Estimating parameters from mixed samples using sample fractional moments

Journal

of Statistical

Planning

and Inference

231

21 (1989) 231-244

North-Holland

ESTIMATING PARAMETERS FROM MIXED SAMPLE FRACTIONAL MOMENTS Steven

USING

G. FROM

Department

K.M.

SAMPLES

of Mathematics,

University

of Nebraska-Omaha,

Omaha,

NE 68182-0243,

U.S.A.

La1 SAXENA

Department

of Mathematics

and Statistics,

University

of Nebraska-Lincoln,

Lincoln,

NE 68588,

U.S.A. Received

19 January

Recommended

Abstract: mixing

Alternate

estimators

two independent

asymptotically estimators,

normal

Subject

Key words:

based

samples

Classification:

moments

for estimating

These estimators

by optimally

In large samples,

in the exponential

Fractional

on fractional

are proposed.

and are obtained

all of closed form.

posed estimators AMS

1987

by S. Zacks

combining

these estimators

scale case by numerically

parameters

are strongly several

methods

are compared computing

consistent

and

of moments

to previously

asymptotic

when

pro-

variances.

62FlO.

moments;

asymptotic

variances;

mixed samples.

1. Introduction Let 9 = {F(x; 8) : BE Cl} be a parametric family of cumulative distribution funcrandom variables tions where x and 8 are both real. Let X,, . . . , X, be n independent with m of the X;‘s distributed according to F(x; 0,) and n-m of the X;‘s distributed according to F(x; St), where m < n, 0, I 02. Assume that m and n are known and that it is not possible to identify the Xl’s which are distributed according to F(x; 0,) and the X;‘s which are distributed according to F(x; 0,). The objective is to estimate et, 19,. This non-identifiability problem may be very real in situations where confidentiality of a subject’s classification in one of the two populations is crucial. This problem has been considered recently, in several different contexts, by Shaked and Tran (1982), Chikkagoudar and Kanchur (1980), Marks and Rao (1978) and Kale (1975). Kale obtained the maximum likelihood estimators of 8, and f+ when F(x; 0) is a normal cdf or an exponential cdf. But these estimators are inconsistent. Shaked and Tran (1982) used a method of moments-type procedure, based on two different linear combinations of order statistics, to obtain estimators for 8, 0378.3758/89/$3.50

‘CC1989, Elsevier

Science

Publishers

B.V. (North-Holland)

232

S. G. From, K.M. L. Saxena / Estimating parameters from mixed samples

and f&, which are strongly consistent scale family of distributions.

and asymptotically

normal,

for location

and

In Section 2 we propose another set of estimators denoted by 0, and & which are based on fractional moments and which are optimal linear combinations of a finite number of estimators calculated from several different sample fractional moments. These estimators are motivated by the following. Soong (1969), when estimating a real parameter 6’from a single random sample, showed that an optimal linear combination of two or three different method of moments estimators has a higher asymptotic relative efficiency (ARE) than any of the ARE of the estimators combined, in the gamma shape parameter family and in a mixture of exponential family. Tallis and Light (1967), using sample fractional moments of the form gains in ARE C’,, KU/n, O
as

m

and

n--too.

2. The estimators The methods of estimation for 8, and &, as is the case with ST estimators, depends on the form of the family of distributions {F(x; S)}. In this paper we consider only the scale parameter family. Let F(x; f?)=F(x/B), x>O, 8>0. Let f(x; 0) = (1 /@Jo/e) be the corresponding density function. Let r be a positive integer and let {(x1, . . . , a,} be a set of positive numbers such that ai # 2aj for any i, j. Assume that ‘rn

\

~*~fl+*~,f(x) dx
1 li,

jlr.

._O

Define -)cn

tj=

t(olj)

x”‘f(x)

=

L0 I

dx,

j = 1, . . . , r,

(2.1)

233

S.G. From, K.M.L. Saxena / Estimating parameters from mixed samples

and

‘co x2ajf(x) dx, I .,O

sj=s(clj)=

j= 1, . . . . r.

