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Biometrics & Biostatistics International Journal

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Received: January 01, 1970 | Published: ,

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Abstract

Length-biased and Area-biased distributions arise in many forestry applications, as well as other environmental, econometric, and biomedical sampling problems. We examine the Length-biased and Area-biased distributions versions of the Exponentiated Weibull distribution (EW). This study introduced a new distribution based on Length-biased Exponentiated Weibull distribution (LBEW) and Area-biased Exponentiated Weibull distribution (ABEW). Some characteristics of the new distributions were obtained. Plots for the cumulative distribution function, pdf and tables with values of skewness and kurtosis were also provided. Height-Diameter (H-D) data on Bombax and Pines (Pinus caribaea) were used to demonstrate the application of the distributions. Estimation of parameters of EW, LBEW and ABEW distributions were done using the maximum likelihood approach and compared across the distributions using criteria like AIC and Loglikelihood. We therefore proposed that similar to Exponentiated Weibull distribution (EW), a better fitting of Bombax and Pines H-D data are possible by LBEW and ABEW distributions. We hope in numerous fields of theoretical and applied sciences, the findings of this study will be useful for the practitioners.

Keywords: length-biased distribution, area-biased distributions, forestry, bombax, pines, height and diameter

Introduction

Trees contribute to the environment by providing oxygen, improving air quality, climate amelioration, conserving water, preserving soil, and supporting wildlife. During the process of photosynthesis, trees take in carbon dioxide and produce the oxygen we breathe. According to the U.S. Department of Agriculture, "One acre of forest absorbs six tons of carbon dioxide and puts out four tons of oxygen. This is enough to meet the annual needs of 18 people." Trees, shrubs and turf also filter air by removing dust and absorbing other pollutants like carbon monoxide, sulphur dioxide and nitrogen dioxide. After trees intercept unhealthy particles, rain washes them to the ground. Trees can add value to your home, help cool your home and neighborhood break the cold winds to lower your heating costs, and provide food for wildlife.

Height-diameter relationships are used to estimate the heights of trees measured for their diameter at breast height (DBH). Such relationship describes the correlation between height and diameter of the trees in a stand on a given date and can be represented by a linear or non-linear statistical model. In forest inventory designs diameter at breast height is measured for all trees within sample plots, while height is measured for only some selected trees, normally the dominant ones in terms of their DBH. In this study, the two species of trees considered explained thus;

  1. Pinus caribaea:‘Pinus’ is from the Greek word ‘pinos’ (pine tree), possibly from the Celtic term ‘pin’ or ‘pyn’ (mountain or rock), referring to the habitat of the pine.  Pinus caribaea is a fine tree to 20-30m tall, often 35m, with a diameter of 50-80cm and occasionally up to 1m; trunk generally straight and well formed; lower branches large, horizontal and drooping; upper branches often ascending to form an open, rounded to pyramidal crown; young trees with a dense, pyramidal crown. Pinus caribaea is rated as moderately fire resistant. It tolerates salt winds and hence may be planted near the coast.
  2. Bombax costatum:'Bombax' is derived from the Greek 'bombux', meaning silk, alluding to the dense wool-like floss covering the inner walls of the fruits and the seeds. Bombax costatum is a fire resisting tree of the savannas and dry woodlands from Senegal to central Africa, from Guinea across Ghana and Nigeria to southern Chad. Its tuberous roots act as water and/or sugar storage facilities during long drought periods. Usually associated with Pterocarpus erinaceus, Daniellia oliveri, Cordyla pinnata, Parkia biglobosa, Terminalia macroptera and Prosopis africana.

Length-biased and area-biased distribution

When the probability of selecting an individual in a population is proportional to its magnitude, it is called length biased sampling. However, when observations are selected with probability proportional to their length, the resulting distribution is called length-biased. When dealing with the problem of sampling and selection from a length-biased distribution, the possible bias due to the nature of data-collection process can be utilized to connect the population parameters to that of the sampling distribution. That is, biased sampling is not always detrimental to the process of inference on population parameters. Inference based on a biased sample of a certain size may yield more information than that given by an unbiased sample of the same size, provided that the choice mechanism behind the biased sample is known. Statistical analysis based on length-biased samples has been studied in detail since the early 70’s. Size-biased distributions have been found to be useful in probability sampling designs for forestry and other related studies. These designs are classified into length-biased methods where sampling is done with probability proportional to some lineal measure and area-biased methods where units are selected into the sample with probability proportional to some real attributes. Hence, area-biased distribution is the square of the random variable of X or the second order power of size-biased distribution

