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Maximum Likelihood Estimation in an Alpha-Power ISSN: 2641-6336 DOI: 10.33552/ABBA.2019.02.000541 Annals of Biostatistics & Biometric Applications Short Communication Copyright © All rights are reserved by Clement Boateng Ampadu Maximum Likelihood Estimation in an Alpha-Power Clement Boateng Ampadu*1 and Abdulzeid Yen Anafo2 1Department of Biostatistics, USA 2African Institute for Mathematical Sciences (AIMS), Ghana Received Date: May 26, 2019 *Corresponding author: Clement Boateng Ampadu, Department of Biostatistics, USA. Published Date: June 03, 2019 Abstract In this paper a new kind of alpha-power transformed family of distributions (APTA-F for short) is introduced. We discuss the maximum likelihood method of estimating the unknown parameters in a sub-model of this new family. The simulation study performed indicates the method of maximum likelihood is adequate in estimating the unknown parameters in the new family. The applications indicate the new family can be used to model real life data in various disciplines. Finally, we propose obtaining some properties of this new family in the conclusions section of the paper. Keywords: Maximum likelihood estimation; Monte carlo simulation; Alpha-power transformation; Guinea Big data Introduction Table 1: Variants of the Alpha Power Transformed Family of Distributions. Mahdavi and Kundu [1] proposed the celebrated APT-F family Year Name of Distribution Author of distributions with CDF 2019 Ampadu APT -G Ampadu [4] αξF − ( x;1) α Gx( ;,αξ) = 1 2019 PT− G Ampadu [4] α −1 e α 1 where 1≠>α 0, x ∈and ξ is a vector of parameters all PT− F 2019 e Within Quantile Ampadu and to appear in [5] of whose entries are positive. By modifying the above CDF, the Distribution (New) Zubair-G family of distributions appeared in [2] with the following 2019 APTA-F (Current Manuscript) Ampadu CDF The rest of this paper is organized as follows. In Section 2 we 2 eαξGx( ; ) −1 introduce and illustrate the new family. In Section 3 we derive the Fx( ;,αξ) = eα −1 quantile function of the APTA-F family of distributions. In Section 4, we discuss approximate random sampling from the APTAF family where α >∈0, x and ξ is a vector of parameters all of of distributions, and maximum likelihood stimation in Section 5. whose entries are positive. Subsequently in [3] we introduced the Section 6 presents simulation results associated with the APTA-F Ampadu-G family of distributions by modifying the above CDF, to family of distributions, assuming F follows the Weibull distribution give the following CDF 2 1− e−λξGx( ; ) as defined in Section 2. Section 7 discusses usefulness of the APTA-F Fx( ;,λξ) = to the conclusions. 1− e−λ family of distributions in fitting real-life data. Section 8 is devoted The New Family where λ >∈0, x , and ξ is a vector of parameters all of whose x ∈∈,α , and Fx( ;ξ ) be a baseline CDF all entries are positive. Consequently, we introduced new variants of of whose parameters are in the vector ξ . We say a random variable the APT-F family of distributions, which are recorded in the table Definition 2.1. Let X follows the alpha-power transformed distribution of the Ampadu below (Table 1). type (APTA-F for short) if the CDF is given by This work is licensed under Creative Commons Attribution 4.0 License ABBA.MS.ID.000541. Page 1 of 5 Annals of Biostatistics & Biometric Applications Volume 2-Issue 4 Proposition 2.4. The PDF of the alpha-power transformed ξ −αξFx( ; ) Fx( ; ) e distribution of the Ampadu type is given by e−α αα− Fx( ; ξ) If the baseline distribution is given by the Weibull distribution e(1;;−αξ Fx( )) fx( ξ) with CDF a x where x ∈∈,α and Fx( ;ξ ) and fx( ;ξ ) are the CDF − 1− e b and PDF, respectively, of some basline distribution all of whose for xab,,> 0, then we have the following from the above parameters are in the vector ξ (Figure 2) The Quantile definitionProposition 2.2. The CDF of the APTA-Weibull distribution is −1 Theorem 3.1. Let QFF (⋅=) ( ⋅) denote the quantile of some given by baseline distribution with CDF F and PDF f, α ∈ and 01<<u . The a x x a quantile function of the alpha-power transformed distribution of − − b 1− eeb αe the ampadu type is given by W(−α eu−α ) Q − F α Where xab,,> 0and α ∈ Notation 2.3. If a random variable V follows the APTA-Weibull Where = gives the principal solution for w distribution, we write W( z) : Pr oductLog( z) in z= wew V APTAW( a,, b α ) (Figure 1) −1 01<<u . Since QFF (⋅=) ( ⋅) the quantile function can be obtained by solving the following equation for y Proof. Let F( ye) −α Fy( ) u = e−α Approximate Random Number