Q2. What is the common form of quantile functions used in applied work?
Many of the quantile functions used in applied work like various forms of lambda distributions (Ramberg and Schmeiser, 1974; Freimer et al., 1988; van Staden and Loots, 2009; Gilchrist, 2000), the power-Pareto distribution (Gilchrist, 2000, Hankin and Lee, 2006), Govindarajulu distribution (Nair et al., 2011) etc. do not have tractable distributions.
Q3. What is the Shannon entropy function for past lifetime?
Using the QF defined in (5) and (6), the Shannon entropy in (1) can be written as ξ = ξ(X) = − 1 0 (log fQ (p))fQ (p)dQ (p),= 1 0 (log q(p))dp. (7)Clearly, by knowing eitherQ (u) or q(u), the expression for ξ(X) is quite simple to compute.
Q4. What is the proof of the following theorem?
1u log 1+u 1−u , holds for all u if and only if X follows a half-logistic distribution with Q (u) = σ log (1+u) (1−u) , σ > 0.Theorem 10.
Q5. What is the proof for the density function x?
For equilibrium distribution with density functionfE(t) = F(t)/µ, (16)we have 1qE (u) = (1−u) µ , or log qE(u) = logµ − log(1 − u), then using (8) the PQE is given by ψE(u) = 1 + logµ + log u + (1−u)u log(1 − u) = ψE(u)+ log u + (1−u)u log(1 − u).
Q6. What is the proof of the relationship between rv x and u?
Theorem 8. The rv X is distributed as power function with Q (u) = αu1/β;α, β > 0, holds for all u if and only if it satisfies the relationship ψ(u) = C − log A(u) where 0 < C < 1.Theorem 9. The relationshipψ(u) = 2− log A(u)−
Q7. What is the inverse of the Weibull?
The nonnegative function φ(x) = xα, x > 0, is concave if 0 < α < 1. Hence due to Theorem 1, the inverted Weibull is DPQE for 0 < α < 1.
Q8. What is the density quantile function for y?
Using fw(t), the corresponding density quantile function is given byfw(Q (u)) = w(Q (u))f (Q (u))/µ, where µ = 1 0 w(Q (p))f (Q (p))d(Q (p)) = 1 0 w(Q (p))dp.
Q9. What is the recent work on quantile functions?
it has been shown by many authors that quantile functionsQ (u) = F−1(u) = inf{t | F(t) ≥ u}, 0 ≤ u ≤ 1 (5)are efficient and equivalent alternatives to the distribution function inmodeling and analysis of statistical data (see Gilchrist, 2000; Nair and Sankaran, 2009).
Q10. What is the hazard rate of a unit?
Denoting a(x) = f (x)/F(x) the reversed hazard rate (see Block et al., 1998), Eq. (3) can be rewritten asξ(X; t) = 1 − 1F(t) t 0 log(a(x))f (x)dx. (4)Given that at time t , a unit is found to be down, ξ(X; t) measures the uncertainty about its past lifetime.
Q11. What is the relationship between (u) and (x)?
For the exponential distribution in the support of (−∞, 0) with F(t) = exp (λt) , λ > 0 (see Block et al., 1998) wehave Q (u) = 1 λ log u, q(u) = 1 λu and A(u) = λ so that ψ(u) = 1 + log(uq(u)) = 1 − log A(u) = 1 − log λ.