2.1. Pareto-type distributions

A general version of Pareto-type distributions, called the Pareto (IV) distribution, is discussed in chapter 3 of Arnold (1983). The cumulative distribution function of this family is

where −∞ < µ < ∞, θ > 0 and γ > 0 are location, scale and inequality parameters, respectively and α > 0 is the shape parameter which characterizes the tail of the distribution. This distribution is denoted by Pareto (IV)(µ, θ, γ, α), and its density function is as follows:

• •

  Setting (α = 1), (γ = 1) and (γ = 1, µ = θ) in relation (2.1) and (2.2), one at a time, leads to the cumulative distribution and probability density functions of Pareto (III), Pareto (II) and Pareto (I) distributions, respectively.

• •

  The Burr (XII) distribution is a special case of the Pareto (IV) distribution in which µ = 0, γ→1γ. If X ∼Pareto(IV)(µ, θ, γ, α), and Zα=1+(x−μθ)1γ, then

  (2.3)FZα(z)=1−z−α, z>1,

  and

  (2.4)fZα(z)=αzα+1, z>1,

  where α > 0. We denote this distribution by Pareto (α) and note that Z_α has the same distribution as Z11α.

2.2. Shannon information

Consider a discrete random variable X such that P{X = x_i} = p_i, i = 1,2,··· and ∑i=1∞Pi=1. Then

is known as the “entropy” or Shannon information (SI) of the random variable X. If X is a random variable having an absolutely continuous cumulative distribution function (cdf) F(x) and probability density function (pdf) f(x), then SI is defined as Suppose that Z be a random variable with cdf F_Z, pdf f_Z and Shannon information H(Z). The following lemma is well-known and easy to check.

The Shannon entropy of the random variable X is a mathematical measure of information which measures the average reduction of uncertainty of X. Because of its descriptive character, analytical expressions for univariate distributions have been obtained, among others, by Cover and Thomas (2006).

2.3. k-record data

Let {X_n : n = 1,2,…} be a sequence of independent and identically distributed (i.i.d) random variables with absolutely continuous cumulative distribution function F(x; θ) and probability density function f(x; θ). We are interested in the Shannon information (SI) contained in the sequence of k-records. A k-record is basically the k-th largest observation in a partial sample. More precisely, let X_i:n denote the i-th order statistic from a random sample of size n. We define upper k-record times T_n,k and upper k-record values R_n,k as follows :

1. (i)

   T_1,k = k and R_1,k = X_1:k,

2. (ii)

   T_n,k = min{j : j > T_n−1,k, X_j > X_{Tn−1,k−k+1:Tn−1,k}} and R_n,k = X_Tn,k − k + 1 : T_n,k, (n ≥ 2).

Suppose N_n,k be the number of k-record values in X₁,…,X_n. Lower k-record values, lower k-record times and the number of lower k-records are similarly defined (Arnold et al., 1998).

In the case of record and k-record data from a random sample of size n, Madadi and Tata (2011, 2014) obtained the SI contained in the last record (maximum), upper k-record values with associated k-record times, denoted by HMU(n) and HRTU(n,k), respectively. The SI for the lower records and k-records are denoted by HML(n) and HRTL(n,k), respectively.

In an inverse sampling plan (ISP), one takes observations until a fixed number m of k-records is reached. Denote the SI contained in all upper k-record values, the last upper k-record and all upper k-record values together with associated k-record times by hRU(m,k), hMU(m,k) and hRTU(m,k), respectively. The corresponding SI for lower k-record data are denoted by hRL(m,k), hML(m,k) and hRTL(m,k), respectively.

Madadi and Tata (2011) showed that the SI contained in the last upper record is

where For lower record values, we have where ϕfL(i,k) is obtained from ϕfU(i,k) by replacing F by 1 −F.

Madadi and Tata (2014) also proved that the SI contained in the data consisting of all upper k-record values and k-record times of a random sample of size n (n ≥ k) is given by

and where γ^* = 0.57721566 is the Euler constant and ψ is the digamma function where The HRTL(n,k) and HRTL(n,1) contained in corresponding lower k-record statistics are obtained by replacing ϕfU(i,k) and ϕfU(i,1) by ϕfL(i,k) and ϕfL(i,1) in (2.8) and (2.9) , respectively.

They also showed that the SI contained in all of k-record values, last upper k-record and all of upper k-record values and times of an ISP are respectively obtained as

and for k > 1, where Here ζ(i,k)=∑j=k∞1ji is the incomplete Zeta function.

Note that for k = 1, i.e. for ordinary records,

Similar formulas hold for lower k-records.

3.1. Shannon information in a finite sample

Since the number of observations n is fixed (and the number of records is random), the last upper record is the maximum in the sample. From (2.5) and (2.6) , we have

and Therefore the SI contained in the last record for the Pareto (IV) are and When α is integer, and Let and denote the n-th differential of the entropy, that is, the change in the entropy when a new observation occurs. From (3.1) and (3.2) , we obtain and Fig. 1. represent behaviours of HMU(n,Z1) and HML(n,Z1) with respect to n.

Fig. 1.

(a) and (b) represent HMU(n,Z1) and HML(n,Z1), respectively.

One can easily see that

1. (i)

   limn→∞ΔHMU(n,Zα)=limn→∞ΔHML(n,Zα)=0.

