Relations for order statistics from non-identical logistic random variables and assessment of the...

Journal of Statistical Planning andInference 136 (2006) 2227–2253

www.elsevier.com/locate/jspi

Relations for order statistics from non-identical logisticrandom variables and assessment of the effect ofmultiple outliers on the bias of linear estimators

Aaron Childs∗, N. BalakrishnanDepartment of Mathematics and Statistics, McMaster University, Hamilton, Ont., Canada L8S 4K1

Available online 13 September 2005

Abstract

In this paper, we establish some relations for the single moments of order statistics arising from n inde-pendent and non-identically distributed logistic random variables. These relations are then used to deduce aset of recurrence relations for the moments of order statistics from the multiple-outlier logistic model (witha slippage of p observations). The moments of order statistics from the p-outlier model are expressed interms of the corresponding quantities from (p − 1)-outlier and (p − 2)-outlier models. We then use theserecurrence relations along with the corresponding i.i.d. results originally derived by Shah [1970. Note onmoments of a logistic order statistics,Ann. Math. Statist. 41, 2151–2152.] and the results for the single outliermodel discussed by Balakrishnan [1992a. Relationships between single moments of order statistics fromnon-identically distributed variables. In: Sen, P.K., Salama, I.A. (Eds.), Order Statistics and Nonparametrics:Theory and Applications. Elsevier, Amsterdam, pp. 65–78.] to compute the bias of various linear estimatorsof the location and scale parameters of the logistic distribution in the presence of multiple outliers.© 2005 Elsevier B.V. All rights reserved.

Keywords: Order statistics; Outliers; Single moments; Product moments; Recurrence relations; Logistic distribution;Permanents; Robust estimation; Bias

1. Introduction

Balakrishnan (1994a) considered the order statistics arising from n independently and non-identically distributed exponential random variables and derived several recurrence relations sat-isfied by the single and the product moments of order statistics by using a differential equationtechnique. He then applied these results to discuss the robustness of various linear estimatorsof the exponential mean with regard to the presence of multiple outliers. In a discussion of this

∗ Corresponding author.E-mail address: [email protected] (A. Childs).

0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jspi.2005.08.026

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mailto:[email protected]

2228 A. Childs, N. Balakrishnan / Journal of Statistical Planning and Inference 136 (2006) 2227–2253

paper, Arnold (1994) presented a direct approach (by utilizing the fact that minima of independentexponential random variables are again exponential) for the computation of the single and theproduct moments. He then remarked that ‘Bala’s specialized differential equation techniques mayperhaps have their finest hour in dealing with logistic Xi’s for which minima and maxima are notnice. His proposed work in this direction will be interesting.’

In this paper, we use the differential equation technique to derive some recurrence relationshipsfor the single moments of order statistics from independent and non-identically distributed logisticrandom variables. We then apply these results to assess the effect of multiple outliers on linearestimators of location and scale. It is observed that the sample median is the most bias-resistantestimator among all L-estimators of the location, a result that has been established analytically byDavid and Ghosh (1985) for the single-outlier model. Similarly, the Winsorized Median AbsoluteDeviation turns out to be the least biased estimator overall among all L-estimators of the scale.

Let X1, X2, . . . , Xn be independent random variables having cumulative distribution functionsF1(x), F2(x), . . . , Fn(x) and probability density functions f1(x), f2(x), . . . , fn(x), respectively.Let X1:n �X2:n � · · · �Xn:n denote the order statistics obtained by arranging the n Xi’s inincreasing order of magnitude. Then the density function of Xr:n (1�r �n) can be written as(David, 1981, p. 22)

fr:n(x) = 1

(r − 1)!(n − r)!∑p

r−1∏a=1

Fia (x)fir (x)

n∏b=r+1

{1 − Fib (x)}, (1.1)

where∑

p denotes the summation over all n! permutations (i1, i2, . . . , in) of (1, 2, . . . , n).

Similarly, if another independent random variable Xn+1d= Xi (that is, with cumulative distri-

bution function Fi(x) and probability density function fi(x)) is added to the original n variablesX1, X2, . . . , Xn, then the density function of Xr:n+1 (1�r �n + 1) can be written, using multi-nomial arguments, as

f[i]+r:n+1(x) = Fi(x)

(r − 2)!(n − r + 1)!∑p

r−2∏a=1

Fia (x)fir−1(x)

n∏b=r

{1 − Fib (x)}

+ fi(x)

(r − 1)!(n − r + 1)!∑p

r−1∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)}

+ 1 − Fi(x)

(r−1)!(n−r)!∑p

r−1∏a=1

Fia (x)fir (x)

n∏b=r+1

{1−Fib (x)} − ∞ < x < ∞, (1.2)

with the conventions that∏s

i=r = 1 if s − r = −1 and∏s

i=r = 0 if s − r = −2, so that the firstterm is omitted if r = 1 and the last term is omitted if r = n + 1. The superscript [i]+ indicatesthat the random variable Xi is repeated.

Alternatively, the densities in (1.1) and (1.2) can be written in terms of permanents of matrices;see Vaughan and Venables (1972).

In recent years, many of the recurrence relations and identities for order statistics from i.i.d.samples and samples containing a single outlier have been generalized to the case where therandom variables are independent and non-identically distributed. See for example, Balakrishnan(1988, 1989a,b, 1992a, 1994a–c), Bapat and Beg (1989a, b), Beg (1991), and Childs and Bal-akrishnan (1997). However, due to the algebraic and computational difficulties that arise whendealing with multiple-outliers from the logistic distribution, the majority of results for the logistic

A. Childs, N. Balakrishnan / Journal of Statistical Planning and Inference 136 (2006) 2227–2253 2229

distribution are for the i.i.d. case, and only a few are for the single-outlier model. For example,in the i.i.d. case, Birnbaum and Dudman (1963) and Gupta and Shah (1965) have derived ex-plicit expressions for the single and product moments of order statistics in terms of digamma andtrigamma functions. Shah (1966, 1970) used the differential equation f (x) = F(x){1 − F(x)}to derive some recurrence relations satisfied by the single and product moments of order statis-tics from the logistic model. These recurrence relations allow one to compute means, variancesand covariances of order statistics from the logistic distribution for all sample sizes n in a sim-ple recursive manner; see also Arnold et al. (1992). For the single-outlier model, David andGhosh (1985) showed that for a certain class of distributions, which includes the logistic dis-tribution, the median has the smallest bias among all L-estimators of the location parameter.Balakrishnan (1992b) calculated the bias and mean square error of various linear estimators ofthe parameters of the logistic distribution in the presence of a single-outlier. These and otherrelated results have been reviewed in the books of Balakrishnan (1992b) and Johnson et al.(1995).

In this paper, we generalize the i.i.d. results for the logistic model established by Shah (1970).We consider the case when the variables Xi’s are independent having logistic distributions withdensity functions

fi(x) = ce−c(x−�i )/�i

�i (1 + e−c(x−�i )/�i )2 , −∞ < x < ∞, −∞ < �i < ∞, �i > 0, (1.3)

and cumulative distribution functions

Fi(x) = 1

1 + e−c(x−�i )/�i, −∞ < x < ∞, −∞ < �i < ∞, �i > 0, (1.4)

for i = 1, 2, . . . , n, where c = �/√

3.From Eqs. (1.3) and (1.4), we see that the distributions satisfy the differential equations

fi(x) = c

�i

Fi(x){1 − Fi(x)}, −∞ < x < ∞, �i > 0, (1.5)

for i = 1, 2, . . . , n.Let us denote the single moments E(Xk

r:n) by �(k)r:n, 1�r �n and k = 1, 2, . . . . Let us also use

�[i](k)r:n−1 and �[i]+(k)

r:n+1 to denote the single moments of order statistics arising from n − 1 variablesobtained by deleting Xi from the original n variables X1, X2, . . . , Xn and the single moments

of order statistics arising from n + 1 variables obtained by adding an independent Xn+1d= Xi

to the original n variables X1, X2, . . . , Xn, respectively. By making use of the differential equa-tions in (1.5), we establish several recurrence relations satisfied by the single moments of orderstatistics. The recurrence relations for the p-outlier model are deduced as special cases. We thenuse the recurrence relations for the p-outlier model to examine the bias of various linear estima-tors of the location and scale parameters of the logistic distribution in the presence of multipleoutliers.

Similar work in the case of the exponential distribution has been carried out by Balakrishnan(1994a), and for the double exponential distribution by Childs and Balakrishnan (1997).


2. Relations for single moments

In this section, we use the differential equations in (1.5) to establish the following recurrencerelations for the single moments.

