Testing NBU(2) class of life distribution based on goodness of fit approach

1 Introduction

Many applications in reliability theory and biostatistics involve the modeling of lifetime data. In these applications the outcome of interest is the time T, until some event occurs. This event may be death, the appearance of a tumor, the development of some disease, recurrence of a disease, conception, cessation of smoking, and so forth. The basic quantity employed to describe time-to-event phenomena is the survival function. This function, also known as the survivor function or survivorship function, is the probability an individual survives beyond time t and it is defined as $\overline{F}(t)={\int }_t^{\infty }f(u)\mathit{du}$ .

An important characteristic of survival distribution is its ageing properties. There are a number of classes that have been suggested in the literature to categorize distributions. The first aging class is the class of new better than used (NBU) distribution. Here a life distribution has the NBU property if, and only if, $$\overline{F}(x+t)⩽\overline{F}(x)\overline{F}(t)\text{,}\text{for all}t>0\text{.}$$

The NBU class of life distributions has been widely studied in the theory of reliability and life-testing since it was introduced earlier (Barlow and Proschan, 1981). The class, which emphasizes that a new item has a stochastically larger life length than does a used one at age t > 0, arises in many situations in which a better maintenance policy is believed to exist. Because it is easy to make a valuable judgment when such a model really exists, and because incorporating this model into an inferential procedure increases its statistical efficiency, it is desirable to recognize the existence of the model and to select a corresponding maintenance policy.

A second aging class is the class of new better than used in expectation (NBUE) distribution. A non-negative random variable T is said to have NBUE distribution if, and only if, $${\int }_x^{\infty }\overline{F}(u)\mathit{du}⩽\mu \overline{F}(x)\text{,}\text{for all}x>0\text{,}$$ where μ = E(T), assumed finite.

A third aging class is the class of new better than used in the increasing concave order (NBU(2)) distributions. A non-negative random variable T is said to have NBU(2) if

Inequality (1.1) has definite physical meaning: the life length of a new item is stochastically larger than that of a used one at age t > 0 in terms of increasing concave ordering. Probabilistic properties of the above three classes of aging distributions as well as many others have been extensively studied in the literature see e.g., Bryson and Siddique (1969), Marshall and Proschan (1972), Hollander and Proschan (1975), Deshpande et al. (1986), Hendi et al. (1999), Franco et al. (2001), Ahmad (2001), Hu and Xie (2002) and Lia and Xie (2006), among others. The relations between the above classes as easily seen are as follows:

In the context of reliability and life testing, its well-known that no ageing property corresponds to the exponential distribution. Hence testing non-parametric classes is done by testing exponentiality versus some kind of ageing classes. In this paper, a new testing procedure for exponentiality against the NBU(2) class is addressed based on the goodness of fit approach showing that it is simpler than most earlier ones and holds high relative efficiency for some commonly used alternatives. In Section 2, we present a procedure to test that T is exponential against that it is NBU(2) and not exponential for non-censored data. The Pitman asymptotic relative efficiency with respect to Hendi et al. (1999) and Hollander and Proschan (1975) are discussed. Also, the power of the test for some commonly used distributions in reliability and the critical values of the proposed test are calculated. In Section 3, we deal with the right censored problem. As a consequence the critical percentiles of the proposed test are calculated and tabulated for sample size n = 5(5)80, 81, 85 and 86 based on 10,000 replications. Finally, in Section 4, some applications are presented for non-censored and censored data to illustrate the theoretical results.

2 Testing against NBU(2) class for non-censored data

In this section a test statistic based on the goodness of fit approach is developed to test H₀:F is fully specified exponential against H₁:F is NBU(2) and not exponential. Since under H₀, F is completely known, we take it's mean to be one.

3 Testing against NBU(2) class for censored data

In this section, a test statistic is proposed to test H₀ versus H₁ with randomly right-censored data. Such a censored data is usually the only information available in a life-testing model or in a clinical study where patients may be lost (censored) before the completion of a study. This experimental situation can formally be modeled as follows.

