On Smoothed Nonparametric MWSD Estimation of Mixing Proportions
- PDF / 597,000 Bytes
- 38 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 185 Views
RESEARCH ARTICLE
On Smoothed Nonparametric MWSD Estimation of Mixing Proportions K. L. Mehra1 • Satish Konda2
•
Y. S. Ramakrishnaiah3
Accepted: 18 September 2020 / Published online: 19 November 2020 The Indian Society for Probability and Statistics (ISPS) 2020
Abstract The paper considers the problem of ‘‘smoothed’’ nonparametric estimation of mixing proportion p, 0 \ p \ 1, in a mixtures of two distributions F1 and F2 , viz., F = pF1 ? (1 - p)F2 , based on an i.i.d. sample of size n from the mixed distribution, using the minimum weighted square distance methodology. Our smoothed estimators are fully nonparametric in the sense that for their validity we need make no model assumptions at all regarding the underlying component distributions, not even the existence of their densities. The relative asymptotics of the proposed ‘smoothed’ versus ‘unsmoothed’ estimators are explored, first when the Fi ’s are known and are then extended to cover the case when they are unknown under Hosmer’s (Biometrics 29:761–770, 1973) both Models I and II. Their large sample properties such as asymptotic normality, rates of strong consistency and mean square errors are established and also the superiority of ‘smoothed’ over ‘unsmoothed’ estimation in the sense of lowering mean square errors. Keywords Mixture of distributions Mixing proportion Smoothed MWSD estimation Optimal band width
Y. S. Ramakrishnaiah: Deceased. & Satish Konda [email protected] K. L. Mehra [email protected] 1
Professor Emeritus, Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada
2
Department of Statistics, Aurora’s Degree and PG College, Chikkadpally, Hyderabad, India
3
Department of Statistics, Osmania University, Hyderabad, India
123
350
Journal of the Indian Society for Probability and Statistics (2020) 21:349–386
1 Introduction Let {X1 , X2 , …, Xn } be independent random observations from an ‘‘identifiable’’ (see Sect. 1.1 for definition) finite mixture of m distributions represented by FðxÞ ¼
m X
pm Fm ðxÞ; 1\x\1;
ð1:1Þ
m¼1
where the mixing component distribution functions (d.f.s) Fm , m = 1, 2, …, m may be fully or partially known or completely P unknown, with the (unknown) mixing proportions pm ’s satisfying 0 \ pm \ 1 and m m¼1 pm = 1. The aim is to estimate the unknown mixing proportions, utilizing the given mixture sample and any other information available regarding the component distributions, say, in the form of ‘‘training’’ samples from them. When m = 2, the mixture model (1.1) reduces to FðxÞ ¼ pF1 ðxÞ þ ð1 pÞF2 ðxÞ; 1\x\1:
ð1:2Þ
In any mixture analysis, the main objective always is to estimate, as effectively as possible, the mixing proportions pm , m = 1, 2, …, m, along with possibly other parameters when the component distributions at best are only partially known. The literature is replete with competing estimation procedures under varying optimality criteria, depending on the extent to which we have knowledge or are willing to make assumptions regarding the nature of
Data Loading...
