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In this communication, we consider and study a generalized two parameters entropy of order statistics and derive bounds for it. The generalized residual entropy using order statistics has also been discussed.

Suppose are independent and identically distributed observations from a distribution, where is differentiable with a density which is positive in an interval and zero elsewhere. The order statistics of the sample is defined by the arrangement of from the smallest to largest denoted as. Then the p.d.f. of the order statistics, is given by

for details refer to [

Order statistics has been studied by statisticians for some time and has been applied to problems of statistical estimation [

In Section 2, we express generalized entropy of order statistics in terms of generalized entropy of order statistics of uniform distribution and study some of its properties. Section 3 provides bounds for entropy of order statistics. In Section 4, we derive an expression for residual generalized entropy of order statistics using residual generalized entropy for uniform distribution.

Let be a random variable having an absolutely continuous cdf and pdf, then Verma [

where

is the Renyi entropy, and

is the Shannon entropy .

We use the probability integral transformation of the random variable where the distribution of U is the standard uniform distribution. If are the order statistics of a random sample from uniform distribution, then it is easy to see using (1) that has beta distribution with parameters and. Using probability integral transformation, entropy (2) of the random variable can be represented as

Next, we prove the following result:

Theorem 2.1 The generalized entropy of can be expressed as

where denotes the entropy of the beta distribution with parameters and, denotes expectation of over and is the beta density with parameters and

.

Proof: Since which implies. Thus, from (3) we have

It is easy to see that the entropy (2) for the beta distribution with parameters and (that is, the order statistics of uniform distribution) is given by

Using (6) in (5), the desired result (4) follows.

In particular, by taking, (4) reduces to

a result derived by Ebrahimi et al. [

Remark: In reliability engineering -outof- systems are very important kind of structures. A -out-of- system functions iff atleast

components out of components function. If denote the independent lifetimes of the components of such system, then the lifetime of the system is equal to the order statistic. The special case of and, that is for sample minima and maxima correspond to series and parallel systems respectively. In the following example, we calculate entropy (4) for sample maxima and minima for an exponential distribution.

Example 2.1 Let be a random variable having the exponential distribution with pdf

Here, and the expectation term is given by

For, from (6), we have

Hence, using (4)

which confirms that the sample minimum has an exponential distribution with parameter, since

where is an exponential variate with parameter. Also

Hence, the difference between the generalized entropy of first order statistics i.e. the sample minimum and the generalized entropy of parent distribution is independent of parameter, but it depends upon sample size. Similarly, for sample maximum, we have

It can be seen easily that the difference between and is

which is also independent of parameter.

In this section, we find the bounds for generalized entropy for order statistics (4) in terms of entropy (2). We prove the following result.

Theorem 3.1 For any random variable with, the entropy of the order statistics is bounded above as

where

and, bounded below as

where, , and is the mode of the distribution and is pdf of the random variable.

Proof: The mode of the beta distribution is. Thus,

For, from (4)

which gives (7).

From (4) we can write

Example 3.1 For the uniform distribution over the interval we have

and from (6),

and

Hence, using (7) we get

Further, for uniform distribution over the interval

,. Using (8) we get

Thus, for uniform distribution, we have

We can check that the bounds for are same as that of.

Example 3.2 For the exponential distribution with parameter, we have and

Thus, as calculated in Example 2.1

Using Theorem 3.1

Here we observe that the difference between upper bound and is, which is an increasing function of n. Thus, for the exponential distribution upper bound is not useful when sample size is large.

In reliability theory and survival analysis, usually denotes a duration such as the lifetime. The residual lifetime of the system when it is still operating at time, given by has the probability density

, where. Ebrahimi [

Obviously, when, it reduces to Shannon entropy.

The generalized residual entropy of the type is defined as

where. When, it reduces to (2).

We note that the density function and survival function of (refer to [

where

and

where

and are known as the beta and incomplete beta functions respectively. In the next lemmawe derive an expression for for the dynamic version of as given by (6).

Lemma 4.1 Let be the order statistics based on a random sample of size from uniform distribution on. Then

Proof: For uniform distribution using (10), we have

Putting values from (11) and (13) in (16), we get the desired result (15).

If we put in (15), we get (6).

Using this, in the following theorem, we will show that the residual entropy of order statistics can be represented in terms of residual entropy of uniform distribution.

Theorem 4.1 Let be an absolutely continuous distribution function with density. Then, generalized residual entropy of the order statistics can be represented as

where

Proof: Using the probability integral transformation

and above lemma, the result follows.

Take in (17), it reduces to (4).

Example 4.1 Suppose that is exponentially distributed random variable with mean. Then,

and we have

For, Theorem 4.1 gives

Also

Hence

So, in the exponential case the difference between generalized residual entropy of the lifetime of a series system and residual generalized entropy of the lifetime of each component is independent of time.

The two parameters generalized entropy plays a vital role as a measure of complexity and uncertainty in different areas such as physics, electronics and engineering to describe many chaotic systems. Using probability integral transformation we have studied the generalized and generalized residual entropies based on order statistics. We have explored some properties of these entropies for exponential distribution.

The first author is thankful to the Center for Scientific and Industrial Research, India, to provide financial assistance for this work.