1. Introduction
The importance of probability distributions in statistical modelling cannot be overemphasized. For example, in order to build parametric analysis of variance and regression models, one is required to ensure that the research data follow certain probability distribution. Consequently, the success of many statistical methods relies on the kind of probability distribution that fits the data under consideration. For quite a number of decades, studies have been undertaken in a bid to developing probability distributions capable of modelling real life data in various fields. To this end, Shanker and Shukla [1] developed a new probability distribution called Ishita distribution. The cumulative distribution function (cdf) and the corresponding probability density function (pdf) of an Ishita distributed random variable X having parameter
are respectively given by
(1)
and
(2)
for
and
. Shanker and Shukla [1] investigated the properties of the Ishita distribution and applied it to two real lifetime datasets from biomedical science and engineering. The distribution was found to outperform the exponential, Lindley and Akash distributions respectively. Another important application of the Ishita distribution is the analysis of quality control data. Al-Nasser et al. [2] developed a single-acceptance sampling plan that uses the Ishita distribution to model the lifetime distribution of a product.
It is obvious that a distribution such as Ishita distribution, which depends on one scale parameter, cannot be flexible in modelling real life data with varieties of tails. To obtain a distribution that would provide better fits to various datasets and show flexibility in statistical modelling, one either needs to add parameters to the Ishita distribution or to compound one or more distributions with Ishita distribution. Fortunately, several extensions of the Ishita distribution have been given in the literature. Notable among them are discrete Poisson-Ishita Distribution due to Shukla and Shanker [3], sized-biased Ishita distribution due to Al-Omari et al. [4], Poisson Ishita distribution proposed by Hassan et al. [5] and Transmuted Ishita distribution developed by Gharaibeh and Al-Omari [6]. Another important extension of the Ishita distribution is the Exponentiated Ishita (EI) distribution introduced by Rather and Subramanian [7] with the cumulative distribution function (cdf) given by
(3)
and the probability density function (pdf) corresponding to (3) is given by
(4)
for
,
,
. Rather and Subramanian [7] equally studied the properties of the exponentiated Ishita (EI) distribution and demonstrated its superiority over the Ishita distribution in modelling lifetime.
In spite of the efforts made by several researchers to generalize the Ishita distribution, there is still need for more generalizations in order to obtain a new form of the Ishita distribution that will allow for the most flexible tails. To contribute in this direction, we adopt the beta-generator introduced by Eugene et al. [8]. Let
denote the cdf of the exponentiated Ishita distribution. Then the cdf of beta-exponentiated distribution is given by
(4)
where
,
are the shape parameters and
is the beta function defined by
(5)
Also,
is the incomplete beta function given by
(6)
Let
denote the pdf of the exponentiated Ishita distribution. Then the pdf of the Beta-exponentiated distribution is given by
(7)
By using (4) and (7), Eugene et al. [8] introduced a beta-normal distribution. Several other beta-generated class of distributions available in the literature are found in [9] - [38], among others.
From the existing literature on beta-generated family of distributions, one observes that the shape parameters a and b increases the skewness and tail weights of the Beta-G distribution, allowing for much flexibility. In view of this advantage, this article aims at introducing a four-parameter beta-exponentiated Ishita (BEI) distribution, which generalizes the exponentiated Ishita distribution. We shall derive the properties of the BEI distribution. The parameters of the new model will be estimated using maximum likelihood method. Two real life applications of the distribution will be give to illustrate the usefulness of the distribution.
2. Beta Exponentiated Ishita Distribution
Let
denote the cdf of the exponentiated Ishita random variable X. Then the cdf the BEI distribution is obtained from (4) as
(8)
for
,
,
,
,
. The corresponding pdf of the BEI distribution is obtained from (7) by using (3) and (4) as
(9)
for
,
,
,
,
. Figure 1 gives the shapes of the pdf of BEI distribution for some parameter values.
2.1. Special Cases of the BEI Distribution
Below are the sub-models of the BEI distribution for selected values of the parameters
, a and b.
1) When
, the BEI distribution reduces to the Beta Ishita (BI) distribution, with pdf given by
(10)
for
,
,
,
,
.
