Some Tests Of Normality Based On Transforms.
Sucharita Ghosh.
Department of Statistics, University of North Carolina, Chapel Hill.
Summary: In this paper, some graphical tests of normality are proposed involving methods
based on empirical transforms. The suggested techniques use derivatives of the empirical moment
generating function and the empirical characteristic function and can also be used as formal testing procedures. Good power properties are indicated in power calculations.
The proposed
methods are invariant under location and scale transformations of the data, the graphical techniques in particular are found to be attractive and intuitively appealing in nature. Weak convergence results are obtained under general conditions. Some optimality problems are handled via
maximisation of Bahadur slopes of test statistics.
Key Words: Testing normality; Empirical Transforms; Probability plots; Bahadur slopes.
Introduction
Let X 1'''',x" denote observations on a univariate random variable X with distribution
function F (x )
Pr (X ~ x), density function
=
f (x), characteristic function ¢( t ) =
moment generating function m (t ) = E (e IX) where t
E
R.
E (e ilX) and
The problem here is to test
H o : X ......., N(Jl,cr) against the alternative HI : negation of H o. Here N(Jl,u 2 ) denotes the nor-
mal distribution with mean Jl and variance u 2 • If either Jl or u is not specified, H 0 is a composite
null hypothesis, otherwise as in many practical situations, these parameters may be known with a
reasonably high degree of accuracy, H 0 may be called a simple null hypothesis. A fairly large
amount of work has been done in this area. Mardia in his 1980 paper discusses the existing tests
beautifully. Other references on this topic are Gnanadesikan (1977) and D'Agostino and Stephens
(1986). However the uniqueness of characteristic function and the moment generating functio!!
(when it exits) suggest goodness-of-fit methods based on these transforms. Define the empirical
characteristic function by ¢. (t) =
by m. (t) =
X l'
...
Je
Iz
Je
itz
dF" (x) and the empirical moment generating function
dF" (x) where F. (x) denotes the empirical distribution function for the data
X". Because of excellent properties available for the empirical characteristic function
(ecf) and the empirical moment generating function (emgf) (see Csorgo 1980, 1981, Feuerverger
and McDunnough 1981a, 1981b, 1984, Feuerverger and Mureika 1977), a number of authors have
considered using these for goodness-of-fit problems (see Heathcote 1972, Feigin and Heathcote
- 2-
1977, Koutrovellis 1980,1981, Koutrovellis and Kellermeier 1981, Kellermeier 1980, Murota and
Takeuchi 1981, Hall and Welsh 1983, Epps and Pulley 1983, Epps, Singleton and Pulley 1982,
Csorgo 1986, Epps 1987). A survey of testing by empirical characteristic functions is available in
Csorgo (1984). In the present paper we develop some graphical methods for testing normality
based on certain stochastic processes which can also be used as formal tests. Several graphs are
exhibited in the appendix along with some power calculations for some of the suggested tests
which indicate good power properties.
[A] Use of the empirical moment generating function:
Section [AI]: A general goodness-of-fit problem:
In this section we consider a more general goodness- of-fit problem; for simplicity we restrict
ourselves to simple hypotheses. Thus assuming that the mgf exists both for the distribution
specified by H 0 as well as the one specified by H I' we consider the following testing problem:
H o: m(t) = mo(t)
(I)
versus
where mo(t) ~ml(t)
HI:m(t) =ml(t)
for at least one tER. Then it is easily seen that under Hi (i=O,l), the
finite dimensional distributions of
T. (t)
=
vn (m" (t)
- m;{ t ))
converge to the finite dimensional distributions of a zero mean Gaussian process having the
covariance structure
[Al.I]: Although, two mgf's may intersect at a finite collection of values of the parameter
t , examination of the empirical mgf at a finite number of t -points provides substantial information regarding the population. For simplicity here, a test based on the statistic T" (t ) at a single
value of t, may be based upon an approximate rejection region given by:
I
T" (t)
I > viK o( t ,t ). z 0/2
where zO/2 is the upper a/2 point of the N(O,l) distribution.
For different situations note that several optimality criteria may be used to at least partially
resolve this problem; in particular, Bahadur slopes of test statistics may be maximized with
respect to t to decide about the values of the parameter t to be chosen. Noting that { T" (t )} is
e-
- 3-
a 'standard sequence' (Bahadur 1960), the approximate slope of the sequence { T"
(t )} may be
calculated as:
0(4){t)
[ml{t) -ml{t)]2
Ko(t,t)
=
On the other hand, since T" (t), for fixed t is just";;; times the sample mean of n iid random
variables Chernoff's large deviation theorem (Chernoff 1952) can be used to calculate the exact
slope of the same sequence { T" (t )} and is given by,
It)}
O(t){t) = :2 S:p{(ml(t) - mo(t))s - 10gMo(s
where M o(sit) is the mgf of e IX under H 0, evaluated at s.
Maximization of O(t)( t) with respect to t may suggest a 'good' choice of t for testing (I),
while simple algebraic manipulations show that for maximizing O(t)( t), an initial value of t may
be taken as the value that maximizes 0(4)( t). This is not surprising since asymptotically in the
neighbourhood of H a, the approximate slope and the exact slope are the same.
[A1.2]: Use of weighted integrals: The above discussion generalizes to the case where one con-
0< T < 00
siders a weight function w (t), [- T ,T]
S"
=
and define the test statistic
J w (t) T" (t) dt
I
where the parameter t may be restricted to a finite interval [- T ,T j, T
<
00.
Asymptotic null distribution of S" follows from central limit theorem, so that under H 0, as
n -
00,
S" ----N{O, u2) where q2
=
JJw{s)w(t)Ko(s,t)dsdt.
Clearly the properties of such a test will depend on the choice of w (t ) and hence a reasonable choice of w (.) may be of interest although such a choice will in general depend on HI'
In particular, Bahadur's approximate slope for the sequence
{S" }is given by:
(fw{t)[ml{t) - mo(t)] dt)2
JJw{s)w{t)Ko{s,t) ds dt
which
when
maximized
with
respect
to
w (.)
J J W(8)W (t)K 0(8 ,t) ds dt = 1. gives rise to the solution:
w
(t) = L -I [ f 0(') ]
f {)
under
the
constraint
(w.l.o.g.)
-4where L -I denotes the inverse Laplace transform operator.
Proof: Define U (x)
=
Jw (t )e tz dt.
Noting that
VarHOU(X)
=
JJw(s)w(t)Ko(s,t)dsdt,
the problem reduces to the maximization of
[EH U(X) - EH U(X)]2
1
VarH 0 U (;.)
with respect to
C(/I) =
w(') such that VarHoU(X) = 1. Equivalently w.I.o.g.,
Maximise EH 1 U(X)
(PI)
subject to:
o
(A)
and
(ii)EHO U 2(X) = 1
(B)
Introducing variants 8U(x) to the solution of (PI) we have:
J8U(x)1 I(X )dx = 0
(a)
J8U(x)1 o(x )dx = 0
(b)
and
J8U(x )U(x)1 o(x )dx
=
0
Equations (a),(b) and (c) together imply that there exists constants a and
(c)
13 such that
II(x) = a U(x)/o(x) + j3lo(x).
(d)
Using (A) and (B),
JU(x)1 I(x)dx
=
a
so that
I 1(x) = U (x )I o(x) J U (x )I 1(x )dx + j3I o( x ).
