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The distribution of student's ratio for samples from anexponential populationA.K.M. Sirajul Hoq a , Mir M. Ali b & J.G.C. Templeton ca Department of Mathematics , University of Riyadh , Riyadh, Saudi Arabiab Department of Mathematics , University of Western Ontario , London, Canadac Department of Industrial Engineering , University of Toronto , Toronto, CanadaPublished online: 27 Jun 2007.

To cite this article: A.K.M. Sirajul Hoq , Mir M. Ali & J.G.C. Templeton (1978) The distribution of student's ratio forsamples from an exponential population, Communications in Statistics - Theory and Methods, 7:9, 837-850, DOI:10.1080/03610927808827675

To link to this article: http://dx.doi.org/10.1080/03610927808827675

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CO4nlUN. STATIST. -THEOR. X T H . , A 7 (9) , 337-850 (1973)

A.K.M. Sirajul Hoq

Department of Industria: Engineering University of Toronro

Toronto, Canada

K e y Fiords and P h r a s e s : r o b u s t n e s s o f t - t e s t ; t a i l p r o b a b i l i t i e s ; g e o m e t r i c a l p r o b a b i l i t y ; s m a l l s a m p l e s t a t i s t i c s ; t e s t s o f l o c a t i o n ,

For a sample of size n drawn from an exponential population

with location parameter p and scale parameter o, let T = &/;;I?-)~)/s

where X is the sample mean and S, the sample standard deviation given by (n-l)s2 = Z (Xi-xi2. Pr {T 2 ti is obtained for values of

1,

t 2 ( (n-1) (n-2)/2)-2. For the special cases n = 2 , 3 , 4 exact expres-

sions for the probability are obtained for all values of t. This

work is motivated by a paper of Hotelling (13611, in which, among

other results, he obtained Pr { ~ 2 t ? for tln-1.

Copyright @ 1978 hy Marcel Dekker . Inc. All Rights Reserved, Neither this work n o r any part m s y be r ep roduced o r t r a n s m ~ t t e d In any f o r m o r by any means, e l e c t r o n ~ c o r mechanical , mcluding pho tocopymg, mic ro f i lm~ng , and recording, o r by a n y in fo rmat lon s torage and r e t r ~ e v a l system, w ~ t h o u t pe rmiss~on in w r ~ t i n g f r o m the publisher .

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' INTXCCCCTIC>T A.

Hotelling (1961) obtained the tail probabilities of the

L - - - 3LaLIJLLC ,-: -&: - r ~ r - -,,-allies ~f t > 3-ll i;:ier. the u.-iderlyir.i; s;~>le 1s

assumed to arise frum the expneiit~al oopuiaiion. i ie e x t e i d t i z i 5

+- :-->..2,. ..-.-~.-+ - A , ( < - - I , - - . ? \ d J 5 L C J U L L LV L ' I I - iLLUC Y a L u C 3 GI. i - ( i , ' i, i i l ii/L) .

In performing a :-test, the relevant portion of tkc discri-

burion of the t-stat-stic is rhe tail probabilities. A - - + - L --ste

in general, is called for when the sample size is small. Hence . . tht2 ~ x a c i thii prc,baiii i iirs, idpL)~~xi~~!-ti~~1~ ofLt!l ric.1 r LICLII 'J

accurate) of the t-distribution for small sam~les when the under-

lying sample is assumed to ar;se from a specifled nonnormal ppU-

lation would enable one to ascertain whether a test valid under

the normal~ty asscmption would also be robust under the sceclfier'

nonnormality assumption.

The present work, wb.ile a r i s i n g out ~ J E problems in robustness

studles, 1s in itself a very specialized and restricted mathematl-

.̂., _--Ll-_ L ~ L ~ L U U L C L L L . Hence. it has not been felt worth iiiiile to survey tte

literature in robustness of tests. However, two excellent re-

ports, one by Posten and Hatch (1966) and the other, an annotated

bibliography of robustness studies by Govindarajulu and Leslie

(1970) survey many of the works that bear on the question of ro-

bustness. Paong several more recent works, we I--+;-P ,,L.,Li,. Sansing and .. Owen ( L Y i r i , ~ u e n and iiurrhy (1974) and Sansing (13761. Hoq, Aii

and Templeton (19761, in a more detailed version of the present

paper, give additional details of proofs and a solution (not in

closed form! for the tail probabilities for ((n-l) (r,-3) / 4 ! 5 < t k

5 ((n-11 (n-2!/2) 2 .

2. THE PROBLEM

A sample Xl,XZ, . . . , X of size n is drawn from an exponential '1

population having the cumulative distribution function F(x) =

1 - { exp( - ( x - s ) / a ) } for x t g , zero otherwise. The sample has - - 2

mean X = Z X . /n and standard deviation S given by (n-1) S' = Z ( X . -!<I .

