In this article, we proposed a new three-parameter distribution called generalized half-logistic Poisson distribution with a failure rate function that can be increasing, decreasing or upside-down bathtub-shaped depending on its parameters. The new model extends the half-logistic Poisson distribution and has exponentiated half-logistic as its limiting distribution. A comprehensive mathematical and statistical treatment of the new distribution is provided. We provide an explicit expression for the
Lifetime data may exhibit a decreasing, increasing, or a bathtub failure rate function when modeling and analyzing random phenomena. This arises in several areas of studies, such as biomedical studies, reliability, actuarial science, computer science, demography, and engineering. There are several lifetime models that have been used successfully for to analyze lifetime data in practical applications such as exponential, Weibull, Gompertz, generalized exponential, and half-logistic. For example, exponential distribution can accommodate lifetime data with a decreasing density despite having only a constant failure rate function. Gompertz distribution has a decreasing and unimodal density but has an increasing failure rate function. However, these distributions are unable to accommodate lifetime data with a non-monotone failure rate such as the bathtub or upside-down bathtub. Many researchers have attempted to provide several methods to generate new lifetime models with the ability to fit data with monotone or non-monotone failure rate to overcome these problems. These methods include the generalization of a distribution by exponentiation procedure.
Method of exponentiation is an of the important and commonly used techniques to add a parameter to a lifetime model, the new model becomes more flexible and can accommodate both monotones as well as non-monotone failure rate functions. For example, Kuş (2007) proposed en exponential Poisson (EP) distribution that possesses a decreasing failure rate function; however, Barreto-Souza and Cribari-Neto (2009) proposed generalized exponential Poisson (GEP) by exponentiation of the EP. The GEP distribution can accommodate decreasing, increasing and upside down bathtub failure rates. Similarly, Adamidis and Loukas (1998) introduced exponential geometric distribution, while Silva
The cumulative distribution function (cdf) of the HLP distribution is given by
with
The first motivation of this study is to propose a new three-parameter probability distribution with increasing, decreasing and upside down bathtub-shaped failure functions. The second motivation of this study is to find the association of the density functions and hazard rate functions in practical applications. Last, we are motivated because the exponentiation method provides additional flexibility to a model especially in both the density and failure rate functions. The HLP distribution cannot fit data set with non-monotone failure rate functions; therefore, we hope that the new distribution that is the generalized half-logistic Poisson (GHLP) will provide solutions to many problems in various fields in practical applications.
The rest of the paper is arranged as follows, in Section 2 we provide the density of the GHLP and consider some important mathematical and statistical properties. Section 3 discusses the maximum likelihood estimate (MLE) and a simulation study. Section 4 provides applications of the new model to a real data set Section 5 provides the conclusions.
The cumulative distribution function of the GHLP distribution with parameters
where the corresponding probability density and hazard rate functions are given by
respectively. The limiting distribution given by (
We obtained the first derivative of log
thus, for
Figure 1 below provide the plot of the pdf (
The limiting behavior of the pdf given by (
The
therefore, we can find the numerical values of the median and other percentiles of
Here, we provide the following lemma which is very useful in computations of several important properties of the GHLP. first we recall that,
See
Table 1 provide some numerical values of
Various characteristics and features of a distribution can be analyzed through its moments such as mean, variance, moment generating function, etc. If
thus,
Therefore, the moments of GHLP can easily be computed using mathematical software such as
The moment generating function of the GHLP distribution can be computed directly using
thus, by putting (
One of the alternative measures for the skewness and kurtosis of a distribution are the Bowley skewness (B) and Moor’s kurtosis (M) defined by
Entropy is defined as a measure of uncertainty of a random variable. Here, we consider the two most important entropies known as the Shannon and Rényi entropies. The Shannon entropy is defined by
Hence,
The Rényi entropy of a random variable
by applying Lemma 1 the integral become
Hence, the Rényi entropy of
In this section, we discuss MLEs, one of the most popular and common methods used in inference. In this method the approximate MLEs are obtained either analytically or numerically using some mathematical packages. Let
Hence, the MLEs of
For the interval estimate and hypothesis tests of the parameters we required
Here, we evaluate the performance of the MLEs given by
In this section, we provide an application of the GHLP distribution to a real data set. We used the Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), and Kolmogorov-Smirnov (K-S) test to compare the GHLP and some other existing distributions. The model with the smallest values of these measures fit the data better than the other distributions. The competing distributions are:
The HLP distribution with cdf given by (
Olapade-generalized half-logistic by Olapade (2014) with
Power-half-logistic by Krishnarani (2016) with
The generalized half-logistic with
Poisson-half-logistic by Abdel-Hamid (2016) with
Generalized exponential by Mudholkar and Srivastava (1993) with
The EP by Kuş (2007) with
Poisson-odd exponential uniform by Muhammad (2016b) with
Nadarajah and Haghighi (2011) exponential type (NH) with
The half-logistic distribution with cdf as
The data set is the remission times (in months) of a random sample of 128 bladder cancer patients provided by Lee and Wang (2003).
