Statistical evaluation of multiple process data in geometric processes with exponential failures

Ömer Altındağ

Makale detayları

Statistical evaluation of multiple process data in geometric processes with exponential failures

Yazarlar:

Ömer Altındağ

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistical Analysis

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.54 Issue 2, pp.738-761

Belge Tipi:

Makale

Yayın Tarihi:

4/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001346371

pISSN:

2651-477X

Anahtar kelimeler:

Özet:

 The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.

 The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.

 The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.

 The geometric process is a monotonic stochastic process commonly used to model some sort of processes having monotonic trend in time. The statistical inference problem for a geometric process has been well studied in the literature. However, existing studies only cover single process data obtained throughout a single realization of a geometric process. This study presents how multiple process data for a geometric process can arise and considers its statistical evaluation by assuming that all processes are homogeneous and the inter-arrival times follow an exponential distribution. Two data structures for multiple process data are introduced: one consists of complete samples, while the other includes both complete and censored samples. The maximum likelihood and modified maximum likelihood estimators for the parameters of the geometric process are derived on the basis of these data structures. The Expectation-Maximization algorithm is used to compute the maximum likelihood estimators in the case of censored data. The asymptotic properties of the estimators are also derived. Test statistics are proposed based on the asymptotic results of the estimators to distinguish a geometric process from a renewal process and to test the homogeneity of the processes. A simulation study is conducted to demonstrate the performance of the inferential procedures. Finally, both artificial and real data analyzes are presented for illustration.

Makale detayları

Statistical evaluation of multiple process data in geometric processes with exponential failures

Yazarlar:

Ömer Altındağ

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistical Analysis

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.54 Issue 2, pp.738-761

Belge Tipi:

Makale

Yayın Tarihi:

4/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001346371

pISSN:

2651-477X

geometric process

maximum likelihood

modified maximum likelihood

EM algorithm

Sayı

Makale detayları

Statistical evaluation of multiple process data in geometric processes with exponential failures

Yazarlar: Ömer Altındağ

Yayıncı: Hacettepe University

Ana Başlıklar: Statistical Analysis

Dergi: Hacettepe Journal of Mathematics and Statistics Vol.54 Issue 2, pp.738-761

Belge Tipi: Makale

Yayın Tarihi: 4/2025

Belge Dili: İngilizce

Document ID (Org No.): 2001346371

pISSN: 2651-477X

geometric process

maximum likelihood

modified maximum likelihood

EM algorithm

Sayı

Yazarlar:

Ömer Altındağ

Yayıncı:

Hacettepe University

Ana Başlıklar:

Statistical Analysis

Dergi:

Hacettepe Journal of Mathematics and Statistics Vol.54 Issue 2, pp.738-761

Belge Tipi:

Makale

Yayın Tarihi:

4/2025

Belge Dili:

İngilizce

Document ID (Org No.):

2001346371

pISSN:

2651-477X