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In this paper, a new distribution with a unit interval named type II half logistic Kumaraswamy (TIIHLKw) distribution is proposed. Its density and distribution functions are presented using alternate expressions. This distribution is obtained by adding an extra parameter in the existing model to rise its ability fitting complex data sets. Some important statistical properties of TIIHLKw distribution are derived. The estimation of the parameters is obtained by numerous well-recognized approaches and simulation study confirmed the efficiencies of estimates such obtained. We apply the related model to practical datasets, and it is concluded that the proposed model is the best by model selection criteria than other competitive models.

Over the last few years, inspired by the increasing demand of probability distributions in many fields, many generalized distributions have been studied. Most of them are proposed by the addition of more parameters to well-known probability distributions that exist in literature to make them flexible. For instance, Haq et al. [

The Kumaraswamy (Kw) distribution was introduced and studied by Kumaraswamy [

The

The behavior of

The

Hassan et al. [

This article is dedicated to both its mathematical and application features. A significant portion is kept for the estimation of the parameters through various methods including maximum likelihood estimation (MLE), least-square estimation (LSE), weighted least square estimation (WLSE), percentiles estimation (PCE), and Cramer-von Mises estimation (CVE).

The core purpose of this research is to suggest a simpler and more flexible model called Type II Half Logistic

In this section, we examine the usefulness, and flexibility of a new associate of type II half-logistic-G family having (0, 1), using

The random variable (r.v.) X follows TIIHLKw distribution if its cdf is obtained by inserting (

The pdf of TIIHLKw distribution is as follows

The survival function of TIIHLKw distribution is

The failure rate function of TIIHLKwD is

The flexibility of TIIHLKw distribution can be illustrated in Figures

Pdf plot of TIIHLKwD.

Plots of some pdfs of TIIHLKwD.

Plots of some hrf of TIIHLKwD.

The properties of TIIHLKwD are derived here. After this, we consider a r.v. X follows the pdf (

The quantile function of

Also,

Simulated values from TIIHLKwD can be utilized in the simulation study. The r. v.

On differentiation, we can have the density of quantile function

Here, we present the expansion of the pdf and cdf for TIIHLKw distribution for further mathematical manipulation.

The binomial theorem, for

Then, by applying (

Another form of pdf (

Inserting this expansion in (

Another formula can be formed from pdf (

Let

The cdf

Once more binomial expansion is applied to

The required result is obtained by combing some expression together, completing the proof.

The probability-weighted moments (PWMs) are used to study some more characteristics of the probability distribution. Under the specified setting discussed above, PWMs are denoted by

The PWMs of TIIHLKw are obtained by substituting (

Then,

The moments have a vital role in the study of the distribution and real data applications. Now, we get the

The mean (

Also, the coefficients of skewness and kurtosis of TIIHLKw distribution are given by

