Generalized Method of Moments and Percentile Method: Estimating Parameters of the Novel Median Based Unit Weibull Distribution

Preprint

Article

Generalized Method of Moments and Percentile Method: Estimating Parameters of the Novel Median Based Unit Weibull Distribution

Altmetrics

Downloads

Views

Comments

Iman Attia^*

This version is not peer-reviewed

This preprints belongs to the Topic

Big Data Intelligence: Methodologies and Applications

Submitted:

28 October 2024

Posted:

29 October 2024

You are already at the latest version

Alerts

Abstract

The Median Based Unit Weibull is a new 2 parameter unit Weibull distribution defined on the unit interval (0,1). Estimation of the parameters using MLE encountered some problems like large variance. Using generalized method of moments (GMMs) and percentile method may ameliorate this condition. This paper introduces GMMs and the percentile methods for estimating the parameters of the new distribution with illustrative real data analysis.

Keywords:

Subject: Computer Science and Mathematics - Probability and Statistics

Introduction

Waloddi Weibull(1951) was the first to introduce the Weibull distribution. It is one of the famous distributions used to model life data and reliability. It can describe the increasing failure rate cases as well as the decreasing failure rate cases. The exponential distribution is a special case of it, when the shape parameter is one. Rayleigh distribution is another special case of it, when the shape parameter is 2. It can also describe and explain the life expectancy of the elements entailed in the fatigue derived failure and can also evaluate the electron tube reliability and load handling machines. It is used in many fields like medicine, physics, engineering, biology, and quality control. As the distribution does not represent a bathtub or unimodal shapes, this enforces many researchers to generalize and transform this distribution in the recent decades. To mention some of these researchers:, (Singla et al., 2012) elucidated beta generalized weibull, (Khan et al., 2017) described in details the transmuted weibull, (Xie et al., 2002) explored the modified weibull, (Lee et al., 2007) clearly explained the beta weibull, (Cordeiro et al., 2010) demonstrated Kumaraswamy weibull, (Silva et al., 2010) expounded beta modified weibull, (Mudholkar & Srivastava, 1993) expatiated the exponentiated weibull, (Zhang & Xie, 2011) interpreted truncated weibull, (Khan & King, 2013) explicated transmuted modified weibull, and (Marshall & Olkin, 1997) handled the extended weibull.

The generalized method of moments has been developed by many authors like (Benson, M.A., 1968) and the United States Water Resources Council (US Water Resources Council, 1967; US Water Resources Council, 1981) , (Bob~e, B., 1975) , (Hoshi, K. and Burges, S.J., 1981) , (Rao, D.V, 1980) , (Ashkar, F. and Bob~e, a., 1986a; Ashkar, F. and Bob~e, B., 1986B) , (Bob~e, B. and Ashkar, F, 1986) used method of SAM. All these authors applied the most famous four methods of generalized method of moments to the Log Pearson type 3 (LP) distribution. But these methods can be applied to any given distribution and not specific to the LP distribution. As shown by (Shenton, L.R. and Bowman, K.O, 1977) , the maximum likelihood estimators are moment estimators in case of LP distribution. (Ashkar et al., 1988) introduced GMMs for estimation of the parameters of generalized gamma distribution.

Estimation using the method of percentiles has been used for a long time. It has been used in estimation by many authors and has been found to be better or equally efficient to MLE and least squares (Bhatti et al., 2018; Dubey, 1967; Marks, 2005; Wang & Keats, 1995).

In this paper, the author will discuss the GMMs and the percentile method to estimate the parameters of the new distribution MBUW. The author will also discuss application of these methods on real data.

The paper is arranged into 4 sections. In section 1, the author will explain the methodology of estimating the parameters using GMMs. In section 2, elaboration of the percentile method of estimation applied to the MBUW distribution. In section 3, analysis of some real data sets to illustrate to what extent the distribution can fit the data. In section 4, conclusion and future work will be declared.