The concept of length-biased was introduced by Cox in 1962.1 This concept is found in various applications in biomedical area such as family history and disease, survival analysis, intermediate events and latency period of AIDS due to blood transfusion. Many works were done to characterize relationships between original distributions and their length-biased versions. Patill and Rao expressed some basic distributions and their length-biased forms such as log-normal, gamma, pareto, beta distributions. Recently, many researches are applied to length-biased for lifetime distribution, length-biased weighted Weibull distribution, and length-biased weighted generalized Rayleigh distribution, length-biased beta distribution, and Bayes estimation of length- biased Weibull distribution.2

Exponentiated weibull distribution

The Weibull distribution was introduced by Wallodi Weibull, Swedish scientist, in 1951. It is perhaps the most widely used distribution to analyze the lifetime data. Gupta & Kundu3 proposed an Exponentiated Exponential distribution which is a special case of the Exponentiated Weibull family. Flaih et al.,4 extended the Inverted Weibull distribution to the Exponentiated Inverted Weibull (EIW) distribution by adding another shape parameter. This study suggested that the EIW distribution can provide a better fit to the real dataset than the IW distribution. Shittu, O I. and Adepoju, K A.5 the exponentiated Weibull was used as an alternative distribution that adequately describe the wind speed and thereby provide better representation of the potentials of wind energy.

Structural properties of exponentiated weibull distribution: According to Mudhokar, et al.,6 the Exponentiated Weibull density function is defined as;

fEW(x;k,λ,α)=αkxk-1λkexp(-(x/λ)k)(1-exp(-(x/λ)k))α-1fEW(x;k,λ,α)=αkxk1λkexp((x/λ)k)(1exp((x/λ)k))α1 …… (1)
and the cdf is;
FEW(x;k,λ,α)=(1exp((x/λ)k))αFEW(x;k,λ,α)=(1exp((x/λ)k))α

αα  and kk are shape parameters; λλ  is a scale parameter.
the rthrth  moment of the exponentiated weibull is given as;
E(xr)=αλrΓ(rk+1)j=0(1)j(α1j)(1+j)rk+1E(xr)=αλrΓ(rk+1)j=0(1)j(α1j)(1+j)rk+1

Where Γ(rk+1)=0xrkexp(x)xΓ(rk+1)=0xrkexp(x)x  at r=1, the first moment of EW is

E(x)=αλΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1E(x)=αλΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1

at r=2 is the second moment and the variance of EW is given thus;

Var(x)=E(x2)-[E(x)]2Var(x)=E(x2)-[E(x)]2
Var(x)=αλ2Γ(2k+1)j=0(1)j(α1j)(1+j)2k+1α2λ2Γ2(1k+1)[j=0(1)j(α1j)(1+j)1k+1]2Var(x)=αλ2Γ(2k+1)j=0(1)j(α1j)(1+j)2k+1α2λ2Γ2(1k+1)j=0(1)j(α1j)(1+j)1k+12
The skewness and kurtosis of EW
k3=αλΓ(3k+1)j=0(1)j(α1j)(1+j)3k+1[3αλΓ(2k+1)j=0(1)j(α1j)(1+j)2k+1][αλΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1]+2αλΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1[Var(x)]3/2

Materials and methods

In this study, we propose two new distributions which are LBEW and ABEW distributions. We first provide a general definition of the Length-biased and Area-biased distributions which we subsequently reveal their pdfs.
Let f(x;θ) be the pdf of the random variable X  and θ  be the unknown parameter. The weighted distribution is defined as;

k4=αλΓ(4k+1)j=0(1)j(α1j)(1+j)4k+1[4αλΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1][αλΓ(3k+1)j=0(1)j(α1j)(1+j)3k+1]+[6α2λΓ(1k+1)j=0(1)j(α1j)(1+j)2k+1][αλΓ(2k+1)j=0(1)j(α1j)(1+j)2k+1]3α4λΓ(1k+1)j=0(1)j(α1j)(1+j)1k+1[Var(x)]2

g(x;θ)=xmf(x;θ)E[f(x)] XR,θ>0 ………….. (2)
The distributions in equation (2) are termed as size-biased distribution of order m. When m=1, it is called size-biased of order 1 or say length biased distribution, whereas for m=2, it is called the area- biased distribution.

Length-biased EW distribution (LBEW)

If X has a lifetime distribution with pdf f(x) and expected value, E[f(x)]< , the pdf of length-biased distribution of X  can be defined as:
gLB(x;θ)=xf(x;θ)E[f(x)] ………….. (3)

Let X be a random variable of an EW distribution with pdf f(x) .