(2.2)

Let kV,i=IV,(a,)=~,~,X~, I

j=l,...,

r,

(2.3)

and W2J=

Wl(Orj)= k ,g, X;‘“/, j=l I

Forj=l,..., the system

r, let 8t,, ~~j denote the solution

9... 3r.

(2.4)

(when it exists) satisfying

e,jI

ozj, to

of equations

(2.5) and

(2.6) obtained by equating Wj’S to their expected &, to be of the form

values.

We propose

estimators,

0, and

(2.7) where C$=t C,j= 1, i= 1,2. It should be pointed out that the Qlj’s, which are given in the following theorem, are of closed form. The choice for the C,‘s is discussed later in this section. Note that if r= 1 and al = 1, then St and & are method of moments estimators given in Shaked and Tran (1982). Theorem 2.1. For any aj>O such that S(Olj)< 03, a solution tions (2.5) and (2.6) exists with probability tending to one. Proof. The system of equations Ot(CZj)5 f?l(aj) provided

(2.5)

and

(2.6)

ofthe system ofequa-

has a real

Dl(~~)=(t(a~)>4s(orj)W*(orj)-(t(a~))2b(olj))2(W~(~j))22O

solution

satisfying

(2.8)

and l/2 D2(aj)=

If both

Wl(aj)t(aj>s(~j)P

(2.8) and (2.9) hold,

4(q) =

m

n-m

Dl(Orj)

2 0.

>

(2.9)

define

D,(aj) (t(aj>)2s(aj>

1/a,

1

(2.10)

234

S. G. From,

K.M. L. Saxena

/ Estimating

parameters

from

and

‘1. l/2

II - tn W(~j)~(@J)~(~j)f

--,(a,)

I Then that 0<8, given

m

(

8,(~j)j)=

mixed samples

(t(aj))2s(aj)

I/a,

(2.11)

it is easily seen that 8,(ai) and Q2(aj) satisfy (2.5) and (2.6). The probability both (2.8) and (2.9) hold tends to 1 as m---t 03, n + 00, m/n +PE(O, 1) for <8,. This can be seen from the fact that the means of W,(~j) and W,(Orj) by

have continuous partial J(B,, 0,) given by J(e,,

0,)

=

derivatives

abhp4 apclza, ia4

I

to 8, and B2. The Jacobian

)

ajep-'

i -

n

n

2ajep-

l

determinant

for 0 < 0, < e2 and all aj > 0 such that .s(aj) < 03. Applying analysis gives

(Implicit

Function

WC"jl

we obtain

matrix

apzcr, /ad,

( m) ( m>

S(OIJ) i -

has non-zero

a.h,/ae2

(

t(aj)

=

with respect

Theorem)

and the strong

~llia,(e,,e2)1,,,:,,=p,

theorem

of

which

i-l,&

as m-w,

PID,(a,)rO,D,(aj)rO]-,l

the inversion

law of large numbers

n-co,

m/n+pE(O,l).

Note that the above theorem also establishes the strong consistency of dj(aj), i-1,2, forO<8,<82andforallor,>Osuchthats(aj)<~. Wecanmake(?,(aj)and B,(ol,) strongly consistent for 0, = e2 if we define them as follows. If D,(oc,)
=

Q2(CXj)

=

L 1 ~

f(orj)

If D,(Olj)ZO and Dz(a,)
S.G. From, K.M.L.

to minimize well known Lemma.

C=(C1,...,

the asymptotic lemma.

Saxena / Estimating parameters from

variances

mixed samples

23s

of 0, and &, for which we need the following

Let .E be a positive definite, symmetric C,.) given by

rxr

matrix.

c=(lT’l)-‘PI,

Then the vector (2.12)

minimizes d%d among all r x 1 vectors d = (d,, . . . , d,)’ such that d’l - 1 = 0, where 1 is an rx 1 vector of ones. Also, the value of C%C is (l’,Z-‘1))‘. From

standard

large sample

fi((e^,,,...,@,,,e,,

theory, ,...,82r)-(8,,...,8,,~2,...,Bz))

has, asymptotically, a multivariate normal distribution. Let Ai = Aj(B1, 02, p) denote the asymptotic covariance matrix of (&;r, . . . , 8,,), i= 1,2. Applying the above lemma and minimizing asymptotic variance, the two sets of coefficients Cr and C, minimizing the asymptotic variances of 6, and &, respectively, are given by Ci=(C,,,

i-1,2.

