Ageing Properties of Curtate Future Life Time

In insurance sector, for determining the premium of an insured aged x, the insurer is interested not only in the complete future lifetime T (x), but also in the individual’s curtate future lifetime K(x) =[T (x)]. In this paper, we derive expressions for the reliability measures of K(x) and explore some of its ageing properties.


Introduction
For a new-born child, the age at death is a continuous random variable X with cumulative distribution function (cdf) F (x) and survival function F (x) = P (X > x) for x > 0. For a non-living object, X represents the age at failure.In actuarial science, F(x) is the survival probability that a person/object survives for at least x years and is denoted by Px .For X with support on {1,2,. . .}, Px = P (X ≥ x) and for support {0,1,2,. . .}, Px = P (X > x).
The notation (x) is used to denote a life aged x and T (x) denotes the complete future lifetime of (x).According to International Actuarial Notations given in 1949, t q x is the cdf of T (x) at t and gives the probability that (x) dies within t years.The survival function of T (x) is The probability that (x) will die between ages x + t and x + t + u is written as t|u q x = t p x − t+u p x = t p x − t p x u p x+t = t p x u q x+t = Px+t − Px+t+u Px .
For u = 1, t|1 q x = t| q x and 0| q x = q x .The curtate future lifetime of (x) is the discrete number of future years completed by (x) prior to death and is written as K(x) = [T (x)] : the greatest integer less than or equal to T (x).K(x) is a discrete random variable whereas T (x) is a continuous random variable.One can refer to Neill (1977), Gerber (1990), Bowers et al. (1997), Slud (2001), Borowiak (2003), Zhu (2007) and Dickson et al. (2009) for the above notations and definitions.
In lifetime analysis, an important aspect is to find a lifetime distribution that can adequately describe the ageing behaviour of the concerned life.Lifetimes are continuous in nature and hence many continuous life distributions have been proposed in the literature.On the other hand, discrete failure data arise in several common situations.For example, reports on insured's deaths are collected half yearly or annually and the observations are the number of deaths without specification of the occurrence of events.Sometimes, it is essential to count the complete number of years for which a patient has survived after going through a severe operation.The curtate future lifetime plays a significant role in assurance contracts and discrete life annuities when the benefit is payable at the end of year of death of the claimant/insured.As compared to continuous failure data, interest in discrete analogue arose relatively late.It was only briefly mentioned by Barlow and Proschan (1981).For earlier works on discrete lifetime distributions, one can see Salvia and Bollinger (1982), Xekalaki (1983), Padgett and Spurrier (1985) and Ebrahimi (1986) 1. the discrete failure rate (FR) is the amount of risk associated with an item at time l and is defined as 2. the discrete reversed failure rate (RFR) is defined as: 3. the discrete failure rate average (FRA) is 4. the discrete mean residual life (MRL) is defined as Some of the discrete ageing properties of Y given by Kelfsjo (1982) and Johnson, Kemp and Kotz (2005) are reproduced below: 1. Discrete IFR (DFR) : A discrete distribution with infinite support has a monotonically increasing (decreasing) failure rate with time according as where 2. Discrete IFRA (DFRA): A discrete lifetime distribution has an increasing or decreasing failure rate average according as or
5. Discrete NBUE (NWUE): Y has a distribution which is new better (worse) than used in expectation according as: 6. Discrete HNBUE (HNBWE): Y is harmonically new better (worse) than used in expectation if where µ = ∑ ∞ j=0 P[Y > j] = ∑ ∞ j=0 Pj .The paper is organised as follows: In Section 2, the expressions for reliability measures of K(x) are derived .The conditions for holding of ageing properties of K(x) have been presented in Section 3. Section 4 investigates the ageing behaviour of K(x) under the assumption that X follows Exponential, Weibull, Pareto or Burr distribution.

2.Reliability Measures of K(x)
In this section, we find the expressions for different reliability measures of K(x).The probability function of K(x) is given by P [K(x) = k] = k p x q x+k = k| q x ( Ref. Bowers et al. (1997)).
1.The failure rate (FR) function of K(x) is given by 2. The reversed failure rate (RFR) of K(x) can be written as

3.
The failure rate average (FRA) of K(x) is given by The ageing properties of K(x) are explored in the next section.

3.Ageing properties of K(x)
The following theorem specifies the conditions required for possessing of various ageing properties by K(x).
Theorem 1: The distribution of curtate future life time Proof: See the appendix In the sequel, the distributions of K(x) will be designated as Curtate Distributions.
Remark 1.For exploring the ageing properties NBU (NWU), NBUE (NWUE) and HNBUE (HNWUE) of curtate distributions, we consider the following differences The conclusions are based on positive or negative signs of the above differences.This is proved either mathematically or through figures in case the mathematical form is not tractable.
In the next section, we explore the ageing properties of K(x) when X, the new born's age at death follows some pre-assumed distribution.

Ageing Properties
The ageing properties of K(x) are investigated when X, the new born's age at death follows Exponential, Weibull, Pareto or Burr distribution.

