1824:
4934:
2508:
1545:
97:
the aim is to find the range of objects (airplanes, boats, etc.) by analyzing the two-way transit timing of received echoes of transmitted pulses. Since the reflected pulses are unavoidably embedded in electrical noise, their measured values are randomly distributed, so that the transit time must be
89:
For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters. Alternatively, it is desired to estimate the probability of a voter voting for
4007:
One of the simplest non-trivial examples of estimation is the estimation of the maximum of a uniform distribution. It is used as a hands-on classroom exercise and to illustrate basic principles of estimation theory. Further, in the case of estimation based on a single sample, it demonstrates
2966:
3690:
2676:
2280:
4288:
324:
1434:
2249:
3559:
2004:
1540:
2754:
1819:{\displaystyle \mathrm {var} \left({\hat {A}}_{2}\right)=\mathrm {var} \left({\frac {1}{N}}\sum _{n=0}^{N-1}x\right){\overset {\text{independence}}{=}}{\frac {1}{N^{2}}}\left)\right]={\frac {1}{N^{2}}}\left={\frac {\sigma ^{2}}{N}}}
3423:
3865:
500:
3933:
3115:
231:
3566:
2517:
4172:
1086:
1206:
240:
3800:
3198:
2743:
377:
1211:
59:
based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An
4131:
2113:
3428:
1886:
419:
5214:
448:
2108:
814:
921:
1441:
2503:{\displaystyle p(\mathbf {x} ;A)=\prod _{n=0}^{N-1}p(x;A)={\frac {1}{\left(\sigma {\sqrt {2\pi }}\right)^{N}}}\exp \left(-{\frac {1}{2\sigma ^{2}}}\sum _{n=0}^{N-1}(x-A)^{2}\right)}
997:
4966:
4343:
3245:
4058:
2275:
764:
3807:
460:
3870:
2971:
101:
As another example, in electrical communication theory, the measurements which contain information regarding the parameters of interest are often associated with a
4316:
135:
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3959:
3218:
2062:
2033:
1881:
1117:
943:
834:
730:
697:
671:
651:
1438:
At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances.
2961:{\displaystyle {\frac {\partial }{\partial A}}\ln p(\mathbf {x} ;A)={\frac {1}{\sigma ^{2}}}\left-A)\right]={\frac {1}{\sigma ^{2}}}\left-NA\right]}
4959:
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3122:
2681:
337:
4877:
4859:
4823:
4708:
5004:
4067:
4514:
3986:
563:
4952:
4483:
4419:
389:
65:
attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered:
1003:
4994:
4915:
4896:
4841:
4786:
4765:
4746:
3685:{\displaystyle {\frac {\partial ^{2}}{\partial A^{2}}}\ln p(\mathbf {x} ;A)={\frac {1}{\sigma ^{2}}}(-N)={\frac {-N}{\sigma ^{2}}}}
2671:{\displaystyle \ln p(\mathbf {x} ;A)=-N\ln \left(\sigma {\sqrt {2\pi }}\right)-{\frac {1}{2\sigma ^{2}}}\sum _{n=0}^{N-1}(x-A)^{2}}
5168:
5149:
4727:
3695:
842:
5090:
4504:
569:
105:
3200:
which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for
4689:
4462:
574:
535:
4801:
4648:
3235:
5035:
4544:
4166:
the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.
4020:
700:
334:(pmf), of the underlying distribution that generated the data must be stated conditional on the values of the parameters:
4283:{\displaystyle {\frac {1}{k}}{\frac {(N-k)(N+1)}{(k+2)}}\approx {\frac {N^{2}}{k^{2}}}{\text{ for small samples }}k\ll N}
5131:
1851:
327:
4938:
42:
5111:
319:{\displaystyle {\boldsymbol {\theta }}={\begin{bmatrix}\theta _{1}\\\theta _{2}\\\vdots \\\theta _{M}\end{bmatrix}},}
424:
116:
For a given model, several statistical "ingredients" are needed so the estimator can be implemented. The first is a
5209:
5204:
5084:
5061:
4534:
4455:
4439:
4346:
2067:
773:
551:
457:(MMSE) estimator, which utilizes the error between the estimated parameters and the actual value of the parameters
454:
1429:{\displaystyle \mathrm {E} \left=\mathrm {E} \left\right]={\frac {1}{N}}\left\right]\right]={\frac {1}{N}}\left=A}
5078:
4975:
4776:
4468:
331:
5025:
4549:
2252:
2244:{\displaystyle p(x;A)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2\sigma ^{2}}}(x-A)^{2}\right)}
673:
592:
70:
3229:
541:
5020:
4478:
3554:{\displaystyle {\frac {\partial }{\partial A}}\ln p(\mathbf {x} ;A)={\frac {1}{\sigma ^{2}}}\left-NA\right]}
1999:{\displaystyle p(w)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {1}{2\sigma ^{2}}}w^{2}\right)}
3937:
Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is
56:
421:
After the model is formed, the goal is to estimate the parameters, with the estimates commonly denoted
951:
5144:
5073:
5056:
4404:
4013:
557:
384:
69:
The probabilistic approach (described in this article) assumes that the measured data is random with
4321:
1828:
It would seem that the sample mean is a better estimator since its variance is lower for every
4524:
4424:
4150:
4002:
3982:
380:
35:
4025:
2258:
1535:{\displaystyle \mathrm {var} \left({\hat {A}}_{1}\right)=\mathrm {var} \left(x\right)=\sigma ^{2}}
5101:
5096:
5068:
4519:
4499:
4443:
4399:
4353:
4009:
3990:
3239:
1847:
1841:
742:
676:
524:
129:
117:
5158:
5116:
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4855:
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4723:
4704:
4685:
4394:
4374:
2511:
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The sample maximum is never more than the population maximum, but can be less, hence it is a
80:
assumes that the measured data vector belongs to a set which depends on the parameter vector.