. . ..C.,)‘=(l’A;‘l)-‘A,~‘l,

(2.13)

These are referred to in the sequel as the optimal coefficients. The formulas for the entries of A, and A, are given by (A.2)-(A.4) of the Appendix. Since C, depends on the unknown parameters 8, and Q2 as well as p, a two-stage estimation procedure is to be employed. Such a procedure is described below for the scale parameter case. Stage 1. Calculate consistent Qp), respectively. Following

initial estimates of 8, and &. Denote are some possibilities.

these by 8/O’ and

i= 1,2,

(4 (b)

B;(O)=median

of @;r,. . . . dj,,

(c)

e;‘o’= e,(l) = method

i= 1,2,

of moments

estimators,

i = 1,2.

Stage 2. If f?~‘0, calculate consistent initial estimates of the optimal coefficients Crj and C2j, j= 1, .._, r, by calculating C,j(8,“‘, @f’, m/n), i= 1,2, j= 1, . . . , r, from (2.13) with d1(0),6r’ and m/n replacing Q,, o2 and p, respectively. Next calculate

2

j=l

Clj(61(0),tJj”), m/n)Blj

(2.14)

and (2.15) J=l

236

S.G. From, K.M.L. Saxena / Estimating parameters from mixed samples

If (2.14) and (2.15) are non-negative, 6, = expression

(2.14)

define and

I& = expression

(2.15).

If (2.14) is negative, define oi = 0. If (2.15) is negative, conclude & =O. If 8,‘o’lO and $@z 0, define 6, = 0 and I$&= 8, -(O). If 8,‘0’< 0 and C?p) Bz in any of the above cases replace both 6, and f$ by +(8, + I?&). Finally, if 8,(0’r or’, conclude 0, = & = i(Q’O’ + #O)) I 2 The probability that 6, and g2 are given by (2.14) and (2.15), respectively, tends to 1 as m, n --t m and m/n +p E (0,l) for 0~ 8i < tY2< 03 by the consistency of 6,c”)and Q(O). 2

It is easily seen that fi(6, - 8i, d2 - 0,) has, asymptotically, a bivariate normal distribution. The formulas for the asymptotic variances in the asymptotic covariance matrix are given in the Appendix. Note that the estimators 0, and o2 are defined for any scale parameter family for which ‘rn x*f(x) dx< 03, I ,O for some estimators

6> 0. An assumption is that

in the scale parameter

case needed

for the ST

‘m

I

x2+‘f(x)

dx< 03.

.O

for some E > 0. Thus 0, and Q2 are defined for a larger class of parametric families than the ST estimators. On the other hand, design subjectivity exists in the proposed estimators 8, and Q2, namely, in the choice of r and (pi, . . . , a,. Some definite design guidelines are given in Section 3 for the exponential scale case and some general design guidelines are given in Section 4. It is emphasized that if enough a-values (r large enough) and a wide enough range of a-values are used, highly efficient estimators are possible in the exponential case. Some of the numerical results are given in the next section. 3. Numerical

comparisons

In this section for the exponential scale family, (a) we present the results of a numerical study to assess the dependence of the asymptotic variances of O;(U) on a and (b) we compare the asymptotic variances of C$and OTT, i = 1,2. (a) Asymptotic variance of &;(a). We have F(x; 0) = 1 - exp(-x/8), x>O, 8> 0. Let r/;(a, t?,, 8,, p), where p = m/n, denote the asymptotic variance of Q;(a). It can be shown that

XC. From, K.M. L. Saxena / Estimating parameters from mixed samples

231

v,(a,8,,e2,~)=e:~(cr,1,p,p),i-42, where p= &/6,. given in Shaked Appendix.

(3.1)

Note that if cr = 1, e,(l) are the method of moments estimators and Tran (1982). Expressions for y((w, 8,, 8,, p) are given in the

Let cXi(p, p) denote

the value of (Y for which

vI(@i(P, P), 1, P, P) = inf vl(a, 1, P, P), CZ>O

i = L2.