2) When
, the BEI distribution reduces to another kind of Beta Ishita (BI) distribution, with pdf given by
;
,
,
(11)
3) When
, the BEI distribution reduces to a new Beta Ishita (BI) distribution, with pdf given by
;
,
,
(12)
4) When
, the BEI distribution reduces to the Ishita (I) distribution, with pdf given by
;
,
(13)
Figure 1. Various shapes of the pdf of the BEI distribution for varying values of the parameters α, θ, a and b.
2.2. Expansion of the CDF of BEI Distribution
The cdf of the BEI distribution in (9) appears complex and cannot be used easily for the derivation of the properties of the distribution. It is therefore necessary to provide expansion of the cdf (9) when
is a real non–integer and when
is an integer value.
If
and
are a real non-integer, we have the following power series expansion
(14)
Applying the series expansion (14) on the cdf of the BEI distribution given in (8), one obtains
(15)
If
is an integer, then index j in (14) stops at
, as a binomial expansion. If
is an integer, the index k stops at
.
2.3. Expansion of the PDF of the BEI Distribution
Applying the series expansion (14) on (9), we get
(16)
2.4. Survival and Hazard Functions of the BEI Distribution
The survival function, denoted by
, may be defined for the BEI distribution as
(17)
The hazard function, denoted by
, may be defined for the BEI distribution as
(18)
Figure 2 gives the shapes of the hazard rate function for the BEI distribution for some parameter values.
3. Properties of the BEI Distribution
3.1. Crude Moments of the BEI Distribution
Let
. Then if
and
are real non-integers, the rth crude moments of X, denoted by
, is given by
(19)
Putting
into (19) and noting that
yields
Figure 2. Plots of hazard function of the BEI distribution for selected values of the parameters α, θ, a and b.
(20)
The first four crude moments of the BEI distribution are obtained by putting
and 4 into (20). Thus, we obtain
(21)
(22)
(23)
(24)
3.2. Cumulants and Central Moments of the BEI Distribution
The rth cumulants
and rth central moments
of
can be determined from (20) using the relations
and
(25)
Putting
and 4 into (25), one obtains the first four cumulants of
as follows
(26)
(27)
(28)
(29)
Similarly, the following central moments of
are obtained from (26)-(29):
(30)
(31)
(32)
3.3. Skewness and Kurtosis of the BEI Distribution
Let
, the skewness
and Kurtosis
of X can be calculated from the third and fourth standardized cumulants. Thus,
(33)
(34)
3.4. MGF and CF of the BEI Distribution
Suppose
, then the moment generating function (mgf) of X is given by
(35)
Putting
into (35) and noting that
gives
(36)
where
. For
, the characteristic function (cf) of
is given by
(37)
3.5. Mean Deviations of the BEI Distribution
The mean deviation is a special kind of the incomplete moments commonly used to measure the amount of scatter in a population. The mean deviation about the mean (when the distribution is symmetric) and the mean deviation about the median (when the distribution is skewed) are defined for
by
and
(38)
respectively, where
and
denote, respectively the mean and median of the BEI distribution. For ease of computation, one defines
and
respectively by
and
(39)
To evaluate
and
, we first determine the following integrals:
(40)
(41)
where
denote the upper incomplete gamma functions. The
expression for
is given in (21) while the expression for
is deduced from (8) by replacing x with
. Consequently, the mean deviations about the mean and about the median of the BEI distribution becomes
(42)
and
(43)
where
,
and
are expressed as
(44)
(45)
(46)
3.6. Bonferroni and Lorenz Curves of the BEI Distribution
The Bonferroni curve is a plot of
versus q while the Lorenz curve is a plot of
versus q. These two curves proposed independently by Bonferroni [39] and Lorenz [40] respectively, may be defined for
as
and
(47)
where
and
. Hence,
(48)
and
(49)
where
(50)
4. Entropy Measures of the BEI Distribution
One of the measures of the quantity of information contained in a random sample about its parent population is the entropy. In this section, we provide the Rényi entropy proposed by Rényi [41] and the S-entropy proposed by Tsallis [42]. The Rényi entropy for
is given by
(51)
To evaluate (51), we may need the following series expansions:
(52)
(53)
(54)
Substituting the expressions for
,
and
into (51), yields
(55)
In addition, when
and
, the s-entropy for
is given by
(56)
5. Distribution of Order Statistics
Order statistics is very useful in reliability analysis, quality control and lifetime analysis. A better understanding of the order statistics is gotten through its distribution. In view of this, one defines the cdf of the rth order statistic
for the BEI distribution as
(57)
The corresponding pdf of the rth order statistic
for the BEI distribution is given by
(58)
The pdf of the first and nth order statistics are obtained by setting
and
respectively in (58).