Integrating both sides of the last equation and applying (A) we have 13 = 1, so that (d) implies
1
+ aU (x) = I 1(( X)).
lox
Since C( /I) is invariant under transformations of the type aU (x)
these constants can be removed and we may write
U(x)
=
II(x)
I o(x)
or
w
(t)
= L
-I
I I(X )
I o(x)
+
b,
- 5-
On the other hand, Bahadur's exact slope for the same sequence
{s. }may be calculated
as:
cle)
= -2
log In/e-'· Mo(s)
•
where, u =
Jw (t )[ m I( t)
Jw (t )[ e tz
m 0( t)] dt, under H 0, evaluated at s assuming that it exists. In this situation it
-
- m o(t)] dt
and M o( s) is the moment generating function of
can be shown that the function w (t) that maximizes C le ) is given by the inverse Laplace
transform of the logarithm of the ratio
~ :~:~
where as before /
tion of the underlying random variable under H j
j
is the probability density func-
•
One may notice at this point that maximization of the exact slope gives rise to the best
(most powerful) test suggested by the Neyman-Pearson lemma whereas the approximate slope
does not give rise to the most efficient test in the Neyman-Pearson sense.
'We give a simple example for the approximate case.
Example: Let X ........ N(O,0'2). We want to test
vs.
HI : 0' = 0'1 (0'1 > 0'0)·
To find the best test according to the approximate Bahadur slope, we need to find w (.) such that
00
J w (t) e tz dt
-00
where
00
Solution: Under the constraint the
Jw (t)
dt
1, the function w ( t ) maximizing C (4 ) is given
-00
by:
w
1
t2
(t) = \&8 exp[- 2D2]'
Section [A2]: Tests of normality and
-00
< t <
00.
Some Derivative Processes: Simple null
hypothesis of normality.
Let X I 'X 2 ' .... ,X. be a set of iid observations with distribution function F(·) and mgf m (t).
- 6-
Although m (t) itself characterizes the population distribution, as a graphical procedure, examination of mIl ( t) alone is not informative enough since mIl ( t) is a strictly convex function in t for
any data set. However it is possible to characterize the normal distribution through some of the
derivative processes associated with the mgf. Some such processes are described below.
[A2.1]: First Derivative of the Cumulant Generating Function:
The following is a characterization of normality.
A random variable X has N (0,1) distribution iff -!!...m (t) =
dt
t where m (t ) is the mgf of X.
Therefore to test the simple null hypothesis of normality (i.e. Ho:m (t) =
et
2
/2.
departure
00
of m,,' (t) - t from zero may indicate non-normality where mIl (t) =
J e tz
dF" (x ).
-00
We may now state the following
Theorem: A(2.1) Define
Vii'"q" (t)
Vii'" { ;t log
=
mIl (t) - ;t logm (t )}
where m (t) is the population (from which the sample is taken) mgf of the underlying random
variable X and mIl (t ) is its empirical version in a random sample of size n. Then for t t[- T ,T],
the stochastic process
Vii'"q" (t)
converges weakly to a zero mean Gaussian process with covari-
ance function:
1
L(s,,)
1
E E aj (s )ak (t )[mU+kl(s +t)
- mUl(s)m (k l(t)]
j=Ok=O
where
dj
dt'
- . m (t)
=
m U l( t )
o
aj(t) = oUj(t)g(U,,(t))
U" (t)
~t)
I
litl
(m" (t ),m,,' (t))'
(m (t ),m' (t))'
and g (-) is defined by:
g(U.. (t))
=
d
d'tlogm,,(t)
m,,' (t)
mIl (t)
and equivalently
Proof: Define Z" (j l( t )
0,1.
Multivariate central limit
e-
- 7theorem can he used to prove the convergence of the finite dimensional distributions of Z.. (t ) and
Z.. ' (t) to the finite dimensional distributions of the zero mean Gaussian processes Z o( t) and
Z t(t ) respectively with respective covariance structures given by:
Ko(s,t) = m(s+t) - m(s)m(t)
and
Kt(s,t) =m(2)(s+t) -m'(s)m'(t).
Also by the multivariate central limit theorem, the finite dimensional distributions of the vector
Gaussian process (Z o( t ),Z t( t ))' . whose covariance structure is given by:
cov(Zo(s),Zt(t)) =m'(s+t) -m(s)m(t)
and
COV(Z;(s),Z;(t)) = m(2i)(s+t) - m(i)(s)m(i)(t)
where i
0,1. Now for fixed t , a Taylor's series expansion can be used to write
vnq.. (t)
t
EY;(t) +R.. (t)
i=O
where
and
1
where I~..
t
a2
t
R.. (t) = ~i~ol~/;(t )Zj (t) aU;(t )aUj (t) g (U.. (t))
' (t) - s{ t ) I < I u.. (t )-s{ t ) I .
First of all
I fft 'It)
Sup R.. (t ) converges to zero in probability. To see this note that
t<I-T,TI
t
IR.. (T)
Sup
t<I-T,TI
I < E
t
EVi"(t) v;'*(t) uij(t)
i=Oj=O
where
Vi"(t) =
Sup Z;(t)
t<[-T,Tj
and
a2
uij(t) = t<~~~TI aU;(t )aU, (t) g (U.. (t))
I fft 'It)"
Applying the law of iterated logarithm to each of Vi"( t ) and v;'*( t) and noting that uniform convergence of each of Ui (t) over the finite interval [-T, T] implies that uij(t) converges to a constant almost surely over [-T,T],
-
Sup R.. (t) is bounded by
O(n- t / 2 [og[ogn) almost surely.
t<I-T,Tj
Tightness follows from a one-term Taylor expansion of Yi (t ) which easily gives
- 8-
:s K
E( Y(s) - Y(t) )
Is-t
1
2
where K ~O is a constant. The theorem now follows by the application of Slutsky's theorem
along with the application of the multivariate central limit theorem.
Therefore, one may reject the null hypothesis H 0
:
m (t)
=
m o( t ) with an approximate level a
if
I-In q,,(t) 1 >
JL o(t,t)ZO'/2
where L o( t ,t) is now calculated from L (t ,s ) by substituting m (t)
= mo( t)
in the expression
for L (t ,s ).
Now consider also as before a statistic of the type
SIll
=
-In f w(t) q,,(t) dt
t
where the parameter t may be restricted to a finite interval [-T,T], T
The asymptotic null distribution of
f f w (s )w (t )L o( s ,t )dsdt,
S,/
<
00.
is then Gaussian with mean zero and variance
•
by the central limit theorem. Denoting this distribution by PH 0' the
e-
approximate Bahadur's slope for the sequence { S"l} is given by:
If{ -JtIOgml(t) - -Jt IOgm (t)}w(t)dtj2
f f Lo(s ,t)w (s )w (t )dsdt
which is obtained by t.aking the limit in probability of
2
• H { I SIll
--logP
n
0
I >
1
SIll (obs)
1 }
where SIll (obs) denotes an observed value of SIll when the observations are sampled from the
distribution specified by HIA special case: Under the simple null hypothesis of normality (both mean and variance known),
m (t) = e t'l/2 which gives
[f {-JtIOgml(t)
ffeh(l +
-
t} w (t )dt f
3ts)w(t)w(s)dtds
For notational simplicity define
9
(t)
=
d
-Iogm I( t) - t
dt
Note that given simple alternate hypothesis H 1> 9 (t ) is a known function. An optimum choice of
- 9-
w (.)
for this special case can be obtained by the maximisation of
Jg{t)w(t)dt
(P2)
with respect to w (-) such that
JJe 'l {1 + 3ts)w{t)w{s)dtds
<
J w {t )dt
1
00
and
<
Jw{t)etlds
00
for all t lying in an open interval around the origin.
