2.1 Introduction

As mentioned in the last chapter, the reliability of a device is the probability that the device performs its intended function for a given period of time under conditions specified for its operation. When the device does not perform its function satisfactorily, we say that it has failed. When the random variable X represents the lifetime of a device, the observation on X is realized as the time of failure. The primary concern in reliability theory is then to understand the pattern in which failures occur for different devices and under varying operating environments. This is often done by analyzing the observed failure times or ages at failure with the help of a model that satisfactorily represents the predominant features of the data. One direct method is to find a probability distribution that provides a reasonable fit to the observations. Sometimes, there may exist more than one distribution that can pass an appropriate goodness-of-fit test. In any case, it is more desirable to find a probability model that manifests certain physical properties of the failure mechanism. In reliability theory, some basic concepts that help in the study of failure patterns have been developed. The objective of this chapter is to define such concepts and discuss their properties and inter-relationships. Two important aspects that necessitate the study of these concepts are (a) various functions considered in this context determine the life distribution uniquely, so that the knowledge of their functional form is equivalent to that of the distribution itself, and (b) it should be easier to deal with these functions than the distribution function or probability density function of the corresponding distributions.

2.2 Hazard Rate Function

Let X be a discrete random variable assuming values in $N=(0,1,\dots )$ with probability mass function $f(x)$ and survival function $S(x)=P(X⩾x)$. We will think of X as the random lifetime of a device that can fail only at times (ages) in N. The hazard rate function of X is defined as

$h(x)=\frac{f(x)}{S(x)}$

at points x for which $S(x)>0$. Treated as a function of x, the hazard rate is also called failure rate, instantaneous death rate, force of mortality and intensity function in other disciplines such as survival analysis, actuarial science, demography, extreme value theory and bio-sciences. Although in the continuous case, the concept of hazard rate dates back to historical studies in human mortality, its discrete version came up much later in the works of Barlow and Proschan (1965), Cox (1972) and Kalbfleisch and Prentice (2002), to mention a few.

When X has a finite support $(0,1,\dots ,n)$, $n<\infty$, then $h(n)=1$. As a convention we take $h(x)=1$ for $x>n$. The hazard function $h(x)$ is interpreted as the conditional probability of the failure of the device at age x, given that it did not fail before age x. Thus, $0⩽h(x)⩽1$. The interpretation and boundedness of the discrete hazard rate is thus different from that of the continuous case.

We see from (2.1) that $h(x)$ is determined from $f(x)$ or $S(x)$. The converse that the hazard rate function determines the distribution of X uniquely is also true. To see this, we note that

$h(x)=\frac{S(x)-S(x+1)}{S(x)},$

or

$\frac{S(x+1)}{S(x)}=1-h(x).$

So,

$S(x)=\{\begin{array}{cc}{\prod }_{t=0}^{x-1}(1-h(t)),\hfill & x⩾1,\hfill \\ 1,\hfill & x=0.\hfill \end{array}$

Eq. (2.2) reveals also that $h(x)$ can be used as a tool to model the life distribution. When a functional form for $h(x)$ is assumed as model for a given data set, one has to ensure that the assumed form conforms to the hazard rate function of a distribution. The following theorem is useful in this regard.

If the probability mass function is required from (2.1) and (2.2), we see that

$f(x)=h(x)\prod _{t=0}^{x-1}(1-h(t)).$

The above results have appeared repeatedly in several papers; see, for example, Gupta (1979), Shaked et al. (1995) and Kemp (2004).

The sum of the hazard rates from 0 through $x-1$ is of interest in reliability theory and is called the cumulative hazard rate, defined by

$H(x)=\sum _{t=0}^{x-1}h(t).$

Also define $H(0)=0$. Graphically, the cumulative hazard rate represents the area under the step function representing $h(x)$. Definition (2.13) does not satisfy properties analogous to the continuous case in which the cumulative hazard rate satisfies the identity

$H(x)=-\log \overline{F}(x),\overline{F}(x)=P(X>x).$

Therefore, Cox and Oakes (1984) proposed an alternative definition of cumulative hazard rate in the form

$H_1(x)=-\log S(x).$

This means that

$H_1(x)=-\log \sum _{t⩾x}f(t)=-\log \prod _{t=0}^{x-1}(1-h(t)).$

If

$H_1(x)=\sum _{t=0}^{x-1}{h}_1(t),x=1,2,\dots$

then $H_1(x)$ is a cumulative hazard rate corresponding to an alternative hazard rate function defined by