Then gLB(x;θ)=xf(x;θ)E[f(x)] is a pdf of the LBEW distribution with two shape parameters α and k and a scale parameter λ . The notation for X with the LBEW distribution is denoted as X ~LBEW ( α , k , λ ). The pdf of X is given by:

gLBEW(x;k,λ,α)=kxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1 ………….. (4)

Area-biased EW distribution (ABEW)

If X has a lifetime distribution with pdf f(x) and expected value, E[f(X)] < , the pdf of length-biased distribution of X  can be defined as:

gAB(x;θ)=x2f(x;θ)E[f(x)] ………….. (5)

Let X be a random variable of an EW distribution with pdf f(x) . Then gAB(x;θ)=x2f(x;θ)E[f(x)] is a pdf of the ABEW distribution with two shape parameters α and k and a scale parameter λ . The notation for X with the ABEW distribution is denoted as X~ABEW  ( α , k , λ ). The pdf of X is given by:

gABEW(x;k,λ,α)=kxk+1λk+1exp((x/λ)k)(1exp((x/λ)k))α1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1 ………….. (6)

The properties

The LBEW distribution properties are as follows;

0gLBEW(x)x=0kxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1x
=kλk+1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+10xkexp((x/λ)k)(1exp((x/λ)k))α1x
If y=(x/λ)k , then we have that;
=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+10y1k+2exp(y)(1exp(y))α1y
Recall that,
(1exp(y))α1=j=0(1)j(α1j)exp(yj)
Therefore,
0gLBEW(x)x=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+10y1k+2j=0(1)j(α1j)exp(y[1+j])y
; where m =y[1+j]
=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1[j=0(1)j(α1j)(1+j)1k+1]0m1k+2exp(m)m0gLBEW(x)x=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1[j=0(1)j(α1j)(1+j)1k+1]Γ(1k+1)0gLBEW(x)x=1

Therefore, the pdf of LBEW distribution sum to 1. NB: It was also obtainable for the ABEW distribution.
The cdf of LBEW, corresponding to (4) is obtained by

FLBEW(X)=x0kxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1x
FLBEW(X)=kλk+1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1x0xkexp((x/λ)k)(1exp((x/λ)k))α1x
Let y=(x/λ)k;yλk=xk;x=λy1/k
FLBEW(X)=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1x0ykexp(yk)(1exp(yk))α1x
Let m=y(1+j);y=m1+j
FLBEW(X)=1Γ(1k+1)x0m1/kexp(m)mFLBEW(X)=x1/kexp(x)[1+1kx]Γ(1k+1)[11k+1k2] ………………… (7)

So, the reliability function of LBEW is,

R(x)=1FLBEW(x)=1x1/kexp(x)[1+1kx]Γ(1k+1)[11k+1k2]R(x)=Γ(1k+1)[11k+1k2][1+1kx]x1/kexp(x)Γ(1k+1)[11k+1k2] ……………. (8)

And the hazard function is,

h(x)=f(x)R(x)=kxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1[11k+1k2]{Γ(1k+1)[11k+1k2][1+1kx]x1/kexp(x)}j=0(1)j(α1j)(1+j)1k+1 ……. (9)

The moments

The rth raw moment of the LBEW random variable X is

E(Xr)=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+10xrαkxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1xE(Xr)=1λrΓ(1k+1)Γ(r+1k+1)j=0(1+j)(1/k)+1(1+j)((r+1)/k)+1 …………. (10)

at r = 1, the first moment of LBEW is

E(Xr)=1Γ(1k+1)j=0(1)j(α1j)(1+j)1k+10xrαkxkλk+1exp((x/λ)k)(1exp((x/λ)k))α1xE(Xr)=1λrΓ(1k+1)Γ(r+1k+1)j=0(1+j)(1/k)+1(1+j)((r+1)/k)+1 …………. (11)

And the variance is
                                                                                         Var(X)=1λ2Γ(1k+1)Γ(3k+1)j=0(1+j)(1/k)[1λΓ(1k+1)Γ(2k+1)j=0(1+j)(1/k)]2 …….. (12)

The skewness and kurtosis of LBEW;

k3=αλΓ(3k+1)j=0(1)j(α1j)(1+j)3k+1[6α2λΓ(4k+1)j=0(1)j(α1j)(1+j)4k+1][α2λΓ(2k+1)j=0(1)j(α1j)(1+j)2k+1]+8α2λ2Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1[Var(X)]3/2 …..(13)

k4=αλ2Γ(4k+1)j=0(1)j(α1j)(1+j)4k+1[8α2λ2Γ(3k+1)j=0(1)j(α1j)(1+j)3k+1][α2λΓ(4k+1)j=0(1)j(α1j)(1+j)4k+1]+[6α2λ2Γ(2k+1)j=0(1)j(α1j)(1+j)2k+1][Var(X)]2           …..(14)

From equation (7), (8), (9), (10), (11), (12), (13), (14), we established all the properties of LBEW distribution and also that of ABEW was obtained which can be fetch in the body of the work.