Solution: Introducing variants 6w (t ) to the solution of problem (P2) we have
J6w(t)g{t)dt
0
=
(1)
Similarly the constraint (**) suggests, after ignoring the term involving 611' (t )6w (s ),
JJe tl (1 + 3ts)[w(t)6w(s) +w(s)6w(t)]dtds = 0
or
JJe"(1 + 3ts)w{t)6w{s)dtds + JJe 'l {1 + 3ts)w{s)6w(t)dtds
o
(2)
Now (2) implies
J,p(t)6w(t)dt
0
=
where
,p(t)
=
Jw{s)e tl (1 + 3ts)ds.
(1) and (3) together imply that there exists
Q
(3)
such that
Q,p( t )
9 ( t) =
or
..ti!.l
Q
= J w (t)e II ds + 3t I sw (s)e II ds
(4)
d
, [w (s)]
= L, [w (s)] + 3t-L
dt
where
L,[{w (t)]
To solve for
Q
=
J
W
{8 )e" ds
without loss of generality assume that
I
Then taking t =
0 in(4)
W
(8 )ds
=
1
- 10-
o:JW(S)d8 =
9(0)
0:
or
0:
=
d
di logm I( t ) I,
Thus given simple alternate hypothesis HI,
0:
=
0
is known. Thus the solution for problem (P2) is
given by the equation
Ldw(·)] + 3t :tLdw(,)] -
~~~J
0
where
L, [w
JW (8 )e " ds
OJ
Now define
W(t)
=
Lt [w (.)J
and
a(t) =
1J!l
9 (0)
where recall that
9 (t)
=
d
dilogm I( t) - t
is a known function in t provided that HI is completely specified. Then we may rewrite the last
differential equation as:
W(t) + 3tW' (t) - a(t)
=
0
the solution for which also defines the solution for the problem (P2).
[A2.2J: Third derivative of the cumulant generating function:
One of the more striking defining properties of the normal distribution is that all its cumulants of order
2:3
are equal to zero. Therefore it is of natural interest to base a test on the third
derivative of the cumulant generating function since it allows one to obtain a visual representation of all the cumulants beyond the second one.
Thus let K (t) = log m (t ). Then
d
3
dt 3K
(t)
3
d I og
= dt3
m
(t)
t2
t3
= K 3 + K 4·t + K 5'2! + K 03T + ...
Clearly m (t ) corresponds to a normal distribution if and only if d 33 K (t) _
dt
Let us define
O.
e-
- 11 -
r" ( t)
= f (v" (t)) - f
v" (t)
=
(~t ))
where
~t)
!
(m" (t ),m.' (t ),m" (2)(t ),m" (3)(t))'
=EFV,,(t)
(V" (t)) =
d3
- 3 K"
dt
mIl
(t)
(3)(t )m" 2(t) - 3m" (2)(t )m,,' (t) + 2(m,,' )3
(m" (t ))3
and equivalently
f(~t))
and
every
for
fixed
3
=
define
d 3K (t)
dt
a
aj(t) =au:-(t)!(V,,(t)I\it);
where
)"
V j " (t)
=
dj
- . mIl
dt)
(t )
i = 0,1,2,3. We may now state the following
Theorem: A(2.2) For t i
[-
T ,T 1 the stochastic process
.;n r" (t)
converges weakly to a zero
mean Gaussian process with covariance structure
F(s,t) =
tt
[mU+k)(s+t) - mU)(s)m(k)(t)]-aj(s)ad t )
j=Ok =0
The method of proof is the same as in the previous theorem with appropriate changes.
Thus in particular, under the simple null hypothesis of normality,
this case, significant departure of
.;n
!
(~t))
=
0 so that in
3
d K" (t ) from zero will indicate non-normality of the data.
dt 3
.As with any good graphical procedure, the study of
.;n
3
d K" (t) (in a neighbourhood of
dt 3
0), as a function of t offers much other valuable information. For example, the value of r" (t ) at
t = 0 gives the third cumulant which apart from a multiplicative constant is equivalent to the
skewness coefficient of the data, the slope at t =
kurtosis coefficient), the curvature at t
=
0 is the fourth cumulant (equivalent to the
0 is the fifth cumulant an so on. As a point of interest
it should be noted that a graphical procedure or a test based on r" ( t ) may be dominated by the
skewness present in the data set. In particular, if there are extreme observations in the data set in
one particular direction, then the procedure will be sensitive to such outliers. Therefore if detection of outliers is of interest, and the statistician is not concerned with just the majority of the
data set, then this procedure can be particularly useful.
- 12 -
To compare this procedure with the existing normal probability plot method, we consider
the chi square family of distribution with varying degrees of freedom. We take samples of size 200
3
and plot r .. (t)
d K .. (t) along with its ±2
dt 3
=
asymptotic null distribution of r .. (t ), where
C7 ..
and ±1
C7 ..
bounds as calculated from the
denotes s.d., vs. t, on the same graph paper. To
C7..
make comparison, we also study the normal probability plots for the same data sets. Although
we have one data set for each of the degrees of freedom, it is clear that r .. (t ) clearly offers a very
attractive method for detecting departure from normality. The graphs are presented in the appendix (pages AI-A8).
Section [A3]: Composite null hypothesis of normality. The above results easily generalize
to the composite null hypothesis situation. Thus consider the hypothesis (for unknown Il and (7)
1
_1_(z -
/1)2
00
<
negation
0/
= r:::--e 2,,2
H o : / (1')
-
1',1l
<
0
00,
<
C7
<
00.
v2il'C7
vs.
H1
:
Ho
e-
In this case define the studentized mgt and its derivatives as:
m.. (t)
1 "
=
Ee
-
n
x -x
t(_J_ _ )
'
i=1
and
- (i)~(t)
m
where
X and
d
i
= dt i
-
m..
- i
x-X
{Xi
-X}
t(~)
=-E
e
1
(t)
II
n i=1
8
are respectively the sample mean and the sample standard deviation in a sample
8
of size n . Correspondingly we define the respective population quantities as:
m(t)
=
X -
/I
E [ e (-,,-)
]
and
m(i) =
di
- . m(t).
dt'
Restricting the parameter t in a finite interval [- T ,T 1 for an arbitrary positive number T, we
have the following
Lemma A(3.1): Under H 0, and for t f.
[- T
,T], 0< T <
00,
as n
-+
process
Z.. (t)
(Z 0.. (t ),Z 1.. (t ),Z 2.. (t ),Z 3.. (t ))'
00,
the vector stochastic
- 13 -
where
converges weakly to a zero mean vector Gaussian process
Z(t) = (Zo{t ),Z I(t ),Z2(t ),Z3(t ))'
with covariance function defined by:
where
0ii(t,s) =m(2 i )(t+s) -m(i)(t)m(i)(s)
- (m (i)(s)+ jm (i-I)(s))m (i+I)(t )-(m (i)(t)+ jm (i-I)(t))m (i+I)(s )
-(jnl (fl{t )+ tm (i +1)( t ))( m(i +2)( S )-m (i)( s ))/2-(jm (i)( s )+sm (i +1)( s ))( m(i +2)( t )-m (i)( t ))/2
+ (tm (i)( t )+ jm (j -1)( t ))(sm (i)( s )+ jm (i -1)( S))
+ : (tm (i -1)( t )+ jm (i)( t ))(sm (i +I)(s )+ jm (i));
and
Oii(t,s) =m(i+i)(s+t) -m(I)(t)mU)(s)
-
-
~
~
[im(i)(t )+tm(i+I)(t)]. [m(i+2)(s )-m(i)(s)]
[jnl(i)(s)+sm(i+I)(s)]' [m(i+2)(t)-m(i)(t)]
_ [sm (i)( s )+ jm (i -1)( S
+:
)]
m(i +1)( t )_ [ m(i l( t )+im (i -1)( t ) ] m(i +1)( S
)
[j;n(i)(s)+sm(i+I)(s)]
+ [sm(i)(s)+imU-1)(s)]' [tm(i)(t)+im(i-I)(t)]
where under the null hypothesis of normality,
m(t)
=
e /2/2 etc.
