Estimating the Variance of an Exponential Distribution in the Presence ...

AUSTRIAN J OURNAL OF S TATISTICS Volume 37 (2008), Number 2, 207–216

Estimating the Variance of an Exponential Distribution in the Presence of Large True Observations Housila P. Singh and Vankim Chander School of Studies in Statistics, Vikram University, Ujjain, India Abstract: The present paper discusses some classes of shrinkage estimators for the variance of the exponential distribution in the presence of large true observations when some a priori or guessed interval containing the variance parameter is available from some past experiences. Empirical study shows the high efficiency of the developed classes of shrinkage estimators when compared with Pandey and Singh’s estimator, minimum MSE estimator and special classes of shrinkage estimators. Zusammenfassung: Dieser Aufsatz diskutiert einige Klassen von shrinkage Sch¨atzern f¨ur die Varianz der Exponentialverteilung falls große wahre Beobachtungen vorhanden sind und falls ein priori oder mutmaßliches Intervall aus vergangener Erfahrung verf¨ugbar ist, das diesen Varianzparameter enth¨alt. Eine empirische Studie zeigt die Effizienz dieser Klasse von Sch¨atzern verglichen mit dem Sch¨atzer von Pandey und Singh, dem Minimum MSE Sch¨atzer und speziellen Klassen von shrinkage Sch¨atzern. Keywords: Bias, Guessed Interval, Mean Squared Error, Percent Relative Efficiency.

1 Introduction The exponential distribution has its significance due to its variety of applications in reliability engineering and life testing problems. The exponential distribution would be an adequate choice for a situation where failure rate appears to be more or less constant. The problem considered in this paper can be illustrated by the question that a sampler frequently asks himself, particularly if he is working with relatively small samples. The question is, “what do I do with large or extreme observations in the sample?” The sampler first attempt to answer this question by a careful review of the data to see if an outlier has somehow appeared or if in fact the offending observation or observations are actually true observations. It is also noted that in practice the experimenter often possesses some knowledge of the experimental conditions, based on awareness with the performance of the system under investigation or from the past experience or from some extraneous source and thus in opinion to give an adequate guessed interval of the value of the variance. In this paper we suggest some classes of shrinkage estimators for the variance of exponential distribution in the presence of large true observations when some a priori or guessed interval (θ12 , θ22 ), θ12 < θ22 , containing the parameter θ2 , say, is available from some past experiences. Let x1 , . . . , xn be a random sample of size n, drawn from the exponential distribution. The probability density function of which is given by ½1 exp(−x/θ) x ≥ θ , θ > 0 , (1) f (x; θ) = θ 0 otherwise,

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where θ is the mean and acts as a scale parameter and θ2 is the variance. Pandey and Singh (1977) suggested the minimum mean squared error (MMSE) estimator n2 θˆMMSE = x¯2 (2) (n + 2)(n + 3) for θ2 in P the class of estimators of the form M x¯2 , where M is a suitably chosen constant and x¯ = ni=1 xi /n. The bias and mean squared error (MSE) of θˆMMSE are ³ ´ 2(2n + 3)θ2 bias θˆMMSE = − (n + 2)(n + 3) and

³

´ 2(2n + 3)θ4 ˆ MSE θMMSE = . (n + 2)(n + 3)

(3)

Tracy et al. (1996) envisaged a class of shrinkage estimators for θ2 when some point prior information θ02 of θ2 is available, and is given as ξ(h) = θ02 + α(h) (¯ x2 − θ02 ) , where α(h) = n2h

Γ(n + 2h) Γ(n + 4h)