Student's ratio T is defined by T = &(T-U)/S. The problem is the Dow

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S.QPIES FROM A?: ESPONEYTI.-V. POPULATIOS

8 -- wnere tr.e aomarn i ' = : x , , . . . , x v n x / s l t and x . 2 0 for i = l , . . . , n 1

- i - n. wzth x = Lx ,'n ar.5 (n-1)s' = L ! x . - x) .

1

>;e make the transformations (Ti) , (T2) , !Ti! , defined below, > - -,,-,----,-,- L.. ..LA.,-->a-U.,~

(Ti) Order x l P x 2 , . . . , x in ascending order of magnitude O r ui 5 u, -. n A - I... < u , say, then the ordered transformation (T1) : ( x i , . . . , ~ 1 - (U l,...,~ 1 has Jacobian equal to n!. The domain r is changed to

2 0 for i = 2,. . . ,n?. (T2) Thls 1s followed by the --

orthogonal Heimert's transformatzon

havinq Jacobian equal to 1 and

defined by

r -- yl = ( U +u2 + . .. + U ) / g n = jnu

I and

for i = 2, ..., n; say Y = HU in obvious matrix notation, so that

T U = H Y. The domain r is now transformed to Y2 given by

1

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840

and

f o r L = 2 ; ..., n-1 and 1: 2 0.

( T 3 ) F ~ n a i l y t n e g e n e r a i ~ ~ e d p o l a r c r a n a f o r m a c ~ o n ,:3 : - T ~ , . . lrn) + ( r , 8 l , . ,'i ) w l t h O r r C m , 0 - 4 ( 7 f o r l = l , . . . , n-2;

n-1 1

and 0 5 Y c iT g l v e n 3 y n- 1

for i = 2 , . . . . n-1 and

n- 1 h a s t h e J a c o b i a n r J ( b l , . . - 1 where w e define

1 ~ n - 1

n- i-1 ) n- i- 2 J ( 5 , , . - . , = ( s i n 3 . 1 ( s i n 0 . . . . s i n S ~ + 1 3-2 '

( 2 )

E, = { ( o , , . . . , 8 / 9 5 8 . ~ ~ 1 2 ; i = l , . . . , n-i: ( 3 ) L A. 3-1' - i

and

D:, = ( e l , . . . ,9n-l) 1 c o t 5 2 tlJn-1, c o s s r ( c o t ' S ~ ) / G T , - 2

4, f ( n - i ) 2 cos 0 . S ( n - i + 2 ) c o t $ , f o r i = 2 , . . . , n - 2 , ~ n / ' 3 j *

l + l 1 n - l

I n a r r i v i n g a t r 3 , we s e e t h a t s t r a i q h t f o m a r d substitution i n

1,

y, 2 y2 and in-i+2 j l iy 2 - + f o r 1 . 2 , . . . , r-: and 1 2 0

L y i e l d s fi s i n 0 cos 5, 5 c o s 31 , (n - i j 2 s i n 8 , . . . s i n Y i c o s 5 .

1 L - l+l 5

< (n - i+2) s i n 0 . . . s i n 8 , c o s 3 fo r i = 2 , . . . ,n-2 and s i n 8 1-1 i l...

s i n 8 2 J . The i a s t i n e q u a l i t y s with 0 5 6 , 5 7i for i = 1 n- 1

, . . . ,n-2 1

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n - l

..- , - ~ ,.I re .J snr: C .% a re d r f iiied in , L a : 3 . 2:rect i n t e g r a t l a r ,

i

.,. ̂ , i f rcmark a t his p o i n t t h a t t h e domaln r , i n t e rms of

n-dimensional qeometry, can be s h n w n to be t h e ort ti on of a cone

x i t h apex a t t h e o r i g i n , a x l s a s t h e e q u i a n g u i a r Llne t o t h e co- -

o r d l n a c e axes and s e m i v e r t i c a l a n g l e a r c c o t ! t f ln -1) con ta ined i n

t n e p o s r t l v e o r t h a r t . For t z n - 1 it can be shown t h a t t h e h a l f -

cone l i e s wholly l n s i d e t h e o r t h a n t and i n t h i s c a s e t h e evalua-

t i o n of t h e l n t e g r a l i n (1) is e a s i l y accomplished. T h i s case was

s t u d i e d by H o t e l l i n g ( 1 9 6 1 ) . For s m a l l e r v a l u e s of t , t h e pro-

blem is nontrivial and f a r from easy . In g e n e r a l , depending on

t h e v a l u e o f t , t h e fo l lowing c o n f i g u r a t i o n s a r . i s e : t h e cone i n t e r -