4989:
4630:
4570:
4529:
4473:
4413:
3692:
and finding the negative expected value is trivial since it is now a deterministic constant
379:
It is also possible for the parameters themselves to have a probability distribution (e.g.,
31:
4539:
4431:
4389:
621:
587:
530:
102:
4293:
4153:, due to application of maximum estimation to estimates of German tank production during
4634:
4509:
4494:
4409:
4379:
4138:
3964:
3944:
3418:{\displaystyle {\mathcal {I}}(A)=\mathrm {E} \left(\left^{2}\right)=-\mathrm {E} \left}
3203:
2038:
2009:
1857:
1120:
1102:
928:
819:
706:
682:
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627:
581:
503:
77:
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4489:
604:
598:
546:
121:
4364:
Numerous fields require the use of estimation theory. Some of these fields include:
5163:
5154:
4870:
Indefinite
Quadratic Estimation and Control: A Unified Approach to H and H Theories
4384:
4154:
3860:{\displaystyle \mathrm {var} \left({\hat {A}}\right)\geq {\frac {1}{\mathcal {I}}}}
495:{\displaystyle \mathbf {e} ={\hat {\boldsymbol {\theta }}}-{\boldsymbol {\theta }}}
3928:{\displaystyle \mathrm {var} \left({\hat {A}}\right)\geq {\frac {\sigma ^{2}}{N}}}
520:
Commonly used estimators (estimation methods) and topics related to them include:
4807:
17:
5051:
4652:
4435:
4146:
1089:
3110:{\displaystyle 0={\frac {1}{\sigma ^{2}}}\left-NA\right]=\sum _{n=0}^{N-1}x-NA}
5183:
4658:
4369:
2748:
52:
4356:
estimator for the population maximum, but, as discussed above, it is biased.
4163:"The sample maximum plus the average gap between observations in the sample",
4944:
515:
226:{\displaystyle \mathbf {x} ={\begin{bmatrix}x\\x\\\vdots \\x\end{bmatrix}}.}
61:
4933:
5139:
737:
90:
a particular candidate, based on some demographic features, such as age.
3238:(CRLB) of the sample mean estimator, it is first necessary to find the
4803:
Optimal State
Estimation: Kalman, H-infinity, and Nonlinear Approaches
4792:
4149:, sampling without replacement. This problem is commonly known as the
502:
as the basis for optimality. This error term is then squared and the
4908:
Unbiased estimators and their applications. Vol. 2: Multivariate case
4872:. PA: Society for Industrial & Applied Mathematics (SIAM). 1999.
4889:
Unbiased estimators and their applications. Vol. 1: Univariate case
4318:, the (population) average size of a gap between samples; compare
4008:
philosophical issues and possible misunderstandings in the use of
816:). Since the variance is known then the only unknown parameter is
94:
4739:
Fundamentals of
Statistical Signal Processing: Estimation Theory
4061:
1096:
733:
4948:
1081:{\displaystyle {\hat {A}}_{2}={\frac {1}{N}}\sum _{n=0}^{N-1}x}
30:"Parameter estimation" redirects here. Not to be confused with
4621:
Johnson, Roger (1994), "Estimating the Size of a
Population",
3981:. In other words, the sample mean is the (necessarily unique)
27:
Branch of statistics to estimate models based on measured data
3851:
3251:
2073:
1201:{\displaystyle \mathrm {E} \left=\mathrm {E} \left\right]=A}
779:
3795:{\displaystyle -\mathrm {E} \left={\frac {N}{\sigma ^{2}}}}
3193:{\displaystyle {\hat {A}}={\frac {1}{N}}\sum _{n=0}^{N-1}x}
506:
of this squared value is minimized for the MMSE estimator.
4605:
Identification of
Parametric Models from Experimental Data
2738:{\displaystyle {\hat {A}}=\arg \max \ln p(\mathbf {x} ;A)}
372:{\displaystyle p(\mathbf {x} |{\boldsymbol {\theta }}).\,}
3220:
samples of a fixed, unknown parameter corrupted by AWGN.
326:
whose values are to be estimated. Third, the continuous
4126:{\displaystyle {\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1}
554:(MMSE), also known as Bayes least squared error (BLSE)
257:
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2012:
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685:
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630:
463:
427:
392:
340:
243:
138:
4778:
Detection, Estimation, and
Modulation Theory, Part 1
925:
Two possible (of many) estimators for the parameter
5130:
5044:
5013:
4982:
4758:An Introduction to Signal Detection and Estimation
4337:
4310:
4282:
4125:
4052:
3973:
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3859:
3794:
3684:
3553:
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3212:
3192:
3109:
2960:
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2502:
2269:
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2102:
2056:
2027:
1998:
1875:
1818:
1534:
1428:
1200:
1111:
1080:
991:
937:
915:
828:
808:
758:
724:
691:
665:
645:
494:
442:
413:
371:
318:
225:
5215:Mathematical and quantitative methods (economics)
4345:above. This can be seen as a very simple case of
3119:This results in the maximum likelihood estimator
616:Unknown constant in additive white Gaussian noise
2706:
414:{\displaystyle \pi ({\boldsymbol {\theta }}).\,}
4160:The formula may be understood intuitively as;
443:{\displaystyle {\hat {\boldsymbol {\theta }}}}
4960:
4434:or uncertainty and it is through statistical
3941:the Cramér–Rao lower bound for all values of
3804:Finally, putting the Fisher information into
2103:{\displaystyle {\mathcal {N}}(A,\sigma ^{2})}
809:{\displaystyle {\mathcal {N}}(0,\sigma ^{2})}
8:
916:{\displaystyle x=A+w\quad n=0,1,\dots ,N-1}
4967:
4953:
4945:
4430:Measured data are likely to be subject to
450:, where the "hat" indicates the estimate.