The summary measures, mean, variance, skewness (

Mean, variance, skewness (

Mean | Var | ||||
---|---|---|---|---|---|

1 | 2.0 | 0.56556 | 0.03908 | -0.09702 | 2.30905 |

3.0 | 0.50543 | 0.03430 | 0.04856 | 2.38086 | |

4.0 | 0.46478 | 0.03053 | 0.13105 | 2.45360 | |

5.0 | 0.43469 | 0.02758 | 0.18440 | 2.51360 | |

6.0 | 0.41114 | 0.02522 | 0.22179 | 2.56192 | |

7.0 | 0.39198 | 0.02330 | 0.24948 | 2.60112 | |

1.5 | 2.0 | 0.64566 | 0.02830 | -0.26723 | 2.53251 |

3.0 | 0.57974 | 0.02601 | -0.08852 | 2.52495 | |

4.0 | 0.53444 | 0.02373 | 0.01141 | 2.56666 | |

5.0 | 0.50062 | 0.02176 | 0.07559 | 2.61164 | |

6.0 | 0.47399 | 0.02011 | 0.12039 | 2.65173 | |

7.0 | 0.45225 | 0.01871 | 0.15347 | 2.68603 | |

2.0 | 2.0 | 0.69540 | 0.02170 | -0.35350 | 2.69440 |

3.0 | 0.62680 | 0.02080 | -0.15040 | 2.62680 | |

4.0 | 0.57910 | 0.01930 | -0.03790 | 2.64750 | |

5.0 | 0.54310 | 0.01800 | 0.03400 | 2.68340 | |

6.0 | 0.51470 | 0.01680 | 0.08400 | 2.71910 | |

7.0 | 0.49140 | 0.01570 | 0.12090 | 2.75120 | |

2.5 | 2.0 | 0.72980 | 0.01740 | -0.40310 | 2.80610 |

3.0 | 0.66000 | 0.01720 | -0.18150 | 2.69350 | |

4.0 | 0.61070 | 0.01640 | -0.05970 | 2.70010 | |

5.0 | 0.57340 | 0.01540 | 0.01780 | 2.73070 | |

6.0 | 0.54380 | 0.01440 | 0.07160 | 2.76440 | |

7.0 | 0.51950 | 0.01360 | 0.11120 | 2.79570 | |

3.0 | 2.0 | 0.81500 | 0.00830 | -0.48390 | 3.03600 |

3.0 | 0.74510 | 0.00940 | -0.21230 | 2.81430 | |

4.0 | 0.69370 | 0.00950 | -0.06610 | 2.79630 | |

5.0 | 0.65380 | 0.00930 | 0.02610 | 2.82280 | |

6.0 | 0.62160 | 0.00900 | 0.08980 | 2.85860 | |

7.0 | 0.59490 | 0.00860 | 0.13640 | 2.89410 |

Graphs of mean and variance of TIIHLKw distribution.

Graphs of skewness and kurtosis of TIIHLKw distribution.

We see a monotonic variation in these measures caused by variation in parameters

The moment-generating function of

The

Following are expressions used to get mean deviation about mean and median, respectively

Bonferroni and Lorenz curves are important applications of the first incomplete moment. These curves are mostly used in different fields of life such as economics, reliability analysis, demographic studies, life testing, life insurance, and medical technology. The Lorenz and Bonferroni curves are obtained, respectively, as follows

In statistical theory, order statistics is widely applied and practiced. Let

More, the

By substituting (

Then,

Renyi (1961) proposed and used this measure. It can be obtained by

Applying binomial expansion (

So, the Rényi entropy of TIIHLKw distribution is as follows

This subsection deals with the stress-strength parameter of TIIHLKw distribution. Let

Under the assumption discussed above, we have

The reliability is defined by

Then, we can write

This section is dedicated to estimation aspects of TIIHLKw distribution, assuming that population parameters (

For a random sample of

The members of

The LSEs of

The sum is independent of the unknown parameters.

Equivalently,

The following function is minimized with respect to

i.e.,

Under the specification defined above including order statistics having relationship

The CVE is another estimation method based on minimum distance. The CV estimators are obtained by minimizing, with respect to

CV minimum distance estimators provide empirical evidence that the bias of this estimator is smaller than the other minimum distance estimators.

In this section, we present a Monte Carlo (MC) simulation study in order to illustrate the behavior of different estimates. We consider four random sample sizes:

Estimates of parameters and their corresponding MSEs are calculated using the same sample under different methods of estimation for four sets of parameters mentioned above for small sample size (

The entries of Table

Estimates and MSEs of parameters of TIIHLKw Distribution for the Set1 (

MLEs | LSEs | WLSEs | CVEs | PCEs | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSE | |

50 | 2.285 | 0.821 | 1.733 | 0.263 | 1.833 | 0.442 | 1.751 | 0.293 | 1.727 | 0.317 |

1.97 | 0.467 | 2.566 | 0.927 | 2.586 | 1.327 | 2.776 | 1.325 | 2.747 | 1.31 | |

2.092 | 0.077 | 1.961 | 0.258 | 2.035 | 0.23 | 2.099 | 0.268 | 2.078 | 0.372 | |

100 | 2.226 | 0.393 | 1.741 | 0.239 | 1.755 | 0.392 | 1.703 | 0.235 | 1.616 | 0.304 |

1.925 | 0.329 | 2.554 | 0.84 | 2.651 | 1.238 | 2.713 | 1.128 | 2.810 | 1.221 | |

2.056 | 0.031 | 1.961 | 0.123 | 1.952 | 0.093 | 2.014 | 0.099 | 1.929 | 0.078 | |

200 | 1.838 | 0.187 | 1.759 | 0.20 | 1.975 | 0.327 | 1.710 | 0.217 | 1.700 | 0.236 |

