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T-Classes of Linear Estimators and the Thoery of Successive Sampling

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T-Classes of Linear Estimators and the Thoery of Successive Sampling

G.C. Tikkiwal

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The book is concerned with the study of different classes of linear estimators in survey sampling, known as T-classes of linear estimators and the theory of successive sampling. The theory of classification of linear estimators in different classes has been developed mainly by Horvitz and Thompson, Godambe, Koop, Prabhu Ajgaonkar, Tikkiwal and the theory of successive sampling by Jessen, Yates, Paterson, Tikkiwal and others. The book presents a detailed study of all the seven T-classes along with the unified theory of unordering. It also discusses the technique of combined unordering and its applications. The chapter on the theory of successive sampling deals with the theory under less restrictive assumptions for finite population, there by making it possible to obtain the main results given in text books on survey sampling, as a special case of the these results. The theory of T-classes along with the theory of successive sampling provide more serviceable estimation procedure based on the time honoured principles of inference than the one provided by Basu, Godambe and others. The material present in this book is meant for one specialised sample survey course in semester scheme for the post graduate students of statistics. Therefore, it can be used as a text book. The book is also useful for research students and faculty engaged in research on theoretical foundations of inference from finite population.

1.  T-Classes of Linear Estimators and Techniques of Unordering and Combined Unordering

(a) Definition of T-classes of linear estimators

Definition of ordered set, Definition of sample

1.1 Tikkiwal's partition of sample space

1.2 Set of T-classes of linear estimators

1.3 Wideness of the set of seven classes of linear estimators

(b) Techniques of unordering and combined unordering

1.4 Unordering of estimators for sampling with varying probabilities with or without replacement - A unified approach

1.5 Murthy's results of unordering of ordered estimators for sampling with varying probabilities without replacement

1.6 The technique of combined unordering

1.7 Applications of unordering and combined unordering

1.8 Unordering of   -class of linear estimators

1.9 Combined unordering of the classical SRSWR estimator

Exercises

2.   -Class Estimators

2.1 Introduction

2.2 Estimators in  -class for some well known sampling schemes:

(a)  Simple Random sampling schemes: SRSWOR and SRSWR

(b) Midzuno Scheme of sampling with or without replacement

Exercises

3.   -Class Estimators

3.1 Introduction

3.2 A general discussion of  -class estimators

3.3 Sampling with varying probabilities and without  replacement :

3.3.1 The variance of Horvitz-Thompson estimator

3.3.2 The three unbiased estimators of the variance of    and their relative merits

3.3.3 Non-negative variance estimators

3.3.4 Calculation of probabilities of inclusion for certain sampling schemes

3.4 Sampling with varying probabilities and with replacement

3.5 Unequal probability sampling without replacement due to Rao, Hartley and Cochran (a) Certain preliminaries; (b) Rao-Hartley-Cochran results

3.6 The theory of multistage sampling with varying probabilities and with or without replacement

3.7 Comparison of the efficiency of estimators for sampling with varying probabilities without replacement and with replacement in case of multistage designs

 Exercises

4.  Estimation in  -Class

4.1 Introcuction

4.2 A Lemma in quadratic equation

4.3 Non-existence of MVLUE in the class for sampling without replacement

4.4 An estimator in  -class for Midzuno scheme of sampling without replacement

4.5 Non-existence of MVLUE in the class for sampling with replacement

4.6 An estimator in the class for Midzuno scheme of sampling with replacement

4.7 A unified proof of non-existence of MVLUE in the class for sampling with or without replacement

 Exercises

5.  Estimation in   -Class

5.1 Introduction

5.2 Non-emptiness of  ¬¬-class

5.3 Non-existence of MVLUE in the class for  SWOR

5.4 Non-existence of MVLUE in the class for SWR

5.5 Des Raj Ordered Estimator

Exercises

6.  Estimation in   and  -Classes

6.1 Introduction 

6.2 Estimation in  -class for sampling with or without replacement

6.3 Estimation in  -class for sampling with or without replacement

6.4 MVLUE in  -class for SRSWOR

6.5 MVLUE in  -class for SRSWR

6.6 Estimation in  ¬-class

6.7 Some results due to Godambe

Exercises

7.   Theory of Successive Sampling

7.1 Introduction 

7.2 Preliminary results

7.3 Successive Sampling on h ( ) occasions

7.4 A general discussion on unistage successive sampling with varying probabilities

7.5 A general discussion on multistage successive sampling.

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