Maximum likelihood approach

Harter and Moore (1965) were the earliest statisticians to use the maximum likelihood procedure because of its desirable characteristics.
The three distributions in the study (EW, LBEW and ABEW) are solved iteratively by computer algorithm to obtain the maximum likelihood estimates of the parameters α , k and λ .

MLE of EW

Let Xbe a random sample of size n from the EW distribution given by equation (1). Then the log likelihood function comes out to be

L(α,λ,k)=nlnα+nlnk+nklnλ+(k1)lnxi+
(α1)ln[1exp{(xi/λ)k}](xi/λ)k     ….. (15)               
Therefore the MLEs of α , λ , k which maximize (15) must satisfy the normal equations given by
Derivative w.r.t α
αL(α,λ,k)=nα+ln[1exp{(xi/λ)k}]=0

We obtain the MLE of

λ as
Λα=nln[1exp{(xi/λ)k}]
      ….. (16)

Derivative w.r.t λ

λL(α,λ,k)=nkλ+(α1)kλk1exp{(xi/λ)k}1exp{(xi/λ)k}xikλk1xik=0

Multiplying the above equation by

λk we get
n+λk[(α1)exp{(xi/λ)k}1exp{(xi/λ)k}xikxik]=0 …... (17)
Derivative w.r.t k

kL(α,λ,k)=nk+nlnλ+lnxi+(α1)λkexp{(xi/λ)k}1exp{(xi/λ)k}xikln(xi/λ)λk1xikln(xi/λ)=0


then,
nk+lnxi+λk[(α1)exp{(xi/λ)k}1exp{(xi/λ)k}xiklnxixiklnxi]=0      ……… (18)

Using (15) in (17) and (18) we get equations, which are satisfied by the MLEs ˆλ and ˆk  of λ and k , respectively. Because of the complicated form of the likelihood equations, algebraically it is very difficult to prove that the solution to the normal equations give a global maximum or at least a local maximum, though numerical computation during data analysis showed the presence of at least local maximum.

However, the following properties of the log-likelihood function have been algebraically noted:

  1. for given ( λ , k ), log-likelihood is a strictly concave function of α . Further, the optimal value of α , given by (8), is a concave increasing function of λ , for given k ;
  2. for given ( α , k ), and α1 , log-likelihood is a strictly concave function of λ

MLE of LBEW

Taking the log-likelihood and derivative of the equation (4) to obtain the MLEs of parameters α , k and λ

αL(α,λ,k)=ni=1(α1)ln[1exp{(xi/λ)k}]α2α{nln[Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1]}=0

               ……… (19)

kL(α,λ,k)=nk+ni=1klnxik1xiknlnλk(x/λ)kni=1{(α1)exp{(kxik/λk+1)}1exp{(xi/λ)k}}α1α{nln[Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1]}=0

                                                                                                                                                                                       
.… (20)

λL(α,λ,k)=n(k+1)λ+kxikλk+1ni=1{(α1)exp{(kxik/λk+1)}1exp{(xi/λ)k}}α1=0         …. (21)

Equations (19), (20) and (21) are solved iteratively to obtain the maximum likelihood estimates of the parameters α , k and λ .

MLE of ABEW

Taking the log-likelihood and derivative of the equation (6) to obtain the MLEs of parameters α , k and λ

αL(α,λ,k)=ni=1(α1)ln[1exp{(xi/λ)k}]α2α{nln[Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1]}=0

…………..(22)

kL(α,λ,k)=nk+ni=1(k+1)lnxikxik+1nlnλk(x/λ)kni=1{(α1)exp{(kxik/λk+1)}1exp{(xi/λ)k}}α1α{nln[Γ(1k+1)j=0(1)j(α1j)(1+j)1k+1]}=0


.… (23)
…. λL(α,λ,k)=(k+1)λ+kxikλk+1ni=1{(α1)exp{(kxik/λk+1)}1exp{(xi/λ)k}}α1=0  ………… (24)

Also, equations (22), (23) and (24) are solved iteratively to obtain the maximum likelihood estimates of the parameters

α , k and λ .