In Figure 4 the TTT (total time on test)-plot show that the data has upside-down bathtub failure rate function and GHLP distribution has the ability to accommodate upside-down bathtub failure rate curve.
We estimate the unknown parameters of each model by the method of maximum likelihood. The numerical values of the log-likelihood (
Table 3 indicates that our proposed model the GHLP distribution has the smallest values of the AIC, CAIC, and K-S; thus, GHLP provide better fit than the other competing models. Figure 5 provides the plots of the (i) histogram and estimated density (ii) empirical cdf and estimated cdf of the
GHLP distribution. Figure 6 shows (i) empirical and estimated hazard functions (ii) quantile-quantile of the GHLP distribution for the given data set.
We introduced a new three-parameter lifetime distribution with increasing, decreasing and upside-down bathtub-shaped hazard rate functions. We also provide explicit expressions for the
For (
then expanding
we get
Let
thus, we have (
Let
For
then,
On the other side
hence,
thus, the root of
For
Consider that lim
We start with
Numerical values of
0.5 | 1.2 | 2.1 | 2.3 | 0.6 | 3.0 | 0.01595 | 0.5 | 0.2 | 0.1 | 0.3 | 0.6 | 0.3 | 7.24441 |
0.6 | 2.2 | 1.1 | 4.3 | 3.6 | 1.3 | 0.00392 | 0.1 | 2.2 | 4.1 | 3.3 | 3.6 | 1.5 | 0.00065 |
3.0 | 2.0 | 4.1 | 3.3 | 6.0 | 5.0 | 0.00150 | 1.0 | 6.0 | 1.0 | 3.0 | 2.0 | 1.0 | 0.00024 |
1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.23417 | 1.5 | 1.0 | 1.0 | 2.5 | 1.0 | 0.5 | 0.34307 |
1.5 | 7.0 | 1.0 | 2.5 | 1.0 | 1.5 | 0.00018 | 10 | 2.0 | 2.0 | 5.0 | 1.0 | 5.0 | 14.2023 |
10 | 2.0 | 12 | 5.0 | 11 | 15 | 0.00123 | 8.0 | 5.0 | 2.0 | 1.0 | 9.0 | 1.0 | 0.00240 |
MLEs and standard deviations for some various values of parameters
Sample size |
Actual values | Estimated values | Standard deviations | ||||||
---|---|---|---|---|---|---|---|---|---|
sd( |
sd( |
sd( |
|||||||
20 | 0.6 | 1.0 | 0.2 | 0.3484 | 0.6445 | 0.4752 | 0.2779 | 0.7451 | 1.0270 |