(4)

and h is a non-zero real number. In particular, if h = 1 then ξ(h) reduces to the point estimator for the variance, which is given as ξ(1) = θ02 +

n2 (¯ x2 − θ02 ) , (n + 2)(n + 3)

and is due to Tracy et al. (1996). The distribution (1) is positive valued and positive skewed. It has positive probability that the sample may contain one or more observations from right tail of the distribution leading to a larger estimate of the parameter using unbiased estimator. In such a situation, where some “extremely large” values xi > t are present in the sample, Searls (1966) suggested an estimation procedure suitable to estimate the population mean θ, which reduces the effect of such large true observations for the distribution which is unimodal, positive valued, and positively skewed. Searls (1966) defined the estimator for θ as à m ! 1 X x¯t = xj + (n − m)t , m = 0, 1, . . . , n , xj ≤ t , (5) n j=1 which is formulated by replacing all the observations greater than a predetermined cutoff point t by the value of t itself. Searls (1966) has shown that there exists a wide range of t values for which the MSE of x¯t is less than the variance of the usual unbiased estimator x¯. We also refer to Bartholomew (1957), Ojha and Srivastava (1979), Ojha (1982), Srivastava et al. (1985), Srivastava (1986), Singh (1987), Srivastava and Kumar (1990), Upadhyaya et al. (1997), and Singh and Shukla (2002) in this context.

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The estimation problem using some a priori information regarding some population parameters has been investigated by various authors, for example see Mehta and Srinivasan (1971), Jani (1991), Singh and Saxena (2003), Saxena and Singh (2004), Singh et al. (2004), Singh and Saxena (2005), Saxena (2006), Saxena and Singh (2006), and also Singh and Chander (2007). Thompson (1968) considered the problem of shrinking an unbiased estimator ψˆ of ψ towards an interval (ψ1 , ψ2 ) and suggested a shrinkage estimator ψˆ + (1 − p)(ψ1 + ψ2 )/2, where 0 ≤ p ≤ 1 is constant. The objective is to propose a class of shrinkage estimators of θ2 , when a prior or guessed interval of θ2 is available in the form of (θ12 , θ22 ), θ12 < θ22 .

2 The Suggested Classes of Shrinkage Estimators Let the prior information on θ2 be available in form of an interval with end points θ12 and θ22 , where θ12 < θ22 . The arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM) are measures of location, which are used for suggesting different classes of shrinkage estimators for θ2 in model (1). We define the family ¡ ¢ (i) ξ˜(h) = AGH(a, b) + α(h) x¯2 − AGH(a, b) , (6) where α(h) is given by (4) and h is a non-zero real number. Moreover, µ AGH(a, b) =

(θ12 θ22 )a

1 2 (θ + θ22 ) 2 1

¶b ,

and i = 1, 2, 3 corresponds to pairs (a, b) respectively taken as (0, 1), (1/2, 0), or (1, −1) in AGH(a, b). It is interesting to note that • for (a, b) = (0, 1) in (6) we get a class of shrinkage estimators based on the arithmetic mean (θ12 + θ22 )/2, defined as ¶ µ ¢ 1 2 1¡ 2 (1) 2 2 2 ˜ ξ(h) = (θ1 + θ2 ) + α(h) x¯ − θ + θ2 , 2 2 1 • for (a,p b) = (1/2, 0) we get a class of shrinkage estimators based on the geometric mean θ12 θ22 , given as µ ¶ q q (2) 2 ξ˜(h) = θ12 θ22 + α(h) x¯ − θ12 θ22 , • for (a, b) = (1, −1) we get a class of shrinkage estimators based on the harmonic mean 2θ12 θ22 /(θ12 + θ22 ), defined as µ ¶ θ12 θ22 θ12 θ22 (3) 2 ˜ ξ(h) = 2 2 + α(h) x¯ − 2 2 . θ1 + θ22 θ1 + θ22

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Bias and MSE of these estimates are ³ ´ ³ ¡ ¢ α(h) ´ (i) 2 ˜ bias ξ(h) = θ r 1 − α(h) + n and

³ ´ ¡ ¢ (i) 2 MSE ξ˜(h) = θ4 r2 + α(h) ζ1∗ + 2rα(h) ζ2∗ ,

(7)

where ζ1∗ =

1 2 (n + 1)(n + 2)(n + 3) − (r + 1)(n + 1) + (r + 1)2 , 3 n n

ζ2∗ =

1 −r, n

AGH(a, b) − 1. θ2 Replacing x¯ by x¯t in (6), we define the class of estimators ª © (i) ξˆ(h,t) = AGH(a, b) + α(h) x¯2t − AGH(a, b) , r=