s e c t s t h e (n -k) - faces of t h e o r t h a n t , b u t d o e s n o t i n t e r s e c t

lower-dimensional f a c e s f o r k = 1 , ..., n-1. In what f o l l o w s we w i l l

n o t use any g e o m e t r i c a l i n t u i t i o n o r argument, s o we w i l l no t en-

t e r i n t o any d e t a i l e d g e o m e t r i c a l d i s c o u r s e . We ex tend Hotel-

l i n g ' s work t o i n c l u d e t h e c a s e k = l o n l y . The method employed i s

c a p a b l e of e x t e n s i o n t o h i g h e r v a l u e s of k , b u t computat ions be-

come t e d i o u s and unwieldy. F i n a l l y we remark t h a t t h e t r a n s f o r -

mat lons ( T l ) , (T21, (T3) were mot iva ted by g e o m e t r i c a l cons idera -

+ ; ,,,ns. r. ( T i ) maps t h e p o s i t i v e o r than t . i n t o l /n ! p a r t of i t s e l f

and t h e symmetry of x , , - . . , x a p p e a r i n g i n (1) s u g g e s t s t h i s -

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84 2 H O Q , XLI, XUD T L Y P L E T O N

Transformations in n-j+2 dimensions, analogous to

!T3!, applied in suqcession to the above lnteqral

(Tl), (T2) and

n i .in r Y A W L

and since

rhe proof of the lemma is obvious

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obtain

for t - n-1, the result obtalned by Hotelllnq (1961) geometrically.

In Case L , the denslty f (t) = -dG (t)/dt = (n - 1) G (t)/t.

so that arc cot (n-1)lI2 5 8 < arc cot (t/ih-1.). In this case the I -

domain D n D appearing in ( 4 ) and defined in ( 3 ) can be written 1 2

as A, u A 2 , where A, and A2 are disjoint sets defined below, with - -

C2, C defined as in the above lemma. 3

7 - n = j Q , 'E12 j arc cot in-1 8 i arc cot (t/Jn-11, '-2 - 1-

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HOO, A L I , AND TEYPLETO?;

so t h a t

a r c c o t t / d n - 1 - 1 3-2

n n ! - 1 ! ~ ( t ) = i (tan? 1 ) sec2i.d4 i 1 j 0

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SLhPLES FROM ,LY EY?ONI?:TIAL ?GDULATIOS

where x = t 2 / ( n - 1 ) 2 , B(a,b) = : ' ( a ) T . ( b ) / i ' ( a + b ) is t h e o r d i n a r y Be ta

-1 .x a - 1 f u n c t i o n a n d 1 ( a r b ) = [ B ( a , b ) ] J O w ( ~ - w ) ~ - ' d w , 1 2 ~ x 5 1 , is

t h e incomplete Beta f u n c t i o n .

Hence we have f i n a l l y

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846 HDQ, XLI, .LXD TEWLETO?:

Case 2 by

Using the results obtarned above, we can compute tie tall

area G ( t ) and denslty f (t) for n = 2 , 3 , 4 . For all n, G?(t) = i n

G 3 ( t ) = { I -3 /2 -2 -1 /2 -2 -1/2 -1 2 1 / ? j 4 1 ~ 3 t - 4 ( 3 ) t arc cos(t/2) + 3 t ( 4 - t

for 1 - t ' 2 ,

33 /2 ( 2 ) - l n (t-2-t-3) - i f j ) - 1 ( 3 ) - 1 / 2 n for ~ / S S t 5 3

2 1 9 - 3 t I

33 /2 ( 2 1 - I n (t-2-t-3) + 3-1/22-1t-2 1t2-27)arc tan i- i - 2 1 I

i A= J

5 / 2 - 3 +3 t arc tan

-1 /2 -1 2 1 / 2 - ( 6 ) -1 (3) -w7 +2 t ( 3 - t ) Dow

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SXVLES FROff A Y E X T O N E X T I A L P O P U L A T I O N

.)r 1 % - 2 iir,i; 4 ; XE chi; ahth in t:;c expectcitio:: cf T , ,i:hich is

of some i n t e r e s t f o r comparison with asympto t ic formulas o b t a i n e d

54' Shenton and 3oiman (19771, and a l s o q i v e s some a d d i t i o n a l rn-

formarlon about t h e d i s t r i b u t i o n . For n = 2 , t h e e x p e c t a t i o n o t T

does not e x l s t . W e f i n d

i

Table I q i v e s v a l u e s of G i t ) and %bile I I g i v e s v a l u e s of

E ( t ) , f o r n = 2 , 3 , 4 and a few v a l u e s o f t l F i g u r e 1 shows f ( t ) n n

f o r n = 2 , 3 , 4 . Values o f G ( t ) an? f !t) f o r

r 2 i i n - i i ( n - 2 j j 2 j li2 can be compute? using a

g iven by Hoq, A l i and Templeton 11977). The

any r? and any

computer prograii!

fo rmulas f o r G ( t )

aiid f ( t ) ..-,. U a ~ d i n t h e progrzm have been o b t a i n e d from t h o s e g iven n

above by expanding t h e incomplete Gamma f u n c t i o n i n a f i n i t e

s e r i e s of approximately n/2 te rms . D i f f e r e n t s e r i e s a r e used f o r

odd and even v a l u e s o f n.