$h_1(x)=\log \frac{S(x)}{S(x+1)},x=0,1,2,\dots .$

Xie et al. (2002a) advocated the use of (2.17) as the hazard rate function instead of (2.1) by citing the following arguments. In the continuous case, the hazard rate is not a probability, but (2.1) is a conditional probability which is bounded. Consequently, (2.1) cannot increase too fast either linearly or exponentially to provide models of lifetimes of components in the wear-out phase. When cumulative hazard rate is defined as the negative logarithm of the survival function, $-\log S(x)\ne {\sum }_{t=0}^{x-1}h(t)$. This causes problems in defining discrete ageing concepts that are analogues of their continuous counterparts, such as increasing hazard rate average (see Chapter 4). Thus, discrete ageing concepts based on $h(x)$ may not convey the same meaning as those in the continuous case. Similar problems persist with the construction of proportional hazards models and with series systems. Further details about these are provided in Sections 2.10 and 2.11. It may also be noted that unlike $h(x)$, the definition of $h_1(x)$ does not have any interpretation. Xie et al. (2002a) and Kemp (2004) have obtained the following interrelationships among the two hazard rate functions and the other reliability functions discussed so far:

$\begin{array}{cc}\hfill H_1(x)& =-\sum _{t=0}^{x-1}\log (1-H(t)+H(t-1)),\hfill \\ \hfill H(x)& =\sum _{t=0}^x(1-\exp [H_1(x)-H_1(x+1)]),\hfill \\ \hfill S(x)& =\exp [-H_1(x)]=\prod _{t=0}^{x-1}(1-H(t)+H(t-1)),\hfill \\ \hfill f(x)& =\exp [-H_1(x)]-\exp [-H_1(x+1)]\hfill \\ \hfill & =(H(x)-H(x-1))\times \prod _{t=0}^{x-1}(1-H(t)+H(t-1)),\hfill \\ \hfill h(x)& =1-\exp [H_1(x)-H_1(x+1)]=H(x)-H(x-1),\hfill \end{array}$

and

$h(x)=1-e^{-h_1(x)}.$

Thus, the function $h(x)(H(x))$ determines $h_1(x)(H_1(x))$ uniquely and hence is useful in characterizing life distributions. We give some examples that compare the expressions of $h(x)$ and $h_1(x)$.

2.3 Mean Residual Life

The analysis of the lifetime of a device after it has attained age x is of special relevance in reliability and survival analysis. Thus, if X is the original lifetime with survival function $S(x)=P(X⩾x)$, the corresponding residual lifetime after age x is the random variable $X_x=(X-x|X>x)$. From the definition of conditional probability, one can arrive at the distribution of $X_x$ as

$S_x(t)=\frac{S(x+t+1)}{S(x+1)},t=0,1,\dots .$

The mean, variance, partial moments, coefficient of variation and percentiles of the distribution in (2.19) have been discussed extensively in the literature in the continuous case.

The mean residual life of X is defined as

$\begin{array}{cc}\hfill m(x)& =E(X-x|X>x)\hfill \\ \hfill & =\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }(t-x)f(t)\hfill \\ \hfill & =\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }S(t),x=-1,0,1,\dots \hfill \end{array}$

It is easy to see that $m(x)$ is the mean of the distribution in (2.19). When $x=-1$,

$m(-1)=E(X+1)=\sum _{t=0}^{\infty }S(t)=1+\mu .$

Characterizations of the distribution of X in terms of $m(x)$ and the hazard rate have been studied by Nair and Hitha (1989). From (2.20),

$S(x+1)m(x)=\sum _{t=x+1}^{\infty }S(t)$

which leads to

$S(x)m(x-1)-S(x+1)m(x)=S(x)$

and to the identity

$h(x)=\frac{1+m(x)-m(x-1)}{m(x)},x=0,1,2,\dots .$

Now, from (2.2), we have

$S(x)=\{\begin{array}{cc}{\prod }_{t=0}^{x-1}\frac{m(t-1)-1}{m(t)},\hfill & x⩾1,\hfill \\ 1,\hfill & x=0.\hfill \end{array}$

Thus, the three functions $h(x)$, $m(x)$ and $S(x)$ determine each other uniquely. Though $h(x)$ can be determined from $m(x)$ and vice-versa, both have unique features that ensure their necessity in reliability theory. The mean residual life function may exist when the hazard function does not exist and vice-versa, as will be seen in some of the examples that are considered later. While $h(x)$ is a local measure of failure patterns for any x, the mean residual life function depends upon the life history of devices at all ages beyond x and hence the latter is more informative. At the same time, $m(x)$ as a summary measure is highly sensitive to a single long-term survivor in a data set, which is not desirable. There are other interesting properties that distinguish the two concepts as will be seen in the subsequent chapters.