(8)

for the variance in the presence of large true observations, where α(h) and x¯t are respectively given by (4) and (5). Also here m denotes the number of observations less than a predetermined cutoff point t and follows a binomial distribution with parameters n and p, where p = P (x < t) = 1 − exp(−t/θ) and q = 1 − p = exp(−t/θ). The necessary expectations are E(¯ xt ) = pθ ,

E(¯ x2t ) =

¢ θ2 ¡ λ + np2 , n

E(¯ x4t ) =

θ4 (η1 + η2 ) , n3

(9)

where η1 = 3λ2 (n − 2) + 6λ(n2 p2 + 2npq + 2) , η2 = np3 (n2 p + 4q + 8) − 12npq(t/θ)2 − 4q(t/θ)3 , λ = 1 − 2qt/θ − q 2 . Bias and MSE of the estimates defined in (8) are ³ ´ ³ ¡ ¢ α(h) ¡ ¢´ (i) 2 2 ˆ bias ξ(h,t) = θ r 1 − α(h) + λ + np − n n and

³ ´ ¡ ¢ (i) 2 MSE ξˆ(h,t) = r2 θ4 + α(h) E(¯ x4t ) + AGH 2 (a, b) − 2AGH(a, b)E(¯ x2t ) ¢ ¡ +2rθ2 α(h) E(¯ x2t ) − AGH(a, b) .

Substituting (9) in (10) gives ³ ´ ¢ ¡ (i) 2 MSE ξˆ(h,t) = θ4 r2 + α(h) ε1 + 2rα(h) ε2 , where 1 2 (η1 + η2 ) + (r + 1)2 − (r + 1)(λ + np2 ) 3 n n 1 ε2 = (λ + np2 ) − (r + 1) . n

ε1 =

(10)

(11)

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³ ´ ³ ´ (i) (i) As t → ∞, we get p → 1, q → 0, and thus MSE ξˆ(h,t) → MSE ξ˜(h) . From (7) and (10) we have ³ ´ ³ ´ ¡ 2 ¢ (i) (i) MSE ξˆ(h,t) − MSE ξ˜(h) = θ4 α(h) (ε1 − ζ1∗ ) + 2rα(h) (ε2 − ζ2∗ ) ¢ θ4 α(h) ¡ 2 ∗ ∗ 2 = α (X + 2n X ) − 2rX (1 − α )n (h) (h) n3 < 0, if r≥ or if

α(h) (2n2 X ∗ + X) , 2X ∗ (1 − α(h) )n2

α(h) (2n2 X ∗ + X) AGH(a, b) ≥ , θ2 2n2 X ∗ (1 − α(h) )

or if 0 ≤ θ2 ≤

AGH(a, b)2n2 X ∗ (1 − α(h) ) , α(h) (2n2 X ∗ + X)

where X = (η1 + η2 ) − (n + 1)(n + 2)(n + 3) ,

X ∗ = n(1 − p2 ) − λ + 1 .

Thus we established the following theorem: (i) (i) Theorem 2.1: The classes of shrinkage estimators ξˆ(h,t) are more efficient than ξ˜(h) , i = 1, 2, 3, if θ2 is between 0 and 2n2 X ∗ AGH(a, b)(1 − α(h) )/(α(h) (2n2 X ∗ + X)).

3 Special Cases (i) For h = 1 in (6) we get classes of shrinkage estimators ψ˜(1) , i = 1, 2, 3, for θ2 as ¡ ¢ (i) ψ˜(1) = AGH(a, b) + α(1) x¯2 − AGH(a, b) ,

(12)

(i) where α(1) = n2 /(n + 2)(n + 3). The MSE of ψ˜(1) can be easily obtained by putting h = 1 in (10) and is given as ³ ´ ¡ ¢ (i) 2 ˜ MSE ψ(1) = θ4 r2 + α(1) ζ1∗ + 2rα(1) ζ2∗ . (13)

