Mixed estimation methods for Halphen distributions with applications in extreme hydrologic events

Abstract

The Halphen family of distributions is a flexible and complete system to fit sets of observations independent and identically distributed. Recently, it is shown that this family of distributions represents a potential alternative to the generalized extreme value distributions to model extreme hydrological events. The existence of jointly sufficient statistics for parameter estimation leads to optimality of the method of maximum likelihood (ML). Nevertheless, the ML method requires numerical approximations leading to less accurate values. However, estimators by the method of moments (MM) are explicit and their computation is fast. Even though MM method leads to good results, it is not optimal. In order to combine the advantages of the ML (optimality) and MM (efficiency and fast computations), two new mixed methods were proposed in this paper. One of the two methods is direct and the other is iterative, denoted respectively direct mixed method (MMD) and iterative mixed method (MMI). An overall comparison of the four estimation methods (MM, ML, MMD and MMI) was performed using Monte Carlo simulations regarding the three Halphen distributions. Generally, the MMI method can be considered for the three Halphen distributions since it is recommended for a majority of cases encountered in hydrology. The principal idea of the mixed methods MMD and MMI could be generalized for other distributions with complicated density functions.

Appendices

Appendix A: ML estimation

The object of this appendix is to provide some elements related to the ML estimation. These elements are required to the development of the MMD and MMI methods. Hereafter, we present in more details the steps (a) and (b) given in Sect. 3.2. The main steps of the ML method, based on the previous numerical resolution approach for the three Halphen distributions, are illustrated in Fig. 8.

Fig. 8

ML estimation algorithm for parameters of Halphen distributions

A.1 ML for HA

• a. For a fixed ν, α(ν) and m(ν) are solutions of the system of Eqs. 23 and 24 which can be rewritten as:

  $$ D_{A} (\alpha (\nu ),\nu ) = {\frac{A}{H}} $$

  (38)
  $$ m(\nu ) \, = {\frac{A}{{R_{A} (\alpha (\nu ), \, \nu )}}} $$

  (39)

  where

  $$ D_{A} (\alpha ,\nu ) = {\frac{{K_{\nu + 1} (2\alpha )K_{\nu - 1} (2\alpha )}}{{\left[ {K_{\nu } (2\alpha )} \right]^{2} }}} $$

  (40)

  and

  $$ R_{A} (\alpha ,\nu ) = {\frac{{K_{\nu + 1} (2\alpha )}}{{K_{\nu } (2\alpha )}}} $$

  (41)

• b. In order to determine the variation interval of ν, we find signs of l ^′_HA (U) and l ^′_HA (–U) where l ^′_HA (ν) is the derivative of the partial log-likelihood function log L _HA, we have:

  $$ l^{\prime}_{\text{HA}} ( - U) = n\left[ {\log \left( {{\frac{G}{H}}{\frac{1}{U}}} \right) + \Uppsi (U)} \right]\;{\text{and}}\;l^{\prime}_{\text{HA}} (U) = n\left[ {\log \left( {{\frac{G}{A}}U} \right) - \Uppsi (U)} \right] $$

  (42)

  where $\Uppsi (z) = \partial \left[ {\log \Upgamma (z)} \right]/\partial z$ is the digamma function and $U = AH^{ - 1} /\left[ {AH^{ - 1} - 1} \right].$ Note that U > 1, since H ≤ A. Hence:

  • If l _HA ′(U) > 0 and l _HA ′(–U) > 0, then the value of ν that maximizes log L _HA(ν|α, m) is greater than U, and in that case, the sample would better presented by a Gamma distribution;

  • If l _HA ′(U) < 0 and l _HA ′(–U) < 0, then the value of ν that maximizes log L _HA is less than −U, and in that case, the sample would be better presented by an Inverse Gamma distribution.

