Stochastic Discounting, Aggregate Claims, and the Bootstrap

1994 ◽  
Vol 26 (01) ◽  
pp. 183-206 ◽  
Author(s):  
M. Aebi ◽  
P. Embrechts ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Random Variables ◽  
General Point ◽  
Claim Amount ◽  
Theoretical Justification ◽  
Aggregate Claim ◽  
Bootstrap Estimation ◽  
Aggregate Claims

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi ), (Zi ) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.

Stochastic Discounting, Aggregate Claims, and the Bootstrap

1994 ◽  
Vol 26 (1) ◽  
pp. 183-206 ◽  
Author(s):  
M. Aebi ◽  
P. Embrechts ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Random Variables ◽  
General Point ◽  
Claim Amount ◽  
Theoretical Justification ◽  
Aggregate Claim ◽  
Bootstrap Estimation ◽  
Aggregate Claims

Obtaining good estimates for the distribution function of random variables like (‘perpetuity’) and (‘aggregate claim amount’), where the (Yi), (Zi) are independent i.i.d. sequences and (N(t)) is a general point process, is a key question in insurance mathematics. In this paper, we show how suitably chosen metrics provide a theoretical justification for bootstrap estimation in these cases. In the perpetuity case, we also give a detailed discussion of how the method works in practice.

On a point process with independent locations

1975 ◽  
Vol 12 (3) ◽  
pp. 435-446 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Independent Random Variables ◽  
General Conditions

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

On a point process with independent locations

1975 ◽  
Vol 12 (03) ◽  
pp. 435-446 ◽  
Author(s):  
Valerie Isham
Keyword(s):  
Distribution Function ◽  
Point Process ◽  
Point Processes ◽  
Random Variables ◽  
Independent Random Variables ◽  
General Conditions

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

Estimating the cumulative distribution function for the linear combination of gamma random variables

2017 ◽  
Vol 20 (5) ◽  
pp. 939-951
Author(s):  
Amal Almarwani ◽  
Bashair Aljohani ◽  
Rasha Almutairi ◽  
Nada Albalawi ◽  
Alya O. Al Mutairi
Keyword(s):  
Distribution Function ◽  
Linear Combination ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Cumulative Distribution

scholarly journals A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Fa-mei Zheng
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Wiener Process ◽  
Continuous Distribution ◽  
Random Variables ◽  
Standard Wiener Process ◽  
Trimmed Sums ◽  
Fixed Constant ◽  
Random Products ◽  
Trimmed Sum

Let be a sequence of independent and identically distributed positive random variables with a continuous distribution function , and has a medium tail. Denote and , where , , and is a fixed constant. Under some suitable conditions, we show that , as , where is the trimmed sum and is a standard Wiener process.

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