Abstract

Developing compound probability distributions is very important in the field of probability and statistics because there are different datasets from different fields with different features. These features range from high skewness, peakedness (kurtosis), bimodality, highly dispersed, and so on. Existing distributions might not easily fit well to these emerging data of interest. So, there is a need to develop more robust and flexible distributions that are positively skewed, negatively skewed, and bathup shape, to handle some of these features in the emerging data of interest. This paper, therefore, proposed a new four-parameter distribution called the Normal-Power{logistic} distribution. The proposed distribution was characterized by its density, distribution, survival, hazard, cumulative hazard, reversed hazard, and quantile functions. Properties such as the r-th moment, heavy tail property, stochastic ordering, mean inactive time were obtained. A useful transformation of the proposed distribution to normal distribution was shown to help generate its quantiles. The method of Maximum Likelihood Estimation (MLE) was used to estimate the model parameters. A simulation study was carried out to test the consistency of the maximum likelihood parameter estimates. The result of the simulation shows that the biases reduce as the sample size increases for different parameter values. The importance of the new distribution was proved empirically using a real-life dataset of gauge lengths of 10mm. The proposed distribution was compared with five other competing distributions, and the results show that the proposed Normal-Power{logistic} distribution (NPLD) performed favourably than the other five distributions using the AIC, CAIC, BIC, HQIC criteria.

1. Introduction

In the field of statistics and probability, the power function distribution is unimodal, skewed, and peaked depending on the shape parameter value ^{1, 2}. In literature, many known probability distributions such as exponential, Weibull, gamma, and lognormal are used for modeling lifetime data and widely applied in a variety of studies because of their advantages in different situations ³, especially using them as baseline distribution in the T-R{Y} family ^{4, 5, 6}. The power function despite its flexibility as a lifetime distribution that arises in several scientific fields has not been considered in this regard ⁷.

The power function distribution is a special model that can be formed or related to the uniform, Weibull, Kumaraswamy distributions. The power function distribution is considered one of the simplest and handy lifetime distributions. It is a special case of the beta distribution and one may sight the importance of the distribution in statistical tests such as the likelihood ratio test. The simplicity and usefulness of the power function distribution compelled the researchers to explore its further extensions, generalizations, and applications in different areas of science ⁸. It is the inverse of Pareto distribution ⁹. Estimation of the power parameters has been done by various authors, for instance; ^{10, 11} proposed the two-parameter power function distribution as a simple alternative to the exponential distribution when it comes to modeling failure data related to mortality rate and component failures. The Weibull power function distribution was suggested by ¹². ¹³ studied the properties of the Odd Lomax-Exponential distribution. ¹ proposed the modified power function distribution.

The probability density function (pdf) and cumulative distribution function (cdf) of two-parameter power function distribution with scale parameter λ and shape parameter k are, respectively, given as

and

The normal distribution is another lifetime distribution with its various advantages. Many authors have worked on the normal distribution and it has huge applications in various fields. Also, many authors have developed and studied convoluted distributions with normal distribution parameters (µ and σ). These authors include but are not limited to Exponentiated-Normal proposed and studied by ¹⁴, Beta-Normal distribution by ¹⁵, Gamma-Normal (GN) distribution by ¹⁶, Kumaraswamy-Normal distribution by ¹⁷, Weibull-Normal by ¹⁸, Exponentiated-Generalized-Normal (EGN) distribution by ¹⁹, Weibull-Normal{log-logistics} distribution by ²⁰, Lomax-Cauchy {Uniform} by ²¹, Rayleigh Cauchy distribution by ²² and Weibull-Inverse Rayleigh distribution: Classical/ Bayesian approach by ²³.

Many classical distributions have been extensively used for modeling real data in many areas. However, in many situations; there is a clear need for extended forms of these distributions to improve the flexibility and goodness of fit of these distributions. For that reason, generated families of continuous distributions are developed by introducing one or more additional shape parameter(s) to the baseline distribution or by combining two or more distributions to produced new ones. ²⁴ described such new distributions as convoluted distributions.

