On the exact computation of the density and of the quantiles of linear combinations of t and F random variables

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Journal of Statistical Planning and Inference 94 3 www.elsevier.

Sciences, Dubravska cesta 9, 849 Bratislava, Slovak Republic Received 4 June ; received in revised form 5 July ; accepted 7 August Abstract The inversion formula for evaluation of the distribution of

1 Journal of Statistical Planning and Inference On the exact computation of the density and of the quantiles of linear combinations of t and F random variables Viktor Witkovsky Institute of Measurement Science, Slovak Academy of Sciences, Dubravska cesta 9, 849 Bratislava, Slovak Republic Received 4 June ; received in revised form 5 July ; accepted 7 August Abstract The inversion formula for evaluation of the distribution of a linear combination of independent t and F random variables, respectively, is suggested. The method is applied to computing the exact condence intervals for the common mean of several normal populations. This method is compared with the known approximate methods. c Elsevier Science B.V. All rights reserved. MSC: primary 6E5; secondary 6F5 Keywords: Characteristic function of the t distribution; Characteristic function of the F distribution; Linear combinations of t and F random variables; Common mean; Condence interval. Introduction In this paper we discuss a method for numerical evaluation of the distribution function and=or the density function of a linear combination of independent Student s t and Fisher Snedecor s F random variables, respectively. The method is based on the inversion formula which leads to the one-dimensional numerical integration. The characteristic function of the t and the F distribution depends on the special mathematical functions possibly of complex argument. In particular, the characteristic function of the t distribution depends on the modied Bessel function of the second kind and the characteristic function of the F distribution depends on the conuent hypergeometric function of the second kind. The suggested method can be used for computing the exact quantiles of the required distributions. address: umerwitk@savba.sk V. Witkovsky //$ - see front matter c Elsevier Science B.V. All rights reserved. PII: S

2 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 For illustration of the method we consider the problem of computing the exact condence intervals for the common mean of k normal populations. We do not suggest new methods for constructing the exact condence interval nor discuss the optimality properties of the exact condence intervals. Rather than that we employ the method suggested by Fairweather 97 and that suggested by Jordan and Krishnamoorthy 996, respectively. The reader who is interested in the other methods for constructing the exact condence interval for the common mean of several normal populations and their comparison should read the paper by Yu et al The inversion formula Gil-Pelaez 95 derived a version of the inversion formula which is useful for numerical evaluation of a general distribution function by one-dimensional numerical integration: Theorem. Let t = eitx dfx be a characteristic function of the onedimensional distribution function Fx. Then; for x being the continuous point of the distribution; the following inversion relationship holds true: Fx= e itx t e itx t dt it = e itx t Im dt: t Furthermore, it is easy to observe that if the distribution belongs to the continuous type then the density function is given by fx= = e itx t e itx t dt Ree itx t dt: The limit properties of the integrand in are given by the following Lemma : Lemma. Let Fx be a distribution function of a random variable X with expectation EX and its characteristic function t. Then lim Im t e itx t t = EX x; and lim t Im e itx t t =: 3

3 V. Witkovsky / Journal of Statistical Planning and Inference Proof. We will show the rst equality: e itx lim Im t e itx t e itx t = lim t t t i t = i e itx t t= = i ixe itx t+e itx t t= = i t t= ix=ex x: 4 The second equality is a direct consequence of the fact that the function e itx t is bounded in modulus. Consider now X = n k= kx k a linear combination of independent random variables and let Xk t denote the characteristic function of X k, k =;:::;n. The characteristic function of X is X t= X t Xn n t 5 and the distribution function F X x =Pr{X 6x} is given by with t = X t. Notice that e itx lim Im X t = n k EX k x; 6 t t lim Im t e itx X t t k= =: 7 Formula is readily applicable to numerical approximation of the distribution function F X x using a nite range of integration 6t6T, T. In general, a complex-valued function should be numerically evaluated. The degree of approximation depends on the error of truncation and the error of integration method. An interesting application of the above inversion formula was given by Imhof 96 who derived the formula to calculate the distribution of a linear combination of independent non-central chi-squared random variables X = n k= kx k, where X k k k, with k degrees of freedom and the non-centrality parameter k. For review and discussion on numerical inversion of the characteristic function as a tool for obtaining cumulative distribution functions, see Waller et al Characteristic function of the t distribution Theorem. Let X t be a random variable that has Student s t distribution with degrees of freedom with its probability density function given by fx= =+ =+= + x = ; 8 =

