Survival Models and Their Estimation

344 Pages·1988·42.094 MB·English

by  Dick London

0-936031-02-6
Dick London, FSA
ERRATA TEXTBOOK
IN
The following errata have been discovered in the textbook Survival
Models and Their Estimation, Second Edition, by Dick London, FSA.
The errata marked with an asterisk (*) in the list below are present in
both the Second Printing and the Third Printing. The errata without an
asterisk were present in the Second Printing, but have been corrected in
the Third Printing.
Page Location Change
6 Line —7 certian should be (s/b) certain
16 Line 9 distribuions s/b distributions
*34 Five lines after (2.79) Delete the close bracket
34 Six lines after (2.79) Delete the open bracket
36 Exercise 2-7, Line 2 repsectively s/b respectively
69 Table 3.5, Line 3, Column 7 25 s/b 26
90 Line 7 date s/b data
117 Solution to Example 6.5, ?x s/b ix
Line 1
150 Line 4 n s/b n
x
163 Solution to Example 7.7, Delete the apostrophe
Line 3 between In and L
190 Solution to Example 8.10,
Line 7 1.1486 s/b 1.1429
k207 Solution to Example 8.19, First a'z s/b a'z
x t
Line 3
219 Solution to Example 9.1, Figure 9.1 s/b Figure 9.3
Line (ii)
221 Line 7 Second birthday s/b/ birthdays
CJi 1-9 io,
i
SURVIVAL MODELS
AND THEIR ESTIMATION
(Second Edition)
Dick London, FSA
ACTEX
Publications
Winsted and Avon, Connecticut
To all my teachers,
but most of all DGG, B2 and GC
©
Copyright 1988
by ACTEX Publications
No portion of this book may be
reproduced in any form or by any
means without the prior written
permission of the copyright owner.
Requests for permission should be addressed to
ACTEX
Publications
P.O. Box 974
Winsted, CT 06098
Manufactured in the United States ofAmerica
9876543
10
Cover design by MUF
Library of Congress Cataloging-in-Publication Data
London, Dick, 1943-
Survival models and their estimation / Dick London, — 2nd ed.
p. cm.
Bibliography: p.
Includes index.
ISBN 0-936031-02-6 (pbk.) $25.00
:
1. Insurance—Statistical methods. 2. Mortality—Statistical methods.
3. Population forecasting—Statistical methods.
I. Title
HG8782.L57 1988
304.6'4'015195-dc 19 88-22118
CIP
ISBN 0-936031-02-6
CONTENTS
Preface xi
THE NATURE AND PROPERTIES OF
Part -
I
SURVIVAL MODELS
Chapteic 1
INTRODUCTION
1.1 What is a Survival Model 3
1.2 Actuarial Survival Models 4
1.2.1 The Select Model 4
1.2.2 The Aggregate Model 5
1.3 Forms of the Survival Model 6
1.4 Estimation 7
1.5 Study Design 8
1.5.1 Cross-sectional Studies
1.5.2 Longitudinal Studies 9
1.6 Summary and Preview 10
Chapter 2
THE MATHEMATICS OF SURVIVAL MODELS
2.1 Introduction 11
2.2 The Distribution of T 11
2.2.1 The Survival Distribution Function 11
2.2.2 The Cumulative Distribution Function 11
2.2.3 The Probability Density Function 12
2.2.4 The Hazard Rate Function 13
2.2.5 The Moments of the Random Variable T 14
2.2.6 Actuarial Survival Models 15
2.3 Examples of Parametric Survival Models 16
2.3.1 The Uniform Distribution 16
2.3.2 The Exponential Distribution 17
m
iv Contents
2.3.3 The Gompertz Distribution 18