2.287 | 0.247 | 2.566 | 0.83 | 2.295 | 0.869 | 2.611 | 0.917 | 2.634 | 0.905 | |

1.95 | 0.018 | 1.996 | 0.065 | 2.001 | 0.08 | 1.957 | 0.06 | 1.967 | 0.069 | |

500 | 1.901 | 0.079 | 1.709 | 0.205 | 1.872 | 0.26 | 1.764 | 0.173 | 1.706 | 0.207 |

2.175 | 0.129 | 2.572 | 0.804 | 2.417 | 0.853 | 2.503 | 0.712 | 2.564 | 0.783 | |

1.973 | 3.377 | 1.93 | 0.032 | 1.991 | 0.031 | 1.963 | 0.029 | 1.921 | 0.029 |

^{^-3}.

Estimates and MSEs of parameters of TIIHLKwD for the Set2 (

MLEs | LSEs | WLSEs | CVEs | PCEs | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |

50 | 2.209 | 0.417 | 1.941 | 0.291 | 1.95 | 0.369 | 2.069 | 0.365 | 2.12 | 0.327 |

2.833 | 0.492 | 3.461 | 1.365 | 3.499 | 1.871 | 3.454 | 1.49 | 3.19 | 1.119 | |

2.024 | 0.044 | 2.02 | 0.164 | 2.033 | 0.25 | 2.17 | 0.289 | 2.171 | 0.264 | |

100 | 2.258 | 0.377 | 1.973 | 0.218 | 1.978 | 0.329 | 1.976 | 0.245 | 1.974 | 0.24 |

2.689 | 0.449 | 3.335 | 1.12 | 3.504 | 1.752 | 3.35 | 1.127 | 3.3 | 1.01 | |

2.02 | 0.022 | 2.034 | 0.095 | 2.036 | 0.128 | 2.015 | 0.093 | 2.005 | 0.113 | |

200 | 2.207 | 0.132 | 1.986 | 0.198 | 1.938 | 0.329 | 1.946 | 0.212 | 2.031 | 0.218 |

2.728 | 0.2 | 3.285 | 1.006 | 3.559 | 1.708 | 3.491 | 1.213 | 3.277 | 0.99 | |

2.052 | 7.539 | 2.002 | 0.046 | 2.017 | 0.055 | 2.053 | 0.062 | 2.063 | 0.074 | |

500 | 2.14 | 0.03 | 1.95 | 0.182 | 1.961 | 0.323 | 1.979 | 0.189 | 1.958 | 0.186 |

2.78 | 0.07 | 3.317 | 0.925 | 3.454 | 1.707 | 3.251 | 0.896 | 3.294 | 0.902 | |

2.039 | 3.826 | 1.985 | 0.022 | 1.991 | 0.028 | 1.985 | 0.029 | 1.985 | 0.025 |

Estimates and MSEs of parameters of TIIHLKwD for the Set3 (

MLEs | LSEs | WLSEs | CVEs | PCEs | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |

50 | 0.262 | 1.909 | 1.746 | 0.285 | 1.942 | 0.689 | 1.828 | 0.373 | 1.728 | 0.309 |

0.85 | 2.376 | 2.635 | 1.26 | 2.569 | 1.538 | 2.702 | 1.461 | 2.844 | 1.888 | |

0.111 | 2.924 | 2.931 | 0.863 | 3.063 | 0.941 | 3.114 | 0.64 | 3.055 | 1.001 | |

100 | 0.197 | 2.131 | 1.671 | 0.261 | 1.927 | 0.489 | 1.686 | 0.26 | 1.704 | 0.215 |

0.231 | 1.983 | 2.717 | 1.11 | 2.554 | 1.44 | 2.731 | 1.195 | 2.765 | 1.32 | |

0.094 | 3.134 | 2.782 | 0.258 | 3.003 | 0.317 | 2.866 | 0.286 | 2.839 | 0.221 | |

200 | 0.119 | 2.007 | 1.729 | 0.199 | 1.945 | 0.483 | 1.727 | 0.211 | 1.823 | 0.162 |

0.16 | 2.063 | 2.598 | 0.962 | 2.478 | 1.167 | 2.643 | 0.968 | 2.509 | 0.904 | |

0.046 | 2.968 | 2.808 | 0.156 | 3.034 | 0.27 | 2.886 | 0.194 | 2.919 | 0.127 | |

500 | 0.077 | 2.145 | 1.787 | 0.158 | 2.003 | 0.468 | 1.764 | 0.166 | 1.789 | 0.152 |

0.099 | 1.876 | 2.464 | 0.637 | 2.391 | 0.983 | 2.497 | 0.727 | 2.531 | 0.793 | |

0.034 | 3.11 | 2.863 | 0.111 | 3.072 | 0.254 | 2.836 | 0.11 | 2.928 | 0.085 |

^{^-3}.

