International Journal of Science & Technology Volume 2, No 2, 115-121, 2007
Maximum Likelihood Estimators for Fitting The Pearsonian Type II System of Curves Lipi B. MAHANTA and Dilip C. NATH Institute of Advanced Study in Science and Techn., Paschim Boragaon; Garchuk; Guwahati-781035, Assam, INDIA Department of Statistics, Gauhati University, Guwahati-781014, Assam; INDIA
[email protected] (Received: 20.06.2006; Accepted: 03.09.2007)
Abstract: The method of maximum likelihood has been used to estimate parameters for Type II of Pearsonian system of distribution curves. Finally the estimators are used to graduate data to test the goodness of fit. Computer algorithms have been generated and numerical computations based on data available from selected papers have been supplied. The results have been compared with values obtained by using method of moments. Keywords: Pearsonian System of curves, Maximum Likelihood estimators, scoring method.
Eğrilerin Pearsonian Tip II Sistemine Uydurmak İçin En Yüksek Olabilirlik Tahmin Edicileri Özet: Maksimim olabilirlik metodu, dağılım eğrilerinin Pearsonian tip II için parametreleri tahmin etmek için kullanılmıştır. Sonuç olarak tahmin ediciler uyumun güvenilirliğini test etmek için son verileri kullanırlar. Seçilmiş makalelerden ulaşılabilen verilere bağlı olarak sayısal hesaplamalar desteklenmiştir ve bilgisayar algoritmaları üretilmiştir. Anahtar Kelimeler: Eğrilerin Pearsonian Sistemi, en yüksek olabilirlik tahmin edicileri, scoring metot.
AMS Classification: 62 F 05; 62 F 10
1. Introduction and Preliminaries To estimate the parameters involved in a distribution function there are several known methods available for fitting curves. As would be expected of any general method, the method of moments (which is more popularly used) may give difficulty because (i) the equations resulting from
last point can be exemplified by curves, within the Pearson system, which cannot be fitted satisfactorily by moments, e.g. y=
y0 (1 + x 2 )
2. The method of maximum likelihood : k
∫x h
n
f ( x, a, b,....)dx = ν n
cannot be solved to find the constants a,b,… in y = f ( x, a, b,....) or (ii) sufficiently accurate adjustments are not yet available in all circumstances for the statistical moments or (iii) some of the moments are liable to such large standard errors as to make them unreliable. This
This method is due to R.A.Fisher (1921). The method of moments was regarded as efficient in fitting Pearsonian Curves, prior to 1921, when it was shown by Fisher, that its efficiency is restricted to a small region for which β2 lies between the limits 2.65 and 3.42; and for which β1 does not exceed 3.42. Later the same author pointed out that the goodness of fit test would not be accurate if the method of fitting employed is
Lipi B. Mahanta and Dilip C. Nath
inadequate; that is, if the statistics used in the estimation of parameters are inconsistent or inefficient. A statistic satisfying the criterion of efficiency can be found by the method of maximum likelihood as shown by Fisher (1921). Although the general theory, and the principle of its practical application, has thus been available for many years, the teaching and improvement etc. has lagged behind. This method has been successfully used previously by Koshal (1933) to fit the parameters of Type I type of Pearsonian curve to data obtained in an earlier paper by the same author (1930). The χ2 obtained by maximum likelihood method was 5.5375 whereas the χ2 obtained by method of moments was 7.6552. Subsequently the method was used by Des Raj (1953) to estimate the parameters of Type III populations from truncated samples and then by Das (1954) to fit truncated type III curves to daily rainfall data. The MLE method has been used to estimate parameters for a variety of distributions like truncated gamma distribution by Dixit et al(2005), autoregressive process AR(1) by Zhou et al(2005), a new class of families if distributions called W-type families by Atienza etal (2005) and grouped data like Weibull distribution and log normal distribution by Zheng etal(2005). The
Method of maximum likelihood has been used for estimation of parameters by Dimitris (2002), Glutany etal (2002) and Gupta etal (1999) too. 2.1 The scoring method The difficulty of using the method of maximum likelihood in general curve fitting is the same as that mentioned in connection with the least squares; the equations reached cannot be solved directly and the constants have to be found by approximation. Also it sometimes happens that the method leads to complicated equations so that the solutions cannot be obtained directly. In such cases, that is when explicit solution of the equation are not possible, we use the method suggested by Rao (1974), i.e. the method of scoring for approximate calculation of maximum likelihood estimators. Here iterative methods are employed to get the solution. The estimator so obtained is often consistent but less efficient and, therefore, correction is to be applied to bring it nearer to the desired form. The maximum likelihood estimators of the parameters are obtained by solving the following normal equations ( Nath 1984):
δ log L n N r δPr =∑ = 0, j = 1,2,... δθ j r =0 Pr δθ j So, the maximum likelihood estimates of the parameters are estimated from the following matrix equation :
A.δ θ = S ~
where
A = { Ast }, s, t = 1,2,...