In this paper, the aim is to develop a novel univariate continuous probability distribution called the normal-power function {logistic} distribution {NPLD}. It is derived from the T-Power function {Y} family proposed and studied by ⁷. The distribution has four parameters, two from the normal distribution and the other two from the power function distribution. The scope covers different characterizations of the NPLD are to be established such as its density, distribution, survival, hazard, cumulative hazard, reversed hazard, and quantile function. To achieve this aim, different properties of the proposed distribution such as the r-th moment, heavy tail property, stochastic ordering, mean inactive time are to be obtained. The relationship between the proposed distribution and normal distribution is to be established. Thus, the proposed distribution can be positively skewed, negatively skewed, symmetric, and has a bathtub shape. The method of Maximum Likelihood Estimation (MLE) is used to estimate the model parameters. The importance of the new distribution will be proved empirically using a real-life dataset of gauge lengths of 10mm and compared with other convoluted distributions with normal distribution parameters (µ and σ). The proposed distribution will be very useful in engineering, medicine, and all filed of life. It is expected to perform well when normal distribution fails to fit the data of interest.

The rest of the paper contains the following sections. The formation of the pdf and cdf of the NPLD is performed in Section 2. Some statistical properties of NPLD distribution are provided in Section 3. Estimation of parameters, simulation study, and application to real data are presented in Section 4. The article ends with concluding remarks in section 5.

2. Derivation of Normal - Power{logistic} Distribution

Ekum et al. ⁷ defined the cdf and pdf of T-Power {logistic} family as

(1)

and

(2)

respectively, where T can follow any probability distribution that has the same support with the logistic distribution. The quantile function of the logistic distribution is used, which is the log of the odd function of the power function distribution.

In this research, let T follow normal distribution with parameters μ and σ, i.e. T~N(μ, σ). This proposed distribution is called the Normal-Power Function Distribution.

Let T∈(-∞, ∞) be any random variable. Here, T follows the normal distribution with cdf and pdf given by

(3)

and

(4)

respectively.

By substituting equation (3) into (1) we have

(5)

where

is called the error function.

So, equation (5) is the cdf of the proposed normal-power function {logistic} distribution (NPLD) and the corresponding pdf is derived by substituting (4) into (2) and we have

(6)

Equation (6) can be written as

The NPLD is a novel four-parameter distribution. Parameters λ and σ are scale parameters, k is a shape parameter, while μ is a location parameter. The pdf plots in Figure 1 show that NPLD can be positively skewed, negatively skewed, symmetric (see also the histogram in Figure 4), and can have a bath up shape, depending on the parameter values. The cdf plots in Figure 2 shows that for increasing values of x, the cdfs increase as well, increasing from zero and flattened at lambda since lambda is an upper bound. Figure 2 shows the cumulative hazard plots, which rise from zero slowly and asymptotic to the vertical axis at lambda. The reversed hazard decreases with increasing values of x and it is asymptotic to both axes. The reversed hazard plot is adjusted to display on x values less than 2 for clearer curves.

This section considered some major properties of the new proposed distribution.

The survival, hazard, cumulative hazard, and the reversed hazard functions are given in equations (7 – 10) respectively.

(7)

(8)

(9)

(10)

Lemma 1. If X follows the NPLD with parameters μ, σ, k, λ then random variable follows the normal distribution with parameters μ and σ.

Proof.

Let

(11)

Differentiating equation (11) with respect to x gives

Therefore, the pdf of random variable Y is given by

(12)

Equation (12) completes the proof.

The quantile function of NPLD is given by

(13)

where is the quantile function of the normal distribution with parameters μ and σ. Note that is used to generate random variate that is normally distributed with mean μ and variance σ². This can be achieved by using R codes [T = qnorm(p, mean = μ, sd = σ)], where p is vector of probabilities. It can be a sequence of numbers between 0 and 1 or uniformly distributed random values between 0 and 1. We can have in R as p = runif(n), where n is the number of observations to be generated.