2.3.4 The Makeham Distribution 19
2.3.5 The Weibull Distribution 19
2.3.6 Other Distributions 19
2.3.7 Summary of Parametric Models 20
2.4 Conditional Measures and Truncated Distributions 20
2.4.1 Conditional Probabilities and Densities 20
X
2.4.2 Lower Truncation of the Distribution of 22
X
2.4.3 Upper and Lower Truncation of the Distribution of 23
2.4.4 Moments ofTruncated Distributions 25
2.4.5 The Central Rate 25
2.4.6 Use of Conditional Probabilities in Estimation 27
2.5 Transformation of Random Variables 27
2.5.1 Linear Transformations 30
2.5.2 Probability Integral Transformations 31
2.6 Mean and Variance ofTransformed Random Variables 32
2.7 Summary 35
Exercises 35
Chapter 3
THE LIFE TABLE
3.1 Introduction 39
3.2 The Traditional Form of the Life Table 40
3.3 Other Functions Derived from lx 42
3.3.1 The Force of Mortality 43
3.3.2 The Probability Density Function of X 44
3.3.3 Conditional Probabilities and Densities 46
3.3.4 The Central Rate 49
3.3.5 The Concept of Exposure 50
3.3.6 Relationship between nqx and nmx 53
3.4 Summary of Concepts and Notation 55
3.5 Methods for Non-Integral Ages 58
3.5.1 Linear Form of dx+3 58
3.5.2 Exponential Form for £x+3 60
3.5.3 Hyperbolic Form for £r+3 63
3.5.4 Summary 65
3.6 Select Life Tables 67
3.7 Summary 70
Exercises 71
Contents
PART ESTIMATION OF SURVIVAL
II -
MODELS FROM SAMPLE DATA
Chapter 4
TABULAR SURVIVAL MODELS
ESTIMATED FROM COMPLETE DATA SAMPLES
4.1 Introduction 77
4.2 Study Design 77
4.3 Exact Time of Death 78
4.3.1 The Empirical Survival Distribution 79
4.3.2 Analysis of the Empirical Survival Distribution 82
4.4 Grouped Times of Death 83
4.4.1 Notation 83
4.4.2 The Distributions of N and D 84
t t
4.4.3 Estimation of S(t) and t\qQ 85
4.4.4 Estimation of qt and pt 85
4.4.5 Estimation of S(t) from {p } 87
{
4.4.6 Estimation of Other Functions 87
4.5 Summary 89
Exercises 91
Chapter 5
TABULAR SURVIVAL MODELS
ESTIMATED FROM INCOMPLETE DATA SAMPLES
STUDY DESIGN
5.1 Introduction 95
5.2 Interval Estimates 97
5.3 Single- and Double-Decrement Environments 101
5.4 Summary 105
Exercises 106
vi Contents
Chapter 6
TABULAR SURVIVAL MODELS
ESTIMATED FROM INCOMPLETE DATA SAMPLES
MOMENT PROCEDURES
6.1 Introduction 107
6.2 Moment Estimation in a Single-Decrement Environment 107
6.2.1 The Basic Moment Relationship 108
6.2.2 Special Cases 108
6.2.3 Exposure 110
6.2.4 Grouping 111
6.2.5 Properties of Moment Estimators 113
6.2.6 A Different Moment Approach for Special Case C 115
6.3 Moment Estimation in a Double-Decrement Environment 117
6.3.1 The Basic Moment Relationships 117
6.3.2 Uniform Distribution Assumptions 119
6.3.3 Exponential Distribution Assumptions 121
6.3.4 Hoem's Approach to Moment Estimation in a Double-
Decrement Environment 123
6.4 The Actuarial Approach to Moment Estimation 125
6.4.1 The Concept of Actuarial Exposure 125
6.4.2 Statistical Justification 127
6.4.3 Critique of the Statistical Justification 129
6.4.4 Properties of the Actuarial Estimator 129
6.5 Estimation of S(x) 130
