Estimating Functions for Circular Time Series Models

Abstract

In this article, we describe an estimating function (EF) approach for circular time series models. We construct EFs based on conditional trigonometric moments of the circular discrete-time stochastic processes and provide closed form expressions for the optimal EF of the model parameters and its associated Godambe information. When the conditional circular mean and concentration are functions of the same parameters of interest, we show that the combined EF is more informative than its component sine and cosine EFs. We discuss recursive estimation of circular model parameters and illustrate the approach on two well known circular time series models.

Appendix A: The Gobambe EF Approach for Stochastic Processes

Let $\mathbf {y}_{n}=(y_{1},{\ldots } , y_{n})^{\prime }$ and $({\Omega }, \mathcal {F}, P_{\boldsymbol {\theta }})$ denote the underlying probability space, and let $\mathcal {F}_{t}$ be the σ-field generated by {y₁,…,y_t, t ≥ 1}. Let h_t(y_t, 𝜃),1 ≤ t ≤ n be specified q-dimensional martingale differences (MDs). Let ${\mathscr{M}}$ denote the class of zero mean, square integrable k-dimensional martingale EFs of the form

$$ \mathcal{M} = \left\{\mathbf{g} (\mathbf{y}_{n},\boldsymbol{\theta}): \mathbf{g} (\mathbf{y}_{n},\boldsymbol{\theta}) = {\sum}_{t=1}^{n} \mathbf{a}_{t-1} (\boldsymbol{\theta}) \mathbf{h}_{t} (\mathbf{y}_{t},\boldsymbol{\theta})\right\}, $$

(Appendix.1)

where a_t− 1(𝜃) are k × q matrices depending on 𝜃 and on y_t− 1, 1 ≤ t ≤ n. It is assumed that the EFs g(y_n, 𝜃) are almost surely differentiable with respect to the components of 𝜃, $E \!\left (\left .\!\! \frac {\partial \mathbf {g}(\mathbf {y}_{n},\boldsymbol {\theta })}{\partial \boldsymbol {\theta }} \!\right |\mathcal {F}_{n-1} \!\!\right )$ and that $E (\mathbf {g}(\mathbf {y}_{n},\boldsymbol {\theta }) \mathbf {g} (\mathbf {y}_{n},\boldsymbol {\theta })^{\prime }|\mathcal {F}_{n-1})$ is positive definite (p.d.) for all 𝜃 and for each n ≥ 1. The expectations are always taken with respect to P_𝜃. Estimators of 𝜃 can be obtained by solving the EE g(y_n, 𝜃) = 0. To simplify notation in the following equations, we use g and h_t to denote g(y_n, 𝜃) and h_t(y_t, 𝜃) respectively.

Theorem 2.

The optimality result for EFs states that in the class of all zero mean, square integrable martingale EFs ${\mathscr{M}}$, the optimal EF g^∗ maximizes, in the partial order of nonnegative definite matrices, the information matrix

$$ \begin{array}{@{}rcl@{}} \mathbf{I}_{\mathbf{g}} &= & \left( \sum\limits_{t=1}^{n} \mathbf{a}_{t-1} E \left[\left. \frac{\partial \mathbf{h}_{t} }{\partial \boldsymbol{\theta}}\right| \mathcal{F}_{t -1}\right]\right)^{\prime} \left( \sum\limits_{t = 1}^{n} E [(\mathbf{a}_{t - 1} \mathbf{h}_{t} ) (\mathbf{a}_{t - 1} \mathbf{h}_{t})^{\prime}| \mathcal{F}_{t -1}]\right)^{-1} \\ && \times \left( \sum\limits_{t = 1}^{n} \mathbf{a}_{t - 1} E \left[\left. \frac{\partial \mathbf{h}_{t}} {\partial \boldsymbol{\theta}}\right|\mathcal{F}_{t - 1}\right]\right). \end{array} $$