The proof of the lemma is outlined below.
Proof: The proof follows by noting that the stochastic process
Zh (t ) can he written as:
where
Wk{t) =
x -p
(Xi - Jl)k/,(-;-) _m(k)(t)
q
_~ [kin l')(') + ,;nI>+I)(,)]. [(x; u- Pl' - 1]
- (Xi
q-
Jl)[tm(k)(t) +km(k-I)(t)]
- 14 -
and the remainder term R n (t) converges to zero in probability, uniformly in t over the finite
interval [- T ,T]. Tightness follows by a Taylor series expansion of W (8 ) so that
E(W(s) - W(t))2 $ K Is-t
1
2
where K >0 is a finite constant. The lemma follows by the application of the multivariate central
limit along with Slutsky's theorem.
Development of the test statistic:
Define
Un(t) = (U On (t),00.,U 3n (t)).
where U;n(t) =
m(j)~(t), i =
1,2'00 and Uon(t) = mn(t). Similarly define
I
r~0(t)""~3(t))
Sit)
where ~;(t) = inU)(t), } =
1,2'00 and ~o(t) = m(t). Finally define
,; (Un (t)) =
,; (Si t))
d;
dt; log i1l" (t ),
d;
dt; log
=
m(t ),
e-
and
";" (t)
where
i
=
.;;;-
[I ;(Un (t)) - 1; (Sit ))]
is an integer. Using the above lemma we have the following
Theorem A(3.1): Under H 0 for t E
[- T
,T] as n
-+
00,
the stochastic process" 1n (t) weakly
converges to the zero mean Gaussian process Z I( t ) whose covariance function is given by:
E(ZI(8)ZI(t)) =L 1(8,t)
m' (s) 00l(t
s)
m(t )m 2( s )
,
m' (t ) C01 (S , t) + k 0(8 , t) m'
(8 )m' (t ) k 0(8 , t )
( ) _ "()
_ 2( ) _ 2()
1
-
K1(s t) m(t )m (s )
,
=
_
m
8
m·
Theorem A(3.2): Under H o for
t
tE
m
[-T,T], as n
-+
00
t m
8
the stochastic process r 3 n (t) weakly
converges to the zero mean Gaussian process Z 3( t ) whose covariance function is given by:
s
E(ZS(8)Z3(t)) =
E
s
_
Ea;(8)ak(t)0;d 8 ,t)
;=Ok=O
where
a; (t )
where under
HOI
m(t)
=
e1
2
/2
etc.
- 15 -
Several test procedures such as chi-square type statistics (Koutrovellis 1981), maximal deviation statistics (Csorgo 1986), simple projection statistics (Csorgo 1986), Cramer-von Mises type
statistics, may be developed using our basic stochastic processes of Theorems 3.1 and 3.2. However it is of particular interest to obtain graphical procedures for assessing the characteristics of
an underlying population distribution. To study these procedures, a large number of data sets
from a variety of distributions were generated using NAG library routines although only a few of
these results can be shown here. In particular we show graphs for 6 samples of size 50 each generated from N(O,I), Cauchy, and X22 distributions. For each sample, each of the test statistics
(namely,
,.l.. (t), and ,.3.. (t) as in theorems A[3.1] and A[3.2] respectively) and the corresponding
±u.. (t ) and ±2u" (t ) limits were plotted against t . All these calculations were done for the standardized data (standardized with respect to the sample mean and the sample standard deviation).
Departure from normality may be suggested whenever the plot of the statistic, as a function of t
crosses the corresponding ±2u" (t) limits. The graphs are presented in the appendix (pages A9A14).
Not surprisingly the skewness in the data tends to affect these graphs quite markedly. This
is particularly evident from the graph for sample
#
6 from N(O,I). The histogram for this data
(page A15) shows that there are two outliers present in this data set. The presence of these two
large negative observations is creating negative skewness which is clearly depicted in the graph of
3
d 3 /0g in" (t ). Also it is to be noted here that while plotting the graph of dd log
~
t
asymptotic variance vanishes at t
=
mil (t)
- t , the
0 which suggests examination of this process for values of t
away from zero, although, for large values of t , the behavior of this process will be dominated by
the largest (smallest for negative values of t) order statistic in the data set.
[B] Use of the empirical characteristic function.
A test of normality may be conducted in two stages. First one tests for symmetry. If the
hypothesis of symmetry is not rejected, one proceeds to test for the shape of the underlying distribution.
Since a distribution function is symmetric (around zero) iff the characteristic function is real,
a test of symmetry may be based on the imaginary part of the characteristic function (also see
Feuerverger and Mureika 1977).
Thus let
- 16 -
¢>(t) = Ede itX ) = re(¢>(t)) + iim(¢>(t))
and
¢>,,(t) = JeitzdF,,(x) dx = re(¢>,,(t)) + iim(¢>,,(t))
define the characteristic functions in the population and in the sample where F (.) and F" (.) are
respectively the underlying population distribution function and the empirical distribution function based on a set of iid observations X
b ....
,X" .
To develop a test for symmetry for the composite null hypothesis of symmetry, (unknown
location) one could consider (a functional based on)
X ) ] , t ER.
-Y" (t) = -1"
E sin [Xi
t(
s
n i=1
However if the first 2r + 1 moments of the underlying variable X exist, then
3
E(sin X) = t JJ - _t JJ; ... + (-1)
3!
where JJi'
=
2, +1
t 2, +1
,
(2r +1)! JJ2,+1
E (Xi), JJ/ = JJ, j = 2,3, ... which gives
.
,
d3
t2 ,
-3E (Slll X) = -JJ3 + -, JJ5
dt
2.
+...
examination of which in a small neighbourhood of zero (as a function of t) is equivalent to the
examination of the skewness term (or JJ3' ) only, although for values of t away from zero, the
above function provides more information about the population. Therefore one may consider the
stochastic process
X] cos
3
d -Y" (t) = -1 E
" [Xi dt
ni=1
s
3
-3
[
t ( Xi s-.
X)]
Under the null hypothesis of symmetry (about an unknown constant JJ), where we assume
the following
Theorem (Bl) For t E[-T, T], the stochastic process
d3
S" (t)
-
-';;;-3 Y" ( t ) converges weakly
dt
to a zero mean Gaussian process with covariance function
K(s,t)
~
=
['4>(6)(s+t) +
~(6)(s
-
t)] - [3'4>(2)(t)
Proof: The proof follows by a Taylor's series expansion of
-x
x
e
'
x
=e
-x
it(~){
t1
t'4>(3)(t)]~(4)(t)
+ [3'4>(2)(s) -8"4>(3)(s)].[3"4>(2)(t) -t"4>(3)(t)].
- [3'4>(2)(S) -s¢}3)(S)]'4>(4)(S)
it(~)
-
S" (t ) and
-
by noting that
-
X·-X
(·t)2 X·-X U
1+it(-'-)(!!...-1)+-'-(-'-f(--1)20 i
U
8
2!
U
s
}
,
(i)
e-
- 17 -
I 8i I ~ 1 ,
where
vn (.£:.-1)
8
=
n [ .::.!. "
.n 2 ~ [(X·
1
(ii)
,!::.I'
0'(8 + 0') n-l
1
vn (X -
+ n -2u2],
1-1)2
and thus writing
5,,(t) =.!..EQj(t) +R,,(t)
n i=1
where the remainder term R" (t) converges to zero in probability an uniformly in t over an arbitrary finite interval [- T , T 1 and cov (Qj (8 ),Qj (t)) = K (8 ,t). The tightness follows by another
one term Taylor series expansion of Q (t ) which gives
W :S
E (Q (8) - Q (t
K
I 8 -t I 2
where K is a finite positive constant.
