A New Method for Generalizing Burr and Related Distributions

Definition 2.

A continuous random variable X follows generalized Burr (Burr) family of distributions if and only if it has the following CDF:

and F(x) = 0 if x < 0, where the real valued continuous, positive function g: (0, ∞) → ℝ⁺ is differentiable on (0, ∞). The shape parameter α of the Burr distribution is replaced with g(z) = g(x/λ; α, β) = g(x/λ), say, where β is an additional shape parameter and g(z) satisfies the following conditions:

1. The function g(z) is strictly positive and have finite limit at infinity, viz.

   (2.2)limz→∞ g(z)=α(>0).

2. limz→0+ 1+zcg(z)=1 and limz→∞ 1+zcg(z)=∞.

3. g′(z)g(z)≥−czc−11+zclog(1+z), z > 0, where g′(z)=ddz [g(z)].

It is very easy to verify that:

1. F_GBurr(x) is non-decreasing since,

   cxλc−1gxλ+g′xλ 1+xλc log 1+xλ>0,  x>0.

2. limx→ −∞FGBurr(x)=FGBurr(−∞)=0, limx→∞FGBurr(x)=1.

3. F_GBurr(x) is right continuous.

Thus, (2.1) is a standard CDF and it can also be expressed as follows:

The corresponding survival function is given by

The probability density function is

The hazard rate function is given by

The CDF (2.1) of the GBurr family of distributions is a function with regularly varying tails and belongs to the maximum domain of attraction (MDA) of the Frechet distribution with index α > 0, viz. F ∈ MDA(Φ_α), with Φ_α = exp{−x^−α}, x > 0, α > 0 [15].

Theorem 2.1.

GBurr family of distributions are:

1. heavy-tailed,

2. right tail-equivalent to a Pareto distribution, a

3. belongs to the MDA of the Frechet distribution.

Proof.

1. For the GBurr family of distributions, we note that

   limx→∞ exp{kx} (1−F(x))=limx→∞ exp kx−gxλ log 1+xcλc=∞,

   where k, λ, c > 0.

2. To show the tail-equivalent property, we show

   limx→∞ 1−F(x)1−G(x)=limx→∞exp −gxλlog1+xcλcexp −αlog1+xλ=1,   for all λ, c>0,

   where G(x) is the CDF of the Pareto Type-II distribution.

3. It should be noted that any function g as defined in Definition 2 satisfying limz→∞g(z)=α>0, is slowly varying at infinity:

   limz→∞ g(tz)g(z)=1,   for all t>0.

Now,

□

Definition 3.

A continuous random variable X follows GCP family of distributions if and only if it has the following CDF:

and F(x) = 0 if x ≤ γ, γ (> 0) is a scale parameter, m: (1, ∞) → ℝ⁺ is a real, continuous, positive function which is differentiable on (1, ∞), and the function m satisfies the following conditions:

1. m is strictly positive and have finite limit at infinity, viz.

   (2.8)limz→∞ m(z)=α(>0).

2. limz→1+ zm(z)=1 and limz→∞ zm(z)=∞.

3. m′(z)m(z)>−1zlog(z),z>1 .

Definition 4.

Any continuous random variable X with distribution function F(x) follows generalized exponentiated distribution if and only if it has the following CDF

where m: (−∞, ∞) → (0, ∞) is a real, continuous function which is differentiable on (0, ∞). The function m satisfies the following conditions:

1. m is strictly positive and have finite limit at infinity, viz.

   limx→ ∞ m(x)=α(>0).

2. limx→− ∞m(x)=β(>α) .

3. m′(x) < 0.

It is important to note that when m(x) = α, then g(x) corresponds to the exponentiated family of distributions [18, 32]. We give some examples of m(x) to be used for the distributions with X ∈ ℝ.

Example 1.

Let m(x) be a function of the following form satisfying limx→ − ∞ ξ(x)=2 and limx→∞ξ(x)=0:

Some choices of ξ(x) are as follows:

where k is any natural numbers.

Example 2.

Let m(x) be a function of the following form satisfying limx→ − ∞σ(x)=0 and limx→ − ∞σ(x)=1:

Some choices of σ(x) are as follows:

• σ(x)=11+e−xx∈ℝ.

• σ(x)=12+1πarctan(x), x∈ℝ.

• σ(x)=12π∫−∞xexp−y2/2dy, x ∈ ℝ.

It is interesting to see that

is a standard CDF, where Φ(x) is a standard normal CDF. For all the above examples when α = β, we get G(x) = [F(x)]^α.

Theorem 3.1.

The hazard rate function of the NBurr distribution satisfies the following properties:

1. If 0 ≤ c ≤ 1 and β > 0 then for all x > 0, h_NBurr(x) is decreasing in x;

2. If c > 1 and β > 0, then for all x > 0,

   1. h_NBurr(x) is increasing in x, whenc−1>xλc;

   2. h_NBurr(x) is decreasing in x, whenc−1<xλc;

   3. h_NBurr(x) is maximum at x = λ[c − 1]^1/c.

Proof.

Differentiating (3.5) with respect to x, we have

Now, α≥βλx+λ, cλcxc−2xc+λc2≥0 and if β > 0, 2βλ(x+λ)3xc+λc>0.