'
δ θ = (δθ 1 δθ 2 .....), S ' = ( S1 S 2 .....) and
~
n
Ast = N ∑ r =0 n
Ss = ∑ r =0
1 δPr δPr , Pr δθ s δθ t
N r δPr Pr δθ s The variance-covariance matrix of the estimators is the inverse of the information matrix 116
Lipi B. Mahanta and Dilip C. Nath
A. The computation of the elements of the information matrix requires differentiation of the probabilities with respect to the unknown parameters.
The pilot values of parameters under different types, which are required for scoring method, are taken as the values obtained by the method of moments. A computer program has been written in ‘C’ language, for calculating probabilities, scores and the variance covariance matrix of the estimators.
3. Estimation 3.1 Type II The form of the distribution function is m
x2 y = y −a≤ x≤ a where 0 1 − 2 , a N .Γ(2m + 2) NΓ(m + 1.5) = y0 = 2 m +1 2 a.2 {Γ(m + 1)} a π Γ(m + 1) Putting the value of y0 in the above equation, we get, m
Γ(m + 1.5) x 2 1 − , y= a π Γ(m + 1) a 2
−a ≤ x ≤ a
3.2 Parameters The parameter vector is
a
θ = m
3.3 Likelihood Function The likelihood function is
L=
1 a n ( 2 m+1)
( π)
n
(Γ(m + 1 12 ))n n 2 2 m ( a + xi ) (Γ(m + 1))n ∏ i =1
The log-likelihood function is
{
}
n
(
l = log L = − n (2m + 1) log a + log π + log Γ(m + 1) − log Γ(m + 1 12 ) + m ∑ log a 2 − xi
3.4 The Gradient Vector (Score)
116
i =1
2
)
Maximum Likelihood Estimators for Fitting The Pearsonian Type II System of Curves
δPr δ Γ(m + 1.5) x 2 1 − = δa δa a π Γ(m + 1) a 2
Γ(m + 1.5) 1 2mx 2 = 3 π Γ(m + 1) a a
x2 1 − 2 a
δPr δ Γ(m + 1.5) x 2 1 − = δm δm a π Γ(m + 1) a 2
Γ(m + 1.5) 1 x2 = 1 − 2 π Γ(m + 1) a a
m
m
m
m −1
x2 − 1 − 2 a
m 1 2 a
x2 log1 − 2 a
Γ(m + 1.5) 1 x2 + 1 − 2 a a π
m
ψ (m + 1) +
m 1 1 x2 1 − 2 Γ(m + 1.5)ψ (m + 2.5) π Γ(m + 1) a a where Γ ' ( z + 1) = d Γ ( z ) and Γ ' ( zis ) calculated from Di-gamma dz
function (see Abramowitz and Stegun(1972)) defined as ' d Γ ( z) log Γ( z ) = dz Γ( z )
ψ ( z) =
And the asymptotic formula for Di-gamma function is given by
ψ ( z ) = log e z −
1 1 1 1 − + − , ( z ~ ∞, in arg z < π ) 2 4 2 z 12 z 120 z 252 z 6
The gradient vector or score is
δPr δPr δm = δθ δPr δa 3.5 The Information matrix
117
Lipi B. Mahanta and Dilip C. Nath
δ 2 Pr 1 = 2 δm a π
x 2 Γ(m + 1.5) δ x2 log 1 1 − − 2 2 a Γ(m + 1) δm a
δ 1 x2 ( 1 . 5 ) log 1 m Γ + + − 2 Γ(m + 1) δm a
m
x2 + log1 − 2 a
x2 1 − 2 a
x 2 1 − 2 a
m
m δ 1 Γ(m + 1.5) δm Γ(m + 1)
m δ x2 x2 − 1 − 2 Γ(m + 1.5)ψ ' (m + 1) − [Γ(m + 1.5)ψ (m + 1)] 1 − 2 δm a a
m
m m δ Γ(m + 1.5) x2 x2 ' 1 − 2 ψ (m + 2.5) − 1 − 2 ψ (m + 1) Γ(m + 1.5) + a Γ(m + 1) a δm
δ 2 Pr Γ(m + 1.5) 2m(m − 1) 2 x 2 = 4 δa 2 a3 π Γ(m + 1) a
x2 1 − 2 a