Relation 2.1. For n�1 and k = 0, 1, 2, . . . ,

n∑i=1

1

�i

�[i]+(k+1)1:n+1 = − (k + 1)

c�(k)

1:n +(

n∑i=1

1

�i

)�(k+1)

1:n .

Relation 2.2. For 2�r �n and k = 0, 1, 2, . . . ,

n∑i=1

1

�i

�[i]+(k+1)r:n+1 = (k + 1)

c{�(k)

r−1:n − �(k)r:n} −

n∑i=1

1

�i

�[i](k+1)r−1:n−1

+(

n∑i=1

1

�i

){�(k+1)

r−1:n + �(k+1)r:n }.

Relation 2.3. For n�1 and k = 0, 1, 2, . . . ,

n∑i=1

1

�i

�[i]+(k+1)n+1:n+1 = (k + 1)

c�(k)

n:n +(

n∑i=1

1

�i

)�(k+1)

n:n .

Proof of Relation 2.1. We first note that when r = 1, the density function in (1.2) reduces to

f[i]+1:n+1(x) = fi(x)

n!∑p

n∏b=1

{1 − Fib (x)} + 1 − Fi(x)

(n − 1)!∑p

n∏b=2

{1 − Fib (x)}fi1(x),

− ∞ < x < ∞,

which may be rewritten as

f[i]+1:n+1(x) = fi(x)

(n − 1)!∑

p:i1=i

n∏b=1

{1 − Fib (x)}

+ 1 − Fi(x)

(n − 1)!

⎡⎢⎣ ∑

p:i1=i

n∏b=2

{1 − Fib (x)}fi1(x)

+∑

p:i1=i

n∑j=2

n∏b=1b �=j

{1 − Fib (x)}fij (x)

⎤⎥⎦ , −∞ < x < ∞. (2.1)


Now, from Eq. (1.1), let us consider for n�1

(k + 1)

c�(k)

1:n = k + 1

c(n − 1)!∑p

∫ ∞

−∞xkfi1(x)

n∏b=2

{1 − Fib (x)} dx

= k + 1

(n − 1)!∑p

1

�i1

∫ ∞

−∞xkFi1(x)

n∏b=1

{1 − Fib (x)} dx

upon using (1.5). Integrating now by parts treating xk for integration and the rest of the integrandfor differentiation, we obtain

(k + 1)

c�(k)

1:n = 1

(n − 1)!∑p

1

�i1

⎡⎢⎣−

∫ ∞

−∞xk+1fi1(x)

n∏b=1

{1 − Fib (x)} dx

+n∑

j=1

∫ ∞

−∞xk+1Fi1(x)fij (x)

n∏b=1b �=j

{1 − Fib (x)} dx

⎤⎥⎦ .

We now separate out the j = 1 term from∑n

j=1 in the second term above and split the resultingtwo terms by writing Fi1(x) = 1 − {1 − Fi1(x)}. The above equation then becomes

(k + 1)

c�(k)

1:n = 1

(n − 1)!∑p

1

�i1

[−∫ ∞

−∞xk+1fi1(x)

n∏b=1

{1 − Fib (x)} dx

+n∑

j=2

∫ ∞

−∞xk+1fij (x)

n∏b=1b �=j

{1 − Fib (x)} dx

−n∑

j=2

∫ ∞

−∞xk+1{1 − Fi1(x)}fij (x)

n∏b=1b �=j

{1 − Fib (x)} dx

+∫ ∞

−∞xk+1fi1(x)

n∏b=2

{1 − Fib (x)} dx

−∫ ∞

−∞xk+1fi1(x)

n∏b=1

{1 − Fib (x)} dx

].

Comparison of the above equation with (2.1) and (1.1) for r = 1 yields

(k + 1)

c�(k)

1:n = −n∑

i=1

1

�i

�[i]+(k+1)1:n+1 +

(n∑

i=1

1

�i

)�(k+1)

1:n .

Relation 2.1 is obtained simply by rewriting the above equation.


Proof of Relation 2.2. We first note that the density function in (1.2) can be rewritten as

f[i]+r:n+1(x) = Fi(x)

(r − 2)!(n − r + 1)!∑

p:ir−1=i

⎡⎢⎣r−2∑

j=1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)}

+n∑

j=r−1

r−2∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)}⎤⎥⎦

+ fi(x)

(r − 1)!(n − r + 1)!

⎡⎣ ∑

p:ir−1=i

(r − 1)

r−1∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)}

+∑

p:ir=i

(n − r + 1)

r−1∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)}⎤⎦

+ 1 − Fi(x)

(r − 1)!(n − r)!∑

p:ir=i

⎡⎢⎣r−1∑

j=1

r∏a=1a �=j

Fia (x)fij (x)

n∏b=r+1

{1 − Fib (x)}

+n∑

j=r

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)}⎤⎥⎦ , −∞ < x < ∞. (2.2)

Now, from Eq. (1.1), let us consider for 2�r �n, and k = 0, 1, 2, . . . ,

(k + 1)

c{�(k)

r−1:n − �(k)r:n}

= k + 1

c(r − 2)!(n − r + 1)!∑p

∫ ∞

−∞xk

r−2∏a=1

Fia (x)fir−1(x)

n∏b=r

{1 − Fib (x)} dx

− k + 1

c(r − 1)!(n − r)!∑p

∫ ∞

−∞xk

r−1∏a=1

Fia (x)fir (x)

n∏b=r+1

{1 − Fib (x)} dx

= k + 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

∫ ∞

−∞xk

r−1∏a=1

Fia (x)

n∏b=r−1

{1 − Fib (x)} dx

− k + 1

(r − 1)!(n − r)!∑p

1

�ir

∫ ∞

−∞xk

r∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)} dx


upon using (1.5). Integrating now by parts in both integrals above, treating xk for integration andthe rest of the integrand for differentiation, we obtain

(k + 1)

c{�(k)

r−1:n − �(k)r:n}

= 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

⎡⎢⎣−

r−1∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r−1

{1 − Fib (x)} dx +n∑

j=r−1

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)} dx

⎤⎥⎦

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣−

r∑j=1

∫ ∞

−∞xk+1

r∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

+n∑

j=r

∫ ∞

−∞xk+1

r∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx

⎤⎥⎦ . (2.3)

We now split the first term in the first sum above into three by separating out the j = r − 1 termfrom

∑r−1j=1 and splitting the remaining sum,

∑r−2j=1, into two through {1 − Fir−1(x)}. We also

split the first term in the second sum above by separating out the j = r term from∑r

j=1. And wesplit the second term in the second sum above into two through Fir (x) = 1 − {1 − Fir (x)}. Eq.(2.3) now becomes

(k + 1)

c{�(k)

r−1:n − �(k)r:n}

= 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

⎡⎢⎣r−2∑

j=1

∫ ∞

−∞xk+1Fir−1(x)

r−1∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r

{1 − Fib (x)} dx −∫ ∞

−∞xk+1

r−2∏a=1

Fia (x)fir−1(x)

n∏b=r−1

{1 − Fib (x)} dx

−r−2∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

+n∑

j=r−1

∫ ∞


r−2∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)} dx

⎤⎥⎦


− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣−

r−1∑j=1

∫ ∞

−∞xk+1{1 − Fir (x)}

r∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r+1

{1 − Fib (x)} dx −∫ ∞

−∞xk+1fir (x)

r−1∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)} dx

−n∑

j=r

∫ ∞

−∞xk+1{1 − Fir (x)}

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx

+n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx

⎤⎥⎦ .

We now split the second term in the first sum above through {1 − Fir−1(x)} to get

(k + 1)

c{�(k)

r−1:n − �(k)r:n}

= 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

⎡⎢⎣r−2∑

j=1

∫ ∞


r−1∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r

{1 − Fib (x)} dx +∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fir−1(x)

n∏b=r

{1 − Fib (x)} dx

−∫ ∞

−∞xk+1

r−2∏a=1

Fia (x)fir−1(x)

n∏b=r

{1 − Fib (x)} dx

−r−2∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

+n∑

j=r−1

∫ ∞


r−2∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)} dx

⎤⎥⎦

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣−

r−1∑j=1

∫ ∞

−∞xk+1{1 − Fir (x)}

r∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r+1

{1 − Fib (x)} dx −∫ ∞

−∞xk+1fir (x)

r−1∏a=1

Fia (x)

n∏b=r

{1 − Fib (x)} dx


−n∑

j=r

∫ ∞

−∞xk+1{1 − Fir (x)}

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx

+n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx

⎤⎥⎦ .

