Transmuted Lomax Distribution

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3. Moments and Quantiles

In this section we shall present the moments and quantiles for the transmuted Lomax distribution. The r^th moments of a transmuted Lomax random variable X, is given by

(8)

The first three moments can be obtain by taking and 3 in (8) as follows:

(9)

(10)

and

(11)

The variance and coefficient of variation (CV) for TL are

(12)

and

(13)

The skewness and kurtosis measures can be obtained from expressions

Upon substituting in the raw moments. Figure 2 shows the skewness and kurtosis of the TL distribution as a function of for

Figure 2. skewness and kurtosis of TL distribution as a function of

and for

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We can obtain the estimator of and by using the following equations:

i.e.,

(14)

(15)

(16)

We can obtain and by solving nonlinear equation (14), (15) and (16) numerically.

The quantile of the transmuted Lomax distribution can be obtaned from (6) as

(17)

Hence, the distribution median is

(18)

To illustrate the effect of the shape parameter λ on skewness and kurtosis we consider measures based on quantiles. The shortcomings of the classical kurtosis and skewness measures are well known. There are many heavy-tailed distributions for which this measure is infinite, so it becomes uninformative. The Bowley's skewness ^[5] is one of the earliest skewness measures defined by the average of the quartiles minus the median, divided by the half the interquartile range, given by

and the Moors kurtosis ^[7] is based on octiles and is given by

For any distribution symmetrical to 0 the Moors kurtosis reduces to

It is easy to calculate that for standand normal distribution Therefore, Hence, the centered Moor's coefficient is given by

Fiqure 3 displays the Bowley and Moors kurtosis as a function of the parmeter and for and It is evident that both measures depend on the parameter

Figure 3. Behavior of Bowley(B) and Moors(M) kurtosis for TE

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4. Mean Deviation

The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and the median. These are known as the mean deviation about the mean and the mean deviation about the median respectively and are defined by

(19)

and

(20)

where

and

The measures and can be expressed as and where For a transmuted Lomax

(21)

One can easily compute these integrals numerically in software such as Mathcad ^[4] and hence get the mean deviations about the mean and about the median as desired. From the mean deviations we can construct Lorenz and Bonferroni curves, which are used in several areas including economics, reliability, insurance and medicine and others.

Some numerical values of the mean deviation from mean and median for selected value of for and and different values of are listed in the Table 1 below.

Table 1. Mean deviation from the mean and the median for selected values of the parameters

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Tables index

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5. Random Number Generation and Parameter Estimation

Using the method of inversion we can generate random numbers from the transmuted Lomax distribution as

where After simple calculation this yields

(22)

One can use equation (22) to generate random numbers when the parameters and are known. The maximum likelihood estimates, MLEs, of the parameters that are inherent within in the transmuted Lomax probability distribution function is given by the following:

Let be a sample size n from a transmuted Lomax distribution. Then the likelihood function is given by

(23)

Hence, the log-likelihood function becomes

Therefore, the MLEs of and which maximize (23) must satisfy the following normal equations:

The MLE of and is obtained by solving this nonlinear system of equations. Setting these expressions to zero and solving them simultaneously yields the maximum likelihood estimates of the three parameters.

6. Reliability Analysis

The survival function, also known as the reliability function in engineering, is the characteristic of an explanatory variable that maps a set of events, usually associated with mortality or failure of some system onto time. It is the probability that the system will survive beyond a specified time.

The transmuted Lomax distribution can be a useful model to characterize failure time of a given system because of the analytical structure. The reliability function which is the probability of an item not failing prior to sometime is defined by The reliability function of a transmuted Lomax distribution is given by

(24)

The other characteristic of interest of a random variable is the hazard rate function also known as instantaneous failure rate defined by

which is an important quantity characterizing life phenomenon. It can be loosely interpreted as the conditional probability of failure, given it has survived to the time t.

The hazard rate function for a transmuted Lomax distribution is given by

Figure 4. The reliability function of TL distrbution for

when

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It is important to note that the units for is the probability of failure per unit of time, distance or cycles. Figure 5 illustrates the behavior of the hazard rate function of a transmuted Lomax distribution for selected values of the parameters.

Observing the behavior of the hazard rate function it is worth noting that the transmuted Lomax distribution will have more applicability than the Lomax distribution and some of its generalizations.

The hazard rate function of the transmuted Lomax distribution has the following properties:

i. If the failure rate is same as the transmuted Pareto II distribution

ii. If the failure rate is same as the Lomax distribution

iii. If he failure rate is same as the Pareto II distribution.

Figure 5. The Hazard rate function of TL for

when

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