Date of Award

May 2020

Degree Type


Degree Name

Doctor of Philosophy



First Advisor

Vytaras Brazauskas

Committee Members

Jay Beder, Daniel Gervini, Wei Wei, David Spade


actuarial science, claim frequency, discrete distribution, risk measure, Smooth quantiles, statistical models


Statistical models for the claim severity and claim frequency variables are routinely constructed and utilized by actuaries. Typical applications of such models include identification of optimal deductibles for selected loss elimination ratios, pricing of contract layers, determining credibility factors, risk and economic capital measures, and evaluation of effects of inflation, market trends and other quantities arising in insurance. While the actuarial literature on the severity models is extensive and rapidly growing, that for the claim frequency models lags behind. One of the reasons for such a gap is that various actuarial metrics do not possess ``nice'' statistical properties for the discrete models whilst their counterparts for the continuous models do. The objectives of this dissertation to addressing the issue described above are the following:

• Generalize the definitions of ``smoothed quantiles'' for samples and populations of claim counts to vectors of smoothed quantiles. This is motivated by the fact that multiple quantiles are needed for better understanding of insurance risks.

• Investigate large- and small-sample properties of smoothed quantile estimators for vectors, when the underlying claim count distribution has finite support.

• Extend the definition of smoothed quantiles for discrete distributions with infinite support, and study asymptotic and finite-sample properties of the associated estimators.

• Illustrate the appropriateness and flexibility of such tools in solving risk measurement problems.

Smoothed quantiles are defined using the theory of fractional or imaginary order statistics, which was originated by Stigler (1977). To prove consistency and asymptotic normality of sample estimators of smoothed quantiles, we utilize the results of Wang and Hutson (2011) and generalize them to vectors of smoothed quantiles. Further, we thoroughly investigate extensions of this methodology to discrete populations with infinite support (e.g., Poisson and zero-inflated Poisson distributions). Furthermore, large- and small-sample properties of the newly designed estimators are investigated theoretically and through Monte Carlo simulations. Finally, applications of smoothed quantiles to risk measurement (e.g., estimation of distortion risk measures such as value-at-risk, conditional tail expectation, and proportional hazards transform) are discussed and illustrated using actual insurance data.