Let us now turn our attention to testing normality in a symmetric class of distributions with
the first two moments existing. Note that for a general distribution F (.), if the first 2r moments
exist, then
t2 ,
re ¢( t )
1 - -1-12
2!
+
t4
t 2r
(-It (2r )! I-I~r
I
4f1-l4
+
0 (t 2 r+2)
so that
d4
-4
dt
re ¢(t)
t2
2!
I
,
+
1-14
where
1-1;
= EF(xj),
(t 2 r+2)
+ ... + ~O
4
I
- -1-16
1-14
dt
0 (t 2 )
= 1,2, .... This implies that examination of the fourth derivative of
j
the real part of the characteristic function for values of t very near zero is 'equivalent' to a kurtosis test although examination of this function for other values of t potentially offers more information.
Thus using the previous notation, define
O,,(t) =
where
4>" (t)
1 "
=
E
e
n i=1
x
vn{~re
4
dt
4>,,(t)
-~
it (.:.L..:.:.)
'
Using arguments exactly as before, we have the following
Theorem (B2): For t {[- T ,T], where T is an arbitrary finite positive constant, the stochastic
- 18-
process
eft (t ) converges weakly to a zero mean stochastic process with covariance structure:
L(8,t)
=
~['4>(8)(8+t) +
'4>(8)(8-t)] +
- 2'4>(4)( t )'4>(8)(8) - 2'4>(4)(8 )''4>(8)( 8 )
where under the hypothesis of normality,
IJ:u
=
3, '4>( t)
= e -1 2/2 etc.
As before, although several types of test statistics may be constructed based on these stochastic processes whose merrits will depend on the particular functional used, we shall emphasize
graphical procedures particularly.
In particular we consider use of the process described in theorem (B2) for testing for the
shape of the distribution. The same data sets as in the previous examples are used for this purpose and the graphs are presented in the appendix (pages A16-A18). Note that the effect of the
negative outliers in sample
#
6 from the N (0,1) distribution is not visible, which is not so surpris-
ing since the dominating term in the expansion of
d: re '4>(
dt
t ) is the fourth central moment of the
distribution and not the third central moment which accounts for the skewness effect in the distribution.
All the stochastic processes considered above may be used formally for testing purposes. Clearly
•
the power of such tests will depend on the particular functional used. In the appendix (page A19)
we present some Yery modest power calculations based on simulations, for the test based on the
third derivative of the cumulant generating function for the standerdized data. We consider tests
carried out at single t points. Here the null hypothesis of interest is normal with unknown mean
and variance. (Details of these calculations may be obtained from the author). One may refer to
the power calculations done by Shapiro et al (1968) to compare these performances over other
available tests. Based on these limited simulations one can see that the test based on the third
derivative of the cumulant generating function is highly informative and rather promising. Tests
based on the first derivative of the cumulant generating function and the empirical moment generating function were omitted from our simulations study, although we note that the graphs for
these processes also showed promising results.
Acknowledgements: A major part of this work is based on the my Ph.D. thesis at the University of Toronto, Canada. Some new results were added during the my visit to the Department of
•
- 19 -
Statistics at the University of North Carolina at Chapel Hill. I also wish to thank Dr. Andrey
Feuerverger and Dr. Jan Beran for many valuable discussions and suggestions.
References
Bahadur, R. R. (1960) Stochastic comparison of tests. Ann. Math. Stat. 91, 276·295.
Bahadur, R. R. (1967) Rates of convergence of estimates and test statistics. Ann. Math. Stat. 98,
909·924·
Bahadur, R. R. (1971) Some limit theorems in Statistics. Conference Board of the Mathematical
Sciences.
Billingsley, P. (19B) Weak convergence of measures: Applications
In
probability.
Conference
Board of Mathematical Sciences.
Csorgo, S. (1980) The empirical moment generating function.
Colloquia Mathematica Soietatis
Janos Bolyai 92, Nonparametric Statistical Inference, Budapest (Hungary).
Csorgo, S. (1981) Limit behaviour of the empirical characteristic function. Ann. Probab. 9, 190144·
Csorgo, S. (1984) Testing by empirical characteristic function: A survey. Asymptotic Statistics 2,
45-56.
Csorgo, S. (1986) Testing for normality in arbitrary dimension. Ann. Stat. 14, 708-729.
D'Agostino, R. B. and Stephens, M. A. (1986) Goodness-of-fit techniques. Marcel Dekker, Inc.
N.y.
Epps, T. W. (1987) Testing t.hat a stationary time series is Gaussian. Ann. Stat. 15, 1689-1698.
Epps, T. W. and Pulley, L. B. (1983) A test for normality based on the empirical characteristic
function. Biomatrica 70, 729-726.
Epps, T. W. and Pulley, L. B. (1986) A test of exponentiality vs monotone-hazard alternatives
derived from the empirical characteristic function. J. Roy. Stat. Soc. B 48, 206·219.
Epps, T. W., Singleton, K. J. and Pulley, L. B. (1982) A test for separate families of distributions
based on the empirical moment generating function. Biometrika 69, 991-999.
Feigin, P. D. and Heathcote, C. R. (1976) The empirical characteris function and the Cramer-von
~1ises
statistic. Indian J. Stat. A 98, 909-925.
Feuerverger, A. (1987) New derivations of the maximum likelihood estimator and the likelihood
ratio test. Comm. Stat. Theor. Meth. 16, 1241-1251.
Feuerverger, A. and McDunnough, P. (1981) On the efficiency of the empirical characteristic function procedures. J. Roy. Stat. Soc. B 43, 20-27.
- 20-
Feuerverger, A. and McDunnough, P. (1981) On some Fourier methods of inference.
J. Amer.
Stat. Assoc. 76, 979-987.
Feuerverger, A. and Mureika, R. A. (1977) The empirical characteristic function and its applications. Ann. Stat. 5, 88-97.
Gnanadesikan, R. (1977) Methods for statistical data analysis for multivariate observations. John
Wiley & Sons.
Hall, P. and Welsh, A. H. (1983) A test for normality based on the empirical characteristic function. Biometrika 70, 485-489.
Heathcote, C. R. (1972) A test of goodness of fit for symmetric random variables. Austr. J. Stat.
14, 172-171.
Kellermeier, J. (1980) The empirical characteristic function and large sample hypothesis testing.
J. Mult. Anal. 10, 78-87.
Koutrouvelis, I. A. (1980) A goodness of fit test of simple hypotheses based on the empirical
characteristic function. Biometrika 67, 298-240.
Koutrouvelis, I. A. and Kellermeier, J. (1981) A goodness of fit test based on the empirical characteristic function when the parameters must be estimated. J. Roy. Stat. Soc. B 49, 179-176.
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:Murota, K. and Takeuchi, K. (1981) The studentized empirical characteristic function and its
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Serfling, R. J. (1980) Approximation theorems of mathematical statistics.
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N.l'.
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and their application to test for normality. Stat. & Prob. Letters 2, 945-948.
Al
Appendix
Normal Probability Plot
•
•
•
•
••
••
...
•
•
,.
...
,
.,.
••
o
...
•
=-=.:--~-=-':'~~---L.._-_-l.
-3
-2
-1
o
--l
1
1--_ _---1
2
Sample of size 200 from a.chi-square population
Sample' 1:
Chi-square (2 d.f.)
Appendix
THE 2-SIGMA BAND FOR
A2
d 7 /dt 5 (LOG EMGF)
I
/
/
-t-
'. '.
