The random variable Y$Y$ is said to have a two-parameter APE distribution, denoted by APE(α, λ)$\text{APE}(\alpha, \lambda)$, with the shape parameter $\alpha>0$ and scale parameters as α > 0 and λ > 0, respectively, ifparameter $\lambda>0$ if the PDF of Y for y > 0density function is:
f(y; α, λ) = {(log α/ α−1)* λe^(−λy)*α^(1−e^(−λy))} ,if α = 1
= λe^(−λy), if α = 1
$$f_Y(y) = \begin{cases} \log (\frac{\log \alpha}{\alpha-1}) \cdot \lambda e^{-\lambda y} \cdot \alpha^{1-e^{-\lambda y}} & & & \text{for } \alpha \neq 1 \\[8pt] \lambda e^{-\lambda y} & & & \text{for } \alpha = 1 \\[6pt] \end{cases}$$
Let $y_1, y_2,..., y_n$$Y_1, Y_2,..., Y_n \sim \text{IID APE}(\alpha,\lambda)$ be a random sample from the APE(α, λ), then distribution. Then the log-likelihood function becomesis:
l(α, λ) = n log α + n log(log α/( α − 1) )+ n log λ − λsummation (yi) − (log α)[summation(e^(-λy)]
$$\ell_\mathbf{y}(\alpha,\lambda) = n \log \alpha + n \log \bigg( \frac{\log \alpha}{\alpha-1} \bigg) + n \log \lambda - \lambda \sum_i y_i - (\log \alpha) \sum_i e^{-\lambda y_i}.$$
How do iI find MLE of this distribution in RR?
Data given My data is given below:
1 4 4 7 11 13 15 15 17 18 19 19 20 20 22 23 28 29 31 32 36 37 47 48 49 50 54 54 55 59 59 61 61
66 72 72 75 78 78 81 93 96 99 108 113 114 120 120 120 123 124 129 131 137 145 151 156 171
176 182 188 189 195 203 208 215 217 217 217 224 228 233 255 271 275 275 275 286 291 312
312 312 315 326 326 329 330 336 338 345 348 354 361 364 369 378 390 457 467 498 517 566
644 745 871 1312 1357 1613 1630