Section 1

Methodology

Generalized Method of Moments (GMMs):

The BMUW distribution has been discussed in previous work by the author as regards properties and some methods of estimation with applications on real data analysis. The new distribution has the following pdf , cdf and quantilel function respectively,

f (y) = \frac{6}{α^{β}} [1 - y^{\frac{1}{α^{β}}}] y^{(\frac{2}{α^{β}} - 1)}, 0 < y < 1, α > 0, β > 0

(1)

F (y) = 3 y^{\frac{2}{α^{β}}} - 2 y^{\frac{3}{α^{β}}}, 0 < y < 1, α > 0, β > 0

(2)

u = F (y) = 3 y^{\frac{2}{α^{β}}} - 2 y^{\frac{3}{α^{β}}} = - 2 {(y^{\frac{1}{α^{β}}})}^{3} + 3 {(y^{\frac{1}{α^{β}}})}^{2}

(3)

The inverse of the CDF (quantile) is used to obtain y, the real root of this 3rd polynomial function is :

y = F^{- 1} (y) = {\{- .5 (c o s [\frac{c o s^{- 1} (1 - 2 u)}{3}] - \sqrt{3} s i n [\frac{c o s^{- 1} (1 - 2 u)}{3}]) + .5\}}^{α^{β}}

(4)

GMMs uses the raw moments of the distribution. Equating the sample moments with the population moments is the main corner of the method. As the MBUW has 2 parameters, the first and second moments will be used to form a system of equations solved numerically to find the parameters.

Steps of algorithm:

The non-central sample moments of order r for a given sample y₁, y₂,…., y_N are defined as :

m_{r}^{'} (y) = \frac{1}{N} \sum_{i = 1}^{r} y_{i}^{r}

(5)

2.: For this distribution, the non-central population moment has the following formula:

μ_{r}^{'} = E (y^{r}) = \frac{6}{(2 + r α^{β}) (3 + r α^{β})}

(6)

μ_{1}^{'} = E (y) = \frac{6}{(2 + α^{β}) (3 + α^{β})}

(7)

μ_{2}^{'} = E (y^{2}) = \frac{6}{(2 + 2 α^{β}) (3 + 2 α^{β})}

(8)

3.: Equate the sample moment and the population moment for the first and second order.

m_{r}^{'} (y) = μ_{r}^{'}

(9)

m_{1}^{'} (y) = μ_{1}^{'}

, this leads to the following equation:

m_{1}^{'} (y) = μ_{1}^{'} = \frac{1}{N} \sum_{i = 1}^{r} y = \bar{y} = \frac{6}{(2 + α^{β}) (3 + α^{β})} \dots

(10)

m_{2}^{'} (y) = μ_{2}^{'}

, this leads to the following equation:

m_{2}^{'} (y) = μ_{2}^{'} = \frac{1}{N} \sum_{i = 1}^{r} y^{2} = \frac{6}{(2 + 2 α^{β}) (3 + 2 α^{β})}

(11)

4.: Differentiate equation 10 and 11 with respect to both parameters and solve the system of equations using Levenberg-Marquardt algorithm (LM).

\frac{\partial}{\partial α} μ_{1}^{'} = \bar{y} (5 β α^{β - 1} + 2 β α^{2 β - 1}) - 6 = 0

(12)

\frac{\partial}{\partial β} μ_{1}^{'} = \bar{y} (5 α^{β} \ln (α) + 2 α^{2 β} \ln (α)) - 6 = 0

(13)

\frac{\partial}{\partial α} μ_{2}^{'} = m_{2}^{'} (10 β α^{β - 1} + 8 β α^{2 β - 1}) - 6 = 0

(14)

\frac{\partial}{\partial β} μ_{2}^{'} = m_{2}^{'} (10 α^{β} \ln (α) + 4 α^{2 β} \ln (α)) - 6 = 0

(15)

The objective function to be minimized is

E (y^{r}) = \frac{6}{(2 + r α^{β}) (3 + r α^{β})}

θ^{(a + 1)} = θ^{(a)} + {[J^{'} (θ^{(a)}) J (θ^{(a)}) + λ^{(a)} I^{(a)}]}^{- 1} [J' (θ^{(a)})] (y - f (θ^{(a)}))

(16)

Where:

The Jacobian matrix is

[\begin{matrix} e q u a t i o n 12 & e q u a t i o n 14 \\ e q u a t i o n 13 & e q u a t i o n 15 \end{matrix}]

(y - f (θ^{(a)})) = [\begin{matrix} m_{1}^{'} - \frac{6}{(2 + α^{β}) (3 + α^{β})} \\ m_{2}^{'} - \frac{6}{(2 + 2 α^{β}) (3 + 2 α^{β})} \end{matrix}]

Where the parameters used in the first iteration are the initial guess, then they are updated according to the sum of squares of errors.

