4 since μ 4 does not exist for 4. This approximation, however, is not included in our comparative study because of complexity in computing the percentiles of the Pearson Type IV distribution. 6

7 (96) developed bounds for the percentiles of t, ( given by δ t ( 0.5), ( + t, χ, and (.) δ t ( 0.4),, ( + t, χ, where t, and χ, denote 00 - th percentiles of central t and χ distributions respectively. Although approximations (.) in not of great accuracy, it is included here to investigate the sharpness of the bounds. Kraemer and Paik (979) proposed a central t approximation to the noncentral t-distribution by the relationship Pr [ ] δ ( + δ ) t ( ) t Pr t t / δ ( + t / ). Akahira (995) and Akahira et al., (995) derived an higher order approximation formula from the Cornish- Fisher expansion for the statistic based on a linear combination of a normal random variable and a chi-square random variable. The approximate percentile ( derived from the formula is determined by the relationship: b t + t (.),, ( δ ( ( b ) z 4 where b is defined as in (.). t, t, ( ( z ) + { ( )} + t ( b 4,, Approximation (.) gives only an implicit expression and will require an iterative procedure for its solution. Akahira et al., (995) showed that for a fixed such that z and for sufficiently large which is independent of δ, the solution to (.) exists uniquely. The existence of solution is guaranteed when for =, for =, for =, and for 4. Approximation (.) is not included in our comparative study. However, Akahira et al., (995) made a detailed numerical comparison of this along with Jennett-Welch, and ( st Cornish- Fisher) approximations and found that approximation (.) had better numerical precision than others included in the study.. RESULTS The percentiles t, ( calculated for various approximations as well as the exact values, for selected values of, and δ, are given in Table. On comparing the approximate values of t, ( with the exact values one notices some very interesting results. For higher percentiles and small values of, Johnson- Welch and normal approximations perform best; however, for smaller values of δ, the approximation in less accurate than the normal. As δ increases, the accuracy of approximation improves. For moderate to large values of, the Jennett-Welch and approximations are superior to others; the former being better than the latter. For 50 th percentile ( = 0.5), the Jennett-Welch and approximation are equivalent and perform better than others. In this case, the approximation reduces to δ. This fact partially confirms the validity of the computations since they were obtained using the general formulae for all the approximations. For the lower percentiles, the normal approximation performs very poorly. In this case, the approximation perfomrs better except 7

8 when both and δ are small and the Jennett-Welch approximation is superior. For all the cases, when is sufficiently large, all the approximations compare favorably. Both the st and nd Cornish-Fisher approximations provide excellent results for moderate to large values of and small values of δ. However, both approximations progressively degenerate as λ increases, especially for small values of and extreme lower and upper percentiles. In this regard, the performance of the is much worse than the nd one. The three approximations due to Azorín perform poorly, especially for small values of. Azorín s nd and rd approximations show even much poorer performance than the st approximation and their results are not included in Table. Similarly, Laubscher s approximations, especially rd and 4 th, show a rather poor performance, particularly for higher percentiles. For large values of, both Azorín and Laubscher type approximations show a spectacular improvement, but are still not to be recommended. There is not much difference among three approximations of Harley, based on distinct approximations of percentiles of the sample correlation coefficient and the approximations deteriorate rather rapidly for large values of δ. As expected, s bounds are not too sharp, and are to be recommended only as a crude approximation. Finally, Kraemer-Pike approximation gives a uniformly poor performance, especially for small values of and large values of δ. For lower percentiles, the approximation gets better as increases, but is still not to be recommended. ACKNOWLEDGMENTS The exact percentiles of the noncentral t-distribution were calculated using AMOSLIB, a special function library prepared at Sandia Laboratories, Alburquerque, New México. We are indebted to Dr. Donald E. Amos for his courtesy in providing the pertinent information about the routiness including the necessary software to perform the requisite computations. We also thank Professor Constance for reading a preliminary draft of the manuscript and making some helpful comments and suggestions. This work would not have been possible without the generous dedication of time and efforts by Rafael Guajardo Panes and Lorena López whose contributions are gratefully acknowledged. Table. Approximate and exact percentiles of the noncentral t-distribution APPROXIMATIONS = 0.05 δ nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values. Table. (Continued) 8

9 0 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values. 9

10 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values. 0

11 Table. (Continued) 0 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values.

12 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values.