The optimal EF and corresponding optimal information are given by

$$ \begin{array}{@{}rcl@{}} \mathbf{g}^{\ast} &=& \sum\limits_{t = 1}^{n} \mathbf{a}_{t-1}^{\ast} \mathbf{h}_{t} = \sum\limits_{t = 1}^{n} \left( E \left[\left. \frac{\partial \mathbf{h}_{t} }{\partial \boldsymbol{\theta}}\right| \mathcal{F}_{t - 1}\right]\right)^{\prime} (E [\mathbf{h}_{t} \mathbf{h}_{t}^{\prime}| \mathcal{F}_{t - 1}])^{-1} \mathbf{h}_{t}, \\ \mathbf{I}_{\mathbf{g}^{\ast}} &= & \sum\limits_{t = 1}^{n} \left( \text{E} \left[\left. \frac{\partial \mathbf{h}_{t} }{\partial \boldsymbol{\theta}}\right| \mathcal{F}_{t-1}\!\right]\!\right)^{\prime} \left( E [\mathbf{h}_{t} \mathbf{h}_{t}^{\prime}| \mathcal{F}_{t-1}]\right)^{-1} \!\left( \!\!E \!\left[\left. \frac{\partial \mathbf{h}_{t} }{\partial \boldsymbol{\theta}}\right| \mathcal{F}_{t-1}\!\right]\!\right).\\ \end{array} $$

(Appendix.2)

From Eq. Appendix.2, the recursive estimate of 𝜃 becomes

$$ \begin{array}{@{}rcl@{}} \widehat{\boldsymbol{\theta}}_{t} &= & \widehat{\boldsymbol{\theta}}_{t-1} +\mathbf{K}_{t} \mathbf{a}_{t-1}^{\ast} (\widehat{\boldsymbol{\theta}}_{t-1}) h_{t} (\widehat{\boldsymbol{\theta}}_{t-1}), \\ \mathbf{K}_{t} &= & \mathbf{K}_{t-1} \!\left[ \mathbf{I}_{p} \!- \!\left( \mathbf{a}_{t-1}^{\ast}(\widehat{ \boldsymbol{\theta}}_{t-1}) \frac{\partial h_{t} (\widehat{\boldsymbol{\theta}}_{t-1})}{\partial \boldsymbol{\theta}^{\prime}} + \frac{\partial \mathbf{a}_{t-1}^{\ast} (\widehat{\boldsymbol{\theta}}_{t-1})}{\partial \boldsymbol{\theta}} h_{t} (\widehat{\boldsymbol{\theta}}_{t-1}) \!\right)\! \mathbf{K}_{t-1} \!\right]^{-1}, \\ \end{array} $$

(Appendix.3)

for t = 1,…,n, where I_k is the k-dimensional identity matrix. When 𝜃 is a scalar parameter, its recursive estimate is given by

$$ \begin{array}{@{}rcl@{}} \widehat{\theta}_{t} &= & \widehat{\theta}_{t-1} + K_{t} [a^{\ast}_{t-1} (\widehat{\theta}_{t-1}) h_{t} (\widehat{\theta}_{t-1})], \\ K_{t} &= & K_{t-1} -A_{t} K_{t-1} K_{t}, \end{array} $$

(Appendix.4)

where

$A_{t} = a^{\ast }_{t-1} (\widehat {\theta }_{t-1}) \frac {\partial h_{t} (\widehat {\theta }_{t-1})}{\partial \theta } +\frac {\partial a^{\ast }_{t-1} (\widehat {\theta }_{t-1})}{\partial \theta } h_{t} (\widehat {\theta }_{t-1})$.

When the conditional mean and variance are functions of the same parameter, the following Lemma 1 gives the form of the combined EF based on two non-orthogonal martingale differences, which is more informative than each of the component EFs; see Thavaneswaran et al. (2015) for more details. Consider a discrete time stochastic process {y_t, t = 1,2,…} with first two conditional moments given by

$$ \begin{array}{@{}rcl@{}} \mu_{t}(\boldsymbol \theta) &= & E \left( y_{t}|\mathcal{F}_{t-1}\right), \text{ and } {\sigma_{t}^{2}}(\boldsymbol \theta) = \text{Var} \left( y_{t}| \mathcal{F}_{t-1}\right). \end{array} $$

To estimate the parameter 𝜃 based on the observations $\mathbf {y}{_{n}}=(y_{1}, \ldots , y_{n})^{\prime }$, consider two martingale differences (MDs) for t = 1,…,n, i.e., m_t(𝜃) = y_t − μ_t(𝜃) and $s_{t} (\boldsymbol \theta ) = {m_{t}^{2}} (\boldsymbol \theta ) - {\sigma _{t}^{2}} (\boldsymbol \theta )$), with quadratic variations and covariation given by 〈m〉_t,〈s〉_t and 〈m, s〉_t.