m− 2
8mx 2 − 5 a
x2 1 − 2 a
m −1
1 δ x2 1 − 2 − 2 a δa a
m
2 x2 + 3 1 − 2 a a
δ 2 Pr 1 Γ(m + 1.5) 1 x2 1 2x 2 1 x 2 δ x2 + 1 − 2 = + 1 − 1 − 2 2 3 δaδm a a 2 δa a 2 π Γ(m + 1) a a (1 − x a ) a m
x 2 x2 log1 − 2 1 − 2 a a
m
m δ 1 1 x2 1 δ 1 − 2 − Γ(m + 1.5)ψ (m + 1) π a δa a δa a
1 Γ(m + 1.5) x2 1 δ ψ (m + 2.5) + 1 − 2 π Γ(m + 1) a δa a
m
x2 + 1 − 2 a
−m
δ 1 δa a
where
1 z
ψ ' ( z) = +
1 1 1 1 1 + 3− + − , ( z ~ ∞, in arg z < π ) 2 5 7 2z 6z 30 z 42 z 30 z 9
see Abramowitz and Stegun (1972) The Information matrix is
118
m
m
x2 + 1 − 2 a
−m
δ 1 δa a
m
Maximum Likelihood Estimators for Fitting The Pearsonian Type II System of Curves
δ 2 Pr 2 A−1 = δ2a δ Pr δmδa
δ 2 Pr δaδm 2 δ Pr δm 2
4. Iteration Procedure (Algorithm) 4.1 Initial values The pilot values are obtained by using the Method of Moments, Elderton and Johnson (1969). 4.2 Updating At the sth iteration (s=0,1,2,….), the parameter vector is updated as : where
θˆ( s +1) = θˆ( s ) + δθˆ( s )
δθˆ( s ) =
d log L ÷ A(θ s ) dθ s
= S ( s ) ÷ A(θ s ) where n
Ast = N ∑ r =0 n
Ss = ∑ r =0
1 δPr δPr , Pr δθ s δθ t
N r δPr Pr δθ s
4.3 Convergence Criteria Given the convergence criteria ε > 0 , the iteration is considered converged when the following criteria are satisfied :
θˆs − θˆ( s ) < ε 5. Results : Type II : Data Source : Elderton and Johnson (1969) x 2.500 7.500 12.500 17.500
y 11.000 116.000 274.000 451.000
yest 14.150 110.709 286.478 430.337 119
Lipi B. Mahanta and Dilip C. Nath
22.500 27.500 32.500 37.500
432.000 267.000 116.000 16.000
430.089 285.933 110.272 5.034
Chi square test for fit of data : chi= 4.11 Tabulated chi-square test of goodness of fit at 95% confidence interval for 7 degrees of freedom = 14.067 Kolmogorov-Smirnov test for fit of data : highplus=0.2557 highminus=0.9911 6. Conclusion
The table value of Z obtained from Table of Critical values of D in the Kolmogorov− Smirnov one-sample test (two-tailed Dvalues) for n=8 is 0.457 for 95% level of significance. The value of is greater that the table value and hence we can conclude that there is no evidence against the null hypothesis. Further we may conclude that the fitting of Pearsonian System of Curves using the scoring method of Method of Maximum Likelihood yields results which are better than the usual Method of Moments. Many useful distributions and families of distributions are of Pearsonian type families. Hence the results have broad applications. These methods were tested for only the above mentioned type, which is in common use in actuarial sciences etc. The method may be used for fitting data for the other types in the manner adopted in this paper.