Since $0⩽h(x)⩽1$, we can note from (2.21) that

$1+m(x)⩾m(x-1)$

and $m(x)⩾1$. For many discrete distributions, the expression for $m(x)$ is not of a simple form. In the following examples, we have simple forms for $m(x)$ that have many desirable properties. For more examples of the $m(x)$ function, see Table 2.4.

We can see that $m(x)$ is linear and decreasing in x and $h(x)m(x)=\frac{k}{k+1}<1$.

Nair and Hitha (1989) and Hitha and Nair (1989) have established the following characterizations of the above three models.

Identification of lifetime distributions that are uniquely determined by simple relationships between various reliability concepts has attracted several studies in the past and continues to be a fertile area of research. Many results pertaining to individual distributions and families of distributions are available in this context. Theorem 2.3 belongs to this category concerning three specific distributions and its applications will be discussed later in Chapter 4. A more general result in this connection is the following.

Let $k(x)$ and $c(x)$ be real-valued functions defined on N such that $E(k(X))={\mu }_k$ and $V(k(X))={\sigma }_k^2$. We define

$m_k^{⁎}(x)=E[k(X)|X>x].$

Further, assume that $E{c}^2(X)<\infty$ and $E[\mathrm{\Delta }c(X)g(X)]<\infty$, where Δ is the usual difference operator, $\mathrm{\Delta }a(x)=a(x+1)-a(x)$, and $g(x)$ is some real-valued function.

When $k(x)=x$, we write

$m^{⁎}(x)=E(X|X>x)$

and call it the vitality function, which is in fact a conditional mean life function. It satisfies

$m^{⁎}(-1)=\mu \text{ and }{m}^{⁎}(x)=m(x)+x.$

Eq. (2.25) becomes a relationship between the mean residual life function and hazard rate function of the form

$m(x)=x+\mu +\sigma \frac{h(x)g(x)}{1-h(x)}$

that characterizes the class of discrete distributions satisfying

$\frac{f(x+1)}{f(x)}=\frac{\sigma g(x)}{\sigma g(x+1)-\mu +x+1},$

where ${\sigma }^2=V(x)$. It is further observed in Nair and Sudheesh (2008) that

Specialization of the identity in (2.25) for various distributions and their implications will be considered later in Chapter 3 wherein we discuss discrete lifetime models. The variance inequality is also useful in the unbiased estimation of functions of X.

Though closely related to the mean residual life function, the vitality function introduced in (2.28) has some importance in its own right in lifetime studies. By definition,

$m^{⁎}(x)=\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }tf(t),x=-1,0,1,\dots .$

Some properties of $m^{⁎}(x)$ that are of interest in the sequel are given below. From the definition

$m^{⁎}(x-1)S(x)-m^{⁎}(x)S(x+1)=xf(x).$

Dividing by $S(x)$ and simplifying, we get

$m^{⁎}(x)-m^{⁎}(x-1)=h(x)[m^{⁎}(x)-x]=h(x)m(x).$

Further,

$m^{⁎}(x)>x\text{ for all }x\text{ in }N\text{.}$

Analogous to the presentation on vitality function given in Kupka and Loo (1989), we note that in the integer interval $[0,x]$, $m([0,x])=m^{⁎}(-1)-m^{⁎}(x-1)$ represents the increment in the conditional mean life which is achieved by surviving from age 0 to x. Thus, low (high) values of $m^{⁎}([0,x])$ means that the device is ageing rapidly (slowly) during $[0,x]$ and this justifies the name vitality function of the lifetime X. However, it may be noticed that vitality function is always non-decreasing, unlike the mean residual life function which can be either decreasing or increasing.

Returning to the reliability functions based on real-valued functions $k(X)$ of X, more characterizations and useful relationships can be seen. Let $F(x)$ be the distribution function of X such that $F(A)=1$ for some $A>0$.