Lemma 1.

In the class of all combined EFs of the form $\mathcal {G}_{C} =\{\mathbf {g}_{C} (\boldsymbol \theta ): \mathbf {g}_{C} (\boldsymbol \theta ) = {\sum }_{t=1}^{n} \left (\mathbf {a}_{t-1} m_{t} + \mathbf {b}_{t-1} s_{t}\right )\}$,

(a) the optimal EF is given by $\mathbf {g}_{C}^{\ast } (\boldsymbol \theta ) = {\sum }_{t=1}^{n} \left (\mathbf {a}_{t-1}^{*} m_{t} + \mathbf {b}_{t-1}^{*} s_{t}\right )$, where

$$ \begin{array}{@{}rcl@{}} \mathbf{a}_{t-1}^{\ast} &=& \left( 1 - \frac{\langle m,s{\rangle_{t}^{2}}}{\langle m\rangle_{t}\langle s\rangle_{t}}\right)^{-1} \left( -\frac{\partial \mu_{t}}{\partial \boldsymbol \theta}\frac{1}{\langle m\rangle_{t}} + \frac{\partial \langle s(\boldsymbol \theta) \rangle_{t} }{\partial \boldsymbol \theta} \frac{\langle m, s\rangle_{t}}{\langle m\rangle_{t}\langle s\rangle_{t}}\right), \text{ and } \\ \mathbf{b}_{t-1}^{\ast} &=& \left( 1 - \frac{\langle m,s{\rangle_{t}^{2}}}{\langle m\rangle_{t}\langle s\rangle_{t}}\right)^{-1} \left( \frac{\partial \mu_{t}}{\partial \boldsymbol \theta} \frac{\langle m, s(\boldsymbol \theta) \rangle_{t}}{\langle m\rangle_{t}\langle s\rangle_{t}}-\frac{\partial \langle s(\boldsymbol \theta) \rangle_{t} }{\partial \boldsymbol \theta}\frac{1}{\langle s\rangle_{t}}\right); \end{array} $$

(b) the information $\mathbf {I}_{\mathbf {g}_{C}^{\ast }} (\boldsymbol \theta )$ is given by

$$ \begin{array}{@{}rcl@{}} \mathbf{I}_{\mathbf{g}_{C}^{\ast}} (\boldsymbol \theta) &=& \sum\limits_{t=1}^{n}\left( 1 - \frac{\langle m, s{\rangle_{t}^{2}}}{\langle m\rangle_{t}\langle s\rangle_{t}}\right)^{-1} \left( \frac{\partial \mu_{t}}{\partial \boldsymbol \theta} \frac{\partial \mu_{t}}{\partial \boldsymbol \theta^{\prime}} \frac{1}{\langle m\rangle_{t}} + \frac{\partial \langle s(\boldsymbol \theta) \rangle_{t}}{\partial \boldsymbol \theta} \frac{\partial \langle s(\boldsymbol \theta) \rangle_{t}}{\partial \boldsymbol \theta^{\prime}} \frac{1}{\langle s\rangle_{t}} \right.\\ && \left.-\left( \frac{\partial \mu_{t}}{\partial \boldsymbol \theta}\frac{\partial \langle s(\boldsymbol \theta) \rangle_{t}}{\partial \boldsymbol \theta^{\prime}} + \frac{\partial \langle s(\boldsymbol \theta) \rangle_{t}}{\partial \boldsymbol \theta} \frac{\partial \mu_{t}}{\partial \boldsymbol \theta^{\prime}}\right) \frac{\langle m, s\rangle_{t}}{\langle m\rangle_{t}\langle s\rangle_{t}}\right). \end{array} $$

The proof of this lemma is similar to the proof for combining linear and quadratic EFs shown in Thavaneswaran et al. (2015) and is omitted here.