Thus, hNBurr′(x)≥ (≤)0 if

1. A(x)=cxc−1x+λxc+λc−log1+xcλc≥(≤)0;

2. c−1−xcλc≥(≤)0.

Now,

We can see that A′(x) ≤ 0 if 0 ≤ c ≤ 1. Therefore, A(x) is decreasing in X and again A(0) = 0. Thus, we have A(x) ≤ 0.

Similarly, we can prove that if c > 1, then A(x) ≥ 0 when c−1>xλc and A(x) ≤ 0 when c−1<xλc. This implies that for 0 ≤ c ≤ 1, h_NBurr(x) is decreasing in x; for c > 1, h_NBurr(x) is increasing in x, when c−1>xλc and h_NBurr(x) is decreasing in x, when c−1<xλc. Thus, the hazard function h_NBurr(x) attain its maximum at X = λ [c − 1]^1/c. □

The following example shows that the NBurr distribution does not preserves the likelihood ratio ordering. It is useful to recall that a random variable X is said to be larger than another random variable Y in likelihood ratio ordering (written as X ≥_LRY) if f_X(x)/f_Y(x) is an increasing function for X > 0.

Example 3.

Let X and Y be two random variables following NBurr distributions with parameters α₁, β₁, λ₁, c₁and α₂, β₂, λ₂, c₂respectively. Then, for all x > 0, the ratio of the corresponding density functions of X and Y is given by

Now, for λ₁ = λ₂ = 1,

Case I, when c₁ = c₂ = 1, (3.6) reduces to

We can see that for α₁ = 1.5, α₂ = 2, β₁ = 0.5 and β₂ = 1.5, P₁(0.1) = 1.366, P₁(2) = 0.889 and P₁(6) = 1.426, which implies that fXNBurr(x)fYNBurr(x) is not monotone. Thus X≥LRY.

Case II, when c₁ ≠ c₂, (3.6) reduces to

Again, we can see that for c₁ = 2, c₂ = 4, α₁ = 1.5, α₂ = 2, β₁ = 0.5 and β₂ = 1.5, P₂(0.5) = 1.5753, P₂(1) = 0.4843 and P₂(2) = 2.8757, which implies that fXNBurr(x)fYNBurr(x) is not monotone. Thus X≥LRY. Hence, it can be concluded that the proposed NBurr distribution does not preserves the likelihood ratio ordering. □

In the next theorem, we show that the NBurr distribution preserves the usual stochastic ordering. It is useful to remind that a random variable X is said to be larger than another random variable Y in usual stochastic ordering (written as X ≥_STY) if S_X(x) ≥ S_Y(x), for all x > 0.

Theorem 3.2.

Let X and Y be two random variables following NBurr distribution with parameters α₁, β₁, λ₁, c₁and α₂, β₂, λ₂, c₂, respectively. Then, X ≥_ST (≤_ST)Y provided

1. λ₁ ≥ (≤)λ₂,

2. c₁ ≤ (≥)c₂,

3. α₁ ≤ (≥)α₂, and

4. β₁ ≤ (≥)β₂for β > 0.

Proof.

X ≥_ST (≤_ST)Y if and only if for all x > 0, S_XNBurr(x) ≥ (≤)S_YNBurr(x), which is equivalent to

This holds if (i) λ₁ ≥ (≤)λ₂, (ii) c₁ ≤ (≥)c₂, (iii) α₁ ≤ (≥)α₂and (iv) β₁ ≤ (≥)β₂, for β > 0. □

The following theorem gives the condition under which the NBurr distribution preserves the hazard rate ordering. It is well known that a random variable X is said to be larger than another random variable Y in hazard rate ordering (written as X ≥_HRY) if, for all x > 0, h_X(x) ≤ h_Y(x).

Theorem 3.3.

Let X and Y be two random variables following NBurr distribution with parameters α₁, β₁, λ₁, c₁and α₂, β₂, λ₂, c₂, respectively.Then, X ≥_HR (≤_HR)Y provided

1. λ₁ ≥ (≤)λ₂,

2. c₁ ≤ (≥)c₂,

3. α₁ ≤ (≥)α₂, and

4. β₁ ≤ (≥)β₂for β > 0.

Proof.

X ≥_HR (≤_HR)Y if and only if for all x > 0, h_XNBurr(x) ≤ (≥)h_YNBurr(x), which is equivalent to

This holds if (i) λ₁ ≥ (≤)λ₂, (ii) c₁ ≤ (≥)c₂, (iii) α₁ ≤ (≥)α₂ and (iv) β₁ ≤ (≥)β₂, for β > 0. □

Remark 1.

It is interesting to note that the hazard rate function of the NBurr distribution can be both increasing and decreasing under certain conditions as given in Theorem 3.1. Also, NBurr distribution preserves stochastic and hazard rate orderings as shown in Theorems 3.2 and 3.3, respectively.

The Shannon entropy is an important and well-known concept in information theory as well as engineering sciences. Let X be a random variable that follows NBurr distribution with parameters α, β, λ, c. Then Shannon’s entropy for the NBurr distribution is defined as

When we want to study the system that survived up to an age t, then Shannon’s entropy function is not useful in measuring the uncertainty about the residual lifetime of the system. Ebrahimi [3] has introduced residual entropy and defined as

Then the residual entropy for NBurr distribution with parameters α, β, λ, c is given by