(3.2)

A preliminary numerical study showed that for o-values larger than 4, the asymptotic variances of f?;(cx), i = 1,2, are very large, unless et and f& are very close, i.e., /?= 1. For this reason, in actual computations we have taken the infimum for 0< ~~54. Tables 1 and 2 give the comparisons of asymptotic variances for 8;:(a) for values 1 and a;(/$~) for a. From Tables 1 and 2 several observations can be made: (1) The choice of CI seems to be more crucial to the estimation of 8, than to the estimation of 19~. (2) Values of a smaller than 1.O are optimal as long as p = B/f?, is not too close to 1.O; values of (Ylarger than 1 .O are optimal if /I is close to 1 .O. (3) For any given p E (0, l), the choice of a is most crucial for large /3. (4) For a given p, the choice of (x appears to be most crucial for values of p close to 1, especially when estimating 8,. (5) The relative efficiency of o;(l) versus 8,(a;(p, p)), can be compared by comparing Table

Y(l, 1, P, P) with

K(a;(P,

P), l,P,

P),

i=

1,2.

1

Values of ru,(D,p)=a,,

V’,(LLD,pp)=b~

and

v,(al,l.D,~)=cl

P P 0.1

0.3

0.5

0.7

0.9

1.5

2.0

3.0

5.0

10.0

a1 b,

1.13

0.96

0.77

0.62

0.49

2.85

1.88

1.81

4.92

Cl

2.19

1.88

1.64

2.33 1.58

al h

0.91

0.75

0.35

4.23

0.59 4.62

0.46

5.96

1.43

Cl

5.88

3.77

3.06

2.78

20.33 2.63

a, b,

0.80

0.64

0.49

0.37

0.27

14.12

12.25

21.56

Cl

13.19

10.50 8.08

6.16

5.28

63.13 4.12

0, b, Cl

46.04 40.18

0.57 35.28

0.42 42.92

0.29 79.34

0.20 243.40

23.52

16.80

13.41

11.06

al b,

0.50 372.50

0.34 466.25

0.22

475.62

0.11 2787.81

Cl

386.64

215.88

142.60

0.72

0.66

889.06 : 02.44

1.58

72.80

S.G. From, K.M.L.