Finally, the qth moment of the rth order statistic
of the BEI distribution is given by
(59)
6. Stress Strength Reliability
Let X be the strength of a component and Y, the stress applied to the component. Essentially, a component fails at the instant when the stress applied to it exceeds the strength
, and the component will function satisfactorily whenever
. Thus, a measure of component’s reliability, denoted by
, is often required whenever a new distribution is proposed. To achieve this, we let
and
be independent random variables. Thus,
(60)
Using the binomial expansion in (60), we obtain the following
Hence,
(61)
7. Maximum Likelihood Estimation of Parameters of BEI Distribution
Let
be a random sample of sizen from the BEI distribution with parameters
,
, a and b, then the likelihood function is given by
(62)
The log-likelihood function is given by
(63)
Differentiating (63) with respect to parameters
,
, a and b and equating the resulting derivatives to zero, one obtains
(64)
(65)
(66)
(67)
where
is the digamma function. It may be observed that the log-likelihood equations do not admit any explicit solution for the maximum likelihood estimates (MLEs) of the BEI distribution due its non-linear structure. Therefore, the MLEs of the parameters of the BEI distribution, denoted by
are obtained using the Newton-Raphson’s numerical approach with the aid of R software.
8. Asymptotic Confidence Intervals of the Parameters of BEI Distribution
Let
be the MLE of
for the BEI distribution. To construct the confidence intervals, we need the Fisher information matrix, denoted by
. Thus,
(68)
The elements of (68) are the second derivatives of (68) with respect to the parameters of the BEI distribution. As pointed out in Lehmann and Casella [43], the asymptotic distribution of
is
, under certain regularity conditions. Consequently, the approximate
two-sided confidence intervals for
,
, a and b are given, respectively, by
,
,
and
(69)
where
,
,
and
are the diagonal elements of the matrix
and
is the upper
percentile of a standard normal distribution.
9. Applications
The performance of the BEI vis-à-vis other related distributions considered in this study in fitting two real data sets is demonstrated in this section. The estimation of the parameters of each of the distributions is made using the maximum likelihood method of estimation. The standard error of each of the parameters estimated is enclosed in brackets. The performance measures obtained include Log-likelihood (LL), Akaike’s information criterion (AIC), Kolmogorov-smirnov (K-S) statistic, and the corresponding probability value (p-value). The comparison of the proposed distribution is conducted with some well-known lifetime distributions such as the Ishita distribution (ID), Exponentiated Ishita distribution (EID), Akash distribution (AD), Exponentiated Akash distribution (EAD), Exponential distribution (ED) and Exponentiated Exponential distribution (EED) respectively.