Suppose n objects are put on test, and X₁, X₂, … , X_n denote their true life time. We assume that X₁, X₂, … , X_n be independent, identically distributed (i.i.d.) according to a continuous life distribution F. Let Y₁, Y₂, … , Y_n be (i.i.d.) according to a continuous life distribution G and assume that X's and Y's are independent. In the randomly right-censored model, we observe the pairs (Z_j, δ_j), j = 1, … , n, where Z_j = min(X_j, Y_j) and $${\delta }_j=\left\{\begin{array}{ccc}1\text{,}\hfill & \text{if}\hfill & Z_j=X_j\left(j\text{th}\text{observation is uncensored}\right)\hfill \\ 0\text{,}\hfill & \text{if}\hfill & Z_j=Y_j\left(j\text{th}\text{observation is censored}\right).\hfill \end{array}\right.$$

Let Z(0) = 0 < Z(1) < Z(2) < ⋯ < Z(n) denote the ordered Z's and δ_(j) is the δ_j the corresponding to Z_(j). Using the censored data (Z_j, δ_j), j = 1, … , n. Kaplan and Meier (1958) proposed the product limit estimator, $$\begin{array}{cc}\hfill & {\overline{F}}_n\left(X\right)=1-F_n\left(X\right)\text{,}\hfill \\ \hfill & =\prod _{j:Z_{(j)}⩽1}{\left[\frac{n-j}{n-j+1}\right]}^{{\delta }_{(j)}}\text{,}X\in \left[0\text{,}{Z}_{(n)}\right]\text{.}\hfill \end{array}$$

Now for testing H₀:Δ_F = 0, against H₁:Δ_F > 0, using randomly right-censored data, we propose the following test statistic: $$\hat{\mathrm{\Delta }}={\int }_0^{\infty }{\int }_0^{\infty }{e}^{-x}{e}^{-t}v(x)\overline{F}(t)\mathit{dxdt}+{\int }_0^{\infty }{e}^{-x}v(x)\mathit{dx}-{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-x}{e}^{-t}v(x+t)\mathit{dxdt}$$ $${\hat{\mathrm{\Delta }}}_F^c=\left\{\sum _{i=1}^n\sum _{j=1}^n\sum _{k=1}^{j-1}{e}^{-z_j}{e}^{-z_i}\left(\prod _{m=1}^{i-1}{C}_m^{{\delta }_m}\prod _{m=1}^{j-1}{C}_m^{{\delta }_m}\right)\times \left(z_{(k)}-z_{(k-1)}\right)\left(z_{(j)}-z_{(j-1)}\right)\left(z_{(i)}-z_{(i-1)}\right)\right\}+\sum _{j=1}^n\sum _{k=1}^{j-1}{e}^{-z_j}\left(\prod _{m=1}^{k-1}{C}_m^{{\delta }_m}\right)\left(z_{(k)}-z_{(k-1)}\right)\left(z_{(j)}-z_{(j-1)}\right)-\left\{\sum _{i=1}^n\sum _{j=1}^n\sum _{k=1}^{\ell }{e}^{-z_j}{e}^{-z_i}\left(\prod _{m=1}^{k-1}{C}_m^{{\delta }_m}\right)\times \left(z_{(k)}-z_{(k-1)}\right)\left(z_{(j)}-z_{(j-1)}\right)\left(z_{(i)}-z_{(i-1)}\right)\right\}\text{,}$$ where C_k = n − m/n − m + 1, and $$\ell =\left\{\begin{array}{cc}\hfill \#Z’\text{s}⩽Z_{(i)}+Z_{(j)}\text{,}\hfill & \hfill \text{if}{Z}_{(i)}+Z_{(j)}<Z_{(n)}\hfill \\ \hfill 0\text{,}\hfill & \hfill \text{if}{Z}_{(i)}+Z_{(j)}⩾Z_{(n)}\hfill \end{array}\right.\text{.}$$

Table 3.1 gives the critical percentiles of ${\hat{\mathrm{\Delta }}}^c$ test for sample sizes n = 5(5)80, 81, 85 and 86 based on 5000 replications.

From the above table, it is noted that the critical values of the test increases by increasing of the sample size n.

4 Some applications

In this section, we apply the test on some data-sets to elucidate the applications of the NBU(2) in the both non-censored and censored data at 95% confidence level.