By Estate V Khmaladze; Leigh Roberts; Mzia Khmaladze

ISBN-10: 1466505745

ISBN-13: 9781466505742

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**Additional resources for Statistical methods with applications to demography and life insurance**

**Sample text**

Tn of height 1/n each. Let us clarify the question of jump-points. The empirical distribution function has its first jump, from 0 to 1/n, at the point T(1) = min Ti ; its second jump, from 1/n to 2/n, occurs at the point T(2) , which is the second smallest duration of life; and so on until the last jump, from (n − 1)/n to 1, at the point T(n) = max Ti . The random times T(1) , T(2) , . . , T(n) are called order statistics (based on T1 , T2 , . . , Tn ). Although we assumed that random variables T1 , T2 , .

K}. 6) Brownian bridge; χ 2 goodness of fit statistic 48 Although we do not prove this equivalence formally, intuitively it is quite clear. The k dimensional distribution of the random vector {Vn (x j ), j = 1, . . k, } is also called the finite-dimensional distribution of the process Vn . 5) for any collection x1 < . . < xk and for any finite k, we have therefore proved the following: if lifetimes T1 , T2 , . . , Tn are independent and all have the same distribution function F, then all finite-dimensional distributions of the empirical process Vn converge to the corresponding finite-dimensional distributions of the Brownian bridge V .

N[Fn (xk ) − F(xk )]. 3) 35 Instead of these random variables it is more convenient to consider their differences √ √ n[∆Fn (x j ) − ∆F(x j )] = n∆[Fn (x j ) − F(x j )], j = 0, 1, . . 4) where ∆Fn (x j ) = Fn (x j+1 ) − Fn (x j ) is the relative frequency of deaths between ages x j and x j+1 , while ∆F(x j ) = F(x j+1 )−F(x j ) is the probability of a death occurring between these ages. 3), which we will do in the second part of this lecture. We start by observing that the joint distribution of the differences ∆zn (x j ), j = 0, 1, .

### Statistical methods with applications to demography and life insurance by Estate V Khmaladze; Leigh Roberts; Mzia Khmaladze

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