Estimates and MSEs of parameters of TIIHLKw Distribution for the Set4 (

MLEs | LSEs | WLSEs | CVEs | PCEs | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | Estimates | MSEs | |

50 | 1.251 | 0.2 | 1.321 | 0.614 | 1.353 | 0.817 | 1.331 | 0.645 | 1.471 | 1.469 |

1.885 | 0.408 | 1.822 | 0.74 | 1.883 | 1.103 | 1.838 | 0.792 | 1.865 | 0.874 | |

2.104 | 0.099 | 2.246 | 0.571 | 2.236 | 0.742 | 2.379 | 1.046 | 2.5 | 1.88 | |

100 | 1.232 | 0.157 | 1.182 | 0.195 | 1.229 | 0.5 | 1.226 | 0.333 | 1.341 | 0.501 |

1.674 | 0.239 | 1.828 | 0.463 | 1.97 | 0.793 | 1.954 | 0.701 | 1.75 | 0.751 | |

2.131 | 0.069 | 2.145 | 0.294 | 2.116 | 0.181 | 2.247 | 0.506 | 2.307 | 0.403 | |

200 | 1.178 | 0.047 | 1.164 | 0.11 | 1.156 | 0.229 | 1.23 | 0.132 | 1.181 | 0.1 |

1.66 | 0.157 | 1.79 | 0.34 | 1.962 | 0.486 | 1.713 | 0.315 | 1.81 | 0.319 | |

2.09 | 0.011 | 2.062 | 0.095 | 2.117 | 0.119 | 2.169 | 0.123 | 2.165 | 0.125 | |

500 | 1.127 | 0.022 | 1.152 | 0.077 | 1.194 | 0.13 | 1.141 | 0.066 | 1.112 | 0.058 |

1.753 | 0.089 | 1.746 | 0.222 | 1.746 | 0.261 | 1.781 | 0.201 | 1.849 | 0.163 | |

2.068 | 5.923 | 2.092 | 0.038 | 2.098 | 0.056 | 2.066 | 0.035 | 2.065 | 0.034 |

^{^-3}.

Ranks of all the methods of estimation for different parametric specifications.

Parameters | ML | LS | WLS | PC | CV | |
---|---|---|---|---|---|---|

50 | 2 | 1 | 4.5 | 3 | 4.5 | |

100 | 1 | 3.5 | 5 | 2 | 3.5 | |

200 | 1 | 2 | 5 | 3 | 4 | |

500 | 1 | 4 | 5 | 2 | 3 | |

50 | 2 | 1 | 4.5 | 4.5 | 3 | |

100 | 1.5 | 1.5 | 5 | 4 | 3 | |

200 | 1 | 2 | 5 | 3.5 | 3.5 | |

500 | 1 | 2 | 5 | 4 | 3 | |

50 | 1 | 2 | 5 | 3 | 4 | |

100 | 1 | 3 | 5 | 4 | 2 | |

200 | 1 | 3 | 5 | 4 | 2 | |

500 | 1 | 3 | 5 | 4 | 2 | |

50 | 1 | 2 | 4 | 3 | 5 | |

100 | 1 | 2 | 3.5 | 3.5 | 5 | |

200 | 1 | 2 | 5 | 3.5 | 3.5 | |

500 | 1 | 4 | 5 | 3 | 2 | |

18.5 | 38 | 76.5 | 54 | 53 | ||

Overall rank | 1 | 2 | 5 | 4 | 3 |

MLEs and goodness-of-fit measures for the first data set.