Ruiz and Navarro (1995) considered discrete random variables taking values in $C=(x_a,x_{a+1},\dots ,x_b)$, $x_i<x_{i+1}$, and defined for a monotonic $k(x)$, the doubly truncated mean function

$\begin{array}{cc}\hfill M(x,y)& =E[k(X)|x⩽X⩽y]\hfill \\ \hfill & =\frac{1}{F(y)-F(x-)}\sum _{x⩽x_i⩽y}f(x_i)k(x_i)\hfill \end{array}$

for $(x,y)\in D=\{(s,t)|F(t-)<F(s)\}$, $F(x)=P(X<x)$ and $f(x_i)=F(x_i)-F(x_i-1)$. Then,

$F(x_i)=\frac{P_1(x_i,x_i)}{P_1(x_i,x_i)+P_2(x_i,x_i)-1},$

where

$P_1(x,y)=\prod _{x_i<x}\frac{M(x_{i+1},y)-k(x_i)}{M(x_i,y)-k(x_i)}$

and

$P_2(x,y)=\prod _{y<x_i}\frac{k(x_i)-M(x,x_{i-1})}{k(x_i)-M(x,x_i)}.$

They also obtained a necessary and sufficient condition for a given function in $(x,y)$ to be a doubly truncated mean function. This result is of relevance in reliability modelling since $M(x,y)$ tends to the mean residual life as $x\to \infty$ and to the reversed mean residual life (Section 2.7) as $x\to 0$ when $k(X)=X$ and the range of X is N.

The conditional moments

$m_r^{⁎}(x)=E(X^r|X>x)=\frac{1}{S(x+1)}\sum _{x+1}^{\infty }{t}^{r}f(t)$

satisfy

$h(x)=\frac{m_r^{⁎}(x-1)-m_r^{⁎}(x)}{x^r-m_r^{⁎}(x)}$

so that the distribution of X is determined from $m_r^{⁎}(x)$ by virtue of (2.2). Further, since the left hand side of (2.30) is independent of r,

$\frac{m_r^{⁎}(x-1)-m_r^{⁎}(x)}{x^r-m_r^{⁎}(x)}=\frac{m_{r-1}^{⁎}(x-1)-m_{r-1}^{⁎}(x)}{x^{r-1}-m_{r-1}^{⁎}(x)}$

is a recurrence relation connecting two consecutive moments.

Glanzel et al. (1984) considered $k(x)$ to be a monotonic function such that $k(x)\ne$ a constant in $(n,n+1,\dots )\cap N$ for any finite n. Then,

$f(x)=\{\begin{array}{cc}\left({\prod }_{t=0}^{x-1}\frac{m_k(t)-k(t)}{m_k(t+1)-k(t)}\right)\frac{m_k(x+1)-m_k(x)}{m_k(x+1)-k(x)},\hfill & \text{for }x\in N\text{ or }x<n\hfill \\ {\prod }_{t=0}^{n-1}\frac{m_k(t)-k(t)}{m_k(t+1)-k(t)},\hfill & \text{for }x=n\hfill \end{array}$

Further, if $k(x)$ and $g(x)$ are two functions defined on N such that $l(x)=\frac{m_0(x)}{m_k(x)}$ is strictly monotonic, then the distribution of X has probability mass function $f(x)$ satisfying

$A(x)=\frac{f(x)}{f(0)}\frac{l(x+1)-l(x)}{l(1)-l(0)}\prod _{t=1}^x\frac{l(t-1)k(t-1)-g(t-1)}{l(t+1)k(t)-g(t)}$

and $f(0)={(1+{\sum }_{x=1}^{\infty }A(x))}^{-1}$. For a further refinement of the results on conditional expectations, see Su and Huang (2000). For characterizations of the geometric, Waring and negative hyper-geometric laws by doubly truncated mean residual life, one may see to Khorashadizadeh et al. (2012).