For h = 1 in (8) we get shrinkage estimators in the presence of large true observations as ¡ ¢ (i) ψˆ(1,t) = AGH(a, b) + α(1) x¯2t − AGH(a, b) . (14) Putting h = 1 in (11), we get ³ ´ ¢ ¡ (i) 2 (15) ε1 + 2rα(1) ε2 . MSE ψˆ(1,t) = θ4 r2 + α(1) ³ ´ ³ ´ (i) (i) As t → ∞, we get p → 1, q → 0, and MSE ψˆ(1,t) → MSE ψ˜(1) . Thus we proved the following theorem: (i) (i) Theorem 2.2: The classes of shrinkage estimators ψˆ are more efficient than ψ˜ , (1,t)

(1)

i = 1, 2, 3, if θ2 is between 0 and 2X ∗ AGH(a, b)(5n + 6)/(2(n + 2)(n + 3)X ∗ + X).

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4 Numerical Illustrations and Comparisons To have a concrete idea about the performance of the proposed estimators ψˆ(1,t) , we have computed their percentage relative efficiency (PRE) with respect to θˆMMSE given in (2) and (i) ψ˜(1) given by (12) for various values of n, θ12 /θ2 , θ22 /θ2 , and for different cutoff points t. The following formulae are used for this calculation: (i)

³

(i) PRE ψˆ(1,t) , θˆMMSE

´

³ ´ MSE θˆMMSE ³ ´, = 100 (i) ˆ MSE ψ(1,t)

³

(i) (i) PRE ψˆ(1,t) , ψ˜(1)

´

³ ´ (i) MSE ψ˜(1) ³ ´, = 100 (i) ˆ MSE ψ(1,t)

³

´ ³ ´ ³ ´ (i) (i) ˆ ˜ ˆ where MSE θMMSE , MSE ψ(1) , and MSE ψ(1,t) are given by (3), (13) and (15). From Table 1 we observe that the suggested classes of estimators show better efficiency than θˆMMSE if • t/θ = 1, n ≤ 11, and ∆ ∈ [0.7, 2.7], where ∆ = AGH(0, 1)/θ2 , • 2 ≤ t/θ ≤ 10, n ≤ 30, and ∆ ∈ [0.3, 2.0]. From extended computation and from Table 1, we further observe that (1) (2) (3) • for ∆ < 1, ψˆ(1,t) (based on the AM) is more efficient than θˆMMSE , ψˆ(1,t) , and ψˆ(1,t) , (3) (2) (1) • for ∆ > 1, ψˆ(1,t) (based on the HM) is more efficient than θˆMMSE , ψˆ(1,t) , and ψˆ(1,t) , (1) (2) • for ∆ = 1 and 1 ≤ t/θ ≤ 3, ψˆ(1,t) shows higher efficiency than θˆMMSE , ψˆ(1,t) , and (3) (3) (2) ψˆ , whereas for ∆ = 1 and 3 < t/θ ≤ 10, ψˆ is better than θˆMMSE , ψˆ , and (1,t)

(1,t)

(1,t)

(1) ψˆ(1,t) . (i) (i) Further, it reveals from Table 2 that ψˆ(1,t) shows better efficiency than ψ˜(1) if

• t/θ = 1, n ≤ 5, and ∆ ∈ [0.7, 2.0], where ∆ = AGH(0, 1)/θ2 , • t/θ = 1, 5 < n ≤ 25, and ∆ ∈ [1.2, 2.5], • 2 ≤ t/θ < 6, n ≤ 25, and ∆ ∈ [0.7, 2.0]. It is also observed that (1) (i) (2) • for ∆ ≤ 1, the estimator ψˆ(1,t) (based on the AM) is more efficient than ψ˜(1) , ψˆ(1,t) , (3) and ψˆ , (1,t)

(3) (i) (2) • for ∆ > 1, the estimator ψˆ(1,t) (based on the HM) is more efficient than ψ˜(1) , ψˆ(1,t) , (1) and ψˆ , (1,t)

• for t/θ ≥ 6, the classes of estimators ψˆ(1,t) and ψ˜(1) are equally efficient. (i)

(i)