4 4 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 where x. Then the characteristic function of X t is t t= = = = t = K = { = t }; 9 where K {z} denotes the modied Bessel function of second kind. Proof. The characteristic function of X t is t t=ee itx = =+ =+= = e itx + x dx = = =+ = = = = =+ = = = = =+ = = = = =+ = = = e itx + x dx =+= [ e itx + x dx + =+= e itx +e itx + x dx =+= ] e itx + x dx =+= costx + x dx: =+= In accordance with 9:6:5 in Abramowitz and Stegun 965, p. 376 we have K = {tz} = =+ z= = t = costx x + z dx =+= for Re, t, and arg z. Choosing z = = and using the fact that costx = cos tx we get the result. Lemma. The characteristic function of the random variable X t can be calculated by using 9 and the recurrence relation K k=+ {z} = K k= {z} + k z K k={z}; where z = = t. If is even; =n for some integer n; then k =; 4;:::; ; given K {z} and K {z}. If is odd; =n + for some integer n; then k =3; 5;:::; ; given K = {z} and K 3= {z}. Proof. The proof is a direct consequence of recurrence relation 9:6:6 in Abramowitz and Stegun 965, p. 376 Z {z} Z + {z} = z Z {z}; 3 where Z denotes e i K {z}. The following lemma is in accordance with the result given by Mitra 978. For more details see Johnson et al. 995, p. 367.

5 V. Witkovsky / Journal of Statistical Planning and Inference Lemma 3. If is odd; =n + for some integer n; then the characteristic function of the random variable X t is t t= n t exp{ = t }; 4 where n t is given by the recurrence relation k+ t= t k + k k t+ k t; 5 k =;:::;n ; where t=; and t=+ = t. Proof. Eq. ::7 in Abramowitz and Stegun 965, p. 444 states that = K = {z} = exp{ z}; z = K 3= {z} = exp{ z} + z ; z = K 5= {z} = exp{ z}+3z +3z : 6 z Moreover, if we dene f k z= k+ = Kk+= {z}; 7 z then according to Eq. ::8 in Abramowitz and Stegun 965, p. 444 we get f k z f k+ z=k +z f k z 8 for k =; ±; ±;::: : Let z = = t. Then from 7 we get K = z=k n+= z= n+ = fn z: 9 z Using 9 and 6 and after some further simple algebraical manipulations we get the result. Corollary. In particular; the characteristic function t t of the random variable X t with =n + degrees of freedom where n =; ; ; 3 is t t = exp{ t }; t 3t=+ 3 t exp{ 3 t }; t 5t= + 5 t + 53 t exp{ 5 t }; t 7t= + 7 t t t 3 exp{ 7 t }:

6 6 V. Witkovsky / Journal of Statistical Planning and Inference Characteristic function of the F distribution Theorem 3. Let X F ; be a random variable that has central Fisher Snedecor F distribution with and degrees of freedom with its probability density function given by fx= = =+ = x = + =+ = x ; = = where x. Then the characteristic function of X F ; is F ; t= =+ = = ; ; it ; where a; c; z denotes the conuent hypergeometric function of the second kind dened by the integral equation a; c; z= a Proof. See Phillips 98. e zt t a + t c a dt: 3 Lemma 4. Denote a = =; c= =; and z = it =. The series representation of can be obtained from the following formulae: a; c; z= c a c + F a; c; z+ for non-integral c. For c = n with n =; ;:::; c z c F a c +; c; z a 4 a; n; z=z n a + n; n +;z = n z n [ F a + n; n +;z log z + a + n k n! a k= n + k k! ] { a + n + k + k + n + k}z k n! n a k + a + n k k= n k k! zk ; 5 where F a; c; z denotes the conuent hypergeometric function of the rst kind; x = x= x is the logarithmic derivative of the gamma function; and where a n = aa + a + a + n ; a =. The nal term in 5 is omitted if n =. Proof. See Phillips 98. See also Eqs. 3::3 and 3::6 in Abramowitz and Stegun 965, p. 54. Corollary. When is odd then for small t we have the following expansion: F ; t= A k it = B k ; 6 A k=