LM algorithm is an iterative algorithm.

J^{'} (θ^{(a)})

: this is the Jacobian function which is the first derivative of the objective function evaluated at the initial guess

θ^{(a)}

λ^{(a)}

is a damping factor that adjust the step size in each iteration direction, the starting value usually is 0.001 and according to the sum square of errors (SSE) in each iteration this damping factor is adjusted:

S S E^{(a + 1)} \geq S S E^{(a)} s o u p d a t e : λ_{u p d a t e d} = 10 * λ_{o l d}

S S E^{(a + 1)} < S S E^{(a)} s o u p d a t e : λ_{u p d a t e d} = \frac{1}{10} * λ_{o l d}

f (θ^{(a)})

: is the objective function (non-central population moment, first and second order) evaluated at the initial guess.

y

: is the non-central sample moments, the first and second order.

Steps of algorithm:

Start with the initial guess of parameters (alpha and beta).
Substitute these values in the objective function and the Jacobian.
Choose the damping factor, say lambda=0.001
Substitute in equation (LM equation (16)) to get the new coefficients.
Calculate the SSE at these parameters and compare this SSE value with the previous one when using initial parameters to adjust for the damping factor.
Update the damping factor accordingly as previously explained.
Start new iteration with the new parameters and the new updated damping factor, i.e , apply the previous steps many times till convergence is achieved or a pre-specified number of iterations is accomplished.

The value of this quantity:

{[J^{'} (θ^{(a)}) J (θ^{(a)}) + λ^{(a)} I^{(a)}]}^{- 1}

can be considered a good approximation to the variance – covariance matrix of the estimated coefficients. Standard errors for the estimated coefficients are the square root of the diagonal of the elements in this matrix.

Section 2

The Percentile Method

The percentile method used in this paper is applied to the 25th and 75th percentiles. The MBUW has the quantile or the percentile function

y = {\{- .5 (c o s [\frac{c o s^{- 1} (1 - 2 u)}{3}] - \sqrt{3} s i n [\frac{c o s^{- 1} (1 - 2 u)}{3}]) + .5\}}^{α^{β}}

c = - .5 (c o s [\frac{c o s^{- 1} (1 - 2 u)}{3}] - \sqrt{3} s i n [\frac{c o s^{- 1} (1 - 2 u)}{3}]) + .5, 0 < u < 1

(17)

y_{25} = {\{- .5 (c o s [\frac{c o s^{- 1} (1 - 2 * 0.25)}{3}] - \sqrt{3} s i n [\frac{c o s^{- 1} (1 - 2 * 0.25)}{3}]) + .5\}}^{α^{β}}

(18)

y_{75} = {\{- .5 (c o s [\frac{c o s^{- 1} (1 - 2 * .75)}{3}] - \sqrt{3} s i n [\frac{c o s^{- 1} (1 - 2 * .75)}{3}]) + .5\}}^{α^{β}}

(19)

y_{25} = c_{25}^{α^{β}}, y_{75} = c_{25}^{α^{β}}

(20)

The objective function to be minimized in LM algorithm is the quantile function. Differentiation of this objective function with respect to both parameters gives these equations:

\frac{\partial}{\partial α} y_{25} = c_{25}^{α^{β}} [\ln (c_{25})] β α^{β - 1}

(21)

\frac{\partial}{\partial β} y_{25} = c_{25}^{α^{β}} [\ln (c_{25})] (α^{β}) [\ln (α)]

(22)

\frac{\partial}{\partial α} y_{75} = c_{75}^{α^{β}} [\ln (c_{75})] β α^{β - 1}

(23)

\frac{\partial}{\partial β} y_{75} = c_{75}^{α^{β}} [\ln (c_{75})] (α^{β}) [\ln (α)]

(24)

The Jacobian matrix is

[\begin{matrix} e q u a t i o n 21 & e q u a t i o n 23 \\ e q u a t i o n 22 & e q u a t i o n 24 \end{matrix}]

(y - f (θ^{(a)})) = [\begin{matrix} y_{25} - c_{25}^{α^{β}} \\ y_{75} - c_{75}^{α^{β}} \end{matrix}]

Then use LM algorithm as previously explained.

Any percentiles can be used, provided 2 consecutive percentiles. The most common to use are 25th and 75th percentiles.