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(i) Table 1: Percentage relative efficiencies, PRE, of the estimators ψˆ(1,t) , i = 1, 2, 3, with respect to θˆMMSE for different n, θ2 /θ2 , θ2 /θ2 , t/θ. 1

³

θ12 θ22 , θ2 θ2

t/θ = 1

´

(0.2, 1.0) i=1 i=2 i=3 (0.4, 1.2) i=1 i=2 i=3 (0.5, 1.3) i=1 i=2 i=3 (0.6, 1.4) i=1 i=2 i=3 (0.7, 1.5) i=1 i=2 i=3 (0.8, 1.6) i=1 i=2 i=3 (0.9, 1.7) i=1 i=2 i=3 (1.0, 2.2) i=1 i=2 i=3 (1.0, 2.8) i=1 i=2 i=3

2

t/θ = 5

t/θ = 10

n=5

n = 15

n = 25

n=5

n = 15

n = 25

n=5

n = 15

n = 25

196.25 143.04 116.22

71.86 62.42 56.53

43.70 39.92 37.40

199.45 175.65 156.25

137.90 131.75 125.93

125.67 122.03 118.53

175.00 157.48 142.36

125.33 120.94 116.51

115.22 112.78 110.24

324.66 244.23 196.25

87.80 78.67 71.86

49.52 46.27 43.70

217.85 210.45 199.45

142.09 140.45 137.90

128.24 127.19 125.67

187.01 182.54 175.00

127.72 126.95 125.33

116.48 116.09 115.22

440.46 330.49 263.08

97.81 88.38 81.03

52.88 49.72 47.13

218.50 218.07 213.11

142.28 142.14 141.04

128.49 128.27 127.56

186.50 187.11 184.25

127.36 127.73 127.28

116.25 116.48 116.26

627.90 465.26 364.91

109.61 99.62 91.61

56.58 53.46 50.83

213.00 218.00 218.86

141.17 142.19 142.33

128.04 128.47 128.42

181.53 185.97 187.36

125.95 127.20 127.70

115.44 116.15 116.46

955.23 691.77 530.70

123.64 112.84 104.00

60.68 57.55 54.85

202.22 210.77 216.23

138.81 140.70 141.84

126.90 127.82 128.33

172.79 179.69 184.31

123.58 125.45 126.72

114.09 115.16 115.88

1579.4 1107.6 823.6

140.51 128.59 118.68

65.25 62.06 59.27

187.71 197.99 206.31

135.33 137.83 139.73

125.11 126.41 127.36

161.34 169.44 176.06

120.34 122.65 124.47

112.22 113.55 114.60

2845.7 1946.6 1394.9

160.99 147.60 136.32

70.34 67.06 64.15

171.18 181.75 191.30

130.89 133.79 136.22

122.71 124.29 125.58

148.36 156.67 164.16

116.39 118.96 121.16

109.89 111.41 112.69

3352.8 5626.4 4410.2

257.45 211.84 179.38

89.71 81.33 74.55

122.30 140.32 158.38

113.88 120.96 127.06

112.69 117.02 120.57

109.16 123.80 138.23

101.74 107.78 113.04

100.78 104.64 107.88

697.80 2146.2 5687.1

466.93 293.75 208.65

118.19 95.65 80.69

85.47 140.32 141.87

95.18 120.96 121.52

100.16 117.02 117.35

78.32 100.68 125.05

85.83 97.84 108.26

89.84 98.21 104.94

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(i) (i) Table 2: Percentage relative efficiencies, PRE, of the estimators ψˆ(1,t) with respect to ψ˜(1) , i = 1, 2, 3, for different n, θ12 /θ2 , θ22 /θ2 , t/θ.