4720:Mathematical Statistics and Data Analysis
4654:Getting the Best from Teaching Statistics
4616:
4614:
4325:
4323:
4300:
4295:
4290:so a standard deviation of approximately
4266:
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4248:
4242:
4186:
4176:
4174:
4107:
4071:
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4027:
3966:
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3809:
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3329:
3310:
3283:
3268:
3250:
3249:
3247:
3205:
3166:
3155:
3141:
3127:
3126:
3124:
3074:
3063:
3018:
3007:
2990:
2981:
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2909:
2892:
2883:
2839:
2828:
2811:
2802:
2785:
2758:
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2721:
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2685:
2683:
2662:
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2614:
2601:
2588:
2570:
2533:
2519:
2489:
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2415:
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2230:
2196:
2183:
2156:
2147:
2115:
2091:
2072:
2071:
2069:
2040:
2011:
1985:
1963:
1950:
1923:
1914:
1888:
1859:
1805:
1799:
1785:
1765:
1756:
1719:
1707:
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1631:
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1606:
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1577:
1566:
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1526:
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1336:
1317:
1285:
1274:
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1237:
1226:
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1215:
1213:
1165:
1152:
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1140:
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1128:
1104:
1054:
1043:
1029:
1020:
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1008:
1005:
968:
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844:
821:
797:
778:
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708:
684:
658:
629:
487:
473:
472:
464:
462:
429:
428:
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410:
399:
391:
368:
357:
352:
347:
339:
299:
278:
264:
252:
244:
242:
147:
139:
137:
55:that deals with estimating the values of
4442:solutions are sought to extract as much
2678:and the maximum likelihood estimator is
1119:, which can be shown through taking the
4906:V.G. Voinov & M.S. Nikulin (1996).
4887:V.G. Voinov & M.S. Nikulin (1993).
4595:
4562:
488:
475:
431:
400:
358:
330:(pdf) or its discrete counterpart, the
245:
73:dependent on the parameters of interest
4064:estimator for the maximum is given by
383:). It is then necessary to define the
4649:"Estimating the Size of a Population"
679:that consists of an unknown constant
7:
120:– a set of data points taken from a
4607:. London, England: Springer-Verlag.
4515:Maximum entropy spectral estimation
3987:minimum variance unbiased estimator
564:Minimum variance unbiased estimator
4834:Fundamentals of Adaptive Filtering
4635:10.1111/j.1467-9639.1994.tb00688.x
4484:Expectation-maximization algorithm
4420:Network intrusion detection system
3881:
3878:
3875:
3818:
3815:
3812:
3725:
3715:
3703:
3583:
3573:
3438:
3434:
3368:
3358:
3346:
3289:
3285:
3269:
2764:
2760:
1854:(pdf) of the noise for one sample
1726:
1723:
1720:
1597:
1594:
1591:
1556:
1553:
1550:
1493:
1490:
1487:
1452:
1449:
1446:
1360:
1251:
1216:
1166:
1131:
25:
4756:H. Vincent Poor (16 March 1998).
4603:Walter, E.; Pronzato, L. (1997).
3997:Maximum of a uniform distribution
3989:(MVUE), in addition to being the
1846:Continuing the example using the
839:The model for the signal is then
5005:Nyquist–Shannon sampling theorem
4932:
3754:
3612:
3460:
3397:
3311:
2786:
2722:
2534:
2291:
2263:
1095:Both of these estimators have a
992:{\displaystyle {\hat {A}}_{1}=x}
580:Unbiased estimators — see
465:
348:
140:
5091:Discrete-time Fourier transform
4680:E.L. Lehmann & G. Casella.
4505:Least-squares spectral analysis
2751:of the log-likelihood function
879:
570:Nonlinear system identification
4910:. Kluwer Academic Publishers.
4891:. Kluwer Academic Publishers.
4463:Best linear unbiased estimator
4338:{\displaystyle {\frac {m}{k}}}
4233:
4221:
4216:
4204:
4201:
4189:
3895:
3832:
3764:
3750:
3654:
3645:
3622:
3608:
3534:
3528:
3470:
3456:
3407:
3393:
3321:
3307:
3262:
3256:
3187:
3181:
3132:
3095:
3089:
3039:
3033:
2941:
2935:
2872:
2863:
2857:
2851:
2796:
2782:
2732:
2718:
2691:
2659:
2649:
2643:
2637:
2544:
2530:
2486:
2476:
2470:
2464:
2358:
2349:
2343:
2337:
2301:
2287:
2227:
2217:
2211:
2205:
2141:
2132:
2126:
2120:
2097:
2078:
2051:
2045:
2022:
2016:
1982:
1975:
1908:
1905:
1899:
1893:
1870:
1864:
1745:
1742:
1736:
1730:
1652:
1646:
1571:
1511:
1505:
1467:
1378:
1372:
1306:
1300:
1231:
1184:
1178:
1146:
1075:
1069:
1014:
986:
980:
962:
876:
870:
855:
849:
803:
784:
719:
713:
640:
634:
575:Best linear unbiased estimator
478:
434:
404:
396:
362:
353:
344:
209:
197:
180:
174:
164:
158:
1:
5036:Statistical signal processing
4545:Statistical signal processing
4368:Interpretation of scientific
4268: for small samples
4021:discrete uniform distribution
3563:Taking the second derivative
701:additive white Gaussian noise
601:, and its various derivatives
4053:{\displaystyle 1,2,\dots ,N}
2270:{\displaystyle \mathbf {x} }
1852:probability density function
453:One common estimator is the
328:probability density function
4854:. NJ: Prentice-Hall. 2000.
4775:Harry L. Van Trees (2001).
4446:from the data as possible.