7 V. Witkovsky / Journal of Statistical Planning and Inference where A = sin = A k = =+k kk = it B = =+ = + = ; B k = =+ =+k kk + = ; A k ; it B k : 7 Proof. The proof is a direct consequence of, 4 and the expansion of F a; c; z: F a; c; z=+ az c + a z c! + + a nz n + ; 8 c n n! where a n = aa + a + a + n ; a =. See Eq. 3:: in Abramowitz and Stegun 965, p. 54. Lemma 5. For z large; z= it = ; the characteristic function F ; t given by can be approximated by using ; ; it = it [ = R = k =+ = k it k +O k! it R] ; where O it R k= = = R =+ = R it R R! [ = =+ 4 it = 4 R +O it = it ] : Proof. See Eqs. 3:5: and 3:5:3 in Abramowitz and Stegun 965, p. 58. Lemma 6. The characteristic function of the random variable X F ; calculated by using and the recurrence relations can be k +;c; z= [ k ;c; z+c k z k; c; z]; 9 kc k where c = = and z = it =. If is even; =n for some integer n; then k =3; 4;:::; = ; given ;c; z and ;c; z. If is odd; =n + for some

8 8 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 integer n; then k = 3 ; 5 ;:::; = ; given ;c; z and 3 ;c; z; and a; k ; z= [ k z a; k; z+z a; k +;z]; 3 +a k where a = = and z = it =. If is even; =n for some integer n; then k = ; ;:::; =; given a; ; z and a; ; z. If is odd; =n + for some integer n; then k = ; 3 ;:::; =; given a; ; z and a; ; z. Proof. See the recurrence relations 3:4:5 and 3:4:6 in Abramowitz and Stegun 965, p. 57. Lemma 7. The following series expansions can be used to calculate : F a; c; z= a + a = 4 e z= z n + a a na c { n z } n!c n a I n+a = ; 3 F a; c; z= n= c a + c a = 4 e z= z n n+c+a c a nc a { n z } n= n!c n c a I n+c+a = ; 3 where I {z} denotes the modied Bessel function of the rst kind. Proof. See Eqs. 7:8:7 and 7:8:8 in Luke 98, p Application to computation of the exact condence intervals for the common mean of several normal populations Let us assume that we have k independent populations where the ith population follows N; i distribution with common mean and possibly unequal variances i ;i=;:::;k. Let X ij ;j=;:::;n i n i, be a random sample from the ith population. We dene X i and Si as X i = n i n i j= X ij ; S i = n i n i j= i = ;:::;k. The random variables X i and Si X i N ; i n i X ij X i ; 33 are mutually independent and ; n i Si i n i ; i=;:::;k: 34

