Open Access Short Communication

Maximum Likelihood Estimation in an Alpha-Power

Clement Boateng Ampadu*1 and Abdulzeid Yen Anafo2

1Department of Biostatistics, USA

2 African Institute for Mathematical Sciences (AIMS), Ghana

Corresponding Author

Received Date: May 26, 2019;  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

Mahdavi and Kundu [1] proposed the celebrated APT-F family of distributions with CDF

where 1 ≠α > 0, irispublishers-openaccess-biostatistics-biometric-applicationsand ξ is a vector of parameters all of whose entries are positive. By modifying the above CDF, the Zubair-G family of distributions appeared in [2] with the following CDF

where α > 0, irispublishers-openaccess-biostatistics-biometric-applications and ξ is a vector of parameters all of whose entries are positive. Subsequently in [3] we introduced the Ampadu-G family of distributions by modifying the above CDF, to give the following CDF

where λ > 0, irispublishers-openaccess-biostatistics-biometric-applications , and ξ is a vector of parameters all of whose entries are positive. Consequently, we introduced new variants of the APT-F family of distributions, which are recorded in the table below (Table 1).

Table 1: Variants of the Alpha Power Transformed Family of Distributions.

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The rest of this paper is organized as follows. In Section 2 we introduce and illustrate the new family. In Section 3 we derive the quantile function of the APTA-F family of distributions. In Section 4, we discuss approximate random sampling from the APTAF family of distributions, and maximum likelihood stimation in Section 5. Section 6 presents simulation results associated with the APTA-F family of distributions, assuming F follows the Weibull distribution as defined in Section 2. Section 7 discusses usefulness of the APTA-F family of distributions in fitting real-life data. Section 8 is devoted to the conclusions.

The New Family

Definition 2.1. Let irispublishers-openaccess-biostatistics-biometric-applications,irispublishers-openaccess-biostatistics-biometric-applications, and F (x;ξ ) be a baseline CDF all of whose parameters are in the vector ξ . We say a random variable X follows the alpha-power transformed distribution of the Ampadu type (APTA-F for short) if the CDF is given by

If the baseline distribution is given by the Weibull distribution with CDF

for x, a,b > 0 , then we have the following from the above definition

Proposition 2.2. The CDF of the APTA-Weibull distribution is given by

Where x,a,b > 0 and irispublishers-openaccess-biostatistics-biometric-applications

Notation 2.3. If a random variable V follows the APTA-Weibull distribution, we write

V ~ APTAW (a,b,α ) (Figure 1)

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By differentiating the CDF of the APTA-F distribution we have the following

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Proposition 2.4. The PDF of the alpha-power transformed distribution of the Ampadu type is given by

where irispublishers-openaccess-biostatistics-biometric-applications,irispublishers-openaccess-biostatistics-biometric-applicationsand F (x;ξ ) and f (x;ξ ) are the CDF and PDF, respectively, of some basline distribution all of whose parameters are in the vector ξ (Figure 2)

The Quantile

Theorem 3.1. Let QF(.)=F-1 denote the quantile of some baseline distribution with CDF F and PDF f, irispublishers-openaccess-biostatistics-biometric-applicationsand 0 < u <1. The quantile function of the alpha-power transformed distribution of the ampadu type is given by

Where W (z) := Pr oductLog (z) gives the principal solution for w in z = wew

Proof. Let 0 < u <1. Since QF(.)=F-1 the quantile function can be obtained by solving the following equation for y

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 expansion [7]

Using the first term of this series to approximate W (z) := Pr oductLog (z) , gives us a way to approximate the random sample from the APTA-F family of distributions.

In particular if u U (0,1) , that is, u is a uniform random variable, then an (approximate) random sample from the APTA-F family of distributions can be obtained via

where QF(⋅) = F−1 (⋅) denote the quantile of some baseline distribution with CDF F and PDF f, irispublishers-openaccess-biostatistics-biometric-applications

Parameter Estimation

In this section, we obtain the maximum likelihood estimators (MLEs) for the parameters of the APTA-F family of distributions. For this, let 1 2 , ,.... n X X X be a random sample of size n from the APTA-F family of distributions. The likelihood function from Proposition 2.4 is given by

From the above the log-likelihood function is given by

The MLE’s of ξ and can be obtained by maximizing the equation immediately above.

The derivatives of the equation immediately above with respect to the unknown parameters, are given as follows

Now solving the system below for _ and _ gives the maximum likelihood estimators, αˆ and ˆξ of the unknown parameters:

Simulation Studies

In this section, a Monte Carlo simulation study is carried out to assess the performance of the estimation method, when the baseline distribution is defined as in Section 2.

Simulation Study One

Approximate samples of sizes 200, 350, 500, and 700, are drawn from the APTA-Weibull distribution. The approximate samples have been drawn for (a,b,α ) = (1.3, 48.4, 0) using

Table 2: Result of Simulation Study.

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and the maximum likelihood estimators for the parameters a, b, and α are obtained. The procedure has been repeated 200 times and the mean and mean square error for the estimates are computed, and the results are summarized in Table 2 below.

From Table 2 above, we find that the simulated estimates are close to the true values of the parameters and hence the estimation method is adequate. We have also observed that the estimated mean square errors (MSEs) consistently decrease with increasing sample size as seen in the Figures below.

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Simulation study two

Approximate samples of sizes 400, 550, 700, and 900, are drawn from the APTA-Weibull distribution. The approximate samples have been drawn for (a,b,α ) = (1.3, 1.3, 0) using

and the maximum likelihood estimators for the parameters a, b, and _ are obtained. The procedure has been repeated 400 times and the bias and root mean square error for the estimates are computed, and the results are summarized in Table 3 below.

From Table 3 above, we find that the estimated root mean square errors (RMSEs) consistently decrease with increasing sample size as seen in the Figures below. We also find that the bias is consistently around zero, hence estimating the parameters in the distribution via the method of maximum likelihood is adequate.

Table 3: Result of Simulation Study.

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Application

In this section, we illustrate the usefulness of the APTA-F family of distributions in modelling real-life data. We compare the fit of the APTA-Weibull distribution, with the proposed Weibull distribution appearing in [5] to the dataset on the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli. The dataset can be found in [8]. The estimates of the unknown parameters in both distributions are obtained by the maximum likelihood method using the R language software. The measures of goodness of fit considered included Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC) and Hannan-Quinn information criterion (HQIC) statistics and they are defined as follows:

where k is the number of parameters in the statistical model, n is the sample size, and l (⋅) is the maximized value of the log-likelihood function under the considered model. The proposedWeibull distribution appearing in [5], which we denote, Proposed Weibull (a,b,α ) has CDF given by

where irispublishers-openaccess-biostatistics-biometric-applications

Since the Proposed Weibull (a,b,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 survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli The asymptotic variance-covariance matrix of the MLEs under the APTAW (a,b,α ) , and the Proposed Weibull (a,b,α ) distributions, respectively, are given by

Table 4:Estimated Parameters for the dataset on the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli.

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Hence the approximate 95% confidence intervals for the parameters under the APTAW (a,b,α ) and the Proposed Weibull distributions, respectively, are given in Tables 5-7.

Table 5: Criteria for Comparison.

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Table 6:APTAW (a, b, α) distribution.

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Table 7: Proposed Weibull (a, b, α) distribution.

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Concluding Remarks

In this paper a new kind of alpha power transformed family of distributions is introduced, and members of this family are shown to be effective in fitting real life data. As a further development we propose obtaining some mathematical and statistical properties of the APTA-F family of distributions.

Acknowledgement

None.

Conflict of interest

No conflict of interest.

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