AIC and log-likelihood

We calculate AIC value for each model with the same dataset, and the best model is the one with minimum AIC value. The value of AIC depends on the data Pines and Bombax, which leads to model selection uncertainty.
AIC=2logL(ˆθ|xi)+2k
where

  • L(Λθ|xi) = the maximized value of the likelihood function of the model, and where ˆθ  are the parameter values that maximize the likelihood function;
  • xi = the observed data;
  • k = the number of free parameters to be estimated.

Results and discussion

Summary of the data    

The Bombax and Pine Height-Diameter data were extracted from the Forestry Research Institute of Nigeria’s records, cleaned up and the summary statistics of the data was computed as presented in Tables 1 and 2.

PinesHEIGHT

PinesDBH

Bombaxheight

Bombaxdbh

Mean

13.33333

13.87566

8.82142

16.47857

estimated Stdev

3.38336

3.84751

2.67723

6.37982

estimated skewness

-0.14219

0.11252

0.6269

1.22949

estimated kurtosis

2.683

2.76197

2.85557

5.31719

Table 1 Descriptive statistics of the data

EW

PinesHEIGHT    PinesDBH   Bombaxheight    Bombaxdbh

Mean
estimated stdev
estimated skewness
estimated kurtosis

0.02165                  0.046100        0.008641               0.008940 0.00685                0.013384              0.044389        0.066902
-0.77246              -1.379207       5.268808               7.281359
2.757014              3.945957        29.65974               54.01818

LBEW
Mean
estimated stdev
estimated skewness
estimated kurtosis

 

6.34E-05               7.07E-05        3.32E-05               5.88E-05              4.28E-05              4.78E-05               2.62E-05       4.82E-05
2.408499              2.212744       2.871012              3.017609
10.28893              9.160734       15.27437              15.87016

ABEW
Mean
estimated stdev
estimated skewness
estimated kurtosis

 

8.97E-05               4.76E-05        0.000130              4.88E-05 
5.58E-05               3.51E-05        7.95E-05               4.20E-05
2.299099               2.336502        2.215247             3.142198
9.618972               9.925224        11.11609             16.72621

Table 2 Skewness and kurtosis of EW, LBEW and ABEW distribution

Maximum likelihood approach

The above Table 3-5 shown the parameters estimation of EW, LBEW and ABEW distributions. We observed the comparison of the three distributions by their corresponding AIC and -2log-likelihood of each of the dataset pines and bombax. The ideal distribution is the one with the minimum AIC values (Figures 1-5).7-15

EW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k
α
λ
AIC
-2loglik

4.351
1.053
14.468
1999.52
993.76

3.077
1.635
13.542
2098.4
1043.2

1.433
8.920
4.352
538.40
263.20

1.064
11.725
5.747
719.45
353.72

Table 3 Parameters estimation of EW

LBEW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k

0.05

0.05

0.05

0.05

α

0.264

0.261

0.263

0.261

λ

0.045

0.052

0.01

0.053

AIC

1629.07

1634.36

467.19

484.93

-2loglik

407.117

408.44

116.65

121.08

Table 4 Parameters estimation of LBEW

ABEW

Pines

Bombax

parameters

HEIGHT

DBH

Height

dbh

k
α
λ
AIC
-2loglik

0.050
0.262
0.068
1533.11
383.128

0.050
0.262
0.034
1508.10
376.874

0.050
0.267
0.045
449.67
112.27  

0.050
0.263
0.045
450.24
112.41

Table 5 Parameters estimation of ABEW

Figure 1 Histogram boxplot plot of bombax and pines H-D.

Figure 2 The probability distribution function of the EW, LBEW and ABEW distribution.

Figure 3 The cumulative distribution function of the EW distribution.

Figure 4 The cumulative distribution function of the LBEW distribution.

Figure 5 The cumulative distribution function of the ABEW distribution.

Conclusion

This study introduced a new distribution based on LBEW and ABEW. Some characteristics of the new distributions were obtained. Plots for the cumulative distribution function, pdf and tables with values of skewness and kurtosis were also provided. Height-Diameter (H-D) data on Bombax and Pines (Pinus caribeae) were used to demonstrate the application of the distributions. Estimation of parameters of EW, LBEW and ABEW distributions were done using the maximum likelihood approach and compared across the distributions using criteria like AIC and Log-likelihood. We therefore proposed that similar to Exponentiated Weibull distribution (EW), a better fitting of Bombax and Pines H-D data are possible by LBEW and ABEW distributions.

Acknowledgements

We gratefully acknowledge the suggestions given by the anonymous referees, which have immensely helped to improve the presentation of the paper.

Conflicts of interest

None.

References

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