Generation It is well known that the principal solution for w in z= wew , denoted W( z) := Pr oductLog( z) admit the following Taylor series n−1 expansion∞ ([7]−n) 3 8 125 ∑ zn =−+ zz234 z + z + z 5 −... n=1 n! 2 3 24 Figure 1: The CDF of APTAW (1.28352800, 48.41833872, -0.07850413) fitted to the empirical distribution of the data on W( z) := Pr oductLog( z) , gives us a way to approximate the random patients with breast cancer, Section 5.2 [6]. Using the first term of this series to approximate sample from the APTA-F family of distributions. By differentiating the CDF of the APTA-F distribution we have In particular if uU (0,1) , that is, u is a uniform random the following variable, then an (approximate) random sample from the APTA-F family of distributions can be obtained via −α X= QF ( eu) −1 where QFF (⋅=) ( ⋅) denote the quantile of some baseline distribution with CDF F and PDF f, α ∈ Parameter Estimation In this section, we obtain the maximum likelihood estimators F family of distributions. For this, let XX, ,.... Xbe a random sample of size n from the APTA-F (MLEs) for12 the parametersn of the APTA- family of distributions. The likelihood function from Proposition 2.4 is given by Figure 2: The PDF of APTAW (1.28352800, 48.41833872, n −−ααFx( ; ξ) -0.07850413) fitted to the histogram of the data on patients with = i −αξ ξ L∏{ e (1;;Fx( ii)) fx( )} breast cancer, Section 5.2 [6]. i=1 Citation: Page 2 of 5 2(3): 2019. ABBA.MS.ID.000541. DOI: 10.33552/ABBA.2019.02.000541. Clement Boateng Ampadu, Abdulzeid Yen Anafo. Maximum Likelihood Estimation in an Alpha-Power. Annal Biostat & Biomed Appli. Annals of Biostatistics & Biometric Applications Volume 2-Issue 4 From the above the log-likelihood function is given by and the maximum likelihood estimators for the parameters nn na, b, and α are obtained. The procedure has been repeated 200 lnL=−∑∑α( 1 Fx( i ; ξ)) +− ln( 1 αξFx( ii ;)) + ∑ ln fx( ; ξ) ii=11= i= 1times and the mean and mean square error for the estimates are ξ and can be obtained by maximizing the equation computed, and the results are summarized in Table 2 below. immediately above. The MLE’s of The derivatives of the equation immediately above with respect close to the true values of the parameters and hence the estimation From Table 2 above, we find that the simulated estimates are to the unknown parameters, are given as follows method is adequate. We have also observed that the estimated nn n lnL=−α 1 Fx ; ξ +− ln 1 αξFx ; + ln fx ; ξsample size as seen in the Figures below. ∑∑( ( i)) ( ( ii)) ∑( ) mean square errors (MSEs) consistently decrease with increasing ii=11= i= 1 nn ∂ ln L Fx( i ;ξ ) =−+∑∑ln( 1Fx( i ;ξ )) ∂−αii=11= αξFx( i ;1) ∂∂Fx( ;;ξξ) fx( ) α ii ∂ ln L nn∂Fx( ;ξ ) ∂∂ξξ n =−+∑∑α i + ∑ ∂∂ξii=11 ξ= αξFx( ii;1) − i= 1 fx( ; ξ) Now solving the system below for _ and _ gives the maximum likelihood estimators, α and ξˆ of the unknown parameters: Figure 3: Decreasing MSE of a for increasing sample size. ∂ ln L ∂ ln L = 0 and = 0 ∂α ∂ξ Simulation Studies In this section, a Monte Carlo simulation study is carried out to assess the performance of the estimation method, when the Simulationbaseline distribution Study Oneis defined as in Section 2. Figure 4: Decreasing MSE of b for increasing sample size. from the APTA-Weibull distribution. The approximate samples have Approximate samples of sizes 200, 350, 500, and 700, are drawn been drawn for (ab,,α ) = (1.3, 48.4, 0) using 1 X=−− b( log( 1 eu−α )) a Table 2: Result of Simulation Study. Parameter a Sample Size Average Estimate MSE 200 350 1.276885 0.0378470.019242 Figure 5: Decreasing MSE of α for increasing sample size. 500 1.2694851.261209 0.021077 Simulation study two 700 Parameter1.268349 b 0.016317 Sample Size Average Estimate MSE from the APTA-Weibull distribution. The approximate samples have Approximate samples of sizes 400, 550, 700, and 900, are drawn 200 been drawn for (ab,,α ) = (1.3, 1.3, 0) using 350 150.1065 1 48.55782 198.7113 X=−− b( log( 1 eu−α )) a 500 48.3844 140.3203 and the maximum likelihood estimators for the parameters a, 48.73268 123.0901 b, and _ are obtained. The procedure has been repeated 400 times 700 Parameter48.10531 α and the bias and root mean square error for the estimates are Sample Size Average Estimate MSE computed, and the results are summarized in Table 3 below. 200 350 -0.10479 0.377691 500 -0.16417 0.436073 From Table 3 above, we find that the estimated root mean -0.08719 0.311987 square errors (RMSEs) consistently decrease with increasing 700 -0.06227 0.213638 sample size as seen in the Figures below.
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