Now, comparison with (2.2) shows that the first, second, and fifth terms in the first sum abovecombine with the first, second, and third terms in the second sum above to give

(k + 1)

c{�(k)

r−1:n − �(k)r:n} =

n∑i=1

1

�i

�[i]+(k+1)r:n+1 + 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

×⎡⎢⎣−

∫ ∞

−∞xk+1

r−2∏a=1

Fia (x)fir−1(x)

n∏b=r

{1 − Fib (x)} dx

−r−2∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

⎤⎥⎦

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣ n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

×n∏

b=rb �=j

{1 − Fib (x)} dx

⎤⎥⎦

=n∑

i=1

1

�i

�[i]+(k+1)r:n+1

+ 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

⎡⎢⎣−

r−1∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r

{1 − Fib (x)} dx

⎤⎥⎦

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣ n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

×n∏

b=rb �=j

{1 − Fib (x)} dx

⎤⎥⎦ . (2.4)


We now use the fact that

1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣r−1∑

j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

⎤⎥⎦

= 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

⎡⎢⎣ n∑

j=r

∫ ∞

−∞xk+1

r−2∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)} dx

⎤⎥⎦

to rewrite Eq. (2.4) as follows:

(k + 1)

c{�(k)

r−1:n − �(k)r:n} =

n∑i=1

1

�i

�[i]+(k+1)r:n+1 + 1

(r − 2)!(n − r + 1)!∑p

1

�ir−1

×⎡⎢⎣−

r−1∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r

{1 − Fib (x)} dx

−n∑

j=r

∫ ∞

−∞xk+1

r−2∏a=1

Fia (x)fij (x)

n∏b=r−1b �=j

{1 − Fib (x)} dx

⎤⎥⎦

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎣ n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

×n∏

b=rb �=j

{1 − Fib (x)} dx −r−1∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

×n∏

b=r

{1 − Fib (x)} dx

⎤⎦ .

We now recognize the first sum above as −(∑n

i=11/�i )�(k+1)r−1:n, and we split the second term in

the second sum above into two through {1 − Fir (x)} to obtain

(k + 1)

c{�(k)

r−1:n − �(k)r:n}

=n∑

i=1

1

�i

�[i]+(k+1)r:n+1 −

(n∑

i=1

1

�i

)�(k+1)

r−1:n

− 1

(r − 1)!(n − r)!∑p

1

�ir

⎡⎢⎣ n∑

j=r

∫ ∞

−∞xk+1

r−1∏a=1

Fia (x)fij (x)

n∏b=rb �=j

{1 − Fib (x)} dx


+r−1∑j=1

∫ ∞

−∞xk+1

r∏a=1a �=j

Fia (x)fij (x)

n∏b=r+1

{1 − Fib (x)} dx

−r−1∑j=1

∫ ∞

−∞xk+1

r−1∏a=1a �=j

Fia (x)fij (x)

n∏b=r+1

{1 − Fib (x)} dx

⎤⎥⎦

=n∑

i=1

1

�i

�[i]+(k+1)r:n+1 −

(n∑

i=1

1

�i

)�(k+1)

r−1:n −(

n∑i=1

1

�i

)�(k+1)

r:n +n∑

i=1

1

�i

�[i](k+1)r−1:n−1.

Relation 2.2 is obtained simply by rewriting the above equation.The proof of Relation 2.3 is similar to that of Relation 2.1, and is therefore omitted.

3. Results for the multiple outlier model (with a slippage of p observations)

In this section, we consider the special case when X1, X2, . . . , Xn−p are independent logisticrandom variables with location parameter � and scale parameter �, while Xn−p+1, . . . , Xn areindependent logistic random variables with location parameter �1 and scale parameter �1 (andindependent of X1, X2, . . . , Xn−p). This situation is known as the multiple-outlier model with aslippage of p observations; see David (1979) and Barnett and Lewis (1994, pp. 66–68).

Here, we denote the single moments by �(k)r:n[p], and the results established in Section 2 then

readily reduce to the following recurrence relations(a) for n�1,

�(k+1)1:n+1[p + 1] = �1

p

{(n − p

�+ p

�1

)�(k+1)

1:n [p] − n − p

��(k+1)

1:n+1[p] − (k + 1)

c�(k)

1:n[p]}

,

(b) for 2�r �n,

�(k+1)r:n+1[p + 1] = �1

p

{(n − p

�+ p

�1

){�(k+1)

r:n [p] + �(k+1)r−1:n[p]}

− n − p

�{�(k+1)

r:n+1[p] + �(k+1)r−1:n−1[p]}

+ (k + 1)

c{�(k)

r−1:n[p] − �(k)r:n[p]}

}− �(k+1)

r−1:n−1[p − 1],

(c) for n�1,

�(k+1)n+1:n+1[p + 1] = �1

p

{(n − p

�+ p

�1

)�(k+1)

n:n [p] − n − p

��(k+1)

n+1:n+1[p]

+ (k + 1)

c�(k)

n:n[p]}

.

Note that if we replace p by n − p, we get a set of equivalent relations by regarding the first pXi’s as the outliers.


If we now multiply each of the above relations by p/�1 and then set p = 0 and � = 1 (orsimply set p = n and � = 1), we obtain the following recurrence relations for the case when theXi’s are independent and identically distributed standard logistic random variables (which couldalternatively be obtained by setting �1 = �2 = · · · = �n = 1, �1 = �2 = · · · = �n = 0 in Relations2.1–2.3):

�(k+1)1:n+1 = �(k+1)

1:n − (k + 1)

cn�(k)

1:n, n�1,

�(k+1)r:n+1 = {�(k+1)

r:n + �(k+1)r−1:n} − �(k+1)

r−1:n−1 + (k + 1)

cn{�(k)

r−1:n − �(k)r:n}, 2�r �n,

and

�(k+1)n+1:n+1 = �(k+1)

n:n + (k + 1)

cn�(k)

n:n, n�1.

These relations are equivalent to the results of Shah (1970).Assuming that the moments of order statistics for the single-outlier model are known (they can

be found in Balakrishnan et al. (1992)), setting p = 1 in Relations (a)–(c), along with the abovei.i.d results, will enable one to compute all of the moments of order statistics from a 2-outliermodel. One can then set p = 2 in Relations (a)–(c) to obtain all of the moments of order statisticsfrom a 3-outlier model. Continuing in this manner, we see that Relations (a)–(c), the above i.i.d.relations, and knowledge of the moments of order statistics for the single-outlier model will enableone to compute all of the moments of order statistics for the multiple-outlier model in a simplerecursive manner. Interestingly enough, this particular recursive property was made as a conjectureby Balakrishnan (1994a, pp. 252–253) in his reply to the comments made by Arnold (1994). Weshould note however that since the known results for p = 1 are not exact, then calculations forhigher values of p will also only be approximate, and may be subject to some rounding error.

4. Robustness of location estimators

Balakrishnan et al. (1992) explicitly calculated, by numerical integration, the means, variancesand covariances of order statistics for the single-outlier model. Balakrishnan (1992b) then usedthose results to examine the bias of various linear estimators of the location parameter � under asingle location-outlier logistic model. In this section, we discuss the bias of these linear estimatorsof � under the multiple location-outlier logistic model.

The omnibus estimators of � that are considered in this comparative study are the following:(a) Sample mean:

Xn = 1

n

n∑i=1

Xi:n

(b) Median:

for n odd, Xn+12 :n

for n even, 12 (X[n/2]:n + X[n/2]+1:n)


(c) Trimmed mean:

Tn(r) = 1

n − 2r

n−r∑i=r+1

Xi:n

(d) Winsorized mean:

Wn(r) = 1

n

[(r + 1)(Xr+1:n + Xn−r:n) +

n−r−1∑i=r+2

Xi:n

]

(e) Modified maximum likelihood (MML) estimator (Tiku, 1980; Tiku et al., 1986):

�c = 1

m

[r�(Xr+1:n + Xn−r:n) +

n−r∑i=r+1

Xi:n

],

where m = n − 2r + 2r�, � = (g(h2) − g(h1))/(h2 − h1), h1 = F−1(1 − q − √q(1 − q)/n),

h2 = F−1(1 − q + √q(1 − q)/n), q = r/n, F(h) = ∫ h

−∞ f (z) dz, f (z) = 1√2�

e−z2/2, and

g(h) = f (h)/(1 − F(h)).(f) Linearly weighted means:for n odd,

Ln(r) = 1

2( n−12 − r)2 + (n − 2r)

⎡⎣(n−1)/2−r∑

i=1

(2i − 1)(Xr+i:n + Xn−r−i+1:n)

+(n − 2r)Xn+12 :n

⎤⎦ ,

for n even,

Ln(r) = 1

2( 12n − r)2

⎡⎣(1/2)n−r∑

i=1

(2i − 1)(Xr+i:n + Xn−r−i+1:n)