The quantile function of a normal distribution is given by

(14)

Substitute equation (14) into (13) to have the quantile function of NPLD given by

(15)

The quantile function in (15) can be applied by using the quantile function in (13).

Consider a random variable . The r-th moment of the random variable is defined as

(16)

where is the pdf of the NPLD defined in equation (6). Substituting equation (6) into (16) and using integration by parts, we obtained the r-th moment of the NPLD as

(17)

Thus, equation (17) is the r-th moment of GPLD.

Supposing is a continuous random variable with a probability density function. Then, by definition, is said to be heavy-tailed if and only if

(18)

To examine the heavy-tail property of NPLD, substitute equation (6) into (18) to obtain

Using the infinity property

Thus,

Therefore, is not a heavy-tailed function. This can be seen from the pdf plots in Figure 1.

The comparative study of the behavior of a continuous random variable can be investigated using stochastic ordering ^{25, 26}.

Supposing and be continuous random variables. is said to be smaller than in the ²⁵, if the following conditions hold.

i. Stochastic order if for all

ii. Hazard rate order if for all

iii. Mean residual life order if for all

iv. Likelihood ratio order if decreases in

Theorem 1: Let and ~ NPLD with and respectively. If

then. Hence, and.

Proof

The NPLD will be ordered based on the strongest likelihood ratio order.

(19)

(20)

Thus, for and ,

for and ,

and if and

Hence, , and.

3. Mean Inactivity Time

The mean inactivity time function of a random variable with a known pdf and cdf is defined as

(21)

where is the time, the probability function of the NPLD. From the above definition, we obtain the mean inactive function of the NPLD as

Thus,

(22)

4. Results and Discussion

In this section, we derived the maximum likelihood parameter estimates, carried out simulation studies to test the consistency of the parameters, and applied the distribution to two real datasets.

From the pdf in (6), we derived the likelihood function as

Take the log of the likelihood function to have the log-likelihood given by

(23)

Each of the parameters of the distribution can be estimated by differentiating the with respect to each parameter, and equating the result to zero.

(24)

(25)

(26)

(27)

From equation (24), we have

From equation (25), we have

Parameter k is not in closed form, thus, Newton Rapson numerical method is used to estimate the parameter of k., developed by R package (maxLik or optim) (R Development Core Team 2009) ²⁷.

The parameter λ cannot be estimated using maximum likelihood because it is an upper bound of the distribution. So, λ can be estimated directly from data as

where is the standard error of Max(x_i) is the maximum of x and is the arithmetic mean of x.

In this subsection, we present the simulation study for the NPLD to investigate its flexible behavior and consistency of the parameter estimates. We examined the “mean estimates (ME), variance, biases, and root mean square errors (RMSEs)” of the MLEs. Below are the procedures used to perform the simulation studies:

1. Uniform distribution is used to generate n quantiles, p.

2. The quantile function defined in equation (15) is used to generate NPLD random variates.

3. The sample sizes are drawn as , and

4. The parameters values are set as , , and .

Table 1 to Table 3 show that the maximum likelihood parameter estimates are consistent. The biases, variances, and RMSEs reduce as the sample size increases. This shows that the parameters are consistent.

In this subsection, NPLD is applied to two datasets, one in the engineering field and the other in the medical field. This will enable us to see the usefulness of the data in various fields of study.

Application 1

In this first application, we make use of the data set of gauge lengths of 10mm from Kundu and Raqab ²⁸. This data set consists of 63 observations:

1.901, 2.132, 2.203, 2.228, 2.257,2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614,2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977,2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332,3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886,3.971, 4.024, 4.027, 4.225, 4.395, 5.020.