238

Table Values

2 of c&3, P) = a2. V2(l, 1, h P) = b2 and

P 0.1

0.3

0.5

0.7

0.9

Saxena / Estimating parameters from mixed samples

1.5

V2(az, 1, P, P) = c2

2.0

3.0

5.0

10.0

a2

1.15

0.94

0.49

163.12

102.50

0.73 146.25

0.58

b2

349.06

1308.92

c2

158.93

102.06

132.61

288.12

1037.55

0.90

0.72

0.57

0.47

0.42

4

32.15

28.61

47.36

116.01

436.24

c2

31.73

26.04

38.88

90.65

339.32

a2

0.78

0.62

0.49

0.41

0.39

02

16.62

16.50

28.25

69.56

261.73

c2

15.68

14.08

22.16

53.18

202.33

a2

a2

0.70

0.55

0.43

0.37

0.38

6,

11.06

11.58

20.13

49.67

186.95

c2

9.98

9.42

15.32

37.46

144.04

02 6,

0.65 8.25

0.50

0.39

0.34

0.36

8.92

15.63

38.63

145.41

C2

7.15

6.99

11.61

28.82

111.78

Tables 1 and 2 can be used to come up with quick and fairly efficient estimators of closed form. One possible approach suggests itself. Let p^= 1$,(1)/8i(l). Determine the values of /? and p in the above tables closest to p^ and m/n, respectively. Denote these values by p* and p*. Estimate 0, and I!?~ by 6,(cx,(p*, p*)) and &(cr,(/3*,p*)), respectively. If needed, the above tables can be made finer to include more values of p and p. Alternatively, one can actually compute ai@, m/n), i= 1,2, and estimate Bj by ~!?~((~~(j!&~m/n)),i= 1,2, although a minimization routine would be needed to carry this out. If asymptotic efficiency is of prime concern, however, we suggest the estimators, @, and I$ using r a-values for each discussed below. (b) Asymptotic variances of I$ and i?T’, i= 1,2. Let AV(@;) denote the asymptotic variance of fi(dj0;) and let AV(QlsT) denote the asymptotic variance of &(@r - e,), i = 1,2. Formulas for AV(6,) for the scale parameter case are given in the Appendix, and formulas for AV(dTT) can be found in Shaked and Tran (1982). We compare these asymptotic variances numerically for the exponential scale case. From a preliminary numerical investigation it was found that a good value of r to use is r= 10 and a good set of ajj-values (from both a computational and efficiency standpoint) is: j:l CY,: 0.08

2

3

4

5

6

7

8

9

10

0.19

0.30

0.45

0.57

0.72

0.80

1.00

2.40

4.00

S.G. From, K.M.L. Saxena / Estimating

There is nothing

special about

parameters

these or,-values chosen

from

239

mixed samples

above.

As long as one uses

seven or eight aj-values, more or less equally spaced in (0,1] and a couple of ojvalues in (1, S), more or less equally-spaced, and making sure that CXi#2CXjfor any i, j, then highly efficient estimators will be obtained. Since the relative efficiency, AV(8,)/AV(@‘), depends only on p = &/Qr , i = 1,2, 19,was always set equal to I .O. The other parameter values chosen were: p: 0.05

0,: 1.1 (0.1) 1.9; 2.0 (1.0) 10.0; 20.0.

0.95,

(0.05)

All matrix operations were performed by IMSL (see IMSL reference) routines (LINV2F for matrix inversion, VMULFF for matrix multiplication, and MGAMA to evaluate the gamma function). We present the numerical results in the form of two tables, one for comparing AV(8,), and AV(6fT) (Table 3) and one for comparing AV(&) and AV(etT) (Table 4). Only the results for a small fraction of the above parameter combinations are presented in these tables, however, these tables are representative. Upon inspecting Table 3, we see that 6t outperforms 6FT in terms of higher efficiency (smaller asymptotic variance), especially when p = 19/o, I 4. Occassionally when /I= 10 or 20, e^B’ has slightly higher efficiency but only for values of p close

Table 3 Asymptotic

variances

AV(o,)

and AV(fiF)

for the exponential

scale family

(0, = 1)

P 02

1.1

1.5

2.0

3.0

0.3

0.5

0.7

40.34

95.84

281.12

1345.92

33.05

59.49

125.11

369.26

3505.52

AV(.)

0.1

(jsr CJ;

0.9 36686.8

QST

3.47

8.31

22.40

94.32

2155.2

0;

2.78

5.79

12.91

39.20

316.57

8, -ST

2.05

4.43

10.76

39.92

760.62

0,

1.82

3.58

7.63

22.17

203.43

$7

1.59

3.04

6.56

20.84

306.45

0:

1.53

2.77

5.52

14.98

126.96

0, -57

I .47

2.65

5.34

15.51

191.27

0,

1.45

2.51

4.80

12.43

99.11

5.0

0“ST _’ 0,

I .42 1.41

2.45 2.37

4.75 4.40

12.98 I1 .Ol

140.78 83.57

10.0

e-ST ^’ 0,

1.32 1.33

2.10 2.07

3.69 3.60

8.72 8.20

66.95 52.11

20.0

8-ST 1 0,

1.27 1.28

1.89 1.92

3.12 3.13

6.68 6.51

39.41 35.22

4.0

S.C. From, K.M. L. Saxena / Estimating parameters from mixed samples

240

to 0. The comparison is probably unfair for /3 = Bz = 1.1 since the asymptotic bution theory holds only for 8, < 19~and not for 6, = 19~.The approximations St.Dev.