4. DISCUSSION

R e s u l t s glven l n Table I can be compared w l t h t h e correspond-

rng r e s u l t s fo r t h e T - d ~ s t r l b u t i o n f o r a normal p a r e n t population.

I t can t h u s be seen t h a t t h e upper t a l l o f t h e T - d r s t r l b u t l o n IS

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TABLE I

Probability Density Function of the Student t-Statistic for Samples of Size n from an Exponential Population.

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SLhPLES TROhl ;LU EYPOSEXTIAL POPUWTIOS

FIG 1

PROBABILITY DENSITY FUNCTION f , ( I ) FOR n = 2 3 4

heavier when t h e p a r e n t d i s t r i b u t i o n is e x p o n e n t i a l t h a n when t h e

p a r e n t d i s t r i b u t i o n is normal. T h l s i s h a r d l y s u r p r i s i n g , s i n c e

we know t h a t t h e T - d i s t r i b u t i o n h a s G " ( 1 ) = 1 f o r e x p o n e n t i a l pa-

r e n t and G (1) 5 . 2 5 f o r normal p a r e n t . In work t o be r e p o r t e d

e l sewhere , we have compared t h e t a i l s of t h e T - d i s t r i b u t i o n s w i t h

e x p o n e n t i a l p a r e n t and w i t h half-Gaussian p a r e n t ( d e n s i t y f ( x ) =

2 ( 2 / 7 ~ ) '"exp (-x / 2 ) ) , x 2 0. The ha l f -Gauss ian p a r e n t g i v e s a hea-

vier upper t a i l f o r a l l n , when t > n - l .

ACKNOWLEDGEMENTS

The f i n a n c i a l s u p p o r t of t h e Nat iona l Research Counci l of

Canada and o f t h e Canadian I n t e r n a t i o n a l Development Agency is

acknowledged, T h e computer program was w r i t t e n i n F o r t r a n I V G l

by Ger t rude I p . We acknowledqe h e l p f u l comments by a r e f e r e e . Dow

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850 HOO, A L I , AXD TDIPLETON

Govindarajulu, 5. & Leslie, 2.T. (197Q) . .Lm.otated 5~5l:c~r-pr.y on robustness studies. Tech. Report 7, Dept. of StatlSt., Univ. of Kentucky, Lexington, ken tuck;^.

~ o q , & . K . M " S , ; ~:i, M,:.:. & Templetar:, 2.Z.C. i?377: nc -c~~ iL . UL~L.. ~ t l . _ ~ ~ . ~ ~ *;?-

, . - of Student's ratio based on exponential ?opuidiion. wvrkiny Paper #77-001, Dept. of Industrial Engineering, Univ. of m -- rn -,,

" . I , " ! I O L " L L t S , CazaZz.

Hotelllng, 3 . (1961). The behavlor of some standard statlstlcal tests lunder non-standard conditions, Proc. 4th BerkeLeys.jmp= w ~ t h S t a t i c + P r n h lj 319-60. . . - -. . . - - - - - - - . - - - - . -

Posten, H,O, & Hatch, L,O, (1966). Robustness of the Student- a survey. Research Report 2 c , Ee3t. of Statist.,

Univ. of Connecticut, Storrs, Connecticut.

Sansing, ?..C. (1976). The t-statistic for double expon-ential distribution. S I A M J. Appl. Math. 31, - 634-45.

Sansing, R.C. & Owen, O.B. (1974). The density of the t-statis- tic for non-normal distributions. Commun. Statist. 3(2), 139-55.

Shenton, L.R. & Bowman, K.O. (1977). A new algorithm for summins divergent series. Part 3: Appl~cations. J. Cornput. Appl. Math. 3, 35-51. -

"..-.. L U = L L , X.K. & Yarthy, V.X. (1374). Percentage points of the Zis-

trib'utlon of Student's t when the parent 1s Student's t. Technometrics 16, 495-97.

R e c e i v e d M a r c h , 1977; R e v l s e d J u n e , 1 9 7 8 .

R e f e r e e d b y K a r e n Y u e n , U n l v e r s l t y of W l n d s o r , O n t a r i o .

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