³

θ12 θ22 , θ2 θ2

t/θ = 1

´

(0.2, 1.0) i=1 i=2 i=3 (0.4, 1.2) i=1 i=2 i=3 (0.5, 1.3) i=1 i=2 i=3 (0.6, 1.4) i=1 i=2 i=3 (0.7, 1.5) i=1 i=2 i=3 (0.8, 1.6) i=1 i=2 i=3 (0.9, 1.7) i=1 i=2 i=3 (1.0, 2.2) i=1 i=2 i=3 (1.0, 2.8) i=1 i=2 i=3

t/θ = 5

t/θ = 10

n=5

n = 15

n = 25

n=5

n = 15

n = 25

n=5

n = 15

n = 25

112.51 91.10 81.85

57.44 51.70 48.60

37.98 35.44 33.97

114.35 111.87 110.05

110.22 109.12 108.25

109.22 108.34 107.66

100.33 100.29 100.26

100.17 100.16 100.15

100.14 100.14 100.13

174.25 134.27 112.51

68.87 62.08 57.44

42.58 39.92 37.98

116.92 115.70 114.35

111.45 110.83 110.22

110.26 109.73 109.22

100.37 100.35 100.33

100.19 100.18 100.17

100.15 100.15 100.14

237.05 177.28 143.29

76.94 69.32 63.78

45.56 42.75 40.60

117.60 116.98 116.08

111.93 111.49 111.01

110.70 110.29 109.88

100.37 100.37 100.36

100.19 100.19 100.18

100.15 100.15 100.15

347.16 251.11 195.49

87.19 78.47 71.87

49.09 46.10 43.71

117.76 117.66 117.25

112.29 112.00 111.66

111.09 110.77 110.44

100.37 100.37 100.37

100.19 100.19 100.19

100.15 100.15 100.15

554.77 386.38 289.00

100.24 90.12 82.23

53.27 50.06 47.41

117.44 117.73 117.75

112.54 112.36 112.14

111.41 111.17 110.92

100.35 100.36 100.37

100.19 100.19 100.19

100.16 100.16 100.15

982.20 655.96 469.49

116.97 105.05 95.53

58.23 54.74 51.80

116.73 117.26 117.60

112.66 112.59 112.47

111.66 111.49 111.31

100.33 100.35 100.36

100.19 100.19 100.19

100.16 100.16 100.16

1924.3 1246.6 852.58

138.58 124.31 112.73

64.10 60.29 57.01

115.75 116.39 116.92

112.67 112.68 112.64

111.84 111.73 111.61

100.31 100.33 100.34

100.18 100.19 100.19

100.16 100.16 100.16

3078.7 4556.9 3199.9

253.49 196.90 158.98

89.15 77.84 69.21

112.30 113.64 114.91

112.13 112.42 112.61

111.99 112.00 111.93

100.24 100.27 100.29

100.17 100.18 100.18

100.15 100.15 100.16

892.52 2136.5 4560.1

544.85 300.72 193.07

131.73 97.54 77.01

109.33 111.50 113.76

111.06 111.90 112.44

111.64 111.94 111.99

100.18 100.22 100.27

100.15 100.17 100.18

100.14 100.15 100.15

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5

Conclusion

(i) From the above we conclude that the developed classes of estimators ψˆ(1,t) , i = 1, 2, 3, (i) are to be preferred over θˆMMSE and ψ˜ in practice as they are more efficient than θˆMMSE and (1)

(i) ψ˜(1) with larger gain in efficiency. (1) (2) (3) We also note that ψˆ(1,t) (based on the AM) performed better than θˆMMSE , ψˆ(1,t) , ψˆ(1,t) , (i) and ψ˜ , i = 1, 2, 3, when t/θ ≤ 10, n ≤ 25, and 0.7 ≤ ∆ < 1.0, while for t/θ ≥ 6, (1)

(1) (1) n ≤ 25, and 0.7 ≤ ∆ < 1.0, the proposed estimators ψˆ(1,t) and ψ˜(1) are equally efficient, so one can choose any one of them. It is further observed that the suggested estimator (3) (1) (2) (i) ψˆ(1,t) (based on the HM) has smaller MSE than θˆMMSE , ψˆ(1,t) , ψˆ(1,t) , and ψ˜(1) , i = 1, 2, 3, when t/θ < 6, n ≤ 25, and 1 ≤ ∆ < 2. In such situations we suggest the use of the (3) estimator ψˆ(1,t) . On the other hand it is noted that for t/θ ≥ 6, n ≤ 25, and 0.7 ≤ ∆ < 1 (3) (3) the estimators ψˆ(1,t) and ψ˜(1) approximately have the same MSE and hence any one of them can be chosen in such situation.