".
i '--------"--I--1. "
-G!. 5
/
/
/\
I
,
\
\\
\
//
..
-.. ............ -.- .. - ..- ..,
......I
...LI
G!. "
Ill. 5
Sample of size 200 from a chi-square population
Sample
#
1:
Chi-square (2 d.f.)
~
Appendix
A3
Normal Probability Plot
. . . . . . - - - - - - - - - - - - - - - - - - - - - -......
I
I
•
..,1-
•
•
....
••
••
.."...
,• /
;
•••
•
,"•
I
/
~
I
.-/
•••
• •
7 L
-3
•
----.J------l
-3
-1
--I...
0
--!....
--'--
1
3
Sample of size 200 from a chi-square population
Sample
##
2:
Chi-square (10 d.f.)
__
3
A4
Appendix
7HE 2-S1GMA Br.NC FOR
r• Jr------------
d~ lot.' CLOG EMGF)
-'--
.,
-.
III)
•.
-
.
.
.
..
-.. .................... ..- ...
~,-
III)
"1
/
I
/
I
,
/
,
-------------
/'
/
•
..:----------"------_--1...
L.i.8
-8.5
8.8
......
-
... "
.... ..
"" ...
.
",
",
-"-
,
,
,
,
"'.5
Sample of size 200 from a chi-square population
Sample
#
2:
Chi-square (10 d.f.)
..
1.8
AS
Appendix
Normal Probability Plot
~I
I
•
I
"'r
• •
I
j
I
~r
••••
,
••
/.
•
/
,,;
~I
/
i
I
/
I
•
o
~L
I
/
./
I
.,.,1'"
* *. til
.
.....
•
I
•
7
L.-:--
.l....-
L.--_ _---l.
---..l.
o
1
-1
-3
-.l..
Sample of size 200 from a chi-square population
Sample
#
3:
Chi-square (50 d.f.)
-I
3
Appendix
THE 2-SIGMA BAND FOR
A6
dS/~~
(LOG EMGF)
~r----------------------------'"
...................
e·
, .;-
In
.
-
",
-.."..----- ...... .....
...
/
lSi
, "
/
I
I
I
I
,I
" ..
" ...
"... ,,,,
,,
I
,,
I
I
,,
/
-
,,
,,
I
lSI
L 1. 0
-a. 5
0.0
e.5
Sample of size 200 from a chi-square population
Sample II 3:
Chi-square (SO d.f.)
3..GI
Appendix
A7
Normal Probability Plot
.. r------------------
--,
*
*
N
*
IL...
*
...••••
.,....
I
,*
,,;-
./
I
~r
o
/
~
I
/
/
"
."
../
...•
N
•
I
•
•
•
7--------l-----"------L.-_ _----:..
-3
-1
·z
o
1
--'-_ _
Sample of size 200 from a chi-square population
Sample
~
4:
Chi-square (150 d.f.)
-----J
3
Appendix
iHE 2-S1GMA aAND FeR
A8
d3/d~
(LOG EMGF)
~r
I
"
e.
L"l
./
SI
I
I
I
I
/
"
..
.,- , "
.•.
...
.,...".-------.. ... .....
/ "
. .....
....
""
"" ,
'\
'\
/
,
\
/
\
\
/
\
\,
/
lSI
..:
L..1. "
-a, 5
a, "
C.S
Sample of size 200 from a chi-square population
Sample # 4:
Chi-square (ISO d.f.)
1. "
A9
Appendix
Samples of size 50 from a N(O,I) population
Sample • 1
~ II··.".
!
2:519=a l1mIt ••••
'I
... r..,-~···············..·..1·······················.....
. -~"'" . . t ~_~~. .1.1JD1<
o
.
. . . . . . . _~.;~.~.i-~~
..........
_----='.L.
"
,-,-
I
/'
I
""••·••·•·····
'-~,
j
:----1
Ii
"
-~ s1e;ma 11m1 t
"
'
7'"'----.......- .......----'--~--'---~---'
Sample. 3
~..__-------=---r----------,
.................. ··S1~~· ..~~ml t '
-
--==:==:~:iJ:gm,~
-0.4
Sample fI 5
...................
.
........ "S19ma
'.
..., ..
I
,/,/,/
.....
,,_ .... -
-0.4
,/
.,'
-0.8
-0.4
-
,
0.0 0.2 0.4 0.6 0.8
•
' ... "'....
2 s1gma lim1t
........
.
.,
.. ...
/
,.
.,""
"51;'
•
~~ liiiil~
...... ,
"
.
Sa.mple • 6
""
l1mtt··
0
Ol----------if---=_---I
.
.-4
I
"""',
'
7\.l--~--'----'----'--~--'---"'-----'
-0.8
I
--.. .
i ,/
-. s1gma l1m1t "
,1/ . . . .- . . . . .--'--......I..-~-.......l_.......l'--~'
7Uo--.
..
.Uml t
-~ s1gma l1mit "
/
I
,,-
" - 1 - ' - ....
_-----:
~'
7
-
_-------:il..~~_11Iiil~
................
-,- ..-
....·..·_···$.·;~~·1·~·~;·~· ....··..····..
,,,.-'
F====--~;;t~======1
Stat st1c:
..-
.'...
o t======~~~;:::=:=:=::1
.... .....
....
~,.....-------=--r--.:.:------_
.---------=-~----------,
......
o
.
0.0 0.2 0.4 0.6 0.8
2' s1e;ma 11m1 t
I
2 sigm. 11m1t ••'
.....
.-.--
I
.......
'. ••••
s1cpa .11ml-£
.'
sl00a
••••
Sa:ple • 4
.... r-------~"T!--.:.:------....,
.....
iIiiil~
......._.......l_~
0.0 0.2 0.4 0.6 0.8
......................._............
:I~~>---------~~ ::::>,,,
-0.8
-0.4
-0.8
..' ..-
Opo..,,,----------1f-------'""""i
... Stat
... _.•....
..
......
2, 51e;m. 11m1 t •••'
...........
"
7Uo-----'-----'---'--......I..-~-
0.0 0.2 0.4 0.6 0.8
-0.4
..
.
"
I
I····.........
. -
"",
~~ "
-0.8
1 cpa !1.JD1 t
_. -.. s_..........
01--------+-------1
.... _
'.
-....... 1Iiiil~ ... .
~a
"
,/
~L '.~i:.:~. 1:.:1.~ :: ."
7Uo-----'--.. . . . - .......---'----'--~-.......l---0.4
0.0 O.Z 0.4 0.6 0.8 -0.8
VB.
t
0.0 0.2 0.4 0.6 0.8
A10
Appendix
Samples of size SO from a Cauchy population
Sample I 1
~l
Sa=ple I 2
Sl r-------=---=-------.
I
I
oL
2~,
lI'Ir
2~
i
~I
I,
I
o :~:;;.:.::::~.:;..::. __
t
i
I
oi
I I
~~
I
7~
'----L_--'-_.....I-._-'--_'----'_....J .o.,Y.-_"--_'--_ _- - : _ - - ' _ - - I . _........
0.0 0.2 0.4 0.6 0.1 loO.1
-0.4
-0.4
0.0 0.2 0.4 0.6 0.1
21.1.-
_~
!oO.1
Sa!Ilple I 4
~f"---- - - = - - - - - - = - - - - "
e'
10
.•..........-
'"
~
.
'" .......- . -.-.......
·····1·
oI
7
----.
r-'--
__
-~~.;:::::::.;;~~~~~;.~~~~~~~=.~
..
~u-._"--_'--~'-----''-----''-----'-_--'-_....J
-0.1
-0.4
0.0 0.2 0.4 0.6 0.8 -0.8
SClple I S
.. lr--------=--.-l------~
,
;
i
"' ..
