Bayesian Inference For Exponential Distribution Based On Upper Record Range

This paper deals with Bayesian estimations of scale parameter of the exponential distribution based on upper record range (Rn). This has been done in two steps; point and interval. In the first step the quadratic, squared error and absolute error, loss functions have been considered to obtain Bayesian-point estimations. Also in the next step the shortest Bayes interval (Hight Posterior Density interval) and Bayes interval with equal tails based on upper record range have been found. Therefore, the Homotopy Perturbation Method(HPM) has been applied to obtain the limits of Hight Posterior Density intervals. Moreover, efforts have been made to meet the admissibility conditions for linear estimators based on upper record range of the form mRn+d by obtained Bayesian point estimations. So regarding the consideration of loss functions, the prior distribution between the conjunction family has been chosen to be able to produce the linear estimations from upper record range statistics. Finally, some numerical examples and simulations have been presented

ESTIMASI PARAMETER MODEL SURVIVAL DISTRIBUSI EKSPONENSIAL PRIOR UNIFORM DENGAN METODE BAYESIAN ABSOLUTE ERROR LOSS FUNCTION

Data survival adalah data yang menunjukkan waktu suatu individu atau objek dapat bertahan hidup hingga terjadinya suatu kegagalan atau kejadian tertentu. Pada penelitian ini dibahas mengenai estimasi parameter model survival distribusi eksponensial prior Uniform dengan menggunakan Bayesian absolute error loss function (AELF) dan diterapkan pada kasus penderita kanker paru-paru. Estimasi parameter model survival dimulai dengan mencari fungsi distribusi kumulatif, fungsi survival, kemudian menentukan fungsi likelihood, distribusi prior, dan posterior untuk metode Bayesian. Dari metode Bayesian AELF diperoleh dan fungsi survival kemudian diterapkan pada data pasien penderita kanker paru-paru untuk mengetahui peluang individu dapat bertahan hidup. Berdasarkan hasil estimasi metode Bayesian AELF untuk studi kasus penderita kanker paru-paru dapat diketahui bahwa peluang hidup pasien yang mengidap penyakit kanker paru-paru semakin lama akan semakin kecil (mendekati nol). Nilai mean absolute persentage error (MAPE) yang diperoleh dari fungsi survival dengan menggunakan metode Bayesian AELF adalah sebesar 0,485%. Hal ini berarti bahwa metode Bayesian AELF memiliki kemampuan estimasi yang sangat baik dalam mengestimasi peluang bertahan hidup pasien penderita kanker paru-paru.Kata kunci: Loss Function, Prior Uniform, Absolute Erro

Bayesian Optimal Design for Ordinary Differential Equation Models

Bayesian optimal design is considered for experiments where it is hypothesised that the responses are described by the intractable solution to a system of non-linear ordinary differential equations (ODEs). Bayesian optimal design is based on the minimisation of an expected loss function where the expectation is with respect to all unknown quantities (responses and parameters). This expectation is typically intractable even for simple models before even considering the intractability of the ODE solution. New methodology is developed for this problem that involves minimising a smoothed stochastic approximation to the expected loss and using a state-of-the-art stochastic solution to the ODEs, by treating the ODE solution as an unknown quantity. The methodology is demonstrated on three illustrative examples and a real application involving estimating the properties of human placentas

An LpLp −quantile methodology for estimating extreme expectiles

Quantiles are a fundamental concept in extreme-value theory. They can be obtained from a minimization framework using an absolute error loss criterion. The companion notion of expectiles, based on squared rather than absolute error loss minimization, has recently been receiving substantial attention from the fields of actuarial science, finance and econometrics. Both of these notions can actually be embedded in a common framework of $Lp$-quantiles, whose extreme value properties have been explored very recently. However, and even though this generalized notion of quantiles has shown potential for the estimation of extreme quantiles and expectiles, it has so far not been used in the estimation of extreme value parameters of the underlying distribution of interest. In this paper, we work in a context of heavy tails, which is especially relevant to actuarial science, finance, econometrics and natural sciences, and we construct an estimator of the tail index of the underlying distribution based on extreme $Lp$-quantiles. We establish the asymptotic normality of such an estimator and in doing so, we extend very recent results on extreme expectile and $Lp$-quantile estimation. We provide a discussion of the choice of $p$ in practice, as well as a methodology for reducing the bias of our estimator. Its finite-sample performance is evaluated on simulated data and on a set of real hydrological data