Characterizations of Univariate Continuous Distributions (eBook)

Preface 7
Contents 8
1 Introduction 10
1.1 Distribution of Univariate Continuous Distribution 10
1.2 Moment Generating and Characteristic Functions 10
1.3 Some Reliability Properties 11
1.4 Cauchy Functional Equations 12
1.5 Order Statistics 13
1.6 Record Values 14
1.7 Generalized Order Statistics 19
1.8 Lower Generalized Order Statistics (Lgos) 21
1.9 Some Useful Functions 24
2 Some Continuous Distributions 25
2.1 Beta Distribution 25
2.2 Cauchy Distribution 26
2.3 Chi-Squared Distribution 27
2.4 Exponential Distribution 28
2.5 F-Distribution 30
2.6 Gamma Distribution 31
2.7 Gumbel Distribution 32
2.8 Inverse Gaussian (Wald) Distribution 33
2.9 Laplace Distribution 34
2.10 Logistic Distribution 35
2.11 Lognormal Distribution 36
2.12 Normal Distribution 37
2.13 Pareto Distribution 38
2.14 Power Function Distribution 39
2.15 Rayleigh Distribution 40
2.16 Student’s t-Distribution 41
2.17 Weibull Distribution 41
3 Characterizations of Distributions by Independent Copies 43
3.1 Characterization of Normal Distribution 43
3.2 Characterization of Levy Distribution 52
3.3 Characterization of Wald Distribution 53
3.4 Characterization of Exponential Distribution 54
3.5 Characterization of Symmetric Distribution 56
3.6 Charactetrization of Logistic Disribution 57
3.7 Characterization of Distributions by Truncated Statistics 57
3.7.1 Characterization of Semi Circular Distribution 59
3.7.2 Characterization of Lindley Distribution 59
3.7.3 Characterization of Rayleigh Distribution 61
4 Characterizations of Univariate Distributions by Order Statistics 62
4.1 Characterizations of Student’s t Distribution 62
4.2 Characterizations of Distributions by Conditional Expectations (Finite Sample) 66
4.3 Characterizations of Distributions by Conditional Expectations (Extended Sample) 68
4.4 Characterizations Using Spacings 69
4.5 Characterizations of Symmetric Distribution Using Order Statistics 72
4.6 Characterization of Exponential Distribution Using Conditional Expectation of Mean 73
4.7 Characterizations of Power Function Distribution by Ratios of Order Statistics 74
4.8 Characterization of Uniform Distribution Using Range 76
4.9 Characterization by Truncated Order Statistics 78
5 Characterizations of Distributions by Record Values 79
5.1 Characterizations Using Conditional Expectations 79
5.2 Characterization by Independence Property 82
5.3 Characterizations Based on Identical Distribution 88
6 Characterizations of Distributions by Generalized Order Statistics 95
6.1 Characterizations by Conditional Expectations 95
6.2 Characterizations by Equality of Expectations of Normalized Spacings 101
6.3 Characterizations by Equality of Distributions 101
References 104
Index 130