............._ _........ .
:[
o
- ~.-::.~~
I. _ :.::.::..-~::·.::.:::=·:t·.._.:::..:::.:.::.::::.:
..
---.
--
-0.4
0.0 0.2 0.4 0.6 0.8
Sacple .. 6
lI'I.---------=:..--,,:--------......
:f
'"~ '~~::
"
..
--~~-----
------------.......-.........
....
I
-0.8
-0.4
0.0 0.% 0.4 0.6 0.8 -0.8
Jog
mn (t )
0.0 0.2 0.4 0.6 0.8
-0.4
VS.
t
All
Appendix
Samples of size 50 from a Chi-square (2 d.f.) population
Sample I 1
"'.. . - ------=--..----------,
"',,-------.....:~r-~--_:::--~
...
...
...
...
-- -_._----
.... .............-..
I
,,--
",
,," . . . . _....I.
":I:..._.
-0.4
-0.8
....,
.....
__._.-. .
;--_.-=~-_±_-.::-._.
--............. ..........
---.........
------
",
.... "
.'.'
_...._--.•.._....
Ot--------+-------~
O~-------;-------_I
..
Sa=ple • 1
...."
"
"_
,
"
....---
----............
..
........
"
_
" ,
" ": ......" ........ ......._---''----'-_..:...._-'-_........_~
'--_..L.-_........_ ........
0.0 0.1 0.4 0.6 0.8 -0.8
-0.4
0.0 0.1 0.4 0.6 0.8
_-'-_~
Saople • 4
Sa::Ipl. I 3
..,r:--------':...-oy--'-------.
....
... r---------'=--oy----------,
...
...
-_.
__ _
,-,-------
--
o~-------t-------_I
.
~
..~..~;~..~-_..
..
,,"
,,'"
.
....-.........
.............
........
,,,,"
" : . . : . . _ - - - ' _ _......._
-0.8
_
-------------.... "
ot--------+----------I
--_.... ...... ..
-0.4
........._ - - - '_ _......_ . . L . _....._ ~
": ......_
......_ - - ' ' - - _......._ - L
0.0 0.1 0.4 0.6 0.8 -0.8
-0.4
........._
'"
........_~
0.0 0.1 0.4 0.6 0.8
Sample • 6
Sacple • 5
.., ,..--------=---r--------,
oDr--------=---r--'--------,
..
...
...
_.-----------.....-....... ....
""..----- -------.....
O......--------!--------I
-----.:::.-::-.
,..------
---::::::::-----.
~....
",
.
": __........_....I._---''--_..L.-_........_
........_-'-_.....J
-0.8
-0.4
" ..
.
.
... ......
~.
....
..........,
,,"
,,"
"",
7U--.. . . .- .....----''----'--.. . . .- -'--.. . . .---..
-0.4
0.0 0.1 0.4 0.6 0.8 -0.8
Jog
...
m.. (t)
V8.
t
0.0 0.1 0.4 0.6 0.8
Appendix
A12
Samples of size SO from a N(O,l) population
SUlple .. 1
.---------=------------,
I
1"1
;~\
...
o
--_.~.~
....:...... ~ ...
1 fl9=a
I ....
11~;2:
•.:~.~
:...-
........ ."""
o ......."....
f
,/
f"l
I
I
-1 sigma
"
Sample. 2
r-------=-......
- --------,
.
~
'1 ei r
...
."""...--
....····;;;·"S"tati5tlc:!"',
...
o
l1ml~,""1 ... 1\.
ei ~ .•..
2 519=a
.
1"1
•••••
"'~'.>.:: .... 1
"0.....
I'
'"
f"l
'
.. "
I
/
\.
1 -2 51g.na I1mlt
'
'
-1 51;=a l1m1~,
/
I
I""
repa ~.~~~2:'
I
•••••••• tI'"
o
11ml~,
I
..... >"S"tadstlC:"
~
.
2 slg.na l1m.l,.t
.......
'I
/"
,
I
I,-2
"
\
sllT-la 11m1t .
0 ......-....1---'--.........-.......:...'_-'-_--1._---'_---1
-0.4
0.0 0.2 0.4 0.6 0.8
-=-.-
ro.-o
...
...
1
0.0 0.2 0.4 0.6 0.8
11~~,>:'
2 slg.na
o
-0.4
--,
Sa:pl. .. 3
o
~0.8
e'
19ma l1m~I:",.,../
.....
"""",,"'"
",';'~
.........~.•.........
19ma
-1
o,
,, .....
I1m1~,
,,
\
-2 519=a l1mlt .
0 ......--"--------''---'----0..-,.'--'-----'
01.4--.L.----''------''------'----'----'--~
~0.8
-0.4
0.0 0.2 0.4 0.6 0.8 "0.8
-0.4
0.0 0.2 0.4 0.6 0.8
f"l
o
...
SUlple .. 5
r------.......;;......--------,
1
'. . . .
o
...
1
>
~"
.
o
.~:. ~-~
. .:..-.=,..
1
st ....
'
11gma 11~~2:' .......
. . . . . : '....
o
/
I
i
-1
.
.
19ma 11m11>,
,
~
•
L"\
~ "
I
'.
"'... ....................
".
............. :
.... ....
.
,
",
11m11>,
oI
" ,,
,
",
11m1t .
,"
2 sle;ma
..•....
...
.
,,~
I
ei r-I.-------"--,------..,
o
I ....;:.:'-.' .'"
..... ·;;;·"S"tat1st1c:!"',
...
"~:.>"
SUlple • 6
1"1
\
-2 s1e;ma l1m1t .
-2 519=a
0 ......- ......-'----'----'--.......- ..........- - - . . .
01.1...--'----1.---'--........- .......-"""---......---'1
"0.8
-0.4
0.0 0.2 0.4 0.6 0.8 "0.8
-0."
0.0 0.2 0.4 0.6 0.8
f"l
'
-dtd
Jog 1iin (t) - t
VB.
t
A13
Appendix
Samples of size SO from a Cauchy population
Sample I 1
~
SUlple I 2
";1:-------"------~
~.---------=---------,
o
...
...
o
...
,.,
~LI--.l.---''----'---'---'-----'"-
-0.4
0.0 0.2 0.4 0.6 0.8 -0.8
..
..
Sa.cple I 3
~
....- .....
0.0 0.2 0.4 0.6 0.8
-0.4
....-
S_Ul__=p_l_e_I_4
-,
o
.
o
o
.... ~~
- - . . :.• : " ••.-!, . . . . . .
Sample I 6
Sa:nple I 5
I
I
... 1..
o __ ::
.f'" --- . ..._- ..
.
-
~
,.1
..... ~.....:-: ....
"
-..::..:::
....",.
o
..."0.8
u-_"'----''--~
-0.4
- L - _........_-'-----J
CD
o
..
o
..
oLL...-.l.---''----'---'-.......-----'"---'--0.0 0.2 0.4 0.6 0.8 "0.8
-0.4
0.0 0.2 0.4 0.6 0.8
d
dt log
mn (t)
- t
VB.
t
Appendix
A14
Samples of size SO from a Chi-square (2 d.f.) population
Sample 41 1
CD
o!
sacP!. W 4
.0'
0'I
• .. 1
O!
"I"
0(>
.
01
. ~_:..~ ........
.
---'«'«'1
01
o
.. ;
I
~"_~_~_",------,_--,-
"0.8
-0.4
----,l
0.0 0.2 0.4 0.6 0.8
Sacple 41 3
.0
oj'"------....:....-....;...------
-0.4
0.0 0.2 0.4 0.6 0.8
SaJ:IP1 e tI 4
III
or---------"------....----..,
j
.. I
.... :.:.-.;;:::;,::::::
........