The alternative hazard rate $h_1(x)$ can also be related to $m(x)$, as shown in Xie et al. (2002a) when $x=1,2,3,\dots$. From the definition of $m(x)$ in (2.19) and Eq. (2.2), we have

$\begin{array}{cc}\hfill m(x)& =\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }\prod _{u=0}^{t-1}(1-h(u))\hfill \\ \hfill & =\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }\exp [-\sum _{a=0}^{t-1}{h}_1(u)],\text{ using (}\text{2.18}\text{)}\hfill \\ \hfill & =\sum _{t=x+1}^{\infty }\exp [H_1(x+1)-H_1(t-1)],\text{since }S(x)=e^{-H_1(x)}\text{.}\hfill \end{array}$

Some relationships involving $h(x)$ and $m(x)$ needed in the subsequent discussions are

$E\left(\frac{1}{h(X)}\right)=1+\mu$

and

$E(\frac{1}{h(X)}|X>x)=\frac{1}{S(x+1)}\sum _{t=x+1}^{\infty }S(t)=m(x).$

The expression $h(X)$ is referred to as the hazard rate at random time. We now prove a characteristic property of the mean of $h(X)$.

2.3.1 Modelling Data

Although we have defined the hazard rate function and the mean residual life function in terms of the time to failure of a device, the definitions and properties can also be used to model data on the time to occurrence of an ‘event’ with appropriate modifications in interpretations that best suit the context. Accordingly, the results mentioned so far have been extensively applied in other disciplines as well. A detailed examination of this aspect will be taken up later in Chapter 9. In the present section, we consider some real data and illustrate the application of the characterizations established above in finding suitable models for them.

Example 2.11

In this example, we consider the famous Bortkiewicz data on the number of soldiers of the Prussian army who died of horse-kicks in a period of 20 consecutive years, with the modification of the data value suggested by Cohen (1960). The last observation in Cohen's data is omitted for the analysis; see Table 2.1 for details. From the data, the probability mass function of X, the time of death, is estimated as

$\begin{array}{c}\hfill \hat{f}(0)=\frac{129}{199}=0.6482,\hat{f}(1)=\frac{45}{199}=0.2261,\hfill \\ \hfill \hat{f}(2)=\frac{22}{199}=0.1106,\hat{f}(3)=\frac{3}{199}=0.0151.\hfill \end{array}$

Table 2.1

Model for the deaths of solders by horse-kicks

No. of deaths per year	Observed frequency	$\hat{m}(x)$	Expected frequency
0	129	1.4926	123.08
1	45	1.4002	53.79
2	22	1.1201	18.35
3	3	1.0000	3.78

These values give the estimate of the mean residual life as

$\hat{m}(x)=\frac{1}{\hat{S}(x+1)}\sum _{t=x+1}^3\hat{S}(t),x=-1,0,1,2,$

where $\hat{S}(x)={\sum }_{t⩾x}\hat{f}(t)$. The values of $\hat{m}(x)$ can be seen in column 2 of Table 2.1. The next step is to seek a functional form for $\hat{m}(x)$. Accordingly, a straight line fit is considered for $m(x)$ by the method of least-squares. This yields

$m(x)=1.34112-0.17579x.$

Being a linearly decreasing mean residual life, from Theorem 2.1, we propose the negative hyper-geometric distribution as a model for the data with $n=3$. Comparing with the mean residual life function of the negative hyper-geometric law

$m(x)=\frac{k+n-x}{k+1},$

we obtain the estimate of k as $\hat{k}=4.863$. Thus the proposed model has survival function

$\hat{S}(x)=\left(\begin{array}{c}\hat{k}+n-x\\ n-x\end{array}\right)/\left(\begin{array}{c}\hat{k}+n\\ n\end{array}\right)$

with $\hat{k}=4.863$ and $n=3$. The expected probability mass function is

$\hat{f}(x)=\hat{S}(x)-\hat{S}(x+1).$

Obviously, the chosen model rests on the assumption of linearity of $m(x)$. Hence the model is validated by calculating the expected frequencies and then verifying their closeness by the chi-square goodness-of-fit test. The chi-square value of 2.4949 does not reject the model at 5% level of significance.

Example 2.12

The results of the well-known Rutherford-Geiger experiment is produced in Table 2.2. It describes the number of α-particles remitted from radioactive substances in 2608 fractions of 75 seconds. As in the previous example, the estimates $\hat{f}(x)$ and $\hat{S}(x)$ of the probability mass function and the survival function and there from $\hat{m}(x)$ are found out. In addition, we also require the estimated hazard rate function $\hat{h}(x)=\hat{f}(x)/\hat{S}(x)$. The observed values of $\hat{h}(x)$ and $\hat{m}(x)$ are shown in Table 2.2.