13 Table. (Continued) 0 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values

14 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 4

15 Table. (Continued) nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 5

16 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 6

17 Table. (Continued) 0 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 7

18 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley ''' designates undefined values 8

19 Table. (Continued) 0 6 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 9

20 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley = nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 40

21 Table. (Continued) 0 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values 4

22 Table. (Continued) 50 nd Cornish-Fisher nd Laubscher nd Harley nd Cornish-Fisher nd Laubscher nd Harley '' designates undefined values' REFERENCES AKAHIRA, M. (995): "A higher Order Approximation to a Percentage Point of the Non-Central t-distribution", Communications in Statistics, Part B: Simulation and Computation, 4, AKAHIRA, M.; M. SATO and N. TORIGOE (995): "On the New Approximation to Non-Central t-disribution", Journal of Japan Statistical Society, 5, -8. AZORIN, P.F. (95): "On the Noncentral t-distribution", I. II, Trabajos de Estadística, 4, 7-98 and 07-7 (In Spanish). BAGUI, S.C. (99): CRC Handbook of Percentiles of Noncentral t-distributions. CRC Press, Boca Ratón, Florida. (996): ): CRC Handbook of Percentiles of Noncentral t-distributions. CRC Press, Boca Ratón, Florida. 4

23 CARNAHAN, B.; H.A. LUTHER and J.O. WILKES (969): Applied Numerical Methods. John Wiley and Sons, New York. CORNISH, E.A. and R.A. FISHER (97): "Moments and Cumulants in the Specification of Distribution", Revue de Institute International de Statistique, 4, -4. DURRANT, N.F. (978): "A Nomogram for Confidence Limits on Quantiles of the Normal Distribution with Application to Extreme Value Distribution", Journal of Quality Technology, 0, FISHER, R.A: (9): "Introduction to Table of Hh Functions", In: Table of Hh Function, pp. xxvixxxv. (Ed. J.R. Airey). British Association Mathematical Tables, London ( nd ed. 946, rd ed. 95). GUENTHER, W.C. (975): "A Sample Size Formula for a Non-Central t test", The American Statistician, 9, 0-. HALPERIN, M. (96): "Approximations to the Noncentral t, with Applications", Technometrics, 5, HARLEY, B.I. (957): Relation Between the Distribution of Noncentral t and a Transformed Correlation Coefficient", Biometrika, 44, 9-4. IGLEWICZ, B.; R.H. MYERS and R.B. HOWE (968): "On the Percentage Points of the Sample Coefficient of Variation", Biometrika, 55, JOHNSON, N.L; S. KOTZ and N. BALKRISHNAN (995): Continuous Univariate Distributions,, nd. ed. John Wiley and Sons, New York. and B.L. WELCH (940): "Application of the Noncentral t-distribution", Biometrika,, KABE, G. (976): "On Confidence Bands for Quantiles of a Normal Population", Journal of the American Statistical Association, 7, KRAEMER, H.C. (97): "Improved Approximation to the Non-null Distribution of the Correlation Coefficient", Journal of the American Statistical Association, 68, and M. PAIK (979): "A Central t-approximation to the Noncentral t-distribution", Technometrics,, LAUBSCHER, N.F. (960): "Normalizing the Noncentral t and F Distributions", Annals of Mathematical Statistics,, 05-. LEHMANN, E.L. (986): Testing Statistical Hypotheses, nd. ed. John Wiley and Sons, New York. MERINGTON, M. and E.S. PEARSON (958): "An Approximation to the Distribution of Noncentral t". Biometrika, 45, ODEH, R.E. and D.B. OWEN (980): Tables for Normal Tolerance Limits Sampling Plans and Screening. Marcel Dekker, New York. OWEN, D.B. (968): "A Survey of Properties and Applications of the Noncentral t-distribution", Technometrics, 0,

24 (985): "Noncentral t-distribution", In: Encyclopedia of Statistical Sciences, 6, pp (Eds. S. Kotz and N.L. Johnson). John Wiley and Sons, New York. RALSTON, A. and H. WILF (967): Mathematical Methods for Digital Computers. John Wiley and Sons, New York. RUBEN, H. (966): "Some New Results on the Distribution of the Sample Correlation Coefficient", Journal of the Royal Statistical Society Series B, 8, SAHAI, H. and M.M. OJEDA (998): Approximations to Percentiles of the Noncentral t, χ and F. Distributions. University of Veracruz Press, Xalapa, Veracruz, México. SHIBATA, Y. (98): Normal Distributions. Tokyo University Press, Tokyo. (In Japanese). VAN EEDEN, C. (96): "Some Approximations to the Percentage Points of the Noncentral t-distribution", International Statistical Review, 9, 4-. VANGEL, M.G. (996): "Confidence Intervals for a Normal Coefficient of Variation", The American Statistician, 50, -6. WOLFOWITZ, J. (946): "Confidence Limits for the Fraction of a Normal Population Which Lies Between Two Given Limits", Annals of Mathematical Statistics, 7,

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