= 0.5,\\{} & {} \displaystyle \sum _{k = 0}^{M (t)} \frac{\exp \left( k \widehat{\beta _0} \right) \exp \left[ -\exp \left( \widehat{\beta _0} \right) \right] }{k!} = 0.5,\\{} & {} \displaystyle \sum _{k = 0}^{M (t)} \frac{\exp \left[ k \left( \widehat{\beta _0} + \widehat{\beta _1} t \right) \right] \exp \left[ -\exp \left( \widehat{\beta _0} + \widehat{\beta _0} t \right) \right] }{k!} = 0.5,\\{} & {} \displaystyle \sum _{k = 0}^{M(t)} \frac{\exp \left[ k \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) \right] \exp \left[ -\exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) \right] }{k!} = 0.5,\\{} & {} \displaystyle \sum _{k = 0}^{M(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.5,\\{} & {} \displaystyle \sum _{k = 0}^{M(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.5, \end{aligned}$$and$$\begin{aligned} \displaystyle \sum _{k = 0}^{M(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) + \widehat{\beta _2} P(t-2) \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) + \widehat{\beta _2} P(t-2) \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.5, \end{aligned}$$for \(t = 2022, 2023, \ldots , 2031\). = 0.025, 0.975,\\{} & {} \displaystyle \sum _{k = 0}^{L (t), U(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} t \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.025, 0.975,\\{} & {} \displaystyle \sum _{k = 0}^{L (t), U(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.025, 0.975, \end{aligned}$$and$$\begin{aligned} \displaystyle \sum _{k = 0}^{L (t), U(t)} \left[ {k + \exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) + \widehat{\beta _2} P(t-2) \right) - 1 \atopwithdelims ()k} \left( 1 - {\widehat{\phi }} \right) ^{\exp \left( \widehat{\beta _0} + \widehat{\beta _1} P(t-1) + \widehat{\beta _2} P(t-2) \right) } \left( {\widehat{\phi }} \right) ^k \right] = 0.025, 0.975, \end{aligned}$$for \(t = 2022, 2023, \ldots , 2031\).

Source: The North Africa Journal July 15, 2023 14:57 UTC