.
. ., I
o~:."-_-'-_-"-_-'-_-"-_-'-_........_ ........_....JI
..
"0.8
"0.8
-0.4
0.0 0.2 0.4 0.6 0.8
Sample 41 5
ou--.........-"""-----........- " - - - ' - - - - ' - - - J
-0.4
0.0 0.2 0.4 0.6 0.8
Sample 41 6
III
or--------.::.-----------.
;
"I:
,
~.
..
ou--.......-"---"'----"---"------''----......
0.0 0.2 0.4 0.6 0.8
-dtd
"0.8
-0.4
VB.
0.0 0.2 0.4 0.6 0.8
t
Appendix
Histogram for sample
A15
#
6 from N(O,l)
o:-
""
...'lI'l
...o
~-
;f)-
-
<>-'
I
I
-4
-~
I
o
I
I
~
"
A16
Appendix
Sa~ples
of size SO from a N(O,l) population
oD
l
I
.. L
I
ot--"::::::>O......:::-----i-----:::7""""''---;
.~=.:::.:=.~...............
~
~::.~
...
I
,---
:>--=<:
. ~
19m. }.!m1t
•
-2, ....fqm. l1m1t
,
_ ••••••••••••••••••
~",,-
.-
- '..
St.t st1c
--
:;>-<:.
_.~
-1 s1gca }!m1t
-2.:rfgrr.. l1m1t
' ..... -,'
~ _.L---"-----''-'_'...J.-_-_'-'--_ _& . -..... ~U-_......_....L._--L._---J'-----''-----''-_ _~
-0.4
0.0 0.2 0.4 0.6 0.8 -0.8
-0.4
-0.8
0.0 0.2 0.4 0.6 0.8
"
rr
"'-,
-1
"
'''',-
I
_-
,,-I
Salllple • 4
Sample' 3
.c,...---------=:....,,...---------.
oD r----------=.......,-~------..,
""
.........
...-
-
......'.
51'9=. l1m1 t
a
............."
l~~.t
..........
_......
o!----~===I===--=-----1
1 -.··::·:.........
...
~,
,
I
rr
'"
I
'"
0°
···:i··i·;.
/'
I
' ..... ~
-0.4
...
l1m1t
_--'-_-.l.
-0.8
'
}!~1t
-2 ....fgm.
--'
...._ _L.-_-'-_....L.-_........
:
~
_::~::- .
Stat St1c
"...........
~
0.0 0.2 0.4 0.6 0.8
Sample • 5
oD,...--------=---,---------,
••••••••••• -····;·1'9111. 111111 t
......... __
"._""
_1...,.-e;,~. 1 ~~~
..
~ ::=::.~:
•
_~ .
""
I
'
_--'-_--'-_---JL.-_L.-_L.-_'-----I
-0.8
-0.4
0.0 0.2 0.4 0.6 0.8
Sample' 6
.c,...---------=---,,--------,,
......
I
's.·1911. 111111 t
a 1~1t
.'
.
19m. }.!m1t
- 2 ....fgc. 11III1 t
-,'
.
-1 s1gma
- 2 .:rfgc. 11III1 t
..., -,'
~u.-_......
•
::~.:••.........:;;::~:::~
,.,-- ........
"
"
}!1II1t
,
"
......._....
O ........--'7"'"""----i------'''''''''=::----i
.....
:--.:
N....
I
rL
I
"
"
.
-0.
""
_
'.. . .
.cLl-_......:..._--'-_--'-_---''-----'L.----'L.-_.:...--I
I
-0.4
0.0 0.2 0.4 0.6 0.8
d4
re ¢n (t) - (t .c
dt 4
--
-0.8
0.00.2 0.4 0.6
-0.4
t2
- 6t 2 + 3) e
2
V8.
t
o.e
All
Appendix
Samples of size SO from a Cauchy population
~
o
Sample I 1
~
:1
~r
:
~ r...
I
!
:= ::.-:.
-0.4
., I
I
I
0.0 O.l 0.4 0.6 0.8
Sa:lple I 3
=1
....
\
~tV
21, . .
~0.8
Sample • l
~.--------=---------
...o
I
o
.. "'--""-----"-------"----''----'_.......:._.-j
!.0.8
-0.4
0.0 0.2 0.4 0.6 0.8
S&l:Iple • 4
~.--------=---------......,
,
ID~
...
o
~~
... I
I
o i==~:::::=---.f--+_-_+_-==..:::::===::::.j
0
...
~
I
I
~
I
...o
ID
I
I
=.
~
';'tL...._~_'___'___'------'
-0.8
~
-0.4
0.0 0.2 0.4 0.6 0.8
Sample I 5
-------=-----------,
r-I
I
~r
..
.................
.
Ol-----I----I---~----""--f
~ ILL'_ - - ' ~0.8
...I...-_-'--_"-----'
-0.4
~_-'
0.0 O. l 0.4 0.6 0.8
-0.8
-0.4
_
____'_--=::..__.-j
0.0 O. l 0.4 0.6 0.8
Sample • 6
=.-----------'=---------......,
..
...
~..:::;--'---'---'---'------'-----'---'-~
0.0 O.l 0.4 0.6 0.8
-0.4
-0.8
~n(t) - (t 4
t2
-
6t 2 + 3) e
2
'/l8.
t
Appendix
AlB
Samples of size 50 from a Chi-square (2 d . f . ) popu I
'
atlon
Sa.:npl •
~
..
•
1
Sar:pl• • 2
~
..
.......,
-""
".'
...
...
....._.....
0
-
.............~
,,
'"
':''"'-_......_-''-_......_......._-''-_......
"-
-0.8
-0.4
_
......
_~
':'~-.......--'-----""---'_--''___--''____'___--J
0.0 0.2 0.4 0.6 0.8 -0.8
S&l:!pl. I 3
~~--------=---.----------,
0.0 0.2 0.4 0.6 0.8
-0.4
Sa::ple • "
...
~r--------=--r-------
-.......•.
..
",
'" .,--..-,.......
--"
':'u.-_......._-L._--L_--'_ _'--_.......
-0.8
-0.4
_-'-_~
':' ....._-'-_--L_~ _ _'____'___'___.......---J
0.0 0.2 0.4 0.6 0.8 -0.8
-0.4
0.0 0.2 0.4 0.6 0.8
Sa=pl. I 5
CDr---------=.--~-----_..,
.....
Sa~pl. • 6
~r_--------=___.r__------__.
..
...
Ot----+---t----;:----t
...
.
I
I
'fu.--.. . . . - -'----"---''-----''------''---'---0.4
0.0 0.2 0.4 0.6 0.8 -0.8
re
i>a (t )
0.0 0.2 0.4 0.6 0.8
-0.4
t2
+ 3) e
2
VB.
t
A19
Appendix
d3
[-3 log
Power Calculations For n' dt K (
mn (t)]2
o t, t
)
Level Of Significance = 0.10.
Sample Size
=
50.
=1/= of Samples Used
=
500.
Random Number Generators: G05DHF and G05DJF (from the NAG
subroutine library.)
Distribution
y~
x~
t
0.5
II
y~
II
X~n
X~
X~
II
X~
X~n
X~
X~
X~
X~n
t,
tA
tln
t"
tA
t,n
0.2
••
..
••
0.0
••
••
••
0.0
II
II
0.4
••
••
-
Power
0.Q72
0.872
0.714
0.510
1.000
0.Q84
0.8Q4
0.712
1.000
1.000
0.Q70
0.800
0.Q16
0.460
0.2QO
0.584
0.420
0.224