Acknowledgements The authors are grateful to the editor Professor Herwig Friedl and the learned referee for their valuable suggestions regarding the improvement of the paper.

References Bartholomew, D. J. (1957). A problem of life testing. Journal of the American Statistical Association, 65, 350-355. Jani, P. N. (1991). A class of shrinkage estimators for the scale parameter of the exponential distribution. IEEE Transactions on Reliability, 40, 68-70. Mehta, J. S., and Srinivasan, R. (1971). Estimation of the mean by shrinkage to a point. Journal of the American Statistical Association, 66, 86-90. Ojha, V. P. (1982). A note on estimation of variance in exponential density. Journal of the Indian Society of Agricultural Statistics, 34, 82-88. Ojha, V. P., and Srivastava, S. R. (1979). An estimator of the population variance in the presence of large true observations. Journal of the Indian Society of Agricultural Statistics, 31, 77-84. Pandey, B. N., and Singh, J. (1977). A note on estimation of variance in exponential density. Sankhya, B, 39, 294-298. Saxena, S. (2006). A decision theoretic estimation in exponential product life testing model using guesstimate. Vikalpa, 31, 31-45. Saxena, S., and Singh, H. P. (2004). Estimating various measures in normal population through a single class of estimators. Journal of the Korean Statistical Society, 33, 323-337. Saxena, S., and Singh, H. P. (2006). From ordinary to shrinkage square-root estimators. Communications in Statistics – Theory and Methods, 35, 1037-1058.

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Searls, D. T. (1966). An estimators for a population mean which reduces the effect of large true observations. Journal of the American Statistical Association, 59, 1200-1204. Singh, H. P. (1987). A modified estimator for normal population variance in the presence of large true observations. Gujarat Statistical Review, 14, 15-30. Singh, H. P., and Chander, V. (2007). Some classes of shrinkage estimators for estimating the scale parameter towards an interval of exponential distribution. Journal of Probability and Statistical Science. (accepted) Singh, H. P., and Saxena, S. (2003). An improved class of shrinkage estimators for the variance of a normal population. Statistics in Transition, 6, 119-129. Singh, H. P., and Saxena, S. (2005). Using prior information in estimation of k th exponent of scale parameter in negative exponent population. Statistical Methods, 7, 116126. Singh, H. P., Saxena, S., and Espejo, M. R. (2004). Estimation of standard deviation in normal parent by shrinkage towards an interval. Journal of Statistical Planning and Inference, 126, 479-493. Singh, H. P., and Shukla, S. K. (2002). A class of shrinkage estimators for the variance of exponential distribution with type-I censoring. Indian Association for Productivity, Quality and Reliability Transactions, 27, 119-141. Srivastava, R. S. (1986). On estimation of mean life time from a time censored sample using exponential failure model. Journal of National Academy of Mathematics, India, 4, 107-113. Srivastava, R. S., and Kumar, G. (1990). Efficient estimation of variance of exponential population in the presence of large true observations. Journal of National Academy of Mathematics, India, 8, 61-71. Srivastava, R. S., Pandey, B. N., and Srivastava, S. R. (1985). A modified estimator for the population mean which reduces the effect of large true observations. Journal of the Indian Society of Agricultural Statistics, 37, 71-78. Thompson, J. R. (1968). Accuracy borrowing in the estimation of mean by shrinkage to an interval. Journal of the American Statistical Association, 63, 113-122. Tracy, D. S., Singh, H. P., and Raghuvanshi, H. S. (1996). Some shrinkage estimator for the variance of exponential density. Microelectron Reliability, 36, 651-655. Upadhyaya, L. N., Gangele, R. K., and Singh, H. P. (1997). A shrinkage estimator for the scale parameter of the exponential distribution with type-I censoring. International Journal of Management System, 13, 103-114.

Authors Address: Housila P. Singh and Vankim Chander School of Studies in Statistics Vikram University Ujjain – 456010, M. P. India E-mail: [email protected] and [email protected]