The value of chi-square for the same data when fit with values of parameters obtained by the method of moments, was 7.4. So it is seen that the chi-square as obtained by the method of maximum likelihood is much less than that value indicating a better fit. The tabulated value of chi-square at 95% confidence interval for 7 degree of freedom is 14.067. Also, for the Kolmogorov-Smirnov test as described in Deshpande et.al. (1995) and Gibbons (1971) , the null hypothesis may be assumed as : H0 : Fn(x) < F0(x) or Dn− = max{F0 (t ) − Fn (t )} t i.e. F is smaller than F0.
4.
7. References 1.
Type III population from truncated samples. Journal of American Statistical Association, 48,
Abramowitz, M. and Stegun, I.A. (1972) Handbook
336-349.
of Mathematical functions. Dover Publications, 5.
Inc., New York. 2.
Consistency
of
maximum
New Age International Publishers Limited.
likelihood 6.
estimators in finite mixture models of the union of
Dixit, Ulhas J.; Phal,K.D. (2005) On estimating scale parameter of a truncated gamma distribution.
W-type families. Comm.Statist.Theory Methods 34
Soochow J. Math 31, no. 4, 515-523.
, no.7, 1471-1485. 3.
Deshpande, J.V., Gore, A.P. and Shanubhogue, A. (1995) Statistical Analysis of Nonnormal Data.
Atienza,N.;Garcia-Heras,J.;Muñoz-Pichardo,J. (2005)
Des Raj (1953) Estimation of the parameters of
7.
Das, S.C. (1954) The fitting of truncated Type III
Elderton, W.P. and Johnson, N.L. (1969) Systems of
curves to daily rainfall data. Australian Journal of
Frequency
University Press.
Physics, 7, 298-304. 120
curves.
London,
Cambridge
Maximum Likelihood Estimators for Fitting The Pearsonian Type II System of Curves
8.
Fisher, R.A. (1921) The Mathematical Foundations
12. Koshal, R.S. (1933) Application of the Methods of
Phil. Trans., A., Vol.
Maximum Likelihood to the improvement of curves
of Theoretical Statistics,
9.
222, pp. 309-368.
fitted by the Method of Moments. Journal of Royal
Gibbons J.D. (1971) Nonparametric Statistical
Statistical Society, 96, 303-13.
Inference. McGraw-Hill Kogakusha Ltd. Glutany,
13. Nath D.C. (1984) On some analytical Models for
M.E.;Al-Awadhi, S. (2002) Maximum likelihood
number of births and their applications. Thesis
estimation of Burr XII distribution parameters
submitted for Ph.D., Banaras Hindu University.
under random censoring. J. Appl. Stat 29, no. 7,
14. Rao. C.R. (1974) Linear Statistical Inference and
955-965.
its Applications. Wiley Eastern Private Limited.
10. Gupta A.K., Nguyen T.T.; Wang Yining, (1999)
15. Zheng,Ming;Zhang,Han;Yang,Yi
(2005)
On the maximum likelihood estimation of the
Estimating parameters in some distributions from
binomial parameter n. Canad J. Statist, 27 , no. 3,
grouped data. J.Fudan Univ.Nat Sci., 44 , no.3,
599-606.
466-470.
11. Karlis, Dimitris (2002) An EM type algorithm for
16. Zhou,J.;Basawa,I.V.(2005) Maximum likelihood
maximum likelihood estimation of the normal
estimation
inverse Gaussian distribution. Statist Probab Lett
autoregressive process with exponential errors.
57, no.1, 43-52.
J.Time Series Anal, 26, no.6, 825-842.
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for
a
first-order
bifurcating