1.0 | 1.2 | 1.0 | 1.0751 | 1.4988 | 1.3013 | 0.3698 | 0.8655 | 1.4253 | |
0.1 | 0.2 | 0.5 | 0.3826 | 0.1842 | 0.8930 | 0.3908 | 0.0778 | 6.1164 | |
0.3 | 0.7 | 0.4 | 0.2615 | 0.6311 | 0.7950 | 0.1282 | 0.4109 | 3.7589 | |
2.5 | 1.1 | 2.1 | 3.2405 | 1.2652 | 1.7064 | 1.2870 | 0.6377 | 4.2860 | |
1.0 | 1.0 | 1.0 | 1.0914 | 1.2013 | 1.2527 | 0.3901 | 0.5492 | 1.3846 | |
0.2 | 0.5 | 1.5 | 0.2606 | 0.5550 | 1.2747 | 0.1173 | 0.1842 | 3.9691 | |
30 | 0.6 | 1.0 | 0.2 | 0.7988 | 0.6666 | 0.1397 | 0.2009 | 0.4976 | 0.5541 |
1.0 | 1.2 | 1.0 | 1.0318 | 1.3712 | 1.3004 | 0.3149 | 0.4991 | 1.3962 | |
0.1 | 0.2 | 0.5 | 0.3446 | 0.1829 | 0.6371 | 0.3794 | 0.0691 | 2.8381 | |
0.3 | 0.7 | 0.4 | 0.2514 | 0.6060 | 0.7650 | 0.1140 | 0.3707 | 3.4513 | |
2.5 | 1.1 | 2.1 | 3.0149 | 1.1743 | 1.7622 | 1.0964 | 0.3927 | 2.9208 | |
1.0 | 1.0 | 1.0 | 1.0466 | 1.1260 | 1.2523 | 0.3296 | 0.3831 | 1.3701 | |
0.2 | 0.5 | 1.5 | 0.2446 | 0.5318 | 1.2913 | 0.0969 | 0.1363 | 1.3302 | |
40 | 0.6 | 1.0 | 0.2 | 0.7948 | 0.6463 | 0.1307 | 0.1986 | 0.4601 | 0.5467 |
1.0 | 1.2 | 1.0 | 1.0128 | 1.3138 | 1.2735 | 0.2855 | 0.3892 | 1.3833 | |
0.1 | 0.2 | 0.5 | 0.3207 | 0.1835 | 0.5788 | 0.3691 | 0.0661 | 2.0529 | |
0.3 | 0.7 | 0.4 | 0.2505 | 0.5911 | 0.7099 | 0.1088 | 0.3443 | 2.5766 | |
2.5 | 1.1 | 2.1 | 2.8909 | 1.1424 | 1.8507 | 1.0181 | 0.3141 | 1.5175 | |
1.0 | 1.0 | 1.0 | 1.0285 | 1.0942 | 1.2641 | 0.3036 | 0.3070 | 1.3592 | |
0.2 | 0.5 | 1.5 | 0.2326 | 0.5216 | 1.3513 | 0.0858 | 0.1104 | 1.3274 | |
50 | 0.6 | 1.0 | 0.2 | 0.7946 | 0.6359 | 0.1269 | 0.1964 | 0.4450 | 0.5423 |
1.0 | 1.2 | 1.0 | 1.0037 | 0.6211 | 0.4066 | 0.1727 | 0.5680 | 1.0623 | |
0.1 | 0.2 | 0.5 | 0.3088 | 0.1833 | 0.5842 | 0.3647 | 0.0649 | 1.2393 | |
0.3 | 0.7 | 0.4 | 0.2477 | 0.5806 | 0.6368 | 0.1044 | 0.3362 | 1.6333 | |
2.5 | 1.1 | 2.1 | 2.8522 | 1.1244 | 1.8733 | 0.9546 | 0.2751 | 1.5140 | |
1.0 | 1.0 | 1.0 | 1.0144 | 1.0723 | 1.2502 | 0.2831 | 0.2647 | 1.3529 | |
