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Abstrak :
Tugas akhir ini membahas tentang distribusi Kumaraswamy-geometrik yang merupakan distribusi probabilitas dari peubah acak diskrit yang dibangun dengan menggunakan metode Transformed-Transformer. Distribusi Kumaraswamy dapat membuat distribusi geometrik menjadi lebih fleksibel. Pembahasan meliputi fungsi distribusi, fungsi kepadatan probabilitas, perilaku limit, serta kasus khusus dari distribusi Kumaraswamy-geometrik. Karakteristik-karakteristik dari distribusi Kumaraswamy-geometrik yang meliputi modus, persentil, momen, fungsi pembangkit momen, dan fungsi pembangkit probabilitas juga akan dibahas pada tugas akhir ini. Selanjutnya, Metode Maksimum Likelihood digunakan dalam tugas akhir ini untuk mencari penaksir parameter dari distribusi Kumaraswamy-geometrik. Pada bagian akhir, akan digunakan data tentang jumlah klaim suatu asuransi kendaraan bermotor sebagai ilustrasi penggunaan distribusi Kumaraswamy-geometrik. ......This paper discusses about Kumaraswamy geometric distribution, a distribution of discrete random variable which formed by Transformed Transformer method. Kumaraswamy distribution can cause geometric distribution to be more flexible. This paper studies about distribution function, probability density function, limiting behavior, and special cases of Kumaraswamy geometric distribution. Some properties of Kumaraswamy geometric distribution such as mode, percentile, moments, moment generating function, and probability generating function are studied. Then, Maximum Likelihood method is used to estimate the parameters of Kumaraswamy geometric distribution. Finally, data about number of claims on a motor insurance is used to illustrate the use of Kumaraswamy geometric distribution.

Abstrak :
Waktu survival adalah waktu dimana seorang individu atau suatu objek bertahan hingga suatu kejadian terjadi. Data waktu survival lebih sering digambarkan dengan fungsi hazard karena kurva fungsi hazard dapat memiliki berbagai bentuk, seperti bentuk naik, turun, konstan, bathtub, dan unimodal. Salah satu distribusi yang dapat digunakan untuk memodelkan data waktu survival adalah distribusi Rayleigh. Distribusi Rayleigh memiliki fungsi hazard yang naik secara linier terhadap waktu. Namun pada praktiknya, tidak semua data waktu survival yang hazardnya mengalami peningkatan, terjadi secara linier. Akan tetapi, terdapat data waktu survival yang hazardnya naik dengan tren cekung ke atas maupun cekung ke bawah, turun, dan konstan. Dalam skripsi ini, dibahas pembentukan distribusi Rayleigh Weibull (RW) sebagai generalisasi dari distribusi Rayleigh dengan menggunakan metode Transformed-Transformer atau metode T-X. Generalisasi ini bertujuan untuk menambah fleksibilitas distribusi Rayleigh dengan menambah satu parameter bentuk (shape parameter). Kemudian, dibahas juga beberapa karakteristik dari distribusi RW, seperti fungsi kepadatan peluang, fungsi distribusi kumulatif, fungsi survival, fungsi hazard, dan momen ke-r. Estimasi parameter dari distribusi RW dilakukan dengan menggunakan metode maksimum likelihood. Sebagai ilustrasi, data pasien leukemia dimodelkan dengan distribusi Rayleigh, distribusi Weibull, dan distribusi Rayleigh Weibull. Hasil pemodelan menunjukkan bahwa distribusi Rayleigh Weibull lebih baik dalam memodelkan data dibandingkan dengan distribusi Rayleigh dan distribusi Weibull. ...... Survival time is the time where an individual or object survives until an event occurs. Survival data is more frequently described with a hazard function because the curve of the hazard function can have various shapes, such as increasing, decreasing, constant, bathtub, and unimodal. Rayleigh distribution is one of the distributions that can be used to model survival data. Rayleigh distribution has a linearly increasing hazard function curve. However, in practice, not every survival data shows a linear increase. There are survival data where the hazard increases with a concave up trend or concave down trend, decreasing, and constant. The Transformed-Transformer method, often known as the T-X method, is used to construct Rayleigh Weibull distribution as a generalization of Rayleigh distribution. This generalization aims to increase the flexibility of Rayleigh distribution by adding one shape parameter. Some characteristics of Rayleigh Weibull distribution, such as probability density function, distribution function, survival function, hazard function, and r-th moment are also discussed. Rayleigh Weibull distribution’s parameters were estimated using the maximum likelihood method. As an illustration, leukemia cancer data is modeled with Rayleigh distribution, Weibull distribution, and Rayleigh Weibull distribution. In comparison to Rayleigh distribution and Weibull distribution, Rayleigh Weibull distribution is better at modeling the data.