of 6;= n-“2

IIAv
distri-

i= 1,2,

and St . Dev . of esT= n-“2 I

m),

i-1,2,

are poor if /I = 1.1 and other P-values close to 1. Table 4 shows that o2 performs better than 6:’ when p14.0, approximately. When /3>4.0, & and e,“T perform equally well. Overall, it can be concluded that, in terms of asymptotic variances, 19;outperforms O,sT, i = 1,2. Since czs = 1.O, 67;also outperforms the method of moments estimators. The inefficiency of the ST estimators when /I is close to 1 can be explained by the fact that larger cr-values are optimal for /3= 1. The ST estimators throw away much information carried by the term EYE1X,!, in this case. When 8,~ 8, (p& I), smaller a-values are optimal; this is the reason g;(l), i= 1,2, are inefficient for p& 1. The proposed estimators take advantage of the fact that larger a-values are optimal if j3 is close to 1 and that smaller a-values are optimal if p& 1. Thus, sometimes Cl’=, X,’ should be used and sometimes it should not be used. The ST estimators always throw C:‘=, X,’ away, whereas the method of moments estimators, 8;(l), i = 1,2, always use it, explaining why the ST estimators are efficient for larger p and Table

4

Asymptotic

variances

AV(&)

and AV(Qy)

for the exponential

scale family

(0, = 1)

P

02 1.1

AV(. )

0.1

0.3

0.5

0.7

0.9

3189.45

518.06 320.13

281.54

248.33

454.14

125.53

68.94

44.50

213.32

44.99

24.90

31.24

15.41

19.93 9.80

28.98

157.49 115.66

29.65

16.76

12.43

13.71

97.21

25.06

13.63

9.17

6.83

128.92

38.78

22.56

16.07

13.66

123.82

37.28

21.52

15.00

11.44

189.42

59.96

187.12

59.22

35.34 34.80

25.09 24.53

20.02 18.88

275 .OO

88.90 88.47

52.75 52.40

37.49 37.13

29.39 28.69 111.81 111.63

2599.06 1.5

2.0

3.0

4.0

5.0

273.95 10.0

20.0

1017.21 1017.62

336.97

201.69

143.85

336.85

201.60

143.75

4012.51 4013.47

1335.86 1336.05

801.12 801.08

572.04 571.98

7.03

444.81 444.80

S. G. From,

K.M. L. Saxena / Estimating

parameters

from

mixed samples

the methods of moments estimators are efficient for smaller this will be the case in general for other scale families.

4. Design and computational

p. It is believed

241

that

aspects

Choice of the c-w,‘s.The question of how to pick r and oi, i= 1, . . . , r, for arbitrary scale families remains an open one. As a general guideline, it is recommended that larger a-values (ai> 1 .O) be used if j?= &/et is believed to be near 1 (say 1.5 or less) and that smaller o-values (aj< 1.0) be used if p is believed to be larger than 1 (say >1.5). The use of a,-values less than 1.0 is especially recommended in scale parameter families where j? = &/0, is believed to be quite large (as in the setting of contamination of data by outliers). In the absence of any prior numerical studies of efficiency, these a;‘s should be more or less equally-spaced. It is not known whether two optimal sets of cr;‘s exist which minimize AV(g;), i= 1,2, as functions of (Yt, . . . ,(Y~, given 8,, 8* and p. If they do, they will have to be estimated. Choice of r. From preliminary numerical studies done, it appears that not a lot of extra information is gained when r> 10 as opposed to r= 10, at least for the exponential case. The value of r= 10 seems to strike a good balance between computational ease and asymptotic efficiency. Of course, the asymptotic efficiency can always be increased by increasing r and adding more c-ri-values to the ones already present, but it seems that the gain is minimal. Computational aspects. The calculations of 0,s’ and Qi, i = 1,2, require numerical evaluations of some integrals. The difference is that each time the estimators OFT and e,sT are to be calculated, several integrations must be performed within a numerical root-finding routine, whereas the integrals needed to calculate 6, and & can be calculated once and for all. The calculation of @;, i = 1,2, requires two inversions of an r x r matrix but matrix inversion can be completely avoided and replaced by some simple matrix multiplications which are less time-consuming and much more accurate (see Soong (1969)). Determination of confidence intervals. Since both the ST estimators and estimators et, & are asymptotically normal, large sample confidence intervals can be found. An advantage that 0; has over f$sT, i= 1,2, is that in the process of calculating Qt and Ejl, consistent estimates of AV(o;), i= 1,2, are calculated. When computing confidence intervals for the 8t and Q2 using the ST estimators, one must first compute the values of three double integral expressions in addition to computations required to calculate 81sT and 0;‘.