0.2 | 0.5 | 1.5 | 0.2270 | 0.5144 | 1.3846 | 0.0798 | 0.0969 | 1.3273 | |
100 | 0.6 | 1.0 | 0.2 | 0.7941 | 0.6112 | 0.0926 | 0.1912 | 0.4106 | 0.4077 |
1.0 | 1.2 | 1.0 | 0.9978 | 0.6444 | 0.3096 | 0.1346 | 0.4431 | 0.8330 | |
0.1 | 0.2 | 0.5 | 0.3016 | 0.1764 | 0.5559 | 0.3691 | 0.0669 | 0.6401 | |
0.3 | 0.7 | 0.4 | 0.2461 | 0.5641 | 0.5677 | 0.0952 | 0.3104 | 1.3488 | |
2.5 | 1.1 | 2.1 | 2.6656 | 1.1021 | 2.0777 | 0.8103 | 0.1869 | 1.5002 | |
1.0 | 1.0 | 1.0 | 0.9895 | 1.0286 | 1.2264 | 0.2363 | 0.1790 | 1.2841 | |
0.2 | 0.5 | 1.5 | 0.1559 | 0.2888 | 0.6507 | 0.0743 | 0.2057 | 1.1702 | |
150 | 0.6 | 1.0 | 0.2 | 0.7960 | 0.6006 | 0.0718 | 0.1882 | 0.3997 | 0.3421 |
1.0 | 1.2 | 1.0 | 0.9969 | 0.6429 | 0.3005 | 0.1213 | 0.4317 | 0.8044 | |
0.1 | 0.2 | 0.5 | 0.3456 | 0.1667 | 0.4991 | 0.3974 | 0.0720 | 0.5946 | |
0.3 | 0.7 | 0.4 | 0.2470 | 0.5562 | 0.4975 | 0.0921 | 0.3005 | 0.8660 | |
2.5 | 1.1 | 2.1 | 2.5933 | 1.0939 | 2.1501 | 0.7259 | 0.1521 | 1.4292 | |
1.0 | 1.0 | 1.0 | 0.9887 | 1.0200 | 1.1904 | 0.2093 | 0.1483 | 1.1919 | |
0.2 | 0.5 | 1.5 | 0.1514 | 0.2810 | 0.6537 | 0.0689 | 0.2018 | 1.1586 |
MLEs,
Model | AIC | CAIC | K-S | |||||
---|---|---|---|---|---|---|---|---|
GHLP | 0.0642 | 1.3909 | 4.4696 | −410.24 | 826.48 | 820.53 | 0.0427 | 0.9737 |
OGHL | 0.9217 | 0.1250 | - | −412.18 | 828.36 | 824.40 | 0.0661 | 0.6304 |
PwHL | 0.8880 | 0.2015 | - | −415.10 | 834.19 | 830.22 | 0.0751 | 0.4652 |
HLP | 0.0574 | - | 3.8836 | −413.17 | 830.34 | 826.37 | 0.0963 | 0.1863 |
PHL | 0.8880 | - | 0.2015 | −415.10 | 834.19 | 830.22 | 0.0761 | 0.4489 |
GHL | 0.1440 | 0.9527 | - | −416.64 | 837.27 | 833.30 | 0.0950 | 0.1994 |
GE | 0.1212 | 1.2180 | - | −413.08 | 830.16 | 826.19 | 0.0725 | 0.5113 |
EP | 0.3341 | – | 4.3342 | −417.04 | 835.61 | 831.64 | 0.9533 | 0.0000 |
POEU | 1.5503 | 100 | 6.7528 | −415.15 | 834.30 | 830.33 | 0.0808 | 0.3742 |
NH | 0.3341 | - | 4.3342 | −417.04 | 838.09 | 820.53 | 0.5709 | 0.0000 |
HL | 0.1479 | - | - | −416.73 | 835.45 | 833.47 | 0.0989 | 0.1631 |