  • If l ^′_HA (U) < 0 and l _HA ′(–U) > 0, then the value of ν that maximizes log L _HA is in the interval ]–U;U[ ; and the ML estimators are solutions of the system composed by equations 38 and 39. In that case, the partial log-likelihood function is given by:

    $$ \log L_{\rm HA} (\nu |\alpha ,m) = n\left\{ {\log \left[ {{\frac{{G^{\nu - 1} }}{{2m^{\nu } K_{\nu } (2\alpha )}}}} \right] - {\frac{{\alpha \left[ {K_{\nu + 1} (2\alpha ) + K_{\nu - 1} (2\alpha )} \right]}}{{K_{\nu } (2\alpha )}}}} \right\} $$

    (43)

A.2 ML for HB

• a. For a fixed ν, α(ν) and m(ν) are solutions of the following system obtained from Eqs. 26 and 27:

$$ D_{B} (\alpha (\nu ),\nu ) = {\frac{Q}{{A^{2} }}} $$

(44)

$$ m(\nu ) \, = \, {\frac{A}{{R_{B} (\alpha (\nu ),\;\nu )}}} $$

(45)

where

$$ D_{B} (\alpha ,\nu ) = {\frac{{ef_{\nu + 1} (\alpha )ef_{\nu } (\alpha )}}{{\left[ {ef_{{\nu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} (\alpha )} \right]^{2} }}} $$

(46)

and

$$ R_{B} (\alpha ,\nu ) = {\frac{{ef_{{\nu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} (\alpha )}}{{ef_{\nu } (\alpha )}}} $$

(47)

• b. In order to establish the variation interval of ν, we determine the sign of l ^′_HB (ν), the derivative of the partial log-likelihood function log L _HB, evaluated at $V = 1/\left[ {2\left( {Q/A^{2} - 1} \right)} \right],$ we obtain then:

  $$ l^{\prime}_{\rm HB} (V) = 2n\left[ {\log \left( {{\frac{G}{A}}2V} \right) - \Uppsi (2V)} \right] $$

  (48)

  Hence,

  • If l ^′_HB (V) ≥ 0, then the argument of the maximum of the partial log-likelihood function log L _HB with respect to ν is greater than V. In that case, the Gamma distribution is more appropriate to represent the sample;

  • If l ^′_HB (V) < 0, then the maximum of the partial log-likelihood function log L _HB with respect to ν is reached in the interval ]0;V[ . In that case, the partial log-likelihood function is given by:

    $$ \log L_{\rm HB} (\nu |\alpha ,m) = n\left\{ {\log \left[ {{\frac{{2G^{2\nu - 1} }}{{m^{2\nu } ef_{\nu } (\alpha )}}}} \right] - {\frac{{ef_{\nu + 1} (\alpha )}}{{ef_{\nu } (\alpha )}}} + \hat{\alpha }{\frac{{ef_{{\nu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} (\alpha )}}{{ef_{\nu } (\alpha )}}}} \right\} $$

    (49)

A.3 ML for HIB

• a. For a fixed ν, α(ν) and m(ν) are solutions of the following system obtained from Eq. 29 and 30:

  $$ D_{B} (\alpha (\nu ),\nu ) = {\frac{{H^{2} }}{IQ}} $$

  (50)
  $$ m(\nu ) \, = \, HR_{B} (\alpha (\nu ),\nu ) $$

  (51)

where the functions D _B and R _B are given respectively by expressions 46 and 47.

• b. To establish the interval in which belongs ν, we determine the sign of l ^′_HIB (W)which is the derivative of the partial log-likelihood function log L _HIB evaluated at $W = 1/\left[ {2\left( {H^{2} IQ^{ - 1} - 1} \right)} \right],$ given by:

  $$ l^{\prime}_{\rm HIB} (W) = 2n\left[ {\log \left( {{\frac{H}{G}}2W} \right) - \Uppsi (2W)} \right] $$

  (52)

  Therefore,

  • If l ^′_HIB (W) ≥ 0, then the value of ν that maximizes the log-likelihood function log L _HIB is greater than W. In this case, the Inverse Gamma distribution is more appropriate to represent the sample;

  • If l ^′_HIB (W) < 0, then the maximum of function log L _HIB with respect to ν is reached in the interval]0;W[. In this case, the log-likelihood function is given by:

    $$ \log L_{\rm HIB} (\nu |\alpha ,m) = n\left\{ {\log \left[ {{\frac{{2m^{2\nu } }}{{G^{2\nu + 1} ef_{\nu } (\alpha )}}}} \right] - {\frac{{ef_{\nu + 1} (\alpha )}}{{ef_{\nu } (\alpha )}}} + \hat{\alpha }{\frac{{ef_{{\nu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} (\alpha )}}{{ef_{\nu } (\alpha )}}}} \right\} $$

    (53)