5. Concluding

remarks

In this paper, it has been shown that the use of sample fractional moments can greatly enhance the efficiency of method of moments type estimators when estimating

242

S.G. From, K.M.L.

Saxena / Estimating parameters from mixed samples

parameters from two independent, mixed samples. A procedure for estimating parameters in the exponential scale case based on optimally linearly combining several different method of moments-type estimators has proven to be very efficient, as compared to previously proposed methods. Several possibilities for further research remain: (1) What happens as r + co? (2) Do optimal a,-values, i= 1, . . . , r, exist for estimating each of the parameters? (3) What properties do the following continuous f$ and &) have as estimators of 8, and 19,?

where C,(a), j,” C;(a)da= and C,.

versions

of tT?tand & (denoted

i= 1,2, are given (or estimated) weighting functions satisfying Cr 1, i= 1,2, and play the same role as the vectors of coefficients,

Appendix Let AV(6;) denote these are given by

the asymptotic

variance

of fi(f?;-

f?,), i= 1,2. Formulas

i= 1,2,

AV(6j)=(1’(Aj(01,B,,p)))‘1))1,

for

(A.l)

where A,=A,(~~,~*,P)=G~~G;, the asymptotic

covariance

of (6,,, . . . ,8,,),

matrix

A,=A,(B,,~~,P)=G~~G;, the asymptotic covariance matrix of (B,, , . . . , &), and Gk is an YX 2r matrix, k = 1,2, with as i-th row, j-th column entry g,!:!(e,, t!&,p) given by g$_

r = a&,/a

g$! = a&,/a gl:?=O

and .Z=(alj)

w,, 1g;;:;;;,

(A.2)

w,; 1;;::I$,

(A-3)

and j#2i,

ifj#2i-1

-p)ep’+pe,a’),

Nzi=si((i

-p)e:afl+pe;“q,

~2;,2;=W2a,+2a,

matrix -

a2i-1,2j-I=CUa,ia,-

r, j=l,...,

2r, k-1,2,

i= I,..., r,

N,;=f,((i

is a 2rx2r

i=l,...,

i= 1, . . ..r.

(A.4) (A.3 (‘4.6)

given by

u,,u,,)((i U,,U,,)W

-p)e:ff~+2~+pe~a~+2~~), -pw+ff~+P~~+~9~

~2i-~,2j=~2j,2i-1=~ua,+2~,-u~,u2a,~~~1-P~B~+2a/+PB2ai+2a,~,

(A-7) (‘4.8)

(A.9)

S.G. From, K.M.L.

for 1~ i, jsr,

Saxena / Estimating parameters from mixed samples

243

where ‘co

U,= 0

xX.4 dx,

103

t; =

I

xalf(x) dx

and

m

Si=

x

2a

f(x)dx,

O=l,...,

r.

I


.O

In the exponential scale case, f(x) = ePx, xr0, and UC=l(e+ l), where r(e) is the gamma function. The partial derivatives in (A.2) and (A.3) are found by differentiating either equation (2.10) or (2.11) of Section 2. For example, @ki/~IV,i is found by differentiating equation (2.10) with respect to Wi(a;) if k= 1 and differentiating equation (2.11) with respect to II’, if k = 2. The resulting expressions for these partial derivatives are,

:I

Mlj/aW*j = 12,:’

[

7

-

(&iT2] (“a”-’ (A. 10)

tMTlj/i3 W2j=aj'

L

~j

~

-

(“:s”](“aJ’~’ (A.ll)

c3O2j/a Wlj=a,-'

[:

t-

-

(T;y2](“a’)-’ (A. 12)

dlJ2j/dW2j = aj’

(7;s’;)(“aJ)-’

[? 7

-

(A.13) where E’=Di(aj)

is given by (2.8),

j= 1, . . . . r.

(A.14)

244

S.G. From,

K.M.L.

Saxena

/ Estimating

parameters

from

mixed samples

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