Curtate Exponential Distribution
Hence the probability mass function (pmf) of K(x) is

Curtate Weibull Distribution
The pmf of K(x) is given by From ( 2), ( 3), ( 4) and ( 5) FR, RFR, FRA and MRL for Curtate Weibull distribution are derived as : and for x = 20, 30, 40, 50.The tables are based on hypothetical data which can arise in real life situations.On the basis of Table 1 − 3, it can be concluded that • for each λ > 0, and (i) 0 < α < 1, FR decreases in k as well as x.
(ii) α > 1, FR increases in k as well as x.
Remark (2) : Exponential Distribution is a special case of Weibull distribution for α = 1, and in that case both the distributions exhibit the same ageing behaviour.This gives k p Using ( 2) -( 5), FR, RFR, FRA and MRL for Curtate Pareto distribution are derived as: when θ is an integer and ψ(n, x), the digamma function denotes the n th derivative of gamma function.
Proof See the appendix.
For concluding about HNWUE, we consider • the values of FR decrease as initial age x and k increase for all θ > 0.
• the RFR is a decreasing function of x and k.
• FRA values show a decreasing trend as the values of x and k increase for all values of θ .
• MRL is an increasing function of initial age x and k for all θ .
For Curtate Burr distribution, we conclude about the ageing properties by looking at the plots of different reliability measures as shown in Figures 6−9     On the basis of the above discussion , it is concluded that the considerded ageing properties of the distribution of X are preserved by the curtate future lifetime distributions.

Conclusions
In this paper, we have studied the ageing properties of the curtate future lifetime distributions.It is shown that the ageing properties viz IFR (DFR), IFRA (DFRA), DMRL(IMRL), NBU (NWU), NBUE (NWUE) and HNBUE (HNWUE) are preserved by the curtate future lifetime distributions, when the new born's age at death follows Exponential, Weibull, Pareto and Burr Distributions.
This means that Since k p x = Px+k Px , the above inequality can be expressed as It means that the sum of probabilities of survival upto (x + k) th year is greater or less than k + 1 times the survival probability of life aged (x + k + 1) during the next one year.
(iii) Decreasing (Increasing) mean residual life DMRL (IMRL is non − decreasing or non − increasing in j > 0. (iv) The distribution of K(x) will be new better (worse) than used (NBU (NWU)) if the following condition is satisfied Px+k Px which happens if j+k p x < (>) j p x k p x for j, k = 0, 1, 2, 3, . . .This means that K(x) is NBU (NWU) if the probability that (x) survives for ( j + k) years is less (greater) than the product of survival probabilities of (x) till j and k years.
(v) The distribution of K(x) will have a new better (worse) than used in expectation (NBUE ( NWUE)) if the following condition is satisfied (vi) The distribution of K(x) will be Harmonically new better (worse) than used in expectation (HNBUE (HNWUE)) if Proof of Theorem 2 Using (2) -( 5) the FR, RFR, FRA and MRL for Curtate Exponential distribution are respectively.
(i), (iii) and (iv) follow since FR, FRA and MRL are independent of k.
(ii) follows since hence Curtate Exponential is both NBU and NWU.
(vi) The curtate exponential distribution is both NBUE and NWUE.Since (vii) The ageing properties HNBUE and HNWUE follow since Proof of Theorem 3: This gives Hence f (k) is decreasing for α > 1 and increasing for α < 1.
(ii) Consider For α < 1 and λ > 0, dl(k) dk < 0 and This gives the required conclusion. (iii , being the sum of (k+1) decreasing (increasing) functions, is a (an) decreasing (increasing) function.For α < 1, FRA is the product of two positive decreasing functions.So K (x) has DFRA.For α > 1, it is difficult to conclude as FRA is the product of a decreasing and an increasing function.By considering few parametric combinations for α > 1 and λ > 0, the values of FRA are given evaluated in Table 3.It is obvious from values in Table 3 that FRA is decreasing (increasing) for α < (>) 1 and all λ > 0.

Proof of Theorem 4
Let hence FR is a decreasing function of k for k ≥ 0.
Since each f ( j) is increasing for j=0,1,. . .k, hence ∑ k j=0 (1 − f ( j)), being the sum of k decreasing functions will be decreasing.Since 1 k + 1 is decreasing in k hence FRA being the product of two decreasing functions is decreasing in k.
(iv) Since practically k can take values only upto 100 years, hence we write This gives that (x + k) θ (x + k + 1) θ is an increasing function of k.
It can be shown similarly that, (x + k) θ (x + k + 2) θ is an increasing function of k.
Hence for θ , (x + k) θ (x + j) θ for j = k + 1, k + 2, . . . 100 are increasing functions of k.This implies that MRL, being the sum of a finite number of increasing (non-decreasing) functions of k, is increasing for all θ .
(v) K(x) will be NWU if and j, k ≥ 0.
(vi) Similarly consider the difference Hence D 2 > 0. This implies that K(x) is NWUE for all θ .Tables 1 − 9 displaying values of FR, RFR, FRA and MRL for different curtate distribuitons are presented now.

Table 3 :
Failure Rate Average of curtate Weibull Distribution

Table 7 :
Mean Residual life of curtate Pareto Distribution