DISTRIBUSI APT-PARETO DAN APLIKASINYA (Studi Kasus Data Waktu Bertahan Hidup Pasien Myelogenous Leukemia)

Distribusi Pareto merupakan distribusi peluang kontinu yang sering digunakan dalam analisis data uji hidup. Akan tetapi, distribusi Pareto hanya mempunyai satu bentuk grafik dari fungsi kepadatan peluang yang berbentuk eksponensial. Sehingga karena hal tersebut, distribusi Pareto dianggap kurang fleksibel dalam menggambarkan data. Metode transformasi Alpha Power adalah metode baru yang dapat menambah fleksibilitas pada suatu distribusi dengan menambahkan parameter pada distribusi yang sudah ada. Pada penelitian ini, akan dibahas proses pembentukan distribusi APT-Pareto yang merupakan modifikasi dari distribusi Pareto dengan penambahan parameter alpha melalui transformasi Alpha Power. Karakteristik dari disribusi APT-Pareto juga akan dibahas pada penelitian ini. Karakteristik tersebut meliputi fungsi kepadatan peluang, fungsi distribusi kumulatif, fungsi hazard, fungsi survival, ekspektasi, variansi, fungsi pembangkit momen, dan momen. Metode penaksiran maksimum digunakan untuk mengestimasi parameter dari distribusi APT-Pareto. Penelitian ini menggunakan bantuan software Microsoft Excel dan R. Kata kunci: Distribusi Pareto, Transformasi Alpha Power, Karakteristik, Metode Penaksiran Maksimum. Pareto distribution is one of continuous probability distributions that is often used in analyzing life test data. However, the Pareto distribution only have one exponential graphical form of the probability density function. For that reason, the Pareto distribution is considered less flexible in describing data. Alpha Power transformation method is a new method that can add more flexibility to a distribution by adding parameters to an existing distribution. This research will discuss the process of constructing the APT-Pareto distribution which is a modification of the Pareto distribution by adding alpha parameters through the Alpha Power transformation. The characteristics of the APT-Pareto distribution will also be discussed in this study. These characteristics include the probability density function, cumulative distribution function, hazard function, survival function, expectations, variance, moment generating function, and moment. The maximum estimation method is used to estimate the parameters of the APT-Pareto distribution. This study is using Microsoft Excel and R. Keywords: Pareto distribution, Alpha Power Transformation, Characteristics, Maximum Likelihood Estimation

DISTRIBUSI WEIBULL-NORMAL{LOG-LOGISTIK} DAN APLIKASINYA (Studi Kasus Data Waktu Bertahan Hidup Pasien Penderita Jantung Koroner yang Diberikan Treatment Bypass)

Perkembangan ilmu pengetahuan dalam bidang statistika ditandai dengan adanya minat dalam mengembangkan distribusi statistik baru yang lebih fleksibel yang terbagi atas dua periode waktu yaitu sebelum tahun 1980 dan tahun 1980 dan setelahnya. Metode yang digunakan pada tahun 1980 dan setelahnya dikenal sebagai ‘metode kombinasi’. Salah satu metode yang termasuk ‘metode kombinasi’ adalah metode transformasi tranformator, dan hasil dari metode ini yaitu keluarga distribusi T-R{Y}. Skripsi ini membahas mengenai penggabungan distribusi Weibull, normal, dan log-logistik dengan metode transformasi transformator untuk mendefinisikan distribusi Weibull-normal{log-logistik}. Pada skripsi ini dikaji pula karakteristik, fungsi kuantil, fungsi survival, fungsi Hazard, dan pengaplikasiannya terhadap data waktu bertahan hidup pasien penderita jantung koroner yang diberikan treatment bypass dari distribusi Weibull-normal{log-logistik}. The development of statistical sciences is characterized by an interest in developing new distributions that must be more flexible. Generally speaking, the methods developed prior to 1980s and since 1980s may be categorized as ‘methods of combination’. One of the ‘methods of combination’ is transformed-transformator method, and T-R{Y} families distribution for its product. This paper propose the use of transformed-transformator method to combine Weibull distribution, normal distribution, and log-logistic distribution for defining Weibull-normal{log-logistic} distribution. In this paper also discussed about some properties, quantile function, survival function, Hazard function, and its application for survival data of patients with coronary heart disease were given bypass treatment of Weibull-normal{log-logistic} distribution

Statistical properties of Odd Frѐchet Lomax Distribution

A new lifetime distribution with three parameters, called odd Frѐchet Lomax (OFrL), is introduced. Some statistical properties of the OFrL are provided. Explicit expressions for the quntile, moments, moment generating function, probability weighted moments and order statistics are studied. Maximum likelihood estimation technique is employed to estimate the model parameters are studied. In addition, the superiority of the OFrL distribution is illustrated with applications to one real data set. Keywords: Odd Frѐchet -G family; Lomax distribution, Order statistics; moments. DOI: 10.7176/MTM/9-1-0

An Extended Pareto Distribution

<p>For the first time, a new continuous distribution, called the generalized beta exponentiated Pareto type I (GBEP) [McDonald exponentiated Pareto] distribution, is defined and investigated. The new distribution contains as special sub-models some well-known and not known distributions, such as the generalized beta Pareto (GBP) [McDonald Pareto], the Kumaraswamy exponentiated Pareto (KEP), Kumaraswamy Pareto (KP), beta exponentiated Pareto (BEP), beta Pareto (BP), exponentiated Pareto (EP) and Pareto, among several others. Various structural properties of the new distribution are derived, including explicit expressions for the moments, moment generating function, incomplete moments, quantile function, mean deviations and Rényi entropy. Lorenz, Bonferroni and Zenga curves are derived. The method of maximum likelihood is proposed for estimating the model parameters. We obtain the observed information matrix. The usefulness of the new model is illustrated by means of two real data sets. We hope that this generalization may attract wider applications in reliability, biology and lifetime data analysis.</p

The Beta Generalized Inverse Weibull Geometric Distribution

A new six-parameter distribution called the beta generalized inverse Weibull-geometric distribution is proposed. The new distribution is generated from the logit of a beta random variable and includes the generalized inverse Weibull geometric distribution.Various structural properties of the new distribution including explicit expressions for the moments, moment generating function, mean deviation are derived. The estimation of the model parameters is performed by maximum likelihood method

McDonald log-logistic distribution with an application to breast cancer data

We introduce a five-parameter continuous model, called the McDonald log-logistic distribution, to extend the two-parameter log-logistic distribution. Some structural properties of this new distribution such as reliability measures and entropies are obtained. The model parameters are estimated by the method of maximum likelihood using L-BFGS-B algorithm. A useful characterization of the distribution is proposed which does not require explicit closed form of the cumulative distribution function and also connects the probability density function with a solution of a first order differential equation. An application of the new model to real data set shows that it can give consistently better fit than other important lifetime models