请教高手,已知随机变量之和的数据,求参数估计

有一模型，随机变量Z=X+s*Y,其中，X也是随机变量，符合GAMMA(a,b)分布，Y也是是随机变量，符合t(1)分布(即自由度为1的T分布，也叫Cauchy分布）。此模型中的参数是a,b,s，需要估计这3个参数，请教高手，有何办法？注：Z是可以观察的数据，X和Y是不能观察的。

说说解决的总体方法和思路就行，但请务必是能实际行得通的。为证明你的方法是实际行得通，请根据下面观察到的1000个Z的数据来估计模型中这3个参数（a,b,s）。请告诉我估计值（或95%置信度的估计区间），并说说解决的总体方法和思路，但不必一定要麻烦地写出详细过程。谢谢啦！

1000个Z的数据:
13.637, 1.384, 6.346, -18.518, -2.996, 54.412, 0.673, 22.646, -130.965, 11.524, 49.127, 27.583, 12.375, 10.378, -4.026, 15.382, 14.140, 23.385, 9.657, 1.635, -2.759, 3.164, 9.403, 2.618, 2.054, 5.072, 1.103, 2.965, -0.468, 8.124, 5.767, 28.291, -2.783, 15.347, 7.776, 21.783, 8.490, 24.930, 7.963, 6.653, 3.517, 2.129, 25.785, 16.978, 28.369, 10.727, 4.780, 2.089, -26.135, 21.385, 15.011, 19.676, 4.421, 30.297, 9.471, 6.472, 2.778, 3.964, 1.135, 3.106, 5.568, -1.402, 12.583, 1.895, 13.280, 5.222, 9.095, 20.651, 12.363, 50.274, 9.601, 16.919, -798.597, 18.127, 5.687, 6.899, 19.046, 15.323, 1.477, 4.969, -0.324, 17.606, 6.395, 27.506, 10.420, 3.418, 24.748, 9.485, 65.904, -1.875, 22.644, -86.795, 7.628, 44.268, 10.056, 14.694, 27.173, 3.104, -19.443, 10.406, 1.038, 18.261, 17.080, 0.121, 19.950, 27.173, 91.869, 15.793, 8.328, 17.970, 9.887, 0.791, 3.465, 29.352, -2.077, 14.676, 5.441, 6.492, 11.678, 5.071, -33.289, 8.841, 43.941, 0.844, 5.224, 12.433, 4.276, 0.699, 26.339, -5.359, 4.983, -2.393, -0.994, 1.634, 47.295, 5.162, 11.850, 7.782, -0.380, 19.194, 3.092, -12.955, 2.516, 4.427, 5.212, -5.472, 35.390, 9.897, 33.953, 51.898, 43.270, 11.355, 6.230, 11.368, 558.736, 3.852, 39.390, -16.373, 9.388, 12.106, 3.442, 20.787, 13.789, -14.846, 0.481, 18.314, -0.401, 41.212, 3.629, -0.788, 11.770, 2.638, 3.537, 1.490, 8.443, 0.498, 28.841, 5.314, 27.047, 7.326, 8.527, 12.113, 6.942, 23.273, 0.782, 19.119, 1.751, 45.216, 5.484, 32.334, 6.248, 20.041, 34.155, 32.618, 3.310, 2.562, 9.906, 13.168, 6.462, 25.341, 15.792, 10.875, 1.058, 3.122, 14.840, 7.384, 9.706, 13.527, -1.763, 15.125, 19.795, 30.298, 14.086, 7.468, 7.613, 35.677, 12.668, -63.797, 9.172, 16.087, 1.030, 29.373, 11.362, 54.466, 1.197, 11.697, 1.508, 3.670, -40.305, 3.176, 3.166, 16.003, 26.290, 26.303, 9.562, -7.304, 2.482, 25.479, 11.707, 16.924, 9.875, 0.481, 38.675, 8.138, 8.688, 8.838, 7.218, 7.710, -1.311, 17.567, 10.880, 12.482, 23.050, 1.107, 37.956, -24.044, 10.381, 6.213, -0.854, 2.248, 3.792, 7.921, 20.227, 5.784, 57.864, 6.962, 11.425, 20.986, 3.371, 88.152, 7.531, 18.790, 7.764, 0.397, 6.419, 9.248, 5.369, 3.796, 17.567, 2.805, 102.387, 13.739, 28.116, 3.828, 6.881, -661.616, 8.830, 14.671, 3.663, 2.130, -0.915, 0.031, 9.519, -1.006, 7.713, 14.063, -6.959, 7.287, 8.464, -3.506, 8.106, -3.810, 3.286, 17.839, 11.129, 