Model | L | AIC | BIC | |||||
---|---|---|---|---|---|---|---|---|

TIIHLKw | 8.45118 | 28229.6 | 0.209229 | 57.6664 | -109.333 | -104.597 | 0.336229 | 0.014361 |

Kw | 2.07740 | 33.1374 | — | 56.0687 | -108.137 | -104.313 | 0.688407 | 0.105252 |

Tkw | 2.07740 | 16.5687 | 0.99999 | 56.0687 | -106.137 | -100.401 | 0.688407 | 0.105252 |

SBKw | 1.44716 | 19.9669 | — | 55.2067 | -106.413 | -102.589 | 0.838017 | 0.127612 |

Beta | 2.68257 | 13.8658 | — | 54.6067 | -105.213 | -97.4772 | 0.926850 | 0.155443 |

MLEs and goodness-of-fit measures for the second data set.

Model | L | AIC | BIC | A | W | |||
---|---|---|---|---|---|---|---|---|

TIIHLKw | 0.32008 | 5.84284 | 222.766 | 58.5465 | -111.093 | -105.479 | 0.17531 | 0.025823 |

Kw | 2.71874 | 44.6604 | — | 52.4915 | -100.983 | -97.2407 | 1.31057 | 0.208116 |

TKw | 2.98542 | 43.9205 | 0.651548 | 53.7972 | -101.594 | -95.9808 | 1.02032 | 0.155492 |

SBKw | 2.19509 | 32.9024 | — | 53.6088 | -103.218 | -99.4752 | 1.10420 | 0.174441 |

Beta | 5.94177 | 21.2057 | — | 55.6002 | -107.200 | -99.5868 | 0.78992 | 0.131428 |

Using the entries of Table

The TIIHLKw distribution aims at providing an alternative distribution to fit data on the unit interval to other distributions available in the literature. Here, we used the following probability distributions as competitor models;

The Kumaraswamy (Kw) distribution

Transmuted Kumaraswamy (TKw) distribution

Size biased Kumaraswamy (SBKw) distribution

Beta distribution

We use the following accuracy measures for model comparison: Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), log-likelihood, Anderson-Darling (

^{st} data set.

^{nd} data set.

The total test time (TTT) for both datasets are presented in Figure

Total test time (TTT) for both data sets.

The MLEs for TIIHLKw distribution along with some adequacy measures are presented in Tables

Hence, it is concluded that the new model provides the better fit. Figures

The fitted pdf, cdf, survival, and PP plots of the TIIHLKw distribution for the first data.

The fitted pdf, cdf, survival, and PP plots of the TIIHLKw distribution for second data.

In this article, a new Type-II Half Logistic Kumaraswamy distribution is proposed. Some characteristics of the TIIHLKw distribution including linear combination expressions for the density function, probability weighted moments, moments, incomplete moments, quantile function, mean deviation about mean and about median, Bonferroni and Lorenz curves, order statistics, stress-strength reliability, and Rényi entropy are derived. The ML method is used to estimate model parameters. An extensive simulation study is conducted to compare several well-known estimation methods, including the method of maximum likelihood estimation, methods of least squares and weighted least squares estimation, and method of Cramer-von Mises minimum distance estimation. The simulation study showed the reliability and efficiency of the estimates. Finally, by considering the method of maximum likelihood estimation, the new model is fitted to two practical data sets. The applications on real data sets validated the significance of the new distribution.

0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16.

0.0903296, 0.2036540, 0.2043140, 0.2808870, 0.1976530, 0.3286410, 0.1486220, 0.1623940, 0.2627270, 0.1794550, 0.3266350, 0.2300810, 0.1833120, 0.1509440, 0.2000710, 0.1918020, 0.1541920, 0.4641250, 0.1170630, 0.1481410, 0.1448100, 0.1330830, 0.2760160, 0.4204770, 0.1224170, 0.2285950, 0.1138520, 0.2252140, 0.1769690, 0.2007440, 0.1670450, 0.2316230, 0.2910290, 0.3412730, 0.4387120, 0.2626510, 0.1896510, 0.1725670, 0.2400770, 0.3116460, 0.1635860, 0.1824530, 0.1641270, 0.1534810, 0.1618650, 0.2760160, 0.2538320, 0.2004470.

Data is present in appendix.

The authors declare that they have no conflicts of interest.

This work was funded by the Deanship of Scientific Research (DSR), King Abdul Aziz University, Jeddah, under grant No. (DF-287-305-1441). The authors gratefully acknowledge the DSR technical and financial support.