2. (ii)

   ΔHMU(n,Zα) and ΔHML(n,Zα) are respectively always nonnegative and negative functions i.e. HMU(n,Zα) and HML(n,Zα) are increasing and decreasing functions with respect to sample size for each α. Also, ΔHMU(n,Zα) and ΔHML(n,Zα) are decreasing and increasing functions of n for each α.

3. (iii)

   0≤ΔHMU(n,Zα)<−ln2+1+12α and −ln2−12α<ΔHML(n,Zα)≤0.

4. (iv)

   For each fixed n, ΔHMU(n,Zα) and ΔHML(n,Zα) are decreasing convex and increasing concave functions of α, respectively.

5. (v)

   HMU(n,Zα) and HML(n,Zα) are decreasing convex functions of α for each n.

For ordinary records, these information can be written as

and When α is integer, we have For n = k, k + 1,···, let and denote the n-th SI differential corresponding to entropies in equations (3.3) and (3.4), then it is easy to see that and Fig. 2. displays HRTU(n,k,Z1) and HRTU(n,1,Z1) for several values of k and n. Behaviours of HRTL(n,k,Z1) and HRTL(n,1,Z1) are shown in Fig. 3. Our numerical computations suggest that when k ≥ 2 is fixed, the differential entropy is negative and decreasing in n. Also, for k = 1, we have and The following properties are easy to check and confirm via Figs. 2. and 3.

1. (i)

   limn→∞ΔHRTU(n,1,Z1)=limn→∞ΔHRTL(n,1,Z1)=0.

2. (ii)

   ΔHRTU(n,1,Z1) and ΔHRTL(n,1,Z1) are non-negative decreasing functions of n.

3. (iii)

   0≤ΔHKTL(n,1,Z1)<34.

We note that ψ(n+1)=ψ(n)+1n and limn→∞ψ(n)n=0.

Fig. 2.

(a) and (b) represent HRTU(n,k,Z1) and HRTU(n,1,Z1), respectively.

Fig. 3.

(a) and (b) represent HRTL(n,k,Z1) and HRTL(n,1,Z1), respectively.

3.2. Shannon information in an inverse sampling plan

In an inverse sampling plan, we take observations until a fixed number m of k-records is reached. Here, we compute the SI of k-records for an inverse sampling plan from Pareto-type distributions and discuss their properties.

Let

and denote the change in entropy in the last upper and last lower k-record values, respectively when the new k-record is observed. Then, from (3.7) and (3.8) , we obtain and It is easy to see that

1. (i)

   ΔhMU(m,k,Z1) is a positive function of m for each fixed k. Hence the sequence of functions {hMU(m,k,Z1)} is increasing with respect to m. Also, ΔhML(m,k,Z1) is a negative function of m, for each fixed k. Therefore {hML(m,k,Z1)} is a sequence of decreasing functions of m.

2. (ii)

   for each fixed k, ΔhMU(m,k,Z1) and ΔhML(m,k,Z1) are decreasing and increasing functions of m, respectively.

3. (iii)

   limm→∞ΔhMU(m,k,Z1)=1k and limm→∞ΔhML(m,k,Z1)=−1k.

4. (iv)

   12m+1k<ΔhMU(m,k,Z1)<1m+1k and −1k−2kζ(2,k+1)+γ*<ΔhML(m,k,Z1)<−1k, for each m ≥ 2.

We notice that 12m<lnm−ψ(m)<1m, for m > 0, ψ(1) = −γ^* and lim_m→∞ k^mζ (m + 1, k + 1) = 0. For k = 1, we obtain and It is easy to prove that

1. (i)

   limm→∞ΔhMU(m,1,Z1)=1 and limm→∞ΔhML(m,1,Z1)=−1.

2. (ii)

   12m+1<ΔhMU(m,1,Z1)<1m+1, for each m ≥ 2 and ΔhML(m,1,Z1)>12m−π23+1.

3. (iii)

   ΔhMU(m,1,Z1) is a positive decreasing function of m, therefore hMU(m,1,Z1) is an increasing function of m.

Fig. 4. displays a graphical representation of hMU(m,k,Z1) and hML(m,k,Z1) when k = 1, 2, 3, 5, 10.

Let

and denote the SI difference of (k + 1)-record and k-record in the m-th upper and m-th lower cases, respectively. Then, from (3.7) and (3.8) , it is easy to check and We note that ζ(j,k+1)=1(k+1)j+ζ(j,k+2). Hence

1. (i)

   Δ*hMU(m,k,Z1) is a negative increasing function of k, for each fixed m. Hence, hMU(m,k,Z1) is a monotone decreasing function of k. Similarly, Δ*hML(m,k,Z1) is a positive decreasing function of k and therefore hML(m,k,Z1) is a monotone increasing function of k for each m.

2. (ii)

   limk→∞Δ*hMU(m,k,Z1)=limk→∞Δ*hML(m,k,Z1)=0.

3. (iii)

   −ln2−m2<Δ*hMU(m,k,Z1)<0 and 0<Δ*hML(m,k,Z1)<m2+2−ln2−(12)m−1+∑j=2m(2−2j)  ζ(j,3).

Fig. 4.

(a) and (b) represent hMU(m,k,Z1) and hML(m,k,Z1), respectively.

Fig. 5.

(a) and (b) represent hRU(m,k,Z1) and hRL(m,k,Z1), respectively.

Fig. 6.

(a) and (b) represent hRTU(m,k,Z1) and hRTU(m,1,Z1), respectively.

Fig. 7.

(a) and (b) represent hRTL(m,k,Z1) and hRTL(m,1,Z1), respectively.