29.317, 9.231, 12.962, 5.995, 6.145, 26.515, 20.266, 24.090, 4.124, 3.772, 1.536, 17.356, 2.310, 5.897, 6.105, 9.964, 25.085, 4.800, 8.885, 13.020, 8.051, 2.739, 8.466, 12.263, 15.146, 8.955, -0.555, 11.334, 12.218, -52.627, 6.494, 12.594, 16.305, 11.110, 7.135, 1.008, 11.493, -2.206, 6.494, 16.709, 9.383, 6.424, 1.049, 18.770, 0.493, 0.882, 0.549, 5.956, 8.247, 19.461, 11.380, 8.615, 2.878, 5.123, 13.735, 6.661, 3.264, 16.697, 9.806, 34.663, 4.842, 19.055, 27.883, 7.201, 15.705, 20.920, 17.248, 7.983, 3.208, 3.971, 8.294, 0.550, 1.547, 1.517, 23.455, 18.734, 21.365, 2.929, 29.429, 38.793, 21.936, 5.679, 0.782, -39.772, 7.403, 10.046, 4.459, 0.725, -1.660, 18.081, 18.684, 23.006, 21.137, 15.891, 26.868, 17.984, 26.834, 45.597, -83.572, 679.326, 10.237, 4.215, 9.158, 5.636, 23.137, -6720.426, 52.365, 67.384, 2.061, 30.940, 6.059, 5.144, 1.789, 12.702, 6.049, 11.783, 12.144, 2.378, 15.252, 6.793, 5.949, 2.787, 19.278, 25.042, 18.042, 14.860, -34.952, 8.021, -5.071, 3.642, 7.057, 6.917, 0.403, 6.767, 27.360, 18.120, 2.178, 8.936, 19.202, 2.805, 22.248, 0.381, 7.090, 2.500, 19.076, -0.580, 25.961, 6.483, 10.087, 5.789, 23.689, 14.228, 1.948, 18.726, 69.676, 19.457, -124.490, -7.436, 4.421, 19.164, 3.246, 18.278, 12.058, 5.187, 30.378, 1.981, 12.059, 4.966, 11.032, 16.319, 6.582, 10.043, 3.794, 16.068, 2.778, -0.145, 31.677, 7.859, 8.144, 2.543, 10.961, 26.704, 2.233, 3.638, 3.218, 9.313, -5.457, 0.975, 6.008, 4.422, 3.734, 17.294, 25.353, 6.658, 35.247, -1.224, 14.056, 26.676, 10.022, 4.637, 37.786, 26.627, 50.364, 22.690, 51.024, 8.822, 2.807, 8.217, -0.720, 2.582, 0.320, 14.796, 1.977, 5.438, -1.299, 2.981, 23.453, 9.665, 1.658, 13.094, -2.090, 16.913, 12.492, 1.532, 6.889, 13.058, 2.078, 13.991, 35.080, 4.938, 10.440, 5.631, -0.067, 9.592, 15.669, 13.420, 22.249, 16.688, 5.327, 7.162, 4.393, 1.043, -14.120, 10.602, 57.264, 11.156, 21.402, 7.594, 9.430, 21.560, 15.950, 2.413, 1.323, 99.582, -1.331, 7.541, -3.290, 1.444, 1.155, 2.435, 0.573, 31.092, -93.585, 11.405, 1.269, 14.141, 5.146, 21.210, 17.843, 4.962, 8.996, 8.538, 24.017, 20.107, 19.639, 13.453, 13.361, 7.108, 13.680, -7.516, 0.986, -0.211, 64.061, 15.878, 4.280, 15.771, 29.436, 1.097, 2.857, 16.019, 13.914, 1.434, 3.376, -8.815, 15.301, 22.740, 10.607, 20.031, 4.461, 6.381, 33.035, 13.021, 43.330, 5.329, 14.648, -2.256, 10.292, 4.092, 11.082, 25.338, 11.862, 16.129, 27.426, 6.534, 13.109, 6.329, 19.900, 1.312, 7.374, 41.869, 0.067, 19.318, 10.501, 23.450, 26.266, 22.336, 5.557, -7.003, 37.916, 39.447, -2.745, 8.441, 5.381, 37.239, 11.179, 4.251, 7.310, 34.457, 5.202, 8.719, -2.426, -7.043, 40.711, 5.519, 15.040, 0.778, 9.927, 41.601, -0.236, 16.875, 16.986, 16.883, 2.263, 12.645, 7.120, 5.418, 15.385, 19.468, 8.913, 2.363, 16.318, 3.743, 13.115, 5.578, 8.217, 1.009, -13.150, 25.477, 2.196, 15.328, 18.667, 9.875, 4.199, 13.446, 0.842, 0.416, 17.764, 1.144, -3.564, 31.311, 3.826, 0.953, 3.476, 23.073, 6.050, 19.405, 17.732, 5.765, -0.409, 9.461, -12.021, 5.912, 12.250, 23.028, 17.011, 13.228, 2.966, 0.326, 3.133, -12.696, 11.414, 47.081, 10.863, 13.336, 21.886, 7.865, 0.699, 7.707, 4.154, -2.688, 38.588, 6.237, -1.102, -2.076, 12.966, 13.583, 18.093, 5.852, -2.378, 1.061, 2.434, 18.449, 24.669, 2.929, 8.563, 13.595, 6.619, 5.553, 7.612, 14.503, 9.308, 11.443, 14.793, 7.200, 23.219, 3.119, 10.103, 9.694, 30.915, -75.479, 3.905, 12.113, 1.614, -79.526, 8.315, 13.051, 8.898, 10.518, 3.877, 0.973, 1.405, 11.832, 0.416, 6.645, 12.166, 4.229, 20.308, 3.704, -1.086, -10.310, 3.271, 22.018, 23.643, 12.749, 6.518, 26.650, 4.254, 16.481, 6.312, 6.409, 6.171, 0.990, 7.609, 4.228, 5.575, 13.290, 5.204, 9.719, 7.531, 5.285, 3.559, -1.121, 16.730, 1.582, 3.962, 21.733, 4.433, 5.432, -8.846, 9.405, 38.003, 50.754, 29.785, 36.479, 25.278, 5.536, 5.608, 15.686, 9.746, 7.621, -5.580, 10.458, 23.636, 12.935, 1.166, 18.592, 1.720, 0.943, 4.181, 10.511, 6.080, 11.154, 19.529, -4.294, 6.903, 11.722, 13.167, 9.671, 12.828, 1.944, 9.943, 1.948, 8.002, 3.599, -12.709, 5.823, 11.852, 12.153, 11.543, 6.098, 15.361, 9.885, 21.288, 7.984, 1.013, 14.105, 3.276, 7.929, 15.971, 31.483, 7.143, 23.901, 5.821, 20.928, 5.693, 8.974, -2.517, 26.955, 0.348, 2.870, 41.735, 2.952, 2.461, 30.690, 2.887, 35.049, 20.277, 4.356, 5.514, 12.435, -3.110, 0.108, 46.709, 7.409, 10.245, 16.195, -1.790, 2.180, 4.357, 26.917, -27.702, 4.023, 1.417, -2.210, 10.027, 14.324, 9.345, 6.722, 14.777, 3.760, 20.625, 12.583, 23.815, 7.780, 21.379, 18.303, 19.118, 15.608, 5.509, 7.974, 18.933, -5.783, 15.441, 5.431, 5.743, 16.953, 13.001, 30.358, 28.097, 17.091, 22.857, 4.530, 38.174, 8.210, 1.678, -0.490, 3.185, 10.158, 11.884, 9.019, 5.909, 8.777, 6.424, 2.107, 17.141, 5.909, 13.723, -6.729, 12.400, 8.542, 12.943, 13.449, 12.419, 32.523, 12.486, 4.794, 18.790, 0.617, -0.592, 3.749, -14.900, 7.088, 1.265, 5.585, 4.781, 0.877, 31.726, 29.624, 14.757, 26.491, 9.247, 3.130, 15.907, -0.121, 4.617, 0.336, 3.980, 15.114, 1.662, 6.070, 11.372, -1.770, 0.896, 20.409, 12.204, 11.026, 3.924, 2.506, -38.999, 21.734, 0.849, 4.442, 19.987, -2.145, -0.472, 78.371, 3.449, 8.327, 0.406, -911.014, 8.487, -2.156, 5.339, 21.215, 1.615, 1.571, -17.526, 0.809, 1.776, 4.361, 13.821, -8.811, 4.299, 1.082, 4.474, -27.006, 0.192, 6.306, 17.378, 4.869, 14.928, 15.820, 41.816, 0.252, 7.397

嫌悬赏少了？还是对此类问题没兴趣？

TimeT 发表于 2012-12-29 10:37
嫌悬赏少了？还是对此类问题没兴趣？

我觉得只需要写出Z的density function，然后用MLE就行，数据够多就非常准。

Chemist_MZ 发表于 2013-4-24 02:58
我觉得只需要写出Z的density function，然后用MLE就行，数据够多就非常准。

谢谢提示和查看（似乎大家都对此问题无兴趣）。但是我想，该Z的PDF是个积分形式（我想你说的density function 就是 PDF = probability density function吧），不大能写成closed form的公式吧，所以很难求其最大值。这位高手要么试试用MLE（或其他方法）算出a, b, s 的估计值（或范围）？如果您有时间的话，虽然论坛币对您已不重要了，不过我可以再多给10个（因为我有的不多）以表心意。

TimeT 发表于 2013-5-1 22:30
谢谢提示和查看（似乎大家都对此问题无兴趣）。但是我想，该Z的PDF是个积分形式（我想你说的density func ...

哦，我是说用程序搞定，用数值方法找最大值，比如借助matlab的fminsearch或者fmincon函数，我自己写了一个MLE function，随便什么density都行，因为MLE有自己的渐进理论，所以对于任何pdf g(x)都行。但是，这里确实如你所说，两个随机变量的线性组合的pdf是一个积分，假设问题可以简化，X和Y独立，那就可以直接用卷积公式，这里X和Y比较复杂还不是同一个分布，但是如果能写成积分的样子只是时间耗费可能长一点，做理论上应该还是能做的，只要能写出Z的pdf
不过那样我可以预见非常非常慢，因为gamma function本来就是个积分，卷积又要有积分。如果我像大多数在街上工作的industry guy那样，我会用一个非常不正规，但是也算是比较讨巧的办法，至少非常简单可行。t-distribution的mean和skewness都是0，因此Z的mean和skewness就是X的。而gamma distribution的moments又特别简洁

mean=a*b, skewness=2/sqrt(a); 这样就能解出a和b，再利用variance，解出s, 但是很可惜，Cauchy 的variance是无穷，所以s估计不出，至少用这种方法。因为Cauchy分布的这个性质，用别的方法即使s能估计出来，那想必也非常不准确。

我只是提供一些思路，实际操作还得去查阅一些方法。

我是版主，管理论坛回答问题一定意义上说是职责，所以不收币。

Chemist_MZ 发表于 2013-5-2 00:10
哦，我是说用程序搞定，用数值方法找最大值，比如借助matlab的fminsearch或者fmincon函数，我自己写了一个 ...

万分感谢！学习中。

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