The gauge lengths of 10mm data are fitted by using the NPLD, the Beta-Normal (BN) distribution defined and studied by ¹⁵ Kumaraswamy-Normal by ¹⁷. Exponentiated Generalized-Normal (EGN) distribution by ¹⁹, Weibull-Normal (WN) distribution by ²⁰, and Gamma-Normal (GN) distribution by ¹⁴. The data is skewed to the right (skewness = 0.6328 and kurtosis = 3.2863). The results from the six distributions are presented in Table 4. The Log-Likelihood (LogL), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan Quinn Information Criterion (HQIC), Cramer-von Mises statistic (W), Anderson-Darling statistic (A), the Kolmogorov-Smirnov (KS) goodness of fit statistic with its corresponding p-value are reported in Table 4. From Table 4, all six distributions provide an adequate fit to the data set using W, A, KS statistics, and p-value.

Table 4 shows that the NPLD performed best when compared with the other five distributions based on the model selection criteria using AIC, CAIC, BIC, and HQIC. This is because the smaller the values of the selection criteria, the better the model.

Figure 5 displayed the histogram of the gauge length data with the density curve of NPLD and the other five competing distributions. The NPLD appears to fit the data most and gives an account of the gap to the right of the histogram and rise to take care of the dispersed or extreme value. Figure 6 displayed the cumulative distribution of the data with the six distributions fitted on it. The NPLD shows almost a perfect fit.

Application 2

In this second application, we used Crude Mortality Rate (CMR) from Bradley ²⁹. The data set consists of 65 observations and it is given thus:

2.01, 6.32, 3.52, 2.15, 5.42, 2.04, 2.77, 2.26, 1.95, 1.00, 2.45, 0.74, 0.98, 1.27, 2.77, 3.68, 1.1, 1.09, 1.60, 0.57, 3.33, 0.91, 7.14, 2.08, 3.85, 1.99, 7.76, 2.52, 1.57, 4.67, 4.22, 1.92, 1.59, 4.08, 2.02, 0.84, 6.85, 2.18, 2.04, 1.05, 2.91, 1.37, 2.43, 2.28, 3.74, 1.30, 1.59, 1.83, 3.85, 6.30, 4.83, 0.50, 3.40, 2.33, 4.25, 3.49, 2.12, 0.83, 0.54, 3.23, 4.50, 0.71, 0.48, 2.30, 7.73

The second data is also fitted by using the same set of distributions. The data is positively skewed (skewness = 1.1235 and kurtosis = 3.6889). The results from the six distributions are presented in Table 5 using the same set of criteria.

From Table 5, all six distributions provide an adequate fit to the data using the KS p-value, but NPLD has the best fit. Table 5 shows that the NPLD performed best when compared with the other five distributions based on the model selection criteria used.

Figure 7 displayed the histogram of the CMR data with the density curve of NPLD and the other five competing distributions. The NPLD appears to fit the data most showing the positive skewness on the histogram and rise to take care of the dispersed or extreme values. Figure 8 displayed the cumulative distribution of the CMR data with the six distributions fitted on it. The NPLD performed better on the curve.

5. Conclusion

A new generalization of the power function distribution is defined and studied. The Normal-Power function {log-logistic} distribution (NPLD) can be unimodal or bimodal, positively skewed, negatively skewed, symmetric, or bath up. Its pdf, cdf, survival function, hazard function, cumulative hazard function, and reversed hazard function are used to characterize the distribution. The quantile function is derived and displayed. With useful transformation, the four parameters GPLD reduces to the normal distribution with two parameters. The NPLD is not a heavy-tail function, its r-th moment and the mean inactive function exists. The method of maximum likelihood estimation was used to estimate the parameters of GPLD. The distribution is applied to two datasets, a gauge length of 10mm and a crude mortality rate. The analysis shows that the NPLD is capable of providing adequate fit to the two datasets that are about symmetric or skewed to the right, with little peakedness, and it is found to perform well in fitting the two datasets and compared favourably than other five convoluted distributions with normal distribution parameters. It is recommended that NPLD should be used to fit gauge length, crude mortality rate, and any similar data that slightly deviate from normal and cannot be fitted by the normal distribution. The deviation from normality can be handled by the shape parameter. It should be used in the generalized linear regression model whenever the ordinary linear regression model fails.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All Authors drafted the manuscript together. All authors read and approved the final manuscript.

References