Maximum Likelihood Estimation in an Alpha-Power

ISSN: 2641-6336 DOI: 10.33552/ABBA.2019.02.000541 Annals of Biostatistics & Biometric Applications

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

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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< 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<

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. We also find that the bias Citation: Page 3 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 is consistently around zero, hence estimating the parameters in the Application distribution via the method of maximum likelihood is adequate. In this section, we illustrate the usefulness of the APTA-F family Table 3: Result of Simulation Study.

Parameter a APTA-Weibull distribution, with the proposed Weibull distribution of distributions in modelling real-life data. We compare the fit of the Sample Size Bias RMSE appearing in [5] to the dataset on the survival times (in days) of 400

550 -0.01948298 0.1563376 72 guinea pigs infected with virulent tubercle bacilli. The dataset both distributions are obtained by the maximum likelihood method -0.02780227 0.13558230.1193164 can be found in [8]. The estimates of the unknown parameters in 900 0.1100411 700 -0.01273009 considered included Akaike information criterion (AIC), consistent using the R language software. The measures of goodness of fit Parameter-0.01177412 b Akaike information criterion (CAIC), Bayesian information criterion Sample Size Bias RMSE (BIC) and Hannan-Quinn information criterion (HQIC) statistics 400 550 0.02225872 0.3380576 and they are defined as follows:AIC=22 l + k -0.004792135 0.3157712 BIC=2 l + k log ( n) 700900 0.0076356330.009964405 0.2872791 21kk( + ) Parameter α 0.2334487 CAIC= AIC + Sample Size Bias RMSE nk−−1 400 HQIC = 2log log(n)( k− 2 l) 550 -0.1131752 0.5872724 where k is the number of parameters in the statistical model, n is -0.07089181 0.5248752 the sample size, and l (⋅) is the maximized value of the log-likelihood 700900 -0.09093898-0.03949046 0.505457 function under the considered model. The proposedWeibull 0.430892 distribution appearing in [5], which we denote, Proposed Weibull (ab,,α ) has CDF given by

x a − b −−α 1 e  1− e  1− e−α 11 Where αα≠>, ,ab , > 0, and x > 0 ee Figure 6: Decreasing RMSE of a for increasing sample size. Since the Proposed Weibull (ab,,α ) distribution has the smallest AIC, BIC, CAIC, and HQIC values compared to the other

distribution in Table 4, it can be considered a better fit to the tubercle bacilli The asymptotic variance-covariance matrix of the survival times (in days) of 72 guinea pigs infected with virulent APTAW( a,, b α ) , and the Proposed Weibull (ab,,α ) distributions, respectively, are given by MLEs under the Table 4: Estimated Parameters for the dataset on the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli.

Model Parameter Estimate Standard Error

Figure 7: Decreasing RMSE of b for increasing sample size. APTAW (abˆ,,ˆ αˆ ) (a, b, ) =(2.0665258, (0.1801981, 37.1849579, Proposed α 265.3997306,ˆ 0.7428645) 0.2413815) Weibull (abˆ,,αˆ ) (a, b, ) =(2.208316, (0.1968281, 310.356991, 3.239273) 70.0390762,1.7622610) α 3.247136ee−− 02 0.6552024 − 3.845612 − 05  −−6.552024ee 01 1382.7210969 7.889494+ 00 −−3.845612ee 05 7.8894941 5.826503− 02

0.03874132 -0.7365994 0.03693494  -0.73659941 4905.4721882 116.48517435 Figure 8: Decreasing RMSE of α for increasing sample size. 0.03693494 116.4851744 3.10556397

Citation: Page 4 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

Acknowledgement parameters under the APTAW( a,, b α ) and the Proposed Weibull Hence the approximate 95% confidence intervals for the None. distributions,Table 5: Criteria respectively, for Comparison. are given in Tables 5-7. Conflict of Interest Model AIC BIC CAIC HQIC

ReferencesNo conflict of interest. Proposed Weibull AP T AW (a, b, α) 857.8961 864.7261 858.2491 16.4056116.40831 1. with an application to exponential distribution. Communications in 856.7445 863.5745 857.0974 Table 6:(a, APTAW b, α) (a, b, α) distribution. Mahdavi A, Kundu D (2017) A new method for generating distributions

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(305.9651, 314.7489) Manuscript, In Preparation. α (3.128767 3.349778) Essay, African Institute for Mathematical Sciences (Ghana). Unpublished Concluding Remarks 6. Distributions. Statistica 3. In this paper a new kind of alpha power transformed family of Girish Babu Moolath, Jayakumar K (2017) T-Transmuted X Family of distributions is introduced, and members of this family are shown 7. Wikipedia contributors (2019) “Lambert W function.” Wikipedia, The T Bjerkedal (1960) Acquisition of resistance in guinea pigs infected with propose obtaining some mathematical and statistical properties of Free Encyclopedia. Wikipedia, The Free Encyclopedia. to be effective in fitting real life data. As a further development we the APTA-F family of distributions. 8. diﬀerent doses of virulent tubercle bacilli. American Journal of Hygiene 72(1): 130-148.

Citation: Page 5 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.