⎤⎦ ,

and(g) Gastwirth (1966) Mean:

Tn = 310 (X[n/3]+1:n + Xn−[n/3]:n) + 2

5 X

where X is the median.In addition to these omnibus estimators, we also included the following estimators of �:(h) BLUE (Govindarajulu, 1966):

�(r) = 1′�X1′�1

where 1′ = (1, 1, . . . , 1), � = [(�i,j :n[0]); r + 1� i, j �n − r]−1, and X = (X1+r:n, X2+r:n, . . . ,

Xn−r:n); �i,j :n[0] denotes the covariance between the ith and j th order statistics in a sample ofsize n from the standard logistic distribution (the [0] indicates that there are no outliers),


(i) the approximate best linear unbiased estimator of Blom (1958):

�′ = 6

n(n + 1)(n + 2)

n∑i=1

i(n + 1 − i)Xi:n

which is discussed in Gupta and Gnanadesikan (1966), and(j) RSE (estimator proposed by Raghunandanan and Srinivasan, 1970):

v(i∗) = 12 (Xn−i∗+1:n + Xi∗:n)

where i∗ is chosen to minimize the variance of v(i∗) when the X’s are i.i.d..In this section, the recursive method described in Section 3 is used to examine the bias of the

above estimators of � under the multiple location-outlier model.In Table 1,we have presented the bias of each of the above estimators of the location parameter

� for n=10(5)20, p=1(1)3, and (in the notation of Section 3) �=0, �1 =0.5(0.5)3, 4, �=�1 =1.We have considered here three sample sizes, viz. 10, 15 and 20, in order to examine the behaviourof bias and mean square error of all the estimators in a small sample size 10 as well as a largersample size of 20. In addition, we have considered an odd sample size 15 in which case formsof some of the estimators do change. For estimators such as the trimmed mean, where there arevarious forms of the estimator (for different values of r), we have included the forms with r = 0,10, and 20% of n as well as the form that had the smallest bias.

From Table 1, we see that for each value of n and p the median is the estimator with the smallestbias. For large values of r, the linearly weighted mean and the BLUE are quite comparable in biasto the median, with the linearly weighted mean having a smaller bias than the BLUE. For smallvalues of �1, the modified maximum likelihood estimator, the Gastwirth mean and the Winsorizedmean are also comparable to the BLUE and linearly weighted mean, but for larger values of �1their bias becomes much larger. For small values of r, however, all of these estimators are quitesensitive to the presence of outliers, as one would expect.

The fact that all of the estimators have similar bias for small values of �1 is explained by thefact that these estimators are all unbiased in the i.i.d. case. Therefore all of the biases become thesame as �1 approaches zero.

It is of interest to point out here that David and Ghosh (1985) presented a sufficient conditionto verify whether the sample median is the least-biased estimator of location when a single outlieris present in the sample. They then showed that this sufficient condition is satisfied in the logisticcase. We note from Table 1 that the sample median remains as the least-biased estimator evenwhen multiple outliers are present in the sample. This naturally leaves a question whether thereis a multiple outlier version of David and Ghosh’s (1985) result!

Since all the linear estimators of � considered here are symmetric functions of order statistics,they will all be unbiased under the multiple-scale outlier model. Hence, a comparison of theseestimators under the multiple-scale outlier model would have to be made on the basis of theirvariance. Symmetric functions are most appropriate when the direction of the slippage is notknown. However, if the direction is known then some nonsymmetric unbiased estimators willnaturally perform better than the symmetric ones. For example, any estimator that gives lessweight to the larger order statistics (and more to the smaller ones) will be expected to performbetter if �1 is positive.


Table 1Bias of estimators of the mean in the presence of multiple location-outliers

�1 0.5 1 1.5 2 2.5 3 4

n = 10, p = 1BLUE0 0.0490 0.0927 0.1283 0.1560 0.1777 0.1955 0.2252BLUE1 0.0489 0.0915 0.1243 0.1466 0.1604 0.1683 0.1746BLUE2 0.0485 0.0889 0.1171 0.1339 0.1426 0.1467 0.1493BLUE3 0.0480 0.0859 0.1100 0.1228 0.1289 0.1315 0.1331RSE 0.0482 0.0869 0.1123 0.1261 0.1328 0.1357 0.1375Mean 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.4000Trimm1 0.0491 0.0929 0.1278 0.1525 0.1682 0.1774 0.1849Trimm2 0.0484 0.0883 0.1156 0.1315 0.1397 0.1435 0.1459Median 0.0476 0.0836 0.1051 0.1158 0.1206 0.1226 0.1238Winsor1 0.0495 0.0957 0.1351 0.1651 0.1854 0.1977 0.2084Winsor2 0.0488 0.0906 0.1211 0.1400 0.1501 0.1549 0.1581Winsor3 0.0481 0.0863 0.1108 0.1241 0.1303 0.1331 0.1347MML1 0.0494 0.0954 0.1343 0.1636 0.1834 0.1954 0.2057MML2 0.0487 0.0903 0.1204 0.1388 0.1487 0.1534 0.1564MML3 0.0481 0.0862 0.1106 0.1237 0.1299 0.1326 0.1342LinWei1 0.0483 0.0881 0.1156 0.1322 0.1412 0.1458 0.1491LinWei2 0.0480 0.0859 0.1102 0.1233 0.1297 0.1325 0.1342LinWei3 0.0478 0.0845 0.1069 0.1184 0.1236 0.1259 0.1272Gastw 0.0480 0.0856 0.1094 0.1220 0.1279 0.1305 0.1320Blom 0.0493 0.0947 0.1341 0.1677 0.1968 0.2231 0.2712


n = 10, p = 3BLUE0 0.1488 0.2915 0.4246 0.5483 0.6649 0.7767 0.9935BLUE1 0.1487 0.2905 0.4222 0.5445 0.6599 0.7708 0.9861BLUE2 0.1482 0.2872 0.4124 0.5253 0.6308 0.7332 0.9370BLUE3 0.1476 0.2819 0.3937 0.4791 0.5388 0.5770 0.6116RSE 0.1479 0.2841 0.4013 0.4958 0.5661 0.6134 0.6586Mean 0.1500 0.3000 0.4500 0.6000 0.7500 0.9000 1.2000Trimm1 0.1490 0.2926 0.4293 0.5605 0.6883 0.8146 1.0653


(Table 1 continued)

�1 0.5 1 1.5 2 2.5 3 4

Trimm2 0.1481 0.2860 0.4082 0.5152 0.6112 0.7005 0.8711Median 0.1471 0.2774 0.3776 0.4436 0.4809 0.4996 0.5119Winsor1 0.1495 0.2966 0.4420 0.5876 0.7347 0.8830 1.1819Winsor2 0.1486 0.2902 0.4232 0.5516 0.6813 0.8158 1.0997Winsor3 0.1477 0.2828 0.3965 0.4854 0.5491 0.5907 0.6292MML1 0.1494 0.2962 0.4405 0.5845 0.7294 0.8752 1.1685MML2 0.1485 0.2897 0.4212 0.5467 0.6717 0.8000 1.0685MML3 0.1477 0.2825 0.3957 0.4835 0.5459 0.5865 0.6238LinWei1 0.1480 0.2853 0.4050 0.5062 0.5923 0.6679 0.8033LinWei2 0.1476 0.2818 0.3931 0.4791 0.5433 0.5919 0.6642LinWei3 0.1473 0.2791 0.3835 0.4566 0.5022 0.5281 0.5486Gastw 0.1476 0.2814 0.3918 0.4749 0.5320 0.5679 0.5999Blom 0.1492 0.2938 0.4314 0.5623 0.6880 0.8103 1.0498


n = 15, p = 2BLUE0 0.0655 0.1248 0.1747 0.2153 0.2489 0.2778 0.3289BLUE1 0.0655 0.1244 0.1734 0.2128 0.2450 0.2727 0.3223BLUE3 0.0650 0.1210 0.1629 0.1902 0.2059 0.2139 0.2193BLUE6 0.0640 0.1140 0.1453 0.1615 0.1690 0.1722 0.1740RSE 0.0652 0.1220 0.1647 0.1919 0.2066 0.2137 0.2181Mean 0.0667 0.1333 0.2000 0.2667 0.3333 0.4000 0.5333Trimm1 0.0659 0.1279 0.1834 0.2324 0.2766 0.3178 0.3965Trimm3 0.0648 0.1198 0.1597 0.1849 0.1988 0.2057 0.2103Median 0.0638 0.1132 0.1434 0.1587 0.1657 0.1686 0.1703Winsor1 0.0664 0.1312 0.1934 0.2536 0.3135 0.3747 0.5021Winsor3 0.0654 0.1236 0.1697 0.2016 0.2209 0.2311 0.2381Winsor6 0.0640 0.1141 0.1454 0.1617 0.1692 0.1724 0.1743MML1 0.0663 0.1308 0.1923 0.2513 0.3096 0.3687 0.4909MML3 0.0653 0.1231 0.1683 0.1993 0.2178 0.2275 0.2342MML6 0.0640 0.1141 0.1453 0.1616 0.1691 0.1723 0.1742LinWei1 0.0649 0.1204 0.1619 0.1904 0.2093 0.2222 0.2392