Table 2.2

Estimated values of $\hat{h}(x)$, $\hat{m}(x)$ and $\hat{g}(x)$ for the data in Table 2.3

x	$\hat{h}(x)$	$\hat{m}(x)$	$\hat{g}(x)$
0	0.029	3.9569	2.00
1	0.0795	3.2124	2.06
2	0.1632	2.6437	2.07
3	0.2672	2.2431	1.96
4	0.3695	1.9716	1.87
5	0.4493	1.7642	1.85
6	0.5462	1.6839	1.65
7	0.6615	1.7604	1.61
8	0.5518	1.5576	2.81
9	0.6306	1.5082	2.01
10	0.6223	1.3478	2.23
11	0.6522	1.000	2.12
12	1.000

We also have

$m(-1)=\sum _{t=0}^{\infty }\hat{S}(t)=1+\mu =4.8702$

so that an estimate of the mean becomes

$\hat{\mu }=3.8702.$

Neither an examination of the values of $\hat{h}(x)$ and $\hat{m}(x)$ nor their plots in Figs 2.1 and 2.2 exhibit an obvious choice for the functional forms of $h(x)$ and $m(x)$. In such cases, Theorem 2.4 can be of some assistance in the determination of a suitable model. Since we have the empirical mean residual life from the data, choose $k(x)=x$ so that (2.24) becomes

$m^{⁎}(x)=m(x)+x=\mu +\frac{\sigma h(x)g(x)}{1-h(x)}.$

Hence, we can calculate $\hat{g}(x)$ from

$\hat{g}(x)=\left(\frac{\hat{m}(x)+x-\hat{\mu }}{\hat{\sigma }}\right)(\frac{1}{\hat{h}(x)}-1).$

(2.36)

In (2.36), the estimate of the standard deviation σ is taken as the sample standard deviation. The values of $\hat{g}(x)$ so arrived at are shown in Table 2.2. Notice that the value at $x=8$ is significantly larger than the rest. Barring this value, the rest appears to be clustering around the average $\overline{g}(x)=1.94$. This is the same as assuming $g(x)=c$, a constant for all x, since the least-square estimate of c is $\overline{g}(x)$. Recall from Theorem 2.4 that the value of $g(x)$ uniquely determines the distribution of X and further, $g(x)=c$ if and only if the distribution is Poisson with mean $c^2$. Thus, the data follows a Poisson distribution

$f(x)=e^{-\lambda }\frac{{\lambda }^x}{x!},x=0,1,2,\dots$

with $\lambda =c^2=3.8$. To validate the assumption made on $g(x)$, we compare the observed and expected frequencies, when $\lambda =3.8$, presented in Table 2.3.

Table 2.3

Goodness-of-fit for Rutherford-Geiger data

x	Observed frequency	Expected frequency	Contribution to ${\chi }^2$-value
0	57	58.4	0.0036
1	203	221.7	1.5773
2	383	421.2	3.4644
3	525	533.6	0.1386
4	532	507.0	1.2327
5	408	385.2	1.3495
6	273	244.1	3.4215
7	139	132.5	0.3189
8	45	62.9	5.0934
9	27	26.6	0.0060
10	10	10.27	0.0039
$\begin{array}{c}\hfill \text{11}\\ \hfill \text{12}\end{array}$	$\begin{array}{c}\hfill \text{4 }\\ \hfill \text{2 }\end{array}\}$ 6	4.4	0.5819
Total	2608	2608	16.2216

The chi-square value obtained above does not reject the hypothesis that the data follow the Poisson distribution. The following facts can be further observed:

1. (i)  The characterization theorems used in Examples 2.11 and 2.12 provide some quick estimates of the parameters of the hypothesized models. In more sophisticated models that require iterative procedures to find estimates, the above estimates are good choice may serve as initial values;
2. (ii)  The contribution to the chi-square value or the discrepancy between the observed and expected frequencies at $x=8$ is considerably larger compared to the rest. The plot of the hazard function shows that the function is decreasing markedly at this point which is uncharacteristic of the Poisson model where the hazard rate is increasing. If we use the maximum likelihood estimate $\hat{\lambda }=3.87$ for the Poisson distribution, the difference is still higher with expected frequency 67. Thus, the $\hat{g}(x)$ value at $x=8$ is taken as a discordant one and omitted in the calculation of $\overline{g}(x)$.