Dataset one
The first data represents the sum of skin folds in 202 athletes collected at the Australian Institute of sports and used in a book by Weisberg [44]. The data is given below:
28.0, 98.0, 89.0, 68.9, 69.9, 109.0, 52.3, 52.8, 46.7, 82.7, 42.3, 109.1, 96.8, 98.3, 103.6, 110.2, 98.1, 57.0, 43.1, 71.1, 29.7, 96.3, 102.8, 80.3, 122.1, 71.3, 200.8, 80.6, 65.3, 78.0, 65.9, 38.9, 56.5, 104.6, 74.9, 90.4, 54.6, 131.9, 68.3, 52.0, 40.8, 34.3, 44.8, 105.7, 126.4, 83.0, 106.9, 88.2, 33.8, 47.6, 42.7, 41.5, 34.6, 30.9, 100.7, 80.3, 91.0, 156.6, 95.4, 43.5, 61.9, 35.2, 50.9, 31.8, 44.0, 56.8, 75.2, 76.2, 101.1, 47.5, 46.2, 38.2, 49.2, 49.6, 34.5, 37.5, 75.9, 87.2, 52.6, 126.4, 55.6, 73.9, 43.5, 61.8, 88.9, 31.0, 37.6, 52.8, 97.9, 111.1, 114.0, 62.9, 36.8, 56.8, 46.5, 48.3, 32.6, 31.7, 47.8, 75.1, 110.7, 70.0, 52.5, 67.0, 41.6, 34.8, 61.8, 31.5, 36.6, 76.0, 65.1, 74.7, 77.0, 62.6, 41.1, 58.9, 60.2, 43.0, 32.6, 48.0, 61.2, 171.1, 113.5, 148.9, 49.9, 59.4, 44.5, 48.1, 61.1, 31.0, 41.9, 75.6, 76.8, 99.8, 80.1, 57.9, 48.4, 41.8, 44.5, 43.8, 33.7, 30.9, 43.3, 117.8, 80.3, 156.6, 109.6, 50.0, 33.7, 54.0, 54.2, 30.3, 52.8, 49.5, 90.2, 109.5, 115.9, 98.5, 54.6, 50.9, 44.7, 41.8, 38.0, 43.2, 70.0, 97.2, 123.6, 181.7, 136.3, 42.3, 40.5, 64.9, 34.1, 55.7, 113.5, 75.7, 99.9, 91.2, 71.6, 103.6, 46.1, 51.2, 43.8, 30.5, 37.5, 96.9, 57.7, 125.9, 49.0, 143.5, 102.8, 46.3, 54.4, 58.3, 34.0, 112.5, 49.3, 67.2, 56.5, 47.6, 60.4, 34.9
Using (63), (68), and (69), we obtain the values reported in Table 1, which are the estimated parameters, their standard errors and confidence intervals for data set 1. Table 2 gives the Log-likelihood values, K-S statistic and their p-values, AIC and BIC values for data set 1.
Dataset two
The second data set represents time of failure (103h) of turbocharger of one type of engine (Xu et al., [45] ):
1.6, 3.5, 4.8, 5.4, 6.0, 6.5, 7.0, 7.3, 7.7, 8.0, 8.4, 2.0, 3.9, 5.0, 5.6, 6.1, 6.5, 7.1, 7.3, 7.8, 8.1, 8.4, 2.6, 4.5, 5.1, 5.8, 6.3, 6.7, 7.3, 7.7, 7.9, 8.3, 8.5, 3.0, 4.6, 5.3, 6.0, 8.7, 8.8, 9.0
Table 1. The MLEs of the parameters of the fitted distributions and their confidence intervals.
The values reported in Table 3 are the estimated parameters of the distributions, their standard errors and confidence intervals for data set 2. Table 4 gives the Log-likelihood values, K-S statistic and their p-values, AIC and BIC values for data set 2.
The results reported in Table 1 and Table 3 reveal that, on the average, the values of AIC, BIC and K-S are smaller for the BEI distribution than the other distributions while the values of log-likelihood (LL) and p-values are higher for the BEI than the other distributions. Hence, the BEI distribution outperforms the other distributions with respect to data set one and two. In addition, a close look at Figure 3 and Figure 4 indicates that BEI fits the data sets better than others. In addition, a close look at Figures 3-6 indicates that BEI fits the data sets better than other competing distributions.
Table 2. The results of the log-lik, AIC, BIC, KS statistic and p-value of the fitted distributions.
Table 3. The MLEs of the parameters of the fitted distributions and their confidence intervals.
Figure 3. The histogram and PDFs of fitted models for data set one.
Figure 4. CDFs of fitted models for data set one.
Figure 5. The histogram and PDFs of fitted models for data set two.
Figure 6. CDFs of fitted models for data set two.
Table 4. The results of the log-lik, AIC, BIC, KS statistic and p-value of the fitted distributions.
10. Concluding Remarks
In this article, we propose a new distribution called the Beta-Exponentiated Ishita (BEI) distribution for lifetime analysis. We provide for the new distribution expressions for the distribution function, density function, moments, cumulants, skewness, kurtosis, moment generating function, characteristic function, mean deviations, entropies, Bonferroni and Lorenz curves, stress-strength reliability, distribution of order statistics and its moments. Estimation of the parameters by maximum likelihood method is discussed. Two applications to real lifetime data show that the BEI distribution is more flexible than its competitors and can be used quite effectively in analysing positively skewed and heavy tailed data.