(Table 1 continued)

�1 0.5 1 1.5 2 2.5 3 4

LinWei3 0.0643 0.1165 0.1514 0.1712 0.1812 0.1857 0.1886LinWei6 0.0639 0.1136 0.1442 0.1600 0.1672 0.1702 0.1720Gastw 0.0642 0.1155 0.1487 0.1667 0.1753 0.1791 0.1813Blom 0.0657 0.1265 0.1795 0.2251 0.2650 0.3011 0.3679



n = 20, p = 2BLUE0 0.0491 0.0933 0.1297 0.1584 0.1811 0.1996 0.2303BLUE2 0.0490 0.0925 0.1272 0.1526 0.1698 0.1806 0.1904BLUE4 0.0486 0.0895 0.1182 0.1350 0.1433 0.1470 0.1492


(Table 1 continued)

�1 0.5 1 1.5 2 2.5 3 4

BLUE7 0.0477 0.0838 0.1051 0.1156 0.1203 0.1223 0.1234RSE 0.0485 0.0887 0.1162 0.1318 0.1394 0.1428 0.1449Mean 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.4000Trimm2 0.0491 0.0933 0.1293 0.1562 0.1746 0.1862 0.1968Trimm4 0.0485 0.0887 0.1167 0.1332 0.1418 0.1458 0.1482Median 0.0476 0.0836 0.1048 0.1153 0.1200 0.1219 0.1231Winsor2 0.0496 0.0969 0.1394 0.1751 0.2029 0.2224 0.2420Winsor4 0.0490 0.0920 0.1249 0.1464 0.1584 0.1643 0.1680Winsor8 0.0478 0.0843 0.1064 0.1175 0.1226 0.1247 0.1259MML2 0.0496 0.0964 0.1380 0.1726 0.1991 0.2176 0.2360MML4 0.0489 0.0916 0.1238 0.1446 0.1561 0.1617 0.1652MML8 0.0478 0.0843 0.1064 0.1175 0.1225 0.1246 0.1258LinWei2 0.0484 0.0883 0.1160 0.1328 0.1420 0.1467 0.1500LinWei4 0.0480 0.0861 0.1105 0.1236 0.1299 0.1327 0.1343LinWei8 0.0477 0.0838 0.1053 0.1159 0.1207 0.1227 0.1239Gastw 0.0481 0.0866 0.1116 0.1252 0.1317 0.1345 0.1361Blom 0.0492 0.0939 0.1318 0.1631 0.1892 0.2121 0.2528


5. Robustness of scale estimators

By considering both the single location-outlier and single scale-outlier model, Balakrishnan(1992b) computed the bias of the following linear estimators of the scale parameter �:

(a) BLUE (Govindarajulu, 1966):

�(r) = µ′�X

µ′�µ,


where µ′ = (�1+r:n[0], �2+r:n[0], . . . , �n−r:n[0]), � = [(�i,j :n[0]); r + 1� i, j �n − r]−1, andX = (X1+r:n, X2+r:n, . . . , Xn−r:n); �s:n[0] denotes the mean of the sth order statistic in a sampleof size n from the standard logistic distribution (the [0] indicates that there are no outliers),

and(b) RSE (estimator proposed by Raghunandanan and Srinivasan, 1970):The estimator of the following form:

C

[n/2]∑i=r+1

ai(Xn−i+1:n − Xi:n)

where each ai takes the values 0 or 1, and C is a constant. The ai’s and C are chosen so as to makethe estimator unbiased and have minimum variance when the X’s are I.I.D..

In this section, we use the recursive method described in Section 3 to examine the bias of theabove estimators of � under the multiple location-outlier and multiple scale-outlier models. Wealso consider the following approximate best linear unbiased estimator of Blom (1958):

(c)

�′ =n∑

i=1

�iXi:n

where �i = ci(n + 1 − i)(ci − ci−1)/(d(n + 1)2), ci = (ci(n + 1 − i)/(n + 1)2)�i:n[0] − (c(i +1)(n − i)/(n + 1)2)�i+1:n[0], and d =∑n

i=0c2i , the following modified estimator of Jung (1955):

(d)

� = cn

n∑i=1

�iXi:n

where �i = 9c/(n(n+ 1)2(3 +�2)){−(n + 1)2 + 2i(n + 1) + 2i(n + 1 − i) ln(i/(n + 1 − i))

},

and cn = 1/(∑n

i=1�i�i:n[0]),(Both of the above estimators are discussed in Gupta and Gnanadesikan, 1966.)and(e) Winsorized median absolute deviation (WMAD):for n odd,

�(r) = 1

n − 2r

⎡⎣r(Xn−r:n − Xr+1:n) +

n−r∑i=(n−1)/2+2

Xi:n −(n−1)/2∑i=r+1

Xi:n

⎤⎦ ,

for n even,

�(r) = 1

n − 2r

⎡⎣r(Xn−r:n − Xr+1:n) +

n−r∑i=n/2+1

Xi:n −n/2∑

i=r+1

Xi:n

⎤⎦

which, incidentally, is the MLE of � for a symmetrically Type-II censored sample from a Laplacedistribution; see Balakrishnan and Cutler (1995).

In Table 2, we have presented the bias of the above estimators of the scale parameter � underthe multiple location-outlier model for n= 10(5)20, p = 1(1)3, and (in the notation of Section 3)


Table 2Bias of estimators of sigma in the presence of multiple location-outliers

�1 0.5 1 1.5 2 2.5 3 4

n = 10, p = 1BLUE0 0.0138 0.0524 0.1091 0.1767 0.2502 0.3265 0.4821BLUE1 0.0152 0.0547 0.1035 0.1480 0.1811 0.2023 0.2212BLUE2 0.0162 0.0548 0.0958 0.1261 0.1439 0.1529 0.1589BLUE3 0.0168 0.0543 0.0900 0.1132 0.1253 0.1309 0.1343RSE0 0.0140 0.0526 0.1079 0.1725 0.2416 0.3127 0.4568RSE1 0.0153 0.0547 0.1028 0.1457 0.1769 0.1965 0.2135RSE2 0.0162 0.0549 0.0965 0.1277 0.1463 0.1558 0.1621RSE4 0.0171 0.0540 0.0870 0.1073 0.1173 0.1218 0.1245WMAD0 −0.2780 −0.2506 −0.2118 −0.1667 −0.1188 −0.0697 0.0298WMAD1 −0.2519 −0.2230 −0.1880 −0.1570 −0.1344 −0.1200 −0.1074WMAD2 −0.2351 −0.2062 −0.1758 −0.1536 −0.1407 −0.1342 −0.1299WMAD3 −0.2525 −0.2250 −0.1989 −0.1819 −0.1731 −0.1691 −0.1666Blom 0.0132 0.0516 0.1116 0.1883 0.2767 0.3726 0.5754Jung 0.0137 0.0523 0.1096 0.1790 0.2552 0.3351 0.4991

n = 10, p = 2BLUE0 0.0246 0.0943 0.1987 0.3263 0.4678 0.6169 0.9248BLUE1 0.0275 0.1039 0.2162 0.3540 0.5106 0.6812 1.0469BLUE2 0.0297 0.1086 0.2124 0.3143 0.3962 0.4525 0.5061BLUE3 0.0311 0.1105 0.2041 0.2808 0.3296 0.3557 0.3736RSE0 0.0251 0.0954 0.1995 0.3243 0.4602 0.6013 0.8889RSE1 0.0277 0.1045 0.2164 0.3508 0.4986 0.6535 0.9721RSE2 0.0295 0.1084 0.2135 0.3187 0.4048 0.4650 0.5231RSE4 0.0318 0.1110 0.1988 0.2641 0.3015 0.3196 0.3310WMAD0 −0.2701 −0.2198 −0.1461 −0.0585 0.0360 0.1337 0.3324WMAD1 −0.2427 −0.1862 −0.1049 −0.0083 0.0984 0.2122 0.4521WMAD2 −0.2249 −0.1655 −0.0883 −0.0138 0.0450 0.0849 0.1224WMAD3 −0.2420 −0.1836 −0.1151 −0.0592 −0.0238 −0.0049 0.0079Blom 0.0235 0.0907 0.1932 0.3205 0.4629 0.6137 0.9259Jung 0.0244 0.0936 0.1978 0.3256 0.4678 0.6180 0.9287

n = 10, p = 3BLUE0 0.0324 0.1248 0.2652 0.4392 0.6343 0.8414 1.2721BLUE1 0.0365 0.1414 0.3036 0.5091 0.7443 0.9974 1.5283BLUE2 0.0398 0.1543 0.3317 0.5601 0.8312 1.1371 1.8170BLUE3 0.0420 0.1621 0.3373 0.5275 0.6931 0.8138 0.9351RSE0 0.0330 0.1273 0.2704 0.4473 0.6444 0.8525 1.2815RSE1 0.0368 0.1427 0.3076 0.5207 0.7719 1.0512 1.6591RSE2 0.0395 0.1533 0.3311 0.5645 0.8492 1.1787 1.9289RSE4 0.0432 0.1656 0.3357 0.4996 0.6182 0.6875 0.7382WMAD0 −0.2643 −0.1966 −0.0941 0.0315 0.1701 0.3152 0.6122WMAD1 −0.2359 −0.1574 −0.0369 0.1138 0.2830 0.4621 0.8319WMAD2 −0.2172 −0.1305 0.0027 0.1709 0.3657 0.5806 1.0484WMAD4 −0.4249 −0.3574 −0.2636 −0.1732 −0.1078 −0.0696 −0.0417Blom 0.0307 0.1181 0.2500 0.4122 0.5927 0.7834 1.1777Jung 0.0321 0.1235 0.2623 0.4341 0.6266 0.8309 1.2554

n = 15, p = 1BLUE0 0.0093 0.0350 0.0725 0.1168 0.1646 0.2137 0.3134BLUE1 0.0100 0.0364 0.0705 0.1031 0.1284 0.1454 0.1612BLUE3 0.0110 0.0365 0.0627 0.0809 0.0910 0.0958 0.0987


(Table 2 continued)

�1 0.5 1 1.5 2 2.5 3 4

BLUE6 0.0116 0.0357 0.0561 0.0679 0.0736 0.0761 0.0775RSE0 0.0093 0.0350 0.0725 0.1166 0.1641 0.2131 0.3125RSE1 0.0100 0.0364 0.0701 0.1017 0.1258 0.1416 0.1559RSE3 0.0109 0.0366 0.0634 0.0825 0.0932 0.0983 0.1015RSE6 0.0116 0.0357 0.0561 0.0679 0.0736 0.0761 0.0775WMAD0 −0.2668 −0.2482 −0.2220 −0.1918 −0.1598 −0.1270 −0.0606WMAD1 −0.2453 −0.2257 −0.2011 −0.1782 −0.1607 −0.1491 −0.1384WMAD3 −0.2066 −0.1866 −0.1665 −0.1525 −0.1449 −0.1413 −0.1391WMAD4 −0.1969 −0.1771 −0.1587 −0.1470 −0.1411 −0.1384 −0.1368Blom 0.0088 0.0343 0.0743 0.1257 0.1855 0.2511 0.3911Jung 0.0092 0.0349 0.0728 0.1183 0.1678 0.2193 0.3246

n = 15, p = 2BLUE0 0.0172 0.0656 0.1373 0.2237 0.3184 0.4174 0.6205BLUE1 0.0186 0.0698 0.1431 0.2297 0.3252 0.4275 0.6461BLUE3 0.0208 0.0731 0.1352 0.1880 0.2232 0.2430 0.2571BLUE6 0.0222 0.0731 0.1227 0.1550 0.1716 0.1791 0.1836RSE0 0.0172 0.0657 0.1373 0.2234 0.3172 0.4145 0.6130RSE1 0.0188 0.0701 0.1430 0.2278 0.3188 0.4133 0.6068RSE3 0.0206 0.0730 0.1367 0.1921 0.2299 0.2515 0.2671RSE6 0.0222 0.0731 0.1227 0.1550 0.1716 0.1791 0.1836WMAD0 −0.2609 −0.2255 −0.1747 −0.1152 −0.0517 0.0137 0.1463WMAD1 −0.2386 −0.2004 −0.1471 −0.0861 −0.0205 0.0486 0.1948WMAD3 −0.1989 −0.1579 −0.1099 −0.0698 −0.0433 −0.0286 −0.0182WMAD4 −0.1887 −0.1476 −0.1032 −0.0702 −0.0511 −0.0418 −0.0358Blom 0.0163 0.0633 0.1353 0.2253 0.3266 0.4341 0.6569Jung 0.0171 0.0653 0.1371 0.2243 0.3205 0.4215 0.6293

n = 15, p = 3BLUE0 0.0239 0.0915 0.1930 0.3170 0.4546 0.5997 0.8995BLUE1 0.0260 0.0989 0.2077 0.3418 0.4928 0.6540 0.9909BLUE3 0.0292 0.1077 0.2143 0.3256 0.4230 0.4963 0.5736BLUE6 0.0316 0.1112 0.2011 0.2687 0.3074 0.3261 0.3376RSE0 0.0239 0.0917 0.1934 0.3179 0.4556 0.6001 0.8969RSE1 0.0262 0.0995 0.2088 0.3438 0.4977 0.6659 1.0310RSE3 0.0289 0.1073 0.2156 0.3323 0.4378 0.5195 0.6080RSE6 0.0316 0.1112 0.2011 0.2687 0.3074 0.3261 0.3376WMAD0 −0.2559 −0.2059 −0.1326 −0.0453 0.0492 0.1468 0.3455WMAD1 −0.2330 −0.1781 −0.0976 −0.0012 0.1045 0.2152 0.4430WMAD3 −0.1922 −0.1305 −0.0478 0.0368 0.1094 0.1631 0.2190WMAD(r) −0.1817 −0.1186 −0.0398 −0.0005 0.0357 0.0541 0.0661r 4 4 4 5 5 5 5Blom 0.0226 0.0872 0.1853 0.3065 0.4417 0.5844 0.8795Jung 0.0237 0.0909 0.1919 0.3158 0.4535 0.5990 0.8999

n = 20, p = 1BLUE0 0.0070 0.0263 0.0542 0.0871 0.1223 0.1584 0.2313BLUE2 0.0078 0.0275 0.0511 0.0710 0.0842 0.0914 0.0965BLUE4 0.0082 0.0275 0.0471 0.0606 0.0678 0.0712 0.0733BLUE8 0.0088 0.0266 0.0413 0.0496 0.0535 0.0552 0.0562RSE0 0.0070 0.0262 0.0542 0.0871 0.1226 0.1592 0.2337RSE2 0.0078 0.0275 0.0511 0.0708 0.0838 0.0909 0.0959RSE4 0.0082 0.0274 0.0468 0.0600 0.0671 0.0704 0.0724


(Table 2 continued)

�1 0.5 1 1.5 2 2.5 3 4

RSE9 0.0088 0.0265 0.0410 0.0490 0.0528 0.0544 0.0554WMAD0 −0.2573 −0.2432 −0.2233 −0.2006 −0.1765 −0.1519 −0.1022WMAD2 −0.2213 −0.2061 −0.1885 −0.1739 −0.1645 −0.1594 −0.1558WMAD4 −0.1860 −0.1707 −0.1556 −0.1454 −0.1401 −0.1376 −0.1361WMAD6 −0.1626 −0.1474 −0.1341 −0.1261 −0.1222 −0.1205 −0.1195Blom 0.0066 0.0257 0.0555 0.0940 0.1390 0.1886 0.2955Jung 0.0069 0.0262 0.0545 0.0882 0.1247 0.1625 0.2393

n = 20, p = 2BLUE0 0.0132 0.0502 0.1043 0.1692 0.2397 0.3131 0.4631BLUE2 0.0148 0.0540 0.1058 0.1584 0.2032 0.2363 0.2708BLUE4 0.0158 0.0550 0.0998 0.1357 0.1580 0.1695 0.1770BLUE8 0.0170 0.0543 0.0884 0.1092 0.1195 0.1240 0.1267RSE0 0.0132 0.0501 0.1041 0.1687 0.2389 0.3117 0.4604RSE2 0.0148 0.0541 0.1058 0.1579 0.2021 0.2346 0.2685RSE4 0.0159 0.0549 0.0992 0.1343 0.1560 0.1672 0.1746RSE9 0.0171 0.0542 0.0876 0.1078 0.1176 0.1220 0.1245WMAD0 −0.2526 −0.2253 −0.1865 −0.1415 −0.0936 −0.0445 0.0550WMAD2 −0.2157 −0.1855 −0.1467 −0.1084 −0.0764 −0.0531 −0.0292WMAD4 −0.1798 −0.1484 −0.1137 −0.0870 −0.0708 −0.0627 −0.0573WMAD(r) −0.1560 −0.1244 −0.0935 −0.0730 −0.0623 −0.0235 0.0812r 6 6 6 6 6 1 1Blom 0.0125 0.0485 0.1038 0.1730 0.2511 0.3343 0.5069Jung 0.0131 0.0500 0.1045 0.1702 0.2423 0.3175 0.4717