Abstrak :
Tugas akhir ini membahas tentang distribusi Weibull-Pareto yang merupakan distribusi probabilitas kontinu yang dibangun dengan menggunakan metode Transformed-Transformer. Distribusi Weibull-Pareto dapat menggambarkan data yang menceng kanan, menceng kiri, atau simetris serta dapat menggambarkan data yang mempunyai light-tailed maupun heavy-tailed. Pembahasan meliputi fungsi kepadatan probabilitas, fungsi distribusi, fungsi survival, dan fungsi hazard. Kemudian dicari karakteristik-karakteristik dari distribusi Weibull-Pareto yang meliputi modus, persentil, dan fungsi pembangkit momen. Terakhir dicari taksiran parameter dari distribusi ini dengan menggunakan metode Alternative Maximum Likelihood (AML). Simulasi data juga dilakukan sebagai ilustrasi. ......This paper discusses about Weibull-Pareto distribution, the continuous probability distribution which arised by Transformed-Transformer method. The Weibull-Pareto distribution gives a good fit to right skew, left skew, or symmetric. In particular, Weibull-Pareto distribution can solve light tailed or heavy tailed problem. At first, we study about probability density function, cumulative distribution function, survival function, and hazard function. Then, we find the characteristic of Weibull-Pareto distribution, that is mode, percentile, and moment generating function. Finally, we estimate the parameters of Weibull-Pareto distribution using Alternative Maximum Likelihood (AML) method. Simulation data is used as illustration.

Abstrak :
As mentioned before in the submitted proposal, the aim of this investigation is to study the wide angle reflection characteristic which has both advantage and disadvantage. The advantage we could obtain from such reflection angle is the stronger reflections (i.e S/N ratio) compared to the narrower angle reflections. This strong reflection makes the horizon identification becomes easier and its amplitudes might be used to study the rock properties using inversion technique. Its disadvantage comes from the fact that wide angle reflection involves what co-called chase distortion that affects the travel-time picking and also the amplitude measurement. Some workers have studied the phase distortion effect and proposed some methods to overcome the problem. Choy and Richards [2] who studied seismograms produced by a caustic region found the similarities between event in the original seismograms and the Hilbert transformed seismograms. They observed that, for example, an sS-wave is a Hilbert transform pair of an SS-wave, which suggested a 90° phase distortion. To tackle the problem of obscure travel-time, they applied the matched filter to the distorted signal after Hilbert transforming the reference signal. This method, however, is limited to 913° phase distortion only. The matched filter is a simple and straightforward method to estimate the amplitude ratio and differential travel-time of two different signals. However, it works well only when both signals have similar waveforms. When the phase distortion occurs, i.e. the original waveform is distorted, the matched filter fails to obtain an optimum amplitude and travel-time estimations. Pointer and Neuberg [8] introduced an iterative matched filter for analysing the effect of a phase distortion. First, the reference signal is phase distorted and used to find the amplitude ratio and travel-time by matched filter. The procedure is repeated ti changing the phase with a particular increment until they obtain the best cross-correlation value automatically revealing the amplitude ratio, the differential travel-time, and the phase distortion. However, the method is sensitive to the choice of the phase increment and in some cases it.