9 V. Witkovsky / Journal of Statistical Planning and Inference From that we have ni X i t i = t ni ; F i = n i X i S i S i F ;ni ; 35 i =;:::;k: Fairweather 97 suggested construction of the exact condence interval for using a weighted linear combination of the Student s t i statistics, namely W t = k i= u i t i ; u i = Vart i k i= Vart i = n i 3=n i k j= n j 3=n j : 36 Note that Vart i exists if n i 3. If b = denotes the upper cut-o point of the distribution of W t, such that for given ; P W t 6b = = ; 37 then the exact symmetric, two-sided %-condence interval for is obtained as [ k i= ni u i X k ] i =S i b = i= ni u i X i =S i + b = k ; : 38 i= ni u i =S i ni u i =S i k i= To derive the approximate cut-o point b = Fairweather 97 suggested to approximate the distribution of W t by that of ct where and c are determined by equating the second and fourth moments of ct to those of W t. In particular, if we additionally assume that n i 5 for all i =;:::;k then we get =4+ k i= u i =n i 5 ; c= k i= n i 3=n i : 39 Jordan and Krishnamoorthy 996 suggested using a weighted linear combination of the F i statistics, using weights inversely proportional to variances VarF i, namely W f = k [n i 3 n i 5]=[n i n i ] w i F i ; w i = k i= [n j 3 n j 5]=[n j n j ] : 4 i= Note that VarF i exists if n i 5. If a denotes the cut-o point of the distribution of W f, such that for given ; PW f 6a = ; 4 then the exact symmetric, two-sided %-condence interval for is obtained as [ k ] k p i X i ; p i X i + ; 4 where i= i= p i = w i n i =S i k j= w jn j =S j 43

10 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 Table Percentage of albumin in plasma protein Experiment n i Mean Variance A B C D and = a k i= w in i =S i { k i= k } p i X i p i X i : 44 i= To derive the approximate cut-o point a Jordan and Krishnamoorthy 996 suggested to approximate the distribution of W f by that of cf k; where and c are determined by equating the rst two moments of cf k; to those of W f. In particular, if we assume that n i 5 for all i =;:::;k then we get = 4kM k +M km k +M where and M = EW f = k i= M = EW f =3 k w i n i n i 3 i= ; c= M ; 45 w i n i n i 3n i 5 + i j 46 w i w j n i n j : 47 n i 3n j 3 To compare the exact and the approximate methods suggested for computing the cut-o points we provide two examples considered and analyzed by Jordan and Krishnamoorthy 996 and Yu et al Example. Here we examine the data reported in Meier 953 about the percentage of albumin in plasma protein in human subjects. The data given in Table are based on four independent experiments. It is assumed that the samples are from normal populations. We would like to combine the results of the four experiments in order to construct a %-condence interval for the common mean on the signicance level =:5. We have applied the techniques suggested by Fairweather 97 and Jordan and Krishnamoorthy 996. The approximate upper cut-o point b :5 of the weighted linear combination W t = 4 i= u it ni of the Student s t ni statistics with weights u =[:55; :67; :78; :7]; can be computed according to 39 from the distribution of ct, where =6:3984 and c =:5367. This approximation leads to b :5 =:4 and the resulting condence

11 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 Table Selenium in non-fat milk powder Method n i Mean Variance A B C D interval estimate is [59:8973; 6:9] with the true coverage probability P W t b :5 =:954. The exact value rounded o to the fourth decimal place of the upper cut-o point b :5 is b :5 =:. This was computed numerically based on, 5 and 9. Finally, the exact 95%-condence interval estimate for is [59:8996; 6:899]: The approximate cut-o point a :5 of the weighted linear combination W f = 4 i= w if ;ni of the F ;ni statistics with weights w =[:6; :337; :987; :376]; can be computed according to 45 from the distribution of cf 4;, where =5:68 and c =:548. This approximation leads using a linear interpolation of the critical values of the F distribution to a :5 =3:98 and the resulting condence interval estimate is [59:56; 6:443] with the true coverage probability PW f a :5 =:953. The exact value of the cut-o point a :5 is a :5 =3:853 which was computed numerically based on, 5 and and the exact 95%-condence interval estimate for is [59:564; 6:44]: Example. Eberhardt et al. 989 reported the data on Selenium in non-fat milk powder. The data are given in Table. They are based on four independent measurement methods. It is assumed that the samples are from normal populations. The approximate upper cut-o point b :5 of the weighted linear combination W t = 4 i= u it ni of the Student s t ni statistics with weights u =[:39; :645; :736; :39]; can be computed according to 39 from the distribution of ct, where =:5633 and c =:548. This approximation leads to b :5 =:4 and the resulting con- dence interval estimate is [8:5349; :774] with the true coverage probability P W t b :5 =:954. The exact value of the upper cut-o point b :5 is b :5 =: computed numerically based on, 5 and 9. The exact 95%-condence interval estimate for is [8:5369; :77]:

12 V. Witkovsky / Journal of Statistical Planning and Inference 94 3 The approximate cut-o point a :5 of the weighted linear combination W f = 4 i= w if ;ni of the F ;ni statistics, with weights w =[:683; :39; :3543; :683]; can be computed according to 45 from the distribution of cf 4;, where =3:3466 and c =:778. This approximation leads to a :5 =3:44 and the resulting con- dence interval estimate is [8:453; :668] with the true coverage probability PW f a :5 =:95. The exact value of the cut-o point a :5 is a :5 = 3:3748 computed numerically based on, 5 and and the exact 95%-condence interval estimate for is [8:4588; :66]: 6. Concluding remarks Although the presented methods are not new, we believe that it would be benecial for statisticians to have them presented in one place, with examples of applications. According to our knowledge the closed form of the characteristic function of the Student s t distribution as presented in Theorem was unknown in statistical literature. The recurrence properties of the characteristic functions of t and F distributions were rst discussed in Ifram 97. The above examples show that the approximate methods for computing the con- dence intervals perform very well. However, we believe that the suggested method for computing the exact distribution of a linear combination of the Student s t random variables and=or the Fisher Snedecor s F random variables can be useful in more complicated unbalanced situations. The method is quite general without any restriction on the number of the variables, the values of the coecients and the involved degrees of freedom. The numerical implementation is easy, provided we have an ecient algorithm for numerical evaluation of the modied Bessel function of second kind and the conuent hypergeometric function of the second kind of a complex argument. For more details see e.g. Amos 986 and Nardin et al. 99. Acknowledgements The research has been supported by the grant VEGA =795= from the Scientic Grant Agency of the Slovak Republic. References Amos, D.E., 986. A portable package for bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Software,

13 V. Witkovsky / Journal of Statistical Planning and Inference Abramowitz, M., Stegun, I.A., 965. Handbook of Mathematical Functions. Dover, New York. Eberhardt, K.R., Reeve, C.P., Spiegelman, C.H., 989. A minimax approach to combining means, with practical examples. Chemometrics Intell. Lab. Systems 5, Fairweather, W.R., 97. A method of obtaining an exact condence interval for the common mean of several normal populations. Appl. Statist., Gil-Pelaez, J., 95. Note on the inversion theorem. Biometrika 38, Ifram, A.F., 97. On the characteristic functions of the F and t distributions. Sankhya: Ser. A 3, Imhof, J.P., 96. Computing the distribution of quadratic forms in normal variables. Biometrika 48, Johnson, N.L., Kotz, S., Balakrishnan, N., 995. Continuous Univariate Distributions, Vol., nd Edition. Wiley, New York. Jordan, S.M., Krishnamoorthy, K., 996. Exact condence intervals for the common mean of several normal populations. Biometrics 5, Luke, Y.L., 98. Specialnye matematiceskie funkcii i ich approksimacii Mathematical Functions and their Approximations. Mir, Moscow. Meier, P., 953. Variance of a weighted mean. Biometrics 9, Mitra, S.S., 978. Recursive formula for the characteristic function of Student t distributions for odd degrees of freedom. Manuscript, Pensylvania State University, State College. Nardin, M., Perger, W.F., Bhalla, A., 99. Algorithm 77: solution to the conuent hypergeometric function. ACM Trans. Math. Software 8, Phillips, P.C.B., 98. The true characteristic function of the F distribution. Biometrika 69, Waller, L.A., Turnbull, B.W., Hardin, J.M., 995. Obtaining distribution functions by numerical inversion of characteristic functions with applications. Amer. Statist. 49 4, Yu, P.L.H., Sun, Y., Sinha, B.K., 999. On exact condence intervals for the common mean of several normal populations. J. Statist. Plann. Inference 8,

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