n = 20, p = 3BLUE0 0.0188 0.0715 0.1498 0.2445 0.3488 0.4584 0.6835BLUE2 0.0211 0.0787 0.1603 0.2554 0.3594 0.4717 0.7187BLUE4 0.0226 0.0817 0.1570 0.2281 0.2816 0.3146 0.3405BLUE8 0.0246 0.0828 0.1418 0.1817 0.2026 0.2121 0.2179RSE0 0.0188 0.0715 0.1495 0.2437 0.3472 0.4555 0.6777RSE2 0.0211 0.0788 0.1605 0.2551 0.3573 0.4668 0.7093RSE4 0.0228 0.0817 0.1561 0.2256 0.2775 0.3097 0.3350RSE9 0.0248 0.0827 0.1407 0.1791 0.1989 0.2079 0.2132WMAD0 −0.2484 −0.2091 −0.1523 −0.0857 −0.0142 0.0593 0.2084WMAD2 −0.2108 −0.1662 −0.1045 −0.0348 0.0393 0.1176 0.2875WMAD4 −0.1742 −0.1266 −0.0680 −0.0152 0.0229 0.0459 0.0635WMAD(r) −0.1498 −0.1011 −0.0478 −0.0079 −0.0023 0.0067 0.0122r 6 6 6 6 7 7 7Blom 0.0177 0.0685 0.1457 0.2414 0.3481 0.4609 0.6938Jung 0.0186 0.0711 0.1495 0.2450 0.3503 0.4611 0.6893

� = 0, �1 = 0.5(0.5)3, 4, � = �1 = 1. For the RSE, BLUE, and WMAD we have included r = 0,10 and 20% of n, as well as the value of r that gave the estimator with the smallest bias.

From Table 2, we see that for small values of �1 Blom’s estimator is usually the one with thesmallest bias. For these same small values of �1, the RSE and BLUE both increase in bias as rincreases while the WMAD decreases in bias as r increases. On the other hand, as �1 increases theRSE and BLUE for larger values of r begin to decrease in bias while no clear pattern can be seenfor the WMAD. In this same situation, the estimators of Blom and Jung, being approximations


to the full sample BLUE, have a very large bias as well. For larger values of p and �1, it is theWMAD that has the smallest bias.

In Table 3, we have presented the bias of the above estimators of the scale parameter � underthe multiple scale-outlier model for n = 10(5)20, p = 0(1)3, and (in the notation of Section 3)�=�1 =0, �=1, �1 =0.5(0.5)2, 3, 4. For the RSE, BLUE, and WMAD we have included r =0,10 and 20% of n, as well as the value of r that gave the estimator with the smallest bias.

From Table 3, we see that each estimator except for the WMAD is quite sensitive to the presenceof outliers. As the value of �1 increases from 0.5 to 4.0, the bias of each estimator except for theWMAD increases, although much less so for large values of r. On the other hand the WMADusually decreases in bias as �1 increases. Also, for a given value of �1 and n, the bias of eachestimator increases considerably as p increases.

The bias of the RSE and BLUE are quite comparable, with the forms with large values of r

giving the smallest bias (except for the case �1 = 0.5). But the estimators of Blom and Jung, eachinvolving all of the order statistics, both have very large bias compared with the censored formsof RSE and BLUE.

When there are two or more outliers and �1 is large, the WMAD usually has the smallest bias,whereas the censored forms of the RSE and BLUE have the smallest bias for small values of pand �1.

6. Conclusions

We have established in this paper some recurrence relations for the single moments of orderstatistics arising from n non-identically distributed logistic random variables, thus generalizingthe i.i.d. work of Shah (1970). These results have then been used to deduce a set of recurrencerelations for the case when the order statistics arise from a multiple outlier model (with a slippageof p observations). These results in turn allowed us to generalize the work of Balakrishnan (1992b)by examining the robustness (in terms of bias) of various linear estimators of the location andscale parameters of the logistic distribution to the presence of multiple outliers. We have seen,as was shown by David and Ghosh (1985) for the single-outlier case, that the median continuesto be the linear estimator of � with the smallest bias even when multiple outliers are present. Itwould be interesting to see if in fact the results of David and Ghosh (1985) can be generalizedto the multiple-outlier model. The linearly weighted mean also possesses nearly the same bias asthe median. Among linear estimators of �, we have seen that the WMAD is the estimator of �with the smallest bias when multiple outliers are present.

A more complete comparative study of the linear estimators of � and � would be based notonly on bias, but on variance (or mean square error) as well. Analogous recurrence relations forthe product moments of order statistics will be needed in order to examine the variances (or meansquare errors) of these estimators. Proofs of such recurrence relations for the product momentsare a lot more complicated and work in this direction is currently in progress.

Acknowledgements

The authors thank the Natural Sciences and Engineering Research Council of Canada forfunding this research. The authors also express sincere thanks to the referees for making somesuggestions which led to an improvement in the presentation of this paper.


Table 3Bias of estimators of sigma in the presence of multiple scale-outliers

�1 0.5 1 1.5 2 3 4

n = 10, p = 1BLUE0 −0.0448 0.0000 0.0526 0.1084 0.2238 0.3413BLUE1 −0.0533 0.0000 0.0437 0.0759 0.1173 0.1418BLUE2 −0.0625 0.0000 0.0397 0.0649 0.0938 0.1096BLUE3 −0.0704 0.0000 0.0375 0.0594 0.0833 0.0958RSE0 −0.0464 0.0000 0.0516 0.1049 0.2136 0.3233RSE1 −0.0540 0.0000 0.0433 0.0746 0.1145 0.1379RSE2 −0.0614 0.0000 0.0400 0.0656 0.0952 0.1113RSE4 −0.0749 0.0000 0.0365 0.0570 0.0788 0.0901WMAD0 −0.3218 −0.2881 −0.2517 −0.2144 −0.1390 −0.0631WMAD1 −0.3039 −0.2633 −0.2315 −0.2087 −0.1796 −0.1625WMAD2 −0.2951 −0.2474 −0.2177 −0.1990 −0.1777 −0.1661WMAD3 −0.3168 −0.2649 −0.2374 −0.2213 −0.2039 −0.1947Blom −0.0412 0.0000 0.0560 0.1203 0.2618 0.4114Jung −0.0441 0.0000 0.0533 0.1106 0.2307 0.3540

n = 10, p = 2BLUE0 −0.0910 0.0000 0.1048 0.2154 0.4445 0.6782BLUE1 −0.1072 0.0000 0.0897 0.1625 0.2794 0.3784BLUE2 −0.1237 0.0000 0.0819 0.1382 0.2099 0.2534BLUE3 −0.1371 0.0000 0.0776 0.1262 0.1830 0.2148RSE0 −0.0939 0.0000 0.1029 0.2092 0.4259 0.6448RSE1 −0.1086 0.0000 0.0889 0.1597 0.2707 0.3619RSE2 −0.1220 0.0000 0.0825 0.1398 0.2134 0.2584RSE4 −0.1445 0.0000 0.0755 0.1209 0.1720 0.1998WMAD0 −0.3561 −0.2881 −0.2154 −0.1411 0.0095 0.1612WMAD1 −0.3446 −0.2633 −0.1981 −0.1465 −0.0653 0.0020WMAD2 −0.3417 −0.2474 −0.1861 −0.1444 −0.0917 −0.0600WMAD3 −0.3660 −0.2649 −0.2080 −0.1724 −0.1309 −0.1076Blom −0.0841 0.0000 0.1105 0.2354 0.5060 0.7896Jung −0.0896 0.0000 0.1059 0.2192 0.4559 0.6989

n = 10, p = 3BLUE0 −0.1385 0.0000 0.1564 0.3209 0.6611 1.0086BLUE1 −0.1615 0.0000 0.1376 0.2571 0.4713 0.6737BLUE2 −0.1832 0.0000 0.1267 0.2207 0.3571 0.4587BLUE3 −0.1995 0.0000 0.1203 0.2013 0.3043 0.3675RSE0 −0.1424 0.0000 0.1539 0.3126 0.6364 0.9640RSE1 −0.1634 0.0000 0.1364 0.2531 0.4579 0.6477RSE2 −0.1810 0.0000 0.1275 0.2232 0.3639 0.4704RSE4 −0.2082 0.0000 0.1173 0.1927 0.2838 0.3367WMAD0 −0.3909 −0.2881 −0.1793 −0.0682 0.1572 0.3843WMAD1 −0.3853 −0.2633 −0.1633 −0.0784 0.0705 0.2083WMAD2 −0.3867 −0.2474 −0.1526 −0.0830 0.0169 0.0901WMAD3 −0.4119 −0.2649 −0.1766 −0.1173 −0.0420 0.0040Blom −0.1288 0.0000 0.1637 0.3455 0.7348 1.1394Jung −0.1366 0.0000 0.1578 0.3255 0.6751 1.0334

n = 15, p = 1BLUE0 −0.0302 0.0000 0.0348 0.0713 0.1459 0.2215BLUE1 −0.0339 0.0000 0.0300 0.0529 0.0832 0.1015BLUE3 −0.0429 0.0000 0.0259 0.0418 0.0597 0.0692