Section 3

Some Real Data Analysis

The database, OECD, is used. Some variables are analyzed to discover what distributions fit the data better. The data is available at: https://stats.oecd.org/index.aspx?DataSetCode=BLI

It was used by the author in other works .(Iman M.Attia, 2024)

First data: (Dwelling Without Basic Facilities)

These observations measure the percentage of homes in the involved countries that lack essential utilities like indoor plumbing, central heating, clean drinking water supplies.

0.008	0.007	0.002	0.094	0.123	0.023	0.005	0.005	0.057	0.004
0.005	0.001	0.004	0.035	0.002	0.006	0.064	0.025	0.112	0.118
0.001	0.259	0.001	0.023	0.009	0.015	0.002	0.003	0.049	0.005
0.001	0.03	0.067	0.138	0.359

Second data: (Quality of Support Network)

This data set explores how much the person can rely on sources of support like family, friends, or community members in time of need and disparate. It is represented as percentage of persons who had found social support in times of crises.

0.92	0.93	0.88	0.80	0.82	0.96	0.95	0.96	0.94	0.90
0.78	0.98	0.89	0.92	0.91	0.77	0.94	0.95	0.96	0.85

Third data: (Voter Turnout)

This data set evaluates the percentage of capable and qualified persons for casting a vote in election reflecting the democracy in the country.

0.92	0.76	0.88	0.68	0.47	0.53	0.66	0.62	0.85	0.64
0.69	0.75	0.79	0.58	0.70	0.81	0.63	0.67	0.73	0.53
0.77	0.55	0.57	0.90	0.63	0.79	0.82	0.78	0.68	0.49
0.66	0.53	0.72	0.87	0.45	0.86	0.68	0.65

Fourt data: (Flood Data)

These are 20 observations for the maximum flood level in Susquehanna River at Harrisburg, Penssylvania (Dumonceaux & Antle, 1973).

0.26	0.27	0.3	0.32	0.32	0.34	0.38	0.38	0.39	0.4
0.41	0.42	0.42	0.42	0.45	0.48	0.49	0.61	0.65	0.74

Fifth data: (Time between Failures of Secondary Reactor Pumps)(Maya et al., 2024, 1999) (Suprawhardana and Prayoto)

0.216	0.015	0.4082	0.0746	0.0358	0.0199	0.0402	0.0101	0.0605
0.0954	0.1359	0.0273	0.0491	0.3465	0.007	0.656	0.106	0.0062
0.4992	0.0614	0.532	0.0347	0.1921

Sixth data: (to evaluate the factors concerning the unit capacity, data was collected to compare between algorithms like SC 16 and P3) (Maya et al., 2024, 1999)

0.853	0.759	0.866	0.809	0.717	0.544	0.492	0.403	0.344
0.213	0.116	0.116	0.092	0.07	0.059	0.048	0.036	0.029
0.021	0.014	0.011	0.008	0.006

Fitting the MBUW to the above data revealed results with high variance. The method used was the MLE. These data was previously analyzed by the author for fitting the following unit distributions: Beta distribution, Kumaraswamy distribution, Median Based Unit Rayleigh (BMUR) distribution, and Median Based Unit Weibull (MBUW) distribution. The following table shows the results for fitting the MBUW. This was previously discussed in other author’s works.(Iman M.Attia, 2024),

These are the pdfs of the distributions used in the analysis of these 6 data sets, in addition to two distributions that were used in analysis of other data sets provided by the author in a previous preprint paper discussing the new distribution (MBUR). (Iman M. Attia, 2024)

Beta Distribution:

f (y; α, β) = \frac{Γ (α + β)}{Γ (α) Γ (β)} y^{α - 1} {(1 - y)}^{β - 1}, 0 < y < 1, α > 0, β > 0

2.: Kumaraswamy Distribution:

f (y; α, β) = α β y^{α - 1} {(1 - y^{α})}^{β - 1}, 0 < y < 1, α > 0, β > 0

3.: Median Based Unit Rayleigh:

f (y; α) = \frac{6}{α^{2}} [1 - y^{\frac{1}{α^{2}}}] y^{(\frac{2}{α^{2}} - 1)}, 0 < y < 1, α > 0

Tools of comparison are:

(k) is the number of parameter while (n) is the number of observations.

A I C = - 2 M L L + 2 k, w h e r e k = n u m b e r o f p a r a m e t e r s

A I C - c o r r e c t e d = - 2 M L L + \frac{2 k}{n - k - 1}

H Q I C = 2 \log \{\log (n) [k - 2 M L L]\}

B I C = - 2 M L L + k \log (n)

K S - t e s t = S u p_{n} |F_{n} - F|, F_{n} = \frac{1}{n} \sum_{i = 1}^{n} I_{x_{i} < x}

First, second, third data set:

	First data set , n=35		Second data set n=20	Third data set , n=38
theta	$α = 6.636$		$α = 0.257$	$α = 0.5509$
	$β = 0.8726$		$β = 1.5073$	$β = 1.2779$
Var	$3.5 * 10^{6}$	$- 2.5 * 10^{5}$
	$- 2.5 * 10^{5}$	$1.7 * 10^{4}$
SE	316.227		Cannot be estimated	Cannot be estimated
	22.039		Cannot be estiamted	Cannot be estiamted
AIC	152.585		64.079	48.1377
AIC correc	152.9600		64.7848	48.4805
BIC	155.6957		66.0704	51.4128
HQIC	4.2965		3.927	4.0216
NLL	-74.2925		-30.0395	-22.0688
K-S Value	0.1794		0.1309	0.1364
H₀	Fail to Reject		Fail to Reject	Fail to Reject
P-value	0.1860		0.8399	0.4401

Fourth, fifth, sixth data set:

	Fourth data set , n=20		Fifth data set n=23		Sixth data set , n=23
theta	$α = 1.0932$		$α = 5.1285$		$α = 3.7003$
	$β = 0.9719$		$β = 0.7113$		$β = 0.7415$
Var	$1.66 * 10^{4}$	$- 1.7 * 10^{5}$	$1.02 * 10^{7}$	$- 8.7 * 10^{5}$	$5.2 * 10^{6}$	$- 7.8 * 10^{5}$
	$- 1.7 * 10^{5}$	$1.6 * 10^{6}$	$- 8.7 * 10^{5}$	$7.4 * 10^{4}$	$- 7.8 * 10^{5}$	$1.2 * 10^{5}$
SE	28.81		665		475
	28.81		56.7		72.2
AIC	16.9233		43.862		19.2158
AIC correc	17.6292		44.4620		19.8158
BIC	18.9148		46.1330		21.4867
HQIC	3.4805		3.8543		3.5773
NLL	-6.4617		-19.9310		-7.6079
K-S Value	0.3202		0.1584		0.1518
H₀	Fail to Reject		Fail to Reject		Fail to Reject
P-value	0.0253		0.5575		0.4074

As shown from the above analysis, MLE yielded large variance. GMMs are used as previously described in section 3. The initial guess used in LM algorithm was the values obtained from MLE method as shown in the table for each data set. The variances obtained from GMMs method are dramatically small compared to one used by MLE for each data set. The parameter estimation results were assessed by goodness of fit procedures like: KS-test and visualized by qq-plot and the pp-plot.

The following table shows the results of GMMs:

	First data set , n=35		Second data set n=20		Third data set , n=38
theta	$α = 7.2434$		$α = 0.2381$		$α = 0.5474$
	$β = 0.8744$		$β = 1.4903$		$β = 1.281$
Var	$.1458$	$- 0.7633$	0.0555	-0.0512	0.033	-0.0292
	$- 0.7633$ .	$0.0617$	-0.0512	0.0699	-0.0292	0.0441
SE	0.586		0.053		0.029
	0.042		0.059		0.034
SSE	0.001		0.0003		0.000001
K-S Value	0.1804		0.157		0.1409
H₀	Fail to Reject		Fail to Reject		Fail to Reject
P-value	0.081		0.6515		0.4008
$m_{1}^{'}$	0.0475		0.9005		0.6911
$m_{2}^{'}$	0.0081		0.8148		0.4928

	Fourth data set , n=20		Fifth data set n=23		Sixth data set , n=23
theta	$α = 1.1955$		$α = 5.0368$		$α = 3.4012$
	$β = 1.056$		$β = 0.7077$		$β = 0.7116$
Var	$0.1468$	$- 0.1623$	$1.3712$	$- 0.1323$	$0.2279$	$- 0.0554$
	$- 0.1623$	$0.2404$	$- 0.1323$	$0.0248$	$- 0.0554$	$0.0264$
SE	0.086		0.244		0.099
	0.109		0.175		0.034
SSE	0.0035		0.00007		0.0106
K-S Value	0.2694		0.1682		0.2046
H₀	Fail to Reject		Fail to Reject		Fail to Reject
P-value	0.0899		0.4822		0.2537
$m_{1}^{'}$	0.4225		0.1578		0.2881
$m_{2}^{'}$	0.1932		0.0606		0.1798

The following figures illustrate the ecdf v.s. the theoretical cdf, the pp-plots, and the qq-plots for the MBUW fitted data using the GMMs.

Figure 1. shows the eCDF vs. theoretical CDF of the BMUW distributions for the 1st data set.

Figure 2. shows the pp-plot of the BMUW distributions for the 1st data set.

Figure 3. shows the qq-plot of the BMUW distributions for the 1st data set.

Figure 4. shows the eCDF vs. theoretical CDF of the BMUW distributions for the 2nd data set.

Figure 5. shows the pp-plot of BMUW distributions for the 2nd data set.

Figure 6. shows the qq-plot of BMUW distributions for the 2nd data set.

Figure 7. shows eCDF vs. theoretical CDF of the BMUW distributions for the 3rd data set.

Figure 8. :shows the pp-plot of BMUW distributions for the 3rd data set.

Figure 9. :shows the qq-plot of BMUW distributions for the 3rd data set.

Figure 10. shows eCDF vs. theoretical CDF of the BMUW distributions for the 4th data set.

Figure 11. :shows the pp-plot of BMUW distributions for the 4th data set.

Figure 12. :shows the qq-plot of BMUW distributions for the 4th data set.

Figure 13. shows eCDF vs. theoretical CDF of the BMUW distributions for the 5th data set.

Figure 14. :shows the pp-plot of BMUW distributions for the 5th data set.

Figure 15. :shows the qq-plot of BMUW distributions for the 5th data set.

Figure 16. shows eCDF vs. theoretical CDF of the BMUW distributions for the 6th data set.

Figure 17. shows the pp-plot of BMUW distributions for the 6th data set.

Figure 18. shows the qq-plot of BMUW distributions for the 6th data set.

The figures show the ecdf for the MBUW versus the theoretical MBUW, it is a more or less fit to the tested distribution as h₀ hypothesis cannot be failed. The sum square of error is small. The variances show dramatic reduction as compared to the values obtained from MLE estimation. The values obtained by GMMs can be used as initial guess in other modified MLE methods or other methods like least square or weighted least squares.

Analysis of the Data Sets Using the Percentile Method

The same datasets were used for analysis using the percentile method. The initial guess for the LM algorithm were the values obtained from MLE estimation, this worked for some data sets but it failed to give promising results as a valid KS-test (h0= fail to reject the null hypothesis that assumes the BMUW fits the data) . The data sets that give results of fitting the distribution were the quality support and the voter turnout (The second and third data sets).

The following table shows the results of percentile method:

	Second data set n=20		Third data set , n=38
theta	$α = 0.2577$		$α = 0.6062$
	$β = 1.5071$		$β = 1.2669$
Var	52.4635	219.6857	55.7481	226.1499
	219.6857	949.066	226.1499	945.8367
SE	1.619		1.211
	6.889		4.989
SSE	0.0072		0.0536
K-S Value	0.1297		0.1134
H₀	Fail to Reject		Fail to Reject
P-value	0.8477		0.4106
$y_{25}$	0.865		0.62
$y_{75}$	0.95		0.78

The following figures illustrate the ecdf, pp-plot and qq-plot for the results obtained from the MBUW fitting the 2 data sets using the percentile method.

Figure 19. shows eCDF vs. theoretical CDF of the BMUW distributions for the 2nd data set.

Figure 20. shows the pp-plot of BMUW distributions for the 2nd data set.

Figure 21. shows the qq-plot of BMUW distributions for the 2nd data set.

Figure 22. shows eCDF vs. theoretical CDF of the BMUW distributions for the 3rd data set.

Figure 23. shows the pp-plot of BMUW distributions for the 3rd data set.

Figure 24. shows the qq-plot of BMUW distributions for the 3rd data set.

Discussion of the Results

The analysis of the previous data sets using GMMs with the initial guess for the LM algorithm resulted in values of estimated parameters more or less equal to that given using the MLE but with dramatic reduced variance . This allows standard errors to be more accurately estimated. The percentile method was applicable to only 2 data sets. The variance obtained from this method, although it is less than that estimated from MLE, it is larger than that estimated from GMMs methods. LM algorithm is more useful in estimation as its results are more stable than other optimization algorithm. The estimated parameter values obtained from MLE method, although it may unstable with this large variance, it could be used as an initial guess in other methods like GMMs and percentile method. Moreover, the values obtained from these GMMs and percentile method can be used as an initial guess to modified MLE method and other methods like least square method (LS), weighted least square method (WLS), Anderson Darling Estimator (ADE), Cramer Von Mises Estimator (CVM), and Maximum Product of Spacing (MPS). Because the initial guess is really a challenging task, the GMMs and percentile may help in this task. The KS-test results failed to reject the null hypothesis, so it is most probably that the data follow the BMUW distribution with the p-value reported for each data set in each method. The pp-plot and qq-plot figures can visualize the fitness of the data to the distribution, but they are subjective tests rather than the KS-test which is more objective.

Stage 4:

Conclusions

This new approach of the GMMs method and percentile method can enhance the estimation process for the parameters of BMUW distribution with more stable and reliable parameter estimate and small variance. The estimators can be used as an initial guess in other methods that require numerical computations. The 2 parameters of the MBUW are difficult to be defined in term of each other so the approach described in the paper can compute both simultaneously. This is done by solving differential equation using one of the iterative techniques like Levenberg Marquardt algorithm.

Future work

The author is working on modified maximum likelihood methods of estimation and other methods for this new distribution.

Authors Contribution

AI carried the conceptualization by formulating the goals, aims of the research article, formal analysis by applying the statistical, mathematical and computational techniques to synthesize and analyze the hypothetical data, carried the methodology by creating the model, software programming and implementation, supervision, writing, drafting, editing, preparation, and creation of the presenting work.

Conflicts of Interest

The author declares no competing interests of any type.

References

Ashkar, F. and Bob~e, B.,. (1986a). The generalized method of moments as applied to the log Pearson type 3 distribution with some large sample results. J. J. Hydraul. Eng. Am. Soc. Civ. Eng.
Ashkar, F. and Bob~e, B. (1986b). Variance of the T-year event in the log Pearson type 3 distribution, A comment. J. Hydrol., 84:181 187(84:181 187), 84:181 187.
Ashkar, F., Bobée, B., Leroux, D., & Morisette, D. (1988). The generalized method of moments as applied to the generalized gamma distribution. Stochastic Hydrology and Hydraulics, 2(3), 161–174. [CrossRef]
Benson, M.A.,. (1968). Uniform flood frequency estimating methods for federal agencies. Ater Resour. Res, 4(5): 891-908.(4(5): 891-908.), 4(5): 891-908.
Bhatti, S. H., Hussain, S., Ahmad, T., Aslam, M., Aftab, M., & Raza, M. A. (2018). Efficient estimation of Pareto model: Some modified percentile estimators. PLOS ONE, 13(5), e0196456. [CrossRef]
Bob~e, B. (1975). The log-Pearson type 3 distribution and its application in hydrology. Water Resour. Res, 11(5): 681~689.(11(5): 681~689.), 11(5): 681~689.
Bob~e, B. and Ashkar, F. (1986). Sundry averages method (SAM) for estimating parameters of the log Pearson type 3 distribution. J. Hydraul. Eng. Am. Soc. Civ. Eng.
Cordeiro, G. M., Ortega, E. M. M., & Nadarajah, S. (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347(8), 1399–1429. [CrossRef]
Dubey, S. D. (1967). Some Percentile Estimators for Weibull Parameters. Technometrics, 9(1), 119–129. [CrossRef]
Dumonceaux, R., & Antle, C. E. (1973). Discrimination Between the Log-Normal and the Weibull Distributions. Technometrics, 15(4), 923–926. [CrossRef]
Hoshi, K. and Burges, S.J.,. (1981). Approximate estimation of the derivative of a standard gamma quantile for use in confidence interval estimates. J. Hydrol., 53: 317-325.(53: 317-325.), 53: 317-325.
Iman M. Attita. (2024). A Novel Unit Distribution Named as Median Based Unit Rayleigh (MBUR):Properties and Estimations. Preprints.Org, Preprint, 7 October 2024(7 October 2024). [CrossRef]
Iman M.Attia. (2024). Median Based Unit Weibull (MBUW): ANew Unit Distribution Properties. 25 October 2024, preprint article, Preprints.org(preprint article, Preprints.org). [CrossRef]
Khan, M. S., & King, R. (2013). Transmuted Modified Weibull Distribution: A Generalization of the Modified Weibull Probability Distribution. European Journal of Pure and Applied Mathematics, 6(1), 66–88. Available online: https://www.ejpam.com/index.php/ejpam/article/view/1606.
Khan, M. S., King, R., & Hudson, I. L. (2017). Transmuted Weibull distribution: Properties and estimation. Communications in Statistics - Theory and Methods, 46(11), 5394–5418. [CrossRef]
Lee, C., Famoye, F., & Olumolade, O. (2007). Beta-Weibull Distribution: Some Properties and Applications to Censored Data. Journal of Modern Applied Statistical Methods, 6(1), 173–186. [CrossRef]
Marks, N. B. (2005). Estimation of Weibull parameters from common percentiles. Journal of Applied Statistics, 32(1), 17–24. [CrossRef]
Marshall, A. W., & Olkin, I. (1997). A New Method for Adding a Parameter to a Family of Distributions with Application to the Exponential and Weibull Families. Biometrika, 84(3), 641–652. Available online: https://www.jstor.org/stable/2337585.
Maya, R., Jodrá, P., Irshad, M. R., & Krishna, A. (2024). The unit Muth distribution: Statistical properties and applications. Ricerche Di Matematica, 73(4), 1843–1866. [CrossRef]
Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299–302. IEEE Transactions on Reliability. [CrossRef]
Rao, D.V. (1980). Log Pearson type 3 distribution: Method of mixed moments. J. Hydrol. Div. Am. Soc. Civ. Eng., 106(HY6): 999-1019.(106(HY6): 999-1019.), 106(HY6): 999-1019.
Shenton, L.R. and Bowman, K.O. (1977). Maximum likelihood estimation in small samples. In: A. Stuart (Editor), Grifl~n’s Statistical Monographs and Courses No. 38 McMillan, New York, N.Y, 186 pp(186 pp), 186 pp.
Silva, G. O., Ortega, E. M. M., & Cordeiro, G. M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis, 16(3), 409–430. [CrossRef]
Singla, N., Jain, K., & Kumar Sharma, S. (2012). The Beta Generalized Weibull distribution: Properties and applications. Reliability Engineering & System Safety, 102, 5–15. [CrossRef]
suprawhardana. (1999). Suprawhardana, M.S., Prayoto, S.: Total time on test plot analysis for mechanical components of the RSG-GAS reactor. At. Indones. 25(2), 81–90 (1999). 25(2),81-90(1999), 25(5), 81–90.
US Water Resources Council,. (1967). A uniform technique for determining flood flow frequencies. US Water Resour. Counc., Washington, D.C., Bull, No. 15: 13pp.(No. 15: 13pp.), No. 15: 13pp.
US Water Resources Council. (1981). Guidelines for determining flood-flow frequency. US Water Resour. Counc., Hydrol. Comm., Washington, D.C., Bull., No. 17B: 28pp(No. 17B: 28pp), No. 17B: 28pp.
Wang, F. K., & Keats, J. B. (1995). Improved percentile estimation for the two-parameter Weibull distribution. Microelectronics Reliability, 35(6), 883–892. [CrossRef]
Xie, M., Tang, Y., & Goh, T. N. (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering & System Safety, 76(3), 279–285. [CrossRef]
Zhang, T., & Xie, M. (2011). On the upper truncated Weibull distribution and its reliability implications. Reliability Engineering & System Safety, 96(1), 194–200. [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

MDPI Initiatives

Important Links

Choose an area of interest and we will send you notifications of new preprints at your preferred frequency.

Disclaimer

Generalized Method of Moments and Percentile Method: Estimating Parameters of the Novel Median Based Unit Weibull Distribution

Abstract

Introduction

Methodology

The Percentile Method

Some Real Data Analysis

Analysis of the Data Sets Using the Percentile Method

Discussion of the Results

Conclusions

Future work

Authors Contribution

Conflicts of Interest

References

MDPI Initiatives

Important Links

Subscribe