6.5.1 Expected Value of S(x) 130
6.5.2 Variance of S(x) 131
6.6 Summary 133
Exercises 135
Chapter 7
TABULAR SURVIVAL MODELS
ESTIMATED FROM INCOMPLETE DATA SAMPLES
MAXIMUM LIKELIHOOD PROCEDURES
7.1 Introduction 141
7.2 Single-Decrement Environment, Special Case A 142
7.2.1 Partial Data 142
7.2.2 Full Data 144
7.2.3 Summary of Special Case A 146
7.3 Single-Decrement Environment, General Case 147
7.3.1 General Form for Full Data 147
Contents vii
7.3.2 Full Data, Exponential Distribution 148
7.3.3 Full Data, Uniform Distribution 148
7.3.4 Partial Data, Special Case C 151
7.3.5 Special Case C with Random Censoring 154
7.3.6 Summary of Single-Decrement MLE's 155
7.4 Double-Decrement Environment 155
7.4.1 General Form for Full Data 155
7.4.2 Full Data, Exponential Distribution for Mortality 157
7.4.3 Full Data, Uniform Distribution for Mortality 157
7.4.4 Partial Data, Special Case A 157
7.4.5 Partial Data (Special Case A), Exponential Distributions 158
7.4.6 Partial Data (Special Case A), Uniform Distributions 160
7.4.7 Summary of Double-Decrement MLE's 161
7.5 Properties of Maximum Likelihood Estimators 162
7.5.1 Bias 162
7.5.2 Variance 162
7.5.3 Consistency and Efficiency 164
7.6 The Product Limit Estimator 164
7.6.1 Estimation of qx 164
7.6.2 Properties of the Estimator 167
7.6.3 Estimation of S(i) 167
7.6.4 The Nelson-Aalen Estimator 169
7.7 Summary 171
Exercises 171
Chapter 8
ESTIMATION OF PARAMETRIC SURVIVAL MODELS
8.1 Introduction 177
8.2 Univariate Models, Complete Data 177
8.2.1 Exact Times of Death 177
8.2.2 Grouped Times of Death 185
8.3 Univariate Models, Incomplete Data 188
8.3.1 Maximum Likelihood Approaches 188
8.3.2 Least-Squares Approaches 192
8.4 Hypothesis Testing of Parametric Models 195
8.4.1 Grouped Times of Death 195
8.4.2 Exact Times of Death 196
8.5 Concomitant Variables in Parametric Models 201
8.5.1 Bivariate (Select) Models 201
8.5.2 General Multivariate Models 203
viii Contents
8.5.3 The Additive Model 205
8.5.4 The Multiplicative Model 207
8.6 Summary 209
Exercises 210
Chapter 9
SURVIVAL MODELS
ESTIMATED FROM GENERAL POPULATION DATA
9.1 Introduction 215
9.2 Demographic Notation and Concepts 216
9.2.1 Census Population Available on Each January 1 216
9.2.2 Census Population Available on One Date Only 222
9.2.3 Estimating Infant Mortality Rates 225
9.2.4 Stationary Population 225
9.3 The 1979-81 United States Life Tables 227
9.3.1 Available Data 227
9.3.2 Estimation of Life Table Values Under Age 2 227
9.3.3 Ages 2 to 4 (Age Last Birthday) 230
9.3.4 Ages 5 to 94 (Age Last Birthday) 230
9.3.5 Ages 95 and Over 231
9.3.6 Calculation of Other Values 232
9.4 The 1980-82 Canadian Life Tables 232
9.4.1 Infant Tables 233
9.4.2 Adult Tables, Ages to 4 235
9.4.3 Adult Tables, Ages 5 and Over 236
9.4.4 The Final Table 238
9.5 Summary 238
Exercises 239
PART PRACTICAL ASPECTS OF
111 -
SURVIVAL MODEL ESTIMATION
Chapter 10
EVALUATION OF ESTIMATORS FROM SAMPLE DATA
10.1 Introduction 245
10.2 Actual Ages 245
10.2.1 Decimal Years 246
10.2.2 Exact Ages 247