759:{\displaystyle \sigma ^{2}}
43:Estimation (disambiguation)
5231:
5085:Discrete Fourier transform
5062:Matched Z-transform method
4682:Theory of Point Estimation
4535:Rule of three (statistics)
4453:
4352:The sample maximum is the
4347:maximum spacing estimation
4060:with unknown maximum, the
4000:
3227:
1839:
552:Minimum mean squared error
513:
455:minimum mean squared error
40:
29:
5079:Discrete cosine transform
4976:Digital signal processing
4469:Completeness (statistics)
332:probability mass function
5112:Post's inversion formula
5026:Digital image processing
4701:Systems Cost Engineering
4550:Sufficiency (statistics)
593:Markov chain Monte Carlo
71:probability distribution
5021:Audio signal processing
4647:Johnson, Roger (2006),
4577:the population maximum.
4479:Efficiency (statistics)
4169:This has a variance of
3425:and copying from above
2968:and setting it to zero
2006:and the probability of
78:set-membership approach
4339:
4312:
4284:
4127:
4054:
3975:
3955:
3929:
3861:
3796:
3686:
3555:
3524:
3419:
3236:Cramér–Rao lower bound
3224:Cramér–Rao lower bound
3214:
3194:
3177:
3111:
3085:
3029:
2962:
2931:
2850:
2739:
2672:
2636:
2504:
2463:
2333:
2271:
2245:
2104:
2058:
2029:
2000:
1877:
1820:
1718:
1642:
1536:
1430:
1358:
1296:
1202:
1113:
1082:
1065:
993:
939:
917:
830:
810:
760:
726:
693:
667:
647:
496:
444:
415:
373:
320:
227:
4699:Dale Shermon (2009).
4629:(2 (Summer)): 50–52,
4340:
4313:
4285:
4128:
4055:
3976:
3956:
3930:
3862:
3797:
3687:
3556:
3498:
3420:
3228:Further information:
3215:
3195:
3151:
3112:
3059:
3003:
2963:
2905:
2824:
2740:
2673:
2610:
2505:
2437:
2307:
2272:
2255:, the probability of
2246:
2105:
2059:
2030:
2001:
1878:
1821:
1692:
1616:
1537:
1431:
1332:
1270:
1203:
1114:
1083:
1039:
994:
940:
918:
831:
811:
761:
727:
694:
668:
648:
497:
445:
416:
374:
321:
228:
5145:Anti-aliasing filter
5074:Constant-Q transform
5057:Advanced z-transform
4941:at Wikimedia Commons
4703:. Gower Publishing.
4664:on November 20, 2008
4405:Software engineering
4322:
4294:
4173:
4068:
4026:
4014:likelihood functions
3985:, and thus also the
3965:
3945:
3871:
3808:
3696:
3567:
3429:
3246:
3204:
3123:
2972:
2755:
2682:
2518:
2281:
2259:
2114:
2068:
2064:can be thought of a
2039:
2010:
1887:
1858:
1546:
1442:
1212:
1127:
1103:
1004:
952:
929:
843:
820:
774:
743:
707:
683:
657:
628:
620:Consider a received
558:Maximum a posteriori
461:
425:
390:
385:Bayesian probability
338:
241:
233:Secondly, there are
136:
93:Or, for example, in
41:For other uses, see
4836:. NJ: Wiley. 2003.
4818:. NJ: Wiley. 2008.
4623:Teaching Statistics
4525:Parametric equation
4425:Orbit determination
4311:{\displaystyle N/k}
4151:German tank problem
4003:German tank problem
3983:efficient estimator
1832: > 1.
381:Bayesian statistics
36:Interval estimation
5102:Integral transform
5097:Impulse invariance
5069:Bilinear transform
4718:John Rice (1995).
4573:: it will tend to
4520:Nuisance parameter
4500:Information theory
4400:Project management
4395:Telecommunications
4354:maximum likelihood
4335:
4308:
4280:
4123:
4050:
4010:maximum likelihood
3991:maximum likelihood
3971:
3951:
3925:
3857:
3792:
3682:
3551:
3415:
3240:Fisher information
3210:
3190:
3107:
2958:
2735:
2668:
2500:
2267:
2241:
2100:
2054:
2025:
1996:
1873:
1848:maximum likelihood
1842:Maximum likelihood
1836:Maximum likelihood
1816:
1532:
1426:
1198:
1123:of each estimator
1109:
1078:
989:
935:
913:
826:
806:
756:
722:
689:
663:
643:
525:Maximum likelihood
492:
440:
411:
369:
316:
307:
223:
214:
118:statistical sample
5210:Signal processing
5205:Estimation theory
5192:
5191:
5117:Starred transform
5107:Laplace transform
5031:Speech processing
5000:Estimation theory
4939:Estimation theory
4937:Media related to
4879:978-0-89871-411-1
4861:978-0-13-022464-4
4852:Linear Estimation
4825:978-0-470-25388-5
4722:. Duxbury Press.
4710:978-0-566-08861-2
4456:Estimation theory
4375:Signal processing
4333:
4269:
4264:
4237:
4184:
4115:
4087:
3974:{\displaystyle A}
3954:{\displaystyle N}
3923:
3898:
3855:
3835:
3790:
3739:
3680:
3643:
3597:
3491:
3445:
3382:
3296:
3213:{\displaystyle N}
3149:
3135:
2996:
2898:
2817:
2771:
2747:Taking the first
2694:
2608:
2578:
2512:natural logarithm
2435:
2399:
2386:
2203:
2167:
2164:
2057:{\displaystyle x}
2028:{\displaystyle x}
1970:
1934:
1931:
1876:{\displaystyle w}
1814:
1771:
1685:
1668:
1667:
1614:
1574:
1470:
1402:
1325:
1268:
1234:
1149:
1112:{\displaystyle A}
1037:
1017:
965:
938:{\displaystyle A}
829:{\displaystyle A}
725:{\displaystyle w}
692:{\displaystyle A}
666:{\displaystyle N}
646:{\displaystyle x}
536:Method of moments
481:
437:
49:Estimation theory
18:Estimation Theory
16:(Redirected from
5222:
4990:Detection theory
4969:
4962:
4955:
4946:
4936:
4921:
4902:
4883:
4865:
4847:
4829:
4816:Adaptive Filters
4811:
4806:. Archived from
4796:
4791:. Archived from
4771:
4752:
4733:
4714:
4695:
4666:
4665:
4663:
4657:, archived from
4644:
4638:
4637:
4618:
4609:
4608:
4600:
4578:
4571:biased estimator
4567:
4530:Pareto principle
4474:Detection theory
4414:Adaptive control
4344:
4342:
4341:
4336:
4334:
4326:
4317:
4315:
4314:
4309:
4304:
4289:
4287:
4286:
4281:
4270:
4267:
4265:
4263:
4262:
4253:
4252:
4243:
4238:
4236:
4219:
4187:
4185:
4177:
4132:
4130:
4129:
4124:
4116:
4108:
4088:
4083:
4072:
4059:
4057:
4056:
4051:
3980:
3978:
3977:
3972:
3960:
3958:
3957:
3952:
3934:
3932:
3931:
3926:
3924:
3919:
3918:
3909:
3904:
3900:
3899:
3891:
3884:
3866:
3864:
3863:
3858:
3856:
3854:
3846:
3841:
3837:
3836:
3828:
3821:
3801:
3799:
3798:
3793:
3791:
3789:
3788:
3776:
3771:
3767:
3757:
3740:
3738:
3737:
3736:
3723:
3722:
3713:
3706:
3691:
3689:
3688:
3683:
3681:
3679:
3678:
3669:
3661:
3644:
3642:
3641:
3629:
3615:
3598:
3596:
3595:
3594:
3581:
3580:
3571:
3560:
3558:
3557:
3552:
3550:
3546:
3523:
3512:
3492:
3490:
3489:
3477:
3463:
3446:
3444:
3433:
3424:
3422:
3421:
3416:
3414:
3410:
3400:
3383:
3381:
3380:
3379:
3366:
3365:
3356:
3349:
3338:
3334:
3333:
3328:
3324:
3314:
3297:
3295:
3284:
3272:
3255:
3254:
3230:Cramér–Rao bound
3219:
3217:
3216:
3211:
3199:
3197:
3196:
3191:
3176:
3165:
3150:
3142:
3137:
3136:
3128:
3116:
3114:
3113:
3108:
3084:
3073:
3055:
3051:
3028:
3017:
2997:
2995:
2994:
2982:
2967:
2965:
2964:
2959:
2957:
2953:
2930:
2919:
2899:
2897:
2896:
2884:
2879:
2875:
2849:
2838:
2818:
2816:
2815:
2803:
2789:
2772:
2770:
2759:
2744:
2742:
2741:
2736:
2725:
2696:
2695:
2687:
2677:
2675:
2674:
2669:
2667:
2666:
2635:
2624:
2609:
2607:
2606:
2605:
2589:
2584:
2580:
2579:
2571:
2537:
2509:
2507:
2506:
2501:
2499:
2495:
2494:
2493:
2462:
2451:
2436:
2434:
2433:
2432:
2416:
2400:
2398:
2397:
2392:
2388:
2387:
2379:
2365:
2332:
2321:
2294:
2276:
2274:
2273:
2268:
2266:
2250:
2248:
2247:
2242:
2240:
2236:
2235:
2234:
2204:
2202:
2201:
2200:
2184:
2168:
2166:
2165:
2157:
2148:
2109:
2107:
2106:
2101:
2096:
2095:
2077:
2076:
2063:
2061:
2060:
2055:
2034:
2032:
2031:
2026:
2005:
2003:
2002:
1997:
1995:
1991:
1990:
1989:
1971:
1969:
1968:
1967:
1951:
1935:
1933:
1932:
1924:
1915:
1882:
1880:
1879:
1874:
1825:
1823:
1822:
1817:
1815:
1810:
1809:
1800:
1795:
1791:
1790:
1789:
1772:
1770:
1769:
1757:
1752:
1748:
1729:
1717:
1706:
1686:
1684:
1683:
1671:
1669:
1665:
1661:
1659:
1655:
1641:
1630:
1615:
1607:
1600:
1586:
1582:
1581:
1576:
1575:
1567:
1559:
1541:
1539:
1538:
1533:
1531:
1530:
1518:
1514:
1496:
1482:
1478:
1477:
1472:
1471:
1463:
1455:
1435:
1433:
1432:
1427:
1419:
1415:
1403:
1395:
1390:
1386:
1385:
1381:
1363:
1357:
1346:
1326:
1318:
1313:
1309:
1295:
1284:
1269:
1261:
1254:
1246:
1242:
1241:
1236:
1235:
1227:
1219:
1207:
1205:
1204:
1199:
1191:
1187:
1169:
1161:
1157:
1156:
1151:
1150:
1142:
1134:
1118:
1116:
1115:
1110:
1087:
1085:
1084:
1079:
1064:
1053:
1038:
1030:
1025:
1024:
1019:
1018:
1010:
998:
996:
995:
990:
973:
972:
967:
966:
958:
944:
942:
941:
936:
922:
920:
919:
914:
835:
833:
832:
827:
815:
813:
812:
807:
802:
801:
783:
782:
765:
763:
762:
757:
755:
754:
731:
729:
728:
723:
698:
696:
695:
690:
672:
670:
669:
664:
652:
650:
649:
644:
542:Cramér–Rao bound
531:Bayes estimators
501:
499:
498:
493:
491:
483:
482:
474:
468:
449:
447:
446:
441:
439:
438:
430:
420:
418:
417:
412:
403:
378:
376:
375:
370:
361:
356:
351:
325:
323:
322:
317:
312:
311:
304:
303:
283:
282:
269:
268:
248:
232:
230:
229:
224:
219:
218:
143:
32:Point estimation
21:
5230:
5229:
5225:
5224:
5223:
5221:
5220:
5219:
5195:
5194:
5193:
5188:
5126:
5040:
5009:
4995:Discrete signal
4978:
4973:
4929:
4924:
4918:
4905:
4899:
4886:
4880:
4868:
4862:
4850:
4844:
4832:
4826:
4814:
4799:
4789:
4774:
4768:
4755:
4749:
4737:Steven M. Kay.
4736:
4730:
4717:
4711:
4698:
4692:
4679:
4675:
4670:
4669:
4661:
4646:
4645:
4641:
4620:
4619:
4612:
4602:
4601:
4597:
4592:
4587:
4582:
4581:
4568:
4564:
4559:
4554:
4540:State estimator
4458:
4454:Main category:
4452:
4412:(in particular
4390:Quality control
4380:Clinical trials
4362:
4320:
4319:
4292:
4291:
4254:
4244:
4220:
4188:
4171:
4170:
4164:
4073:
4066:
4065:
4024:
4023:
4012:estimators and
4005:
3999:
3963:
3962:
3943:
3942:
3910:
3885:
3869:
3868:
3822:
3806:
3805:
3780:
3728:
3724:
3714:
3711:
3707:
3694:
3693:
3670:
3662:
3633:
3586:
3582:
3572:
3565:
3564:
3497:
3493:
3481:
3437:
3427:
3426:
3371:
3367:
3357:
3354:
3350:
3288:
3282:
3278:
3277:
3273:
3244:
3243:
3232:
3226:
3202:
3201:
3121:
3120:
3002:
2998:
2986:
2970:
2969:
2904:
2900:
2888:
2823:
2819:
2807:
2763:
2753:
2752:
2680:
2679:
2658:
2597:
2593:
2566:
2562:
2516:
2515:
2485:
2424:
2420:
2411:
2407:
2374:
2370:
2369:
2279:
2278:
2257:
2256:
2226:
2192:
2188:
2179:
2175:
2152:
2112:
2111:
2087:
2066:
2065:
2037:
2036:
2008:
2007:
1981:
1959:
1955:
1946:
1942:
1919:
1885:
1884:
1856:
1855:
1850:estimator, the
1844:
1838:
1801:
1781:
1777:
1773:
1761:
1691:
1687:
1675:
1605:
1601:
1564:
1560:
1544:
1543:
1522:
1501:
1497:
1460:
1456:
1440:
1439:
1408:
1404:
1368:
1364:
1331:
1327:
1259:
1255:
1224:
1220:
1210:
1209:
1174:
1170:
1139:
1135:
1125:
1124:
1101:
1100:
1007:
1002:
1001:
955:
950:
949:
927:
926:
841:
840:
818:
817:
793:
772:
771:
746:
741:
740:
705:
704:
681:
680:
655:
654:
626:
625:
622:discrete signal
618:
613:
588:Particle filter
518:
512:
459:
458:
423:
422:
388:
387:
336:
335:
306:
305:
295:
292:
291:
285:
284:
274:
271:
270:
260:
253:
239:
238:
213:
212:
191:
190:
184:
183:
168:
167:
148:
134:
133:
114:
87:
51:is a branch of
46:
39:
28:
23:
22:
15:
12:
11:
5:
5228:
5226:
5218:
5217:
5212:
5207:
5197:
5196:
5190:
5189:
5187:
5186:
5181:
5176:
5171:
5166:
5161:
5152:
5147:
5142:
5136:
5134:
5128:
5127:
5125:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5088:
5082:
5076:
5071:
5066:
5065:
5064:
5059:
5048:
5046:
5042:
5041:
5039:
5038:
5033:
5028:
5023:
5017:
5015:
5011:
5010:
5008:
5007:
5002:
4997:
4992:
4986:
4984:
4980:
4979:
4974:
4972:
4971:
4964:
4957:
4949:
4943:
4942:
4928:
4927:External links
4925:
4923:
4922:
4916:
4903:
4897:
4884:
4878:
4866:
4860:
4848:
4842:
4830:
4824:
4812:
4810:on 2010-12-30.
4797:
4795:on 2005-04-28.
4787:
4772:
4766:
4753:
4747:
4734:
4728:
4715:
4709:
4696:
4690:
4676:
4674:
4671:
4668:
4667:
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4610:
4594:
4593:
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4586:
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4579:
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4558:
4555:
4553:
4552:
4547:
4542:
4537:
4532:
4527:
4522:
4517:
4512:
4510:Matched filter
4507:
4502:
4497:
4495:Grey box model
4492:
4487:
4486:(EM algorithm)
4481:
4476:
4471:
4466:
4459:
4451:
4448:
4428:
4427:
4422:
4417:
4410:Control theory
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4372:
4361:
4358:
4332:
4329:
4307:
4303:
4299:
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4257:
4251:
4247:
4241:
4235:
4232:
4229:
4226:
4223:
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4215:
4212:
4209:
4206:
4203:
4200:
4197:
4194:
4191:
4183:
4180:
4162:
4139:sample maximum
4122:
4119:
4114:
4111:
4106:
4103:
4100:
4097:
4094:
4091:
4086:
4082:
4079:
4076:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4001:Main article:
3998:
3995:
3970:
3950:
3922:
3917:
3913:
3907:
3903:
3897:
3894:
3888:
3883:
3880:
3877:
3853:
3849:
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3820:
3817:
3814:
3787:
3783:
3779:
3774:
3770:
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3760:
3756:
3752:
3749:
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3743:
3735:
3731:
3727:
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3668:
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3659:
3656:
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3650:
3647:
3640:
3636:
3632:
3627:
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3618:
3614:
3610:
3607:
3604:
3601:
3593:
3589:
3585:
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3303:
3300:
3294:
3291:
3287:
3281:
3276:
3271:
3267:
3264:
3261:
3258:
3253:
3225:
3222:
3209:
3189:
3186:
3183:
3180:
3175:
3172:
3169:
3164:
3161:
3158:
3154:
3148:
3145:
3140:
3134:
3131:
3106:
3103:
3100:
3097:
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3091:
3088:
3083:
3080:
3077:
3072:
3069:
3066:
3062:
3058:
3054:
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3047:
3044:
3041:
3038:
3035:
3032:
3027:
3024:
3021:
3016:
3013:
3010:
3006:
3001:
2993:
2989:
2985:
2980:
2977:
2956:
2952:
2949:
2946:
2943:
2940:
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2934:
2929:
2926:
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2915:
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2895:
2891:
2887:
2882:
2878:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2848:
2845:
2842:
2837:
2834:
2831:
2827:
2822:
2814:
2810:
2806:
2801:
2798:
2795:
2792:
2788:
2784:
2781:
2778:
2775:
2769:
2766:
2762:
2734:
2731:
2728:
2724:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2693:
2690:
2665:
2661:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2634:
2631:
2628:
2623:
2620:
2617:
2613:
2604:
2600:
2596:
2592:
2587:
2583:
2577:
2574:
2569:
2565:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2536:
2532:
2529:
2526:
2523:
2498:
2492:
2488:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2461:
2458:
2455:
2450:
2447:
2444:
2440:
2431:
2427:
2423:
2419:
2414:
2410:
2406:
2403:
2396:
2391:
2385:
2382:
2377:
2373:
2368:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2331:
2328:
2325:
2320:
2317:
2314:
2310:
2306:
2303:
2300:
2297:
2293:
2289:
2286:
2265:
2239:
2233:
2229:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2199:
2195:
2191:
2187:
2182:
2178:
2174:
2171:
2163:
2160:
2155:
2151:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2099:
2094:
2090:
2086:
2083:
2080:
2075:
2053:
2050:
2047:
2044:
2024:
2021:
2018:
2015:
1994:
1988:
1984:
1980:
1977:
1974:
1966:
1962:
1958:
1954:
1949:
1945:
1941:
1938:
1930:
1927:
1922:
1918:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1872:
1869:
1866:
1863:
1840:Main article:
1837:
1834:
1813:
1808:
1804:
1798:
1794:
1788:
1784:
1780:
1776:
1768:
1764:
1760:
1755:
1751:
1747:
1744:
1741:
1738:
1735:
1732:
1728:
1725:
1722:
1716:
1713:
1710:
1705:
1702:
1699:
1695:
1690:
1682:
1678:
1674:
1664:
1658:
1654:
1651:
1648:
1645:
1640:
1637:
1634:
1629:
1626:
1623:
1619:
1613:
1610:
1604:
1599:
1596:
1593:
1589:
1585:
1580:
1573:
1570:
1563:
1558:
1555:
1552:
1529:
1525:
1521:
1517:
1513:
1510:
1507:
1504:
1500:
1495:
1492:
1489:
1485:
1481:
1476:
1469:
1466:
1459:
1454:
1451:
1448:
1425:
1422:
1418:
1414:
1411:
1407:
1401:
1398:
1393:
1389:
1384:
1380:
1377:
1374:
1371:
1367:
1362:
1356:
1353:
1350:
1345:
1342:
1339:
1335:
1330:
1324:
1321:
1316:
1312:
1308:
1305:
1302:
1299:
1294:
1291:
1288:
1283:
1280:
1277:
1273:
1267:
1264:
1258:
1253:
1249:
1245:
1240:
1233:
1230:
1223:
1218:
1197:
1194:
1190:
1186:
1183:
1180:
1177:
1173:
1168:
1164:
1160:
1155:
1148:
1145:
1138:
1133:
1121:expected value
1108:
1093:
1092:
1077:
1074:
1071:
1068:
1063:
1060:
1057:
1052:
1049:
1046:
1042:
1036:
1033:
1028:
1023:
1016:
1013:
999:
988:
985:
982:
979:
976:
971:
964:
961:
934:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
825:
805:
800:
796:
792:
789:
786:
781:
753:
749:
721:
718:
715:
712:
688:
662:
642:
639:
636:
633:
617:
614:
612:
609:
608:
607:
602:
596:
590:
585:
582:estimator bias
578:
572:
567:
561:
555:
549:
544:
539:
533:
528:
514:Main article:
511:
508:
504:expected value
490:
486:
480:
477:
471:
467:
436:
433:
409:
406:
402:
398:
395:
367:
364:
360:
355:
350:
346:
343:
315:
310:
302:
298:
294:
293:
290:
287:
286:
281:
277:
273:
272:
267:
263:
259:
258:
256:
251:
247:
222:
217:
211:
208:
205:
202:
199:
196:
193:
192:
189:
186:
185:
182:
179:
176:
173:
170:
169:
166:
163:
160:
157:
154:
153:
151:
146:
142:
113:
110:
86:
83:
82:
81:
74:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5227:
5216:
5213:
5211:
5208:
5206:
5203:
5202:
5200:
5185:
5182:
5180:
5179:Undersampling
5177:
5175:
5174:Sampling rate
5172:
5170:
5167:
5165:
5162:
5160:
5156:
5153:
5151:
5148:
5146:
5143:
5141:
5138:
5137:
5135:
5133:
5129:
5123:
5122:Zak transform
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5092:
5089:
5086:
5083:
5080:
5077:
5075:
5072:
5070:
5067:
5063:
5060:
5058:
5055:
5054:
5053:
5050:
5049:
5047:
5043:
5037:
5034:
5032:
5029:
5027:
5024:
5022:
5019:
5018:
5016:
5012:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4987:
4985:
4981:
4977:
4970:
4965:
4963:
4958:
4956:
4951:
4950:
4947:
4940:
4935:
4931:
4930:
4926:
4919:
4917:0-7923-3939-8
4913:
4909:
4904:
4900:
4898:0-7923-2382-3
4894:
4890:
4885:
4881:
4875:
4871:
4867:
4863:
4857:
4853:
4849:
4845:
4843:0-471-46126-1
4839:
4835:
4831:
4827:
4821:
4817:
4813:
4809:
4805:
4804:
4798:
4794:
4790:
4788:0-471-09517-6
4784:
4780:
4779:
4773:
4769:
4767:0-387-94173-8
4763:
4759:
4754:
4750:
4748:0-13-345711-7
4744:
4740:
4735:
4731:
4725:
4721:
4716:
4712:
4706:
4702:
4697:
4693:
4687:
4683:
4678:
4677:
4672:
4660:
4656:
4655:
4650:
4643:
4640:
4636:
4632:
4628:
4624:
4617:
4615:
4611:
4606:
4599:
4596:
4589:
4584:
4576:
4575:underestimate
4572:
4566:
4563:
4556:
4551:
4548:
4546:
4543:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4491:
4490:Fermi problem
4488:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4464:
4461:
4460:
4457:
4449:
4447:
4445:
4441:
4437:
4433:
4426:
4423:
4421:
4418:
4415:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4385:Opinion polls
4383:
4381:
4378:
4376:
4373:
4371:
4367:
4366:
4365:
4359:
4357:
4355:
4350:
4348:
4330:
4327:
4305:
4301:
4297:
4277:
4274:
4271:
4259:
4255:
4249:
4245:
4239:
4230:
4227:
4224:
4213:
4210:
4207:
4198:
4195:
4192:
4181:
4178:
4167:
4161:
4158:
4156:
4152:
4148:
4144:
4140:
4136:
4120:
4117:
4112:
4109:
4104:
4101:
4098:
4095:
4092:
4089:
4084:
4080:
4077:
4074:
4063:
4047:
4044:
4041:
4038:
4035:
4032:
4029:
4022:
4017:
4015:
4011:
4004:
3996:
3994:
3992:
3988:
3984:
3968:
3948:
3940:
3935:
3920:
3915:
3911:
3905:
3901:
3892:
3886:
3847:
3842:
3838:
3829:
3823:
3802:
3785:
3781:
3777:
3772:
3768:
3761:
3758:
3747:
3744:
3741:
3733:
3729:
3719:
3708:
3699:
3675:
3671:
3666:
3663:
3657:
3651:
3648:
3638:
3634:
3630:
3625:
3619:
3616:
3605:
3602:
3599:
3591:
3587:
3577:
3561:
3547:
3543:
3540:
3537:
3531:
3525:
3520:
3517:
3514:
3509:
3506:
3503:
3499:
3494:
3486:
3482:
3478:
3473:
3467:
3464:
3453:
3450:
3447:
3441:
3411:
3404:
3401:
3390:
3387:
3384:
3376:
3372:
3362:
3351:
3342:
3339:
3335:
3330:
3325:
3318:
3315:
3304:
3301:
3298:
3292:
3279:
3274:
3265:
3259:
3241:
3237:
3231:
3223:
3221:
3207:
3184:
3178:
3173:
3170:
3167:
3162:
3159:
3156:
3152:
3146:
3143:
3138:
3129:
3117:
3104:
3101:
3098:
3092:
3086:
3081:
3078:
3075:
3070:
3067:
3064:
3060:
3056:
3052:
3048:
3045:
3042:
3036:
3030:
3025:
3022:
3019:
3014:
3011:
3008:
3004:
2999:
2991:
2987:
2983:
2978:
2975:
2954:
2950:
2947:
2944:
2938:
2932:
2927:
2924:
2921:
2916:
2913:
2910:
2906:
2901:
2893:
2889:
2885:
2880:
2876:
2869:
2866:
2860:
2854:
2846:
2843:
2840:
2835:
2832:
2829:
2825:
2820:
2812:
2808:
2804:
2799:
2793:
2790:
2779:
2776:
2773:
2767:
2750:
2745:
2729:
2726:
2715:
2712:
2709:
2703:
2700:
2697:
2688:
2663:
2655:
2652:
2646:
2640:
2632:
2629:
2626:
2621:
2618:
2615:
2611:
2602:
2598:
2594:
2590:
2585:
2581:
2575:
2572:
2567:
2563:
2559:
2556:
2553:
2550:
2547:
2541:
2538:
2527:
2524:
2521:
2513:
2496:
2490:
2482:
2479:
2473:
2467:
2459:
2456:
2453:
2448:
2445:
2442:
2438:
2429:
2425:
2421:
2417:
2412:
2408:
2404:
2401:
2394:
2389:
2383:
2380:
2375:
2371:
2366:
2361:
2355:
2352:
2346:
2340:
2334:
2329:
2326:
2323:
2318:
2315:
2312:
2308:
2304:
2298:
2295:
2284:
2254:
2237:
2231:
2223:
2220:
2214:
2208:
2197:
2193:
2189:
2185:
2180:
2176:
2172:
2169:
2161:
2158:
2153:
2149:
2144:
2138:
2135:
2129:
2123:
2117:
2092:
2088:
2084:
2081:
2048:
2042:
2019:
2013:
1992:
1986:
1978:
1972:
1964:
1960:
1956:
1952:
1947:
1943:
1939:
1936:
1928:
1925:
1920:
1916:
1911:
1902:
1896:
1890:
1867:
1861:
1853:
1849:
1843:
