38:
1720:
1518:
2132:), these are both negatively biased but consistent estimators. With the correction, the corrected sample variance is unbiased, while the corrected sample standard deviation is still biased, but less so, and both are still consistent: the correction factor converges to 1 as sample size grows.
2336:
687:
1097:
1156:
The notion of asymptotic consistency is very close, almost synonymous to the notion of convergence in probability. As such, any theorem, lemma, or property which establishes convergence in probability may be used to prove the consistency. Many such tools exist:
1476:
1715:{\displaystyle {\begin{aligned}&T_{n}+S_{n}\ {\xrightarrow {d}}\ \alpha +\beta ,\\&T_{n}S_{n}\ {\xrightarrow {d}}\ \alpha \beta ,\\&T_{n}/S_{n}\ {\xrightarrow {d}}\ \alpha /\beta ,{\text{ provided that }}\beta \neq 0\end{aligned}}}
1861:
366:
1302:
268:
466:
121:. This means that the distributions of the estimates become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to
2188:
586:
885:
704:, because often one is interested in estimating a certain function or a sub-vector of the underlying parameter. In the next example, we estimate the location parameter of the model, but not the scale:
1523:
2432:
893:
2040:
2381:
146:, and consistency is a property of what occurs as the sample size “grows to infinity”. If the sequence of estimates can be mathematically shown to converge in probability to the true value
1376:
2066:
2180:
2160:
1142:
1755:
2130:
2101:
283:
1170:
218:
2725:
399:
2331:{\displaystyle \Pr(T_{n})={\begin{cases}1-1/n,&{\mbox{if }}\,T_{n}=\theta \\1/n,&{\mbox{if }}\,T_{n}=n\delta +\theta \end{cases}}}
682:{\displaystyle {\underset {n\to \infty }{\operatorname {plim} }}\;T_{n}(X^{\theta })=g(\theta ),\ \ {\text{for all}}\ \theta \in \Theta .}
76:; at the same time, these estimators are biased. The limiting distribution of the sequence is a degenerate random variable which equals
69:, the true value of which is 4. This sequence is consistent: the estimators are getting more and more concentrated near the true value
823:
2627:
2604:
2578:
2464:
718:
475:
is actually unknown, and thus, the convergence in probability must take place for every possible value of this parameter. Suppose
1145:
2695:
2104:
2459:
1092:{\displaystyle \Pr \!\left=\Pr \!\left=2\left(1-\Phi \left({\frac {{\sqrt {n}}\,\varepsilon }{\sigma }}\right)\right)\to 0}
2656:
2386:
110:—having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates
1992:
2651:
2618:(1994). "Chapter 36: Large sample estimation and hypothesis testing". In Robert F. Engle; Daniel L. McFadden (eds.).
2344:
1471:{\displaystyle T_{n}\ {\xrightarrow {p}}\ \theta \ \quad \Rightarrow \quad g(T_{n})\ {\xrightarrow {p}}\ g(\theta )}
2701:
1326:
207:
111:
2076:
1881:
1735:
is given by an explicit formula, then most likely the formula will employ sums of random variables, and then the
1318:
2570:
1487:
can be used to combine several different estimators, or an estimator with a non-random convergent sequence. If
31:
1974:
as it ignores all points but the last), so E = E and it is unbiased, but it does not converge to any value.
165:
2646:
2080:
2045:
2668:
796:
1736:
1484:
173:
2219:
2454:
2443:
1989:
Alternatively, an estimator can be biased but consistent. For example, if the mean is estimated by
736:
2720:
2633:
2448:
1877:
529:
2623:
2600:
2588:
2574:
1314:
2165:
1876:
is defined implicitly, for example as a value that maximizes certain objective function (see
1856:{\displaystyle {\frac {1}{n}}\sum _{i=1}^{n}g(X_{i})\ {\xrightarrow {p}}\ \operatorname {E} }
2677:
1899:
1161:
In order to demonstrate consistency directly from the definition one can use the inequality
491:
27:
Statistical estimator converging in probability to a true parameter as sample size increases
2138:
1127:
17:
2615:
2558:
2072:
361:{\displaystyle \lim _{n\to \infty }\Pr {\big (}|T_{n}-\theta |>\varepsilon {\big )}=0.}
37:
2109:
2592:
2086:
1981:
converges to a value, then it is consistent, as it must converge to the correct value.
1297:{\displaystyle \Pr \!{\big }\leq {\frac {\operatorname {E} {\big }}{h(\varepsilon )}},}
2714:
2663:
2563:
2637:
263:{\displaystyle {\underset {n\to \infty }{\operatorname {plim} }}\;T_{n}=\theta .}
760:
138:, and then imagines being able to keep collecting data and expanding the sample
132:
131:
In practice one constructs an estimator as a function of an available sample of
2705:
88:
2451:— alternative, although rarely used concept of consistency for the estimators
153:, it is called a consistent estimator; otherwise the estimator is said to be
1958:
as the estimator of the mean E. Note that here the sampling distribution of
372:
189:
100:
461:{\displaystyle \Pr {\big (}\lim _{n\to \infty }T_{n}=\theta {\big )}=1.}
2363:
1817:
1670:
1613:
1558:
1446:
1398:
788:. This defines a sequence of estimators, indexed by the sample size
2681:
142:. In this way one would obtain a sequence of estimates indexed by
880:{\displaystyle \scriptstyle (T_{n}-\mu )/(\sigma /{\sqrt {n}})}
1903:
1313:
being either the absolute value (in which case it is known as
2068:, it approaches the correct value, and so it is consistent.
795:
From the properties of the normal distribution, we know the
471:
A more rigorous definition takes into account the fact that
2324:
1120:
of sample means is consistent for the population mean
160:
Consistency as defined here is sometimes referred to as
2289:
2245:
827:
2697:
Econometrics lecture (topic: unbiased vs. consistent)
2389:
2347:
2191:
2168:
2141:
2112:
2089:
2048:
1995:
1758:
1521:
1379:
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896:
826:
589:
402:
286:
221:
1970:
is the same as the underlying distribution (for any
1746:} of random variables and under suitable conditions,
550:) } be a sequence of estimators for some parameter
2562:
2427:{\displaystyle \operatorname {E} =\theta +\delta }
2426:
2375:
2330:
2174:
2154:
2124:
2095:
2060:
2034:
1855:
1714:
1470:
1344:(·) is a real-valued function continuous at point
1296:
1136:
1091:
879:
681:
460:
360:
262:
164:. When we replace convergence in probability with
2035:{\displaystyle {1 \over n}\sum x_{i}+{1 \over n}}
1977:However, if a sequence of estimators is unbiased
1177:
951:
900:
62:, ...} is a sequence of estimators for parameter
2192:
1174:
948:
897:
414:
403:
303:
288:
103:—a rule for computing estimates of a parameter
2376:{\displaystyle T_{n}{\xrightarrow {p}}\theta }
1880:), then a more complicated argument involving
569:observations of a sample. Then this sequence {
2542:
2494:
1269:
1237:
1218:
1180:
993:
970:
447:
408:
347:
308:
8:
177:
1317:), or the quadratic function (respectively
2434:, and the bias does not converge to zero.
808:is itself normally distributed, with mean
608:
240:
2403:
2388:
2358:
2352:
2346:
2300:
2295:
2288:
2275:
2256:
2251:
2244:
2231:
2214:
2202:
2190:
2167:
2146:
2140:
2111:
2088:
2047:
2022:
2013:
1996:
1994:
1849:
1836:
1812:
1800:
1784:
1773:
1759:
1757:
1694:
1683:
1665:
1656:
1647:
1641:
1608:
1599:
1589:
1553:
1544:
1531:
1522:
1520:
1441:
1429:
1393:
1384:
1378:
1268:
1267:
1252:
1236:
1235:
1226:
1217:
1216:
1195:
1179:
1178:
1172:
1129:
1064:
1057:
1054:
1017:
1007:
992:
991:
979:
969:
968:
967:
960:
957:
939:
928:
916:
907:
906:
895:
866:
861:
850:
835:
825:
659:
626:
613:
590:
588:
446:
445:
433:
417:
407:
406:
401:
346:
345:
334:
322:
313:
307:
306:
291:
285:
245:
222:
220:
1902:but not consistent. For example, for an
36:
30:For broader coverage of this topic, see
2530:
2518:
2506:
2482:
2475:
713:Sample mean of a normal random variable
2666:(1988), "Likelihood and convergence",
2083:(that is, when using the sample size
7:
2061:{\displaystyle n\rightarrow \infty }
1309:the most common choice for function
887:has a standard normal distribution:
490:} is a family of distributions (the
393:to the true value of the parameter:
212:to the true value of the parameter:
97:asymptotically consistent estimator
168:, then the estimator is said to be
2390:
2055:
1827:
1229:
1131:
1047:
673:
602:
424:
298:
234:
25:
2622:. Vol. 4. Elsevier Science.
2465:Instrumental variables estimation
1106:tends to infinity, for any fixed
2162:be a sequence of estimators for
2071:Important examples include the
1418:
1414:
2726:Asymptotic theory (statistics)
2460:Statistical hypothesis testing
2409:
2396:
2208:
2195:
2052:
1850:
1846:
1840:
1833:
1806:
1793:
1465:
1459:
1435:
1422:
1415:
1285:
1279:
1264:
1245:
1207:
1188:
1083:
929:
908:
873:
855:
847:
828:
759:observations, one can use the
717:Suppose one has a sequence of
647:
641:
632:
619:
599:
421:
335:
314:
295:
231:
1:
2135:Here is another example. Let
1739:can be used: for a sequence {
1325:Another useful result is the
1148:of the normal distribution).
172:. Consistency is related to
2652:Encyclopedia of Mathematics
1894:Unbiased but not consistent
565:will be based on the first
18:Consistency of an estimator
2742:
2599:(2nd ed.). Springer.
2597:Theory of Point Estimation
1327:continuous mapping theorem
1113:. Therefore, the sequence
751:distribution. To estimate
29:
2543:Newey & McFadden 1994
2495:Lehman & Casella 1998
2077:sample standard deviation
1882:stochastic equicontinuity
1696: provided that
1359:) will be consistent for
719:statistically independent
576:} is said to be (weakly)
2645:Nikulin, M. S. (2001) ,
2620:Handbook of Econometrics
2571:Harvard University Press
1152:Establishing consistency
209:converges in probability
112:converges in probability
32:Consistency (statistics)
2175:{\displaystyle \theta }
1889:Bias versus consistency
1146:cumulative distribution
391:converges almost surely
178:bias versus consistency
166:almost sure convergence
2647:"Consistent estimator"
2428:
2377:
2332:
2176:
2156:
2126:
2097:
2062:
2036:
1857:
1789:
1716:
1472:
1319:Chebyshev's inequality
1298:
1138:
1093:
881:
683:
532:from the distribution
462:
362:
264:
188:Formally speaking, an
84:
2669:Philosophy of Science
2565:Advanced Econometrics
2429:
2378:
2333:
2177:
2157:
2155:{\displaystyle T_{n}}
2127:
2098:
2063:
2042:it is biased, but as
2037:
1985:Biased but consistent
1858:
1769:
1717:
1473:
1299:
1139:
1137:{\displaystyle \Phi }
1094:
882:
797:sampling distribution
692:This definition uses
684:
463:
363:
265:
40:
2387:
2345:
2189:
2166:
2139:
2110:
2087:
2046:
1993:
1898:An estimator can be
1756:
1737:law of large numbers
1519:
1377:
1171:
1128:
894:
824:
700:) instead of simply
587:
400:
284:
219:
93:consistent estimator
2509:, equation (3.2.5).
2485:, Definition 3.4.2.
2455:Regression dilution
2444:Efficient estimator
2367:
2125:{\displaystyle n-1}
2081:Bessel's correction
1821:
1674:
1617:
1562:
1450:
1402:
799:of this statistic:
755:based on the first
387:strongly consistent
170:strongly consistent
83:with probability 1.
2449:Fisher consistency
2424:
2373:
2328:
2323:
2293:
2249:
2172:
2152:
2122:
2105:degrees of freedom
2093:
2058:
2032:
1878:extremum estimator
1853:
1712:
1710:
1468:
1336:is consistent for
1294:
1134:
1089:
877:
876:
679:
606:
458:
428:
358:
302:
260:
238:
128:converges to one.
85:
2368:
2292:
2248:
2096:{\displaystyle n}
2030:
2004:
1826:
1822:
1811:
1767:
1697:
1679:
1675:
1664:
1622:
1618:
1607:
1567:
1563:
1552:
1485:Slutsky's theorem
1455:
1451:
1440:
1413:
1407:
1403:
1392:
1315:Markov inequality
1289:
1072:
1062:
1012:
1002:
965:
871:
666:
662:
658:
655:
591:
528:} is an infinite
413:
287:
273:i.e. if, for all
223:
204:weakly consistent
16:(Redirected from
2733:
2698:
2684:
2659:
2641:
2610:
2584:
2568:
2559:Amemiya, Takeshi
2546:
2540:
2534:
2533:, Theorem 3.2.7.
2528:
2522:
2521:, Theorem 3.2.6.
2516:
2510:
2504:
2498:
2492:
2486:
2480:
2433:
2431:
2430:
2425:
2408:
2407:
2382:
2380:
2379:
2374:
2369:
2359:
2357:
2356:
2341:We can see that
2337:
2335:
2334:
2329:
2327:
2326:
2305:
2304:
2294:
2290:
2279:
2261:
2260:
2250:
2246:
2235:
2207:
2206:
2181:
2179:
2178:
2173:
2161:
2159:
2158:
2153:
2151:
2150:
2131:
2129:
2128:
2123:
2102:
2100:
2099:
2094:
2067:
2065:
2064:
2059:
2041:
2039:
2038:
2033:
2031:
2023:
2018:
2017:
2005:
1997:
1968:
1967:
1957:
1956:
1940:
1939:
1928:
1927:
1917:
1916:
1862:
1860:
1859:
1854:
1824:
1823:
1813:
1809:
1805:
1804:
1788:
1783:
1768:
1760:
1721:
1719:
1718:
1713:
1711:
1698:
1695:
1687:
1677:
1676:
1666:
1662:
1661:
1660:
1651:
1646:
1645:
1635:
1620:
1619:
1609:
1605:
1604:
1603:
1594:
1593:
1583:
1565:
1564:
1554:
1550:
1549:
1548:
1536:
1535:
1525:
1477:
1475:
1474:
1469:
1453:
1452:
1442:
1438:
1434:
1433:
1411:
1405:
1404:
1394:
1390:
1389:
1388:
1303:
1301:
1300:
1295:
1290:
1288:
1274:
1273:
1272:
1257:
1256:
1241:
1240:
1227:
1222:
1221:
1200:
1199:
1184:
1183:
1143:
1141:
1140:
1135:
1124:(recalling that
1112:
1098:
1096:
1095:
1090:
1082:
1078:
1077:
1073:
1068:
1063:
1058:
1055:
1029:
1025:
1021:
1013:
1008:
1003:
998:
997:
996:
984:
983:
974:
973:
966:
961:
958:
944:
940:
932:
921:
920:
911:
886:
884:
883:
878:
872:
867:
865:
854:
840:
839:
820:. Equivalently,
688:
686:
685:
680:
664:
663:
660:
656:
653:
631:
630:
618:
617:
607:
605:
527:
492:parametric model
489:
467:
465:
464:
459:
451:
450:
438:
437:
427:
412:
411:
367:
365:
364:
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338:
327:
326:
317:
312:
311:
301:
269:
267:
266:
261:
250:
249:
239:
237:
162:weak consistency
21:
2741:
2740:
2736:
2735:
2734:
2732:
2731:
2730:
2711:
2710:
2696:
2692:
2662:
2644:
2630:
2613:
2607:
2587:
2581:
2557:
2554:
2549:
2541:
2537:
2529:
2525:
2517:
2513:
2505:
2501:
2493:
2489:
2481:
2477:
2473:
2440:
2399:
2385:
2384:
2348:
2343:
2342:
2322:
2321:
2296:
2286:
2269:
2268:
2252:
2242:
2215:
2198:
2187:
2186:
2164:
2163:
2142:
2137:
2136:
2108:
2107:
2103:instead of the
2085:
2084:
2073:sample variance
2044:
2043:
2009:
1991:
1990:
1987:
1966:
1963:
1962:
1961:
1955:
1952:
1951:
1950:
1938:
1935:
1934:
1933:
1926:
1923:
1922:
1921:
1915:
1912:
1911:
1910:
1896:
1891:
1884:has to be used.
1874:
1796:
1754:
1753:
1744:
1733:
1709:
1708:
1652:
1637:
1633:
1632:
1595:
1585:
1581:
1580:
1540:
1527:
1517:
1516:
1503:
1492:
1425:
1380:
1375:
1374:
1357:
1334:
1275:
1248:
1228:
1191:
1169:
1168:
1154:
1126:
1125:
1118:
1107:
1056:
1050:
1040:
1036:
975:
959:
956:
952:
912:
905:
901:
892:
891:
831:
822:
821:
807:
782:
776:
768:
734:
727:
715:
710:
622:
609:
595:
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584:
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476:
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318:
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217:
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196:
186:
152:
127:
120:
109:
82:
75:
68:
61:
54:
47:
35:
28:
23:
22:
15:
12:
11:
5:
2739:
2737:
2729:
2728:
2723:
2713:
2712:
2709:
2708:
2691:
2690:External links
2688:
2687:
2686:
2682:10.1086/289429
2676:(2): 228–237,
2660:
2642:
2628:
2614:Newey, W. K.;
2611:
2605:
2589:Lehmann, E. L.
2585:
2579:
2553:
2550:
2548:
2547:
2535:
2523:
2511:
2499:
2497:, p. 332.
2487:
2474:
2472:
2469:
2468:
2467:
2462:
2457:
2452:
2446:
2439:
2436:
2423:
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2402:
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2366:
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2314:
2311:
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2299:
2287:
2285:
2282:
2278:
2274:
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2270:
2267:
2264:
2259:
2255:
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2241:
2238:
2234:
2230:
2227:
2224:
2221:
2220:
2218:
2213:
2210:
2205:
2201:
2197:
2194:
2171:
2149:
2145:
2121:
2118:
2115:
2092:
2057:
2054:
2051:
2029:
2026:
2021:
2016:
2012:
2008:
2003:
2000:
1986:
1983:
1964:
1953:
1936:
1930:} one can use
1924:
1913:
1895:
1892:
1890:
1887:
1886:
1885:
1872:
1866:
1865:
1864:
1863:
1852:
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1845:
1842:
1839:
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1829:
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1816:
1808:
1803:
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1795:
1792:
1787:
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1731:
1725:
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1722:
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1644:
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1636:
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1628:
1625:
1616:
1612:
1602:
1598:
1592:
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1576:
1573:
1570:
1561:
1557:
1547:
1543:
1539:
1534:
1530:
1526:
1524:
1511:
1510:
1501:
1490:
1481:
1480:
1479:
1478:
1467:
1464:
1461:
1458:
1449:
1445:
1437:
1432:
1428:
1424:
1421:
1417:
1410:
1401:
1397:
1387:
1383:
1369:
1368:
1355:
1332:
1307:
1306:
1305:
1304:
1293:
1287:
1284:
1281:
1278:
1271:
1266:
1263:
1260:
1255:
1251:
1247:
1244:
1239:
1234:
1231:
1225:
1220:
1215:
1212:
1209:
1206:
1203:
1198:
1194:
1190:
1187:
1182:
1176:
1163:
1162:
1153:
1150:
1133:
1116:
1100:
1099:
1088:
1085:
1081:
1076:
1071:
1067:
1061:
1053:
1049:
1046:
1043:
1039:
1035:
1032:
1028:
1024:
1020:
1016:
1011:
1006:
1001:
995:
990:
987:
982:
978:
972:
964:
955:
950:
947:
943:
938:
935:
931:
927:
924:
919:
915:
910:
904:
899:
875:
870:
864:
860:
857:
853:
849:
846:
843:
838:
834:
830:
803:
780:
774:
770: = (
766:
735:, ...} from a
732:
725:
721:observations {
714:
711:
709:
706:
690:
689:
678:
675:
672:
669:
652:
649:
646:
643:
640:
637:
634:
629:
625:
621:
616:
612:
604:
601:
598:
594:
572:
561:
542:
535:
523:
516:
510:
503:
480:
469:
468:
457:
454:
449:
444:
441:
436:
432:
426:
423:
420:
416:
410:
405:
385:is said to be
377:
369:
368:
357:
354:
349:
344:
341:
337:
333:
330:
325:
321:
316:
310:
305:
300:
297:
294:
290:
271:
270:
259:
256:
253:
248:
244:
236:
233:
230:
226:
202:is said to be
194:
185:
182:
150:
125:
118:
107:
80:
73:
66:
59:
52:
45:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2738:
2727:
2724:
2722:
2719:
2718:
2716:
2707:
2703:
2699:
2694:
2693:
2689:
2683:
2679:
2675:
2671:
2670:
2665:
2661:
2658:
2654:
2653:
2648:
2643:
2639:
2635:
2631:
2629:0-444-88766-0
2625:
2621:
2617:
2612:
2608:
2606:0-387-98502-6
2602:
2598:
2594:
2590:
2586:
2582:
2580:0-674-00560-0
2576:
2572:
2567:
2566:
2560:
2556:
2555:
2551:
2544:
2539:
2536:
2532:
2527:
2524:
2520:
2515:
2512:
2508:
2503:
2500:
2496:
2491:
2488:
2484:
2479:
2476:
2470:
2466:
2463:
2461:
2458:
2456:
2453:
2450:
2447:
2445:
2442:
2441:
2437:
2435:
2421:
2418:
2415:
2412:
2404:
2400:
2393:
2370:
2364:
2360:
2353:
2349:
2318:
2315:
2312:
2309:
2306:
2301:
2297:
2283:
2280:
2276:
2272:
2265:
2262:
2257:
2253:
2239:
2236:
2232:
2228:
2225:
2222:
2216:
2211:
2203:
2199:
2185:
2184:
2183:
2169:
2147:
2143:
2133:
2119:
2116:
2113:
2106:
2090:
2082:
2078:
2074:
2069:
2049:
2027:
2024:
2019:
2014:
2010:
2006:
2001:
1998:
1984:
1982:
1980:
1975:
1973:
1969:
1949:
1945:
1941:
1929:
1909:
1905:
1901:
1893:
1888:
1883:
1879:
1875:
1869:If estimator
1868:
1867:
1843:
1837:
1830:
1818:
1814:
1801:
1797:
1790:
1785:
1780:
1777:
1774:
1770:
1764:
1761:
1752:
1751:
1750:
1749:
1745:
1738:
1734:
1728:If estimator
1727:
1726:
1705:
1702:
1699:
1691:
1688:
1684:
1680:
1671:
1667:
1657:
1653:
1648:
1642:
1638:
1629:
1626:
1623:
1614:
1610:
1600:
1596:
1590:
1586:
1577:
1574:
1571:
1568:
1559:
1555:
1545:
1541:
1537:
1532:
1528:
1515:
1514:
1513:
1512:
1508:
1504:
1497:
1493:
1486:
1483:
1482:
1462:
1456:
1447:
1443:
1430:
1426:
1419:
1408:
1399:
1395:
1385:
1381:
1373:
1372:
1371:
1370:
1366:
1362:
1358:
1351:
1347:
1343:
1339:
1335:
1328:
1324:
1323:
1322:
1320:
1316:
1312:
1291:
1282:
1276:
1261:
1258:
1253:
1249:
1242:
1232:
1223:
1213:
1210:
1204:
1201:
1196:
1192:
1185:
1167:
1166:
1165:
1164:
1160:
1159:
1158:
1151:
1149:
1147:
1123:
1119:
1110:
1105:
1086:
1079:
1074:
1069:
1065:
1059:
1051:
1044:
1041:
1037:
1033:
1030:
1026:
1022:
1018:
1014:
1009:
1004:
999:
988:
985:
980:
976:
962:
953:
945:
941:
936:
933:
925:
922:
917:
913:
902:
890:
889:
888:
868:
862:
858:
851:
844:
841:
836:
832:
819:
815:
812:and variance
811:
806:
802:
798:
793:
791:
787:
783:
773:
769:
762:
758:
754:
750:
748:
744:
740:
731:
724:
720:
712:
707:
705:
703:
699:
695:
676:
670:
667:
650:
644:
638:
635:
627:
623:
614:
610:
596:
592:
583:
582:
581:
579:
575:
568:
564:
557:
553:
549:
545:
538:
531:
526:
519:
509:
502:
498:
493:
487:
483:
474:
455:
452:
442:
439:
434:
430:
418:
396:
395:
394:
392:
388:
384:
381:of parameter
380:
374:
355:
352:
342:
339:
331:
328:
323:
319:
292:
280:
279:
278:
276:
257:
254:
251:
246:
242:
228:
224:
215:
214:
213:
211:
210:
205:
201:
198:of parameter
197:
191:
183:
181:
179:
175:
171:
167:
163:
158:
156:
149:
145:
141:
137:
134:
129:
124:
117:
113:
106:
102:
98:
94:
90:
79:
72:
65:
58:
51:
44:
39:
33:
19:
2673:
2667:
2650:
2619:
2616:McFadden, D.
2596:
2564:
2545:, Chapter 2.
2538:
2531:Amemiya 1985
2526:
2519:Amemiya 1985
2514:
2507:Amemiya 1985
2502:
2490:
2483:Amemiya 1985
2478:
2340:
2134:
2070:
1988:
1978:
1976:
1971:
1959:
1947:
1943:
1931:
1919:
1907:
1897:
1870:
1740:
1729:
1506:
1499:
1495:
1488:
1364:
1360:
1353:
1349:
1345:
1341:
1337:
1330:
1310:
1308:
1155:
1121:
1114:
1108:
1103:
1101:
817:
813:
809:
804:
800:
794:
789:
785:
778:
771:
764:
756:
752:
746:
742:
738:
729:
722:
716:
701:
697:
693:
691:
577:
570:
566:
559:
558:). Usually,
555:
551:
547:
540:
533:
521:
514:
507:
500:
496:
485:
478:
472:
470:
390:
386:
382:
375:
370:
274:
272:
208:
203:
199:
192:
187:
169:
161:
159:
155:inconsistent
154:
147:
143:
140:ad infinitum
139:
135:
130:
122:
115:
104:
96:
92:
86:
77:
70:
63:
56:
49:
42:
2593:Casella, G.
761:sample mean
513:, … :
2715:Categories
2706:Mark Thoma
2552:References
2079:. Without
578:consistent
184:Definition
89:statistics
2721:Estimator
2664:Sober, E.
2657:EMS Press
2422:δ
2416:θ
2394:
2371:θ
2319:θ
2313:δ
2266:θ
2226:−
2170:θ
2117:−
2056:∞
2053:→
2007:∑
1831:
1771:∑
1703:≠
1700:β
1689:β
1681:α
1627:β
1624:α
1575:β
1569:α
1463:θ
1416:⇒
1409:θ
1283:ε
1262:θ
1259:−
1233:
1224:≤
1214:ε
1211:≥
1205:θ
1202:−
1132:Φ
1084:→
1070:σ
1066:ε
1048:Φ
1045:−
1023:σ
1015:ε
1005:≥
1000:σ
989:μ
986:−
937:ε
934:≥
926:μ
923:−
859:σ
845:μ
842:−
674:Θ
671:∈
668:θ
645:θ
628:θ
603:∞
600:→
443:θ
425:∞
422:→
373:estimator
343:ε
332:θ
329:−
299:∞
296:→
255:θ
235:∞
232:→
190:estimator
101:estimator
2638:29436457
2595:(1998).
2561:(1985).
2438:See also
2361:→
2291:if
2247:if
1906:sample {
1900:unbiased
1815:→
1668:→
1611:→
1556:→
1444:→
1396:→
777:+ ... +
708:Examples
539:. Let {
389:, if it
206:, if it
2702:YouTube
1509:, then
1505: →
1494: →
1348:, then
1144:is the
737:normal
661:for all
494:), and
277:> 0
2636:
2626:
2603:
2577:
1918:,...,
1825:
1810:
1678:
1663:
1621:
1606:
1566:
1551:
1498:, and
1454:
1439:
1412:
1406:
1391:
1111:> 0
665:
657:
654:
530:sample
176:; see
99:is an
2634:S2CID
2471:Notes
1329:: if
2624:ISBN
2601:ISBN
2575:ISBN
2075:and
1946:) =
1340:and
593:plim
580:if
340:>
225:plim
174:bias
133:size
91:, a
2704:by
2700:on
2678:doi
1979:and
1904:iid
1321:).
1102:as
499:= {
488:∈ Θ
415:lim
371:An
289:lim
114:to
95:or
87:In
2717::
2674:55
2672:,
2655:,
2649:,
2632:.
2591:;
2573:.
2569:.
2383:,
2193:Pr
2182:.
1972:n,
1367:):
1175:Pr
949:Pr
898:Pr
792:.
784:)/
763::
745:,
728:,
520:~
506:,
484::
456:1.
404:Pr
356:0.
304:Pr
180:.
157:.
55:,
48:,
2685:.
2680::
2640:.
2609:.
2583:.
2419:+
2413:=
2410:]
2405:n
2401:T
2397:[
2391:E
2365:p
2354:n
2350:T
2316:+
2310:n
2307:=
2302:n
2298:T
2284:,
2281:n
2277:/
2273:1
2263:=
2258:n
2254:T
2240:,
2237:n
2233:/
2229:1
2223:1
2217:{
2212:=
2209:)
2204:n
2200:T
2196:(
2148:n
2144:T
2120:1
2114:n
2091:n
2050:n
2028:n
2025:1
2020:+
2015:i
2011:x
2002:n
1999:1
1965:n
1960:T
1954:n
1948:x
1944:X
1942:(
1937:n
1932:T
1925:n
1920:x
1914:1
1908:x
1873:n
1871:T
1851:]
1847:)
1844:X
1841:(
1838:g
1834:[
1828:E
1819:p
1807:)
1802:i
1798:X
1794:(
1791:g
1786:n
1781:1
1778:=
1775:i
1765:n
1762:1
1743:n
1741:X
1732:n
1730:T
1706:0
1692:,
1685:/
1672:d
1658:n
1654:S
1649:/
1643:n
1639:T
1630:,
1615:d
1601:n
1597:S
1591:n
1587:T
1578:,
1572:+
1560:d
1546:n
1542:S
1538:+
1533:n
1529:T
1507:β
1502:n
1500:S
1496:α
1491:n
1489:T
1466:)
1460:(
1457:g
1448:p
1436:)
1431:n
1427:T
1423:(
1420:g
1400:p
1386:n
1382:T
1365:θ
1363:(
1361:g
1356:n
1354:T
1352:(
1350:g
1346:θ
1342:g
1338:θ
1333:n
1331:T
1311:h
1292:,
1286:)
1280:(
1277:h
1270:]
1265:)
1254:n
1250:T
1246:(
1243:h
1238:[
1230:E
1219:]
1208:)
1197:n
1193:T
1189:(
1186:h
1181:[
1122:μ
1117:n
1115:T
1109:ε
1104:n
1087:0
1080:)
1075:)
1060:n
1052:(
1042:1
1038:(
1034:2
1031:=
1027:]
1019:/
1010:n
994:|
981:n
977:T
971:|
963:n
954:[
946:=
942:]
930:|
918:n
914:T
909:|
903:[
874:)
869:n
863:/
856:(
852:/
848:)
837:n
833:T
829:(
818:n
816:/
814:σ
810:μ
805:n
801:T
790:n
786:n
781:n
779:X
775:1
772:X
767:n
765:T
757:n
753:μ
749:)
747:σ
743:μ
741:(
739:N
733:2
730:X
726:1
723:X
702:θ
698:θ
696:(
694:g
677:.
651:,
648:)
642:(
639:g
636:=
633:)
624:X
620:(
615:n
611:T
597:n
573:n
571:T
567:n
562:n
560:T
556:θ
554:(
552:g
548:X
546:(
543:n
541:T
536:θ
534:p
524:θ
522:p
517:i
515:X
511:2
508:X
504:1
501:X
497:X
486:θ
481:θ
479:p
477:{
473:θ
453:=
448:)
440:=
435:n
431:T
419:n
409:(
383:θ
378:n
376:T
353:=
348:)
336:|
324:n
320:T
315:|
309:(
293:n
275:ε
258:.
252:=
247:n
243:T
229:n
200:θ
195:n
193:T
151:0
148:θ
144:n
136:n
126:0
123:θ
119:0
116:θ
108:0
105:θ
81:0
78:θ
74:0
71:θ
67:0
64:θ
60:3
57:T
53:2
50:T
46:1
43:T
41:{
34:.
20:)
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