Estimating the Variance of an Exponential Distribution in the Presence of Large True Observations

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 Schätzern für die Varianz der Exponentialverteilung falls große wahre Beobachtungen vorhanden sind und falls ein priori oder mutmaßliches Intervall aus vergangener Erfahrung verfügbar ist, das diesen Varianzparameter enthält. Eine empirische Studie zeigt die Effizienz dieser Klasse von Schätzern verglichen mit dem Schätzer von Pandey und Singh, dem Minimum MSE Schätzer und speziellen Klassen von shrinkage Schätzern.


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 (θ 2 1 , θ 2 2 ), θ 2 1 < θ 2 2 , containing the parameter θ 2 , say, is available from some past experiences.
Let x 1 , . . ., x n be a random sample of size n, drawn from the exponential distribution.The probability density function of which is given by where θ is the mean and acts as a scale parameter and θ 2 is the variance.Pandey and Singh (1977) suggested the minimum mean squared error (MMSE) esti- for θ 2 in the class of estimators of the form M x2 , where M is a suitably chosen constant and x = n i=1 x i /n.The bias and mean squared error (MSE) of θMMSE are .
(3) Tracy et al. (1996) envisaged a class of shrinkage estimators for θ 2 when some point prior information θ 2 0 of θ 2 is available, and is given as where 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 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 x i > 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 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 xt is less than the variance of the usual unbiased estimator x.

The Suggested Classes of Shrinkage Estimators
Let the prior information on θ 2 be available in form of an interval with end points θ 2 1 and θ 2 2 , where θ 2 1 < θ 2 2 .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 θ 2 in model ( 1).We define the family where α (h) is given by ( 4) and h is a non-zero real number.Moreover, 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 (θ 2 1 + θ 2 2 )/2, defined as where Replacing x by xt in (6), we define the class of estimators for the variance in the presence of large true observations, where α (h) and xt 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 where Bias and MSE of the estimates defined in (8) are Substituting ( 9) in (10) gives where As t → ∞, we get p → 1, q → 0, and thus MSE ξ(i) (h,t) → MSE ξ(i) (h) .From ( 7) and ( 10) we have where Thus we established the following theorem: Theorem 2.1: The classes of shrinkage estimators

Conclusion
From the above we conclude that the developed classes of estimators ψ(i) (1,t) , i = 1, 2, 3, are to be preferred over θMMSE and ψ(i) (1) in practice as they are more efficient than θMMSE and ψ(i) (1) with larger gain in efficiency.

Table 1 :
Percentage relative efficiencies, PRE, of the estimators ψ