(Table 3 continued)

�1 0.5 1 1.5 2 3 4

BLUE6 −0.0534 0.0000 0.0237 0.0366 0.0500 0.0568RSE0 −0.0302 0.0000 0.0348 0.0711 0.1456 0.2211RSE1 −0.0343 0.0000 0.0297 0.0521 0.0814 0.0989RSE3 −0.0417 0.0000 0.0262 0.0425 0.0609 0.0708RSE6 −0.0534 0.0000 0.0237 0.0366 0.0500 0.0568WMAD0 −0.2969 −0.2737 −0.2491 −0.2240 −0.1735 −0.1228WMAD1 −0.2793 −0.2528 −0.2308 −0.2144 −0.1930 −0.1802WMAD3 −0.2495 −0.2152 −0.1950 −0.1827 −0.1690 −0.1617WMAD4 −0.2437 −0.2058 −0.1862 −0.1748 −0.1626 −0.1562Blom −0.0275 0.0000 0.0375 0.0810 0.1776 0.2807Jung −0.0297 0.0000 0.0352 0.0727 0.1505 0.2299

n = 15, p = 2BLUE0 −0.0610 0.0000 0.0695 0.1421 0.2914 0.4428BLUE1 −0.0683 0.0000 0.0609 0.1107 0.1890 0.2532BLUE3 −0.0851 0.0000 0.0530 0.0871 0.1276 0.1505BLUE6 −0.1036 0.0000 0.0485 0.0759 0.1054 0.1209RSE0 −0.0609 0.0000 0.0694 0.1418 0.2905 0.4411RSE1 −0.0691 0.0000 0.0603 0.1089 0.1839 0.2436RSE3 −0.0831 0.0000 0.0535 0.0885 0.1305 0.1544RSE6 −0.1036 0.0000 0.0485 0.0759 0.1054 0.1209WMAD0 −0.3203 −0.2737 −0.2245 −0.1745 −0.0736 0.0278WMAD1 −0.3060 −0.2528 −0.2082 −0.1725 −0.1176 −0.0734WMAD3 −0.2831 −0.2152 −0.1740 −0.1476 −0.1165 −0.0990WMAD4 −0.2802 −0.2058 −0.1656 −0.1414 −0.1141 −0.0993Blom −0.0557 0.0000 0.0743 0.1594 0.3459 0.5428Jung −0.0601 0.0000 0.0702 0.1447 0.2994 0.4574

n = 15, p = 3BLUE0 −0.0922 0.0000 0.1039 0.2126 0.4359 0.6630BLUE1 −0.1030 0.0000 0.0926 0.1723 0.3108 0.4378BLUE3 −0.1266 0.0000 0.0811 0.1361 0.2057 0.2481BLUE6 −0.1504 0.0000 0.0744 0.1183 0.1672 0.1938RSE0 −0.0923 0.0000 0.1038 0.2121 0.4344 0.6599RSE1 −0.1042 0.0000 0.0918 0.1697 0.3025 0.4218RSE3 −0.1238 0.0000 0.0819 0.1383 0.2107 0.2552RSE6 −0.1504 0.0000 0.0744 0.1183 0.1672 0.1938WMAD0 −0.3439 −0.2737 −0.2000 −0.1251 0.0261 0.1781WMAD1 −0.3327 −0.2528 −0.1849 −0.1278 −0.0309 0.0564WMAD3 −0.3160 −0.2152 −0.1520 −0.1096 −0.0563 −0.0240WMAD(r) −0.3153 −0.2058 −0.1442 −0.1053 −0.0594 −0.0053r 4 4 4 4 4 2Blom −0.0847 0.0000 0.1105 0.2353 0.5060 0.7894Jung −0.0910 0.0000 0.1049 0.2160 0.4465 0.6820

n = 20, p = 1BLUE0 −0.0229 0.0000 0.0260 0.0529 0.1079 0.1633BLUE2 −0.0273 0.0000 0.0212 0.0360 0.0539 0.0640BLUE4 −0.0318 0.0000 0.0194 0.0313 0.0445 0.0516BLUE8 −0.0416 0.0000 0.0175 0.0269 0.0365 0.0414RSE0 −0.0230 0.0000 0.0260 0.0532 0.1089 0.1654RSE2 −0.0273 0.0000 0.0212 0.0359 0.0537 0.0638RSE4 −0.0323 0.0000 0.0193 0.0311 0.0441 0.0510


(Table 3 continued)

�1 0.5 1 1.5 2 3 4

RSE9 −0.0424 0.0000 0.0174 0.0267 0.0361 0.0409WMAD0 −0.2804 −0.2626 −0.2439 −0.2251 −0.1871 −0.1490WMAD2 −0.2493 −0.2273 −0.2112 −0.2002 −0.1869 −0.1795WMAD4 −0.2197 −0.1927 −0.1773 −0.1681 −0.1580 −0.1526WMAD6 −0.2015 −0.1698 −0.1548 −0.1465 −0.1377 −0.1332Blom −0.0208 0.0000 0.0281 0.0609 0.1343 0.2129Jung −0.0225 0.0000 0.0263 0.0540 0.1112 0.1693

n = 20, p = 2BLUE2 −0.0548 0.0000 0.0430 0.0743 0.1152 0.1404BLUE4 −0.0633 0.0000 0.0395 0.0645 0.0935 0.1095BLUE8 −0.0811 0.0000 0.0357 0.0552 0.0759 0.0866RSE0 −0.0462 0.0000 0.0519 0.1061 0.2174 0.3301RSE2 −0.0547 0.0000 0.0430 0.0741 0.1147 0.1397RSE4 −0.0643 0.0000 0.0393 0.0640 0.0926 0.1084RSE9 −0.0824 0.0000 0.0355 0.0548 0.0750 0.0855WMAD0 −0.2984 −0.2626 −0.2253 −0.1876 −0.1116 −0.0355WMAD2 −0.2713 −0.2273 −0.1946 −0.1713 −0.1412 −0.1228WMAD4 −0.2462 −0.1927 −0.1614 −0.1420 −0.1198 −0.1077WMAD6 −0.2321 −0.1698 −0.1393 −0.1218 −0.1029 −0.0929Blom −0.0419 0.0000 0.0559 0.1202 0.2624 0.4133Jung −0.0452 0.0000 0.0524 0.1077 0.2218 0.3380

n = 20, p = 3BLUE0 −0.0695 0.0000 0.0777 0.1584 0.3236 0.4909BLUE2 −0.0823 0.0000 0.0654 0.1149 0.1854 0.2352BLUE4 −0.0946 0.0000 0.0601 0.0997 0.1476 0.1754BLUE8 −0.1183 0.0000 0.0544 0.0852 0.1185 0.1361RSE0 −0.0696 0.0000 0.0778 0.1589 0.3253 0.4941RSE2 −0.0823 0.0000 0.0653 0.1146 0.1845 0.2338RSE4 −0.0959 0.0000 0.0598 0.0989 0.1461 0.1734RSE9 −0.1200 0.0000 0.0541 0.0844 0.1171 0.1342WMAD0 −0.3165 −0.2626 −0.2067 −0.1501 −0.0363 0.0779WMAD2 −0.2932 −0.2273 −0.1776 −0.1406 −0.0890 −0.0530WMAD4 −0.2722 −0.1927 −0.1450 −0.1144 −0.0778 −0.0569WMAD(r) −0.2614 −0.1698 −0.1233 −0.0958 −0.0651 −0.0129r 6 6 6 6 6 1Blom −0.0635 0.000 0.0832 0.1781 0.3853 0.6034Jung −0.0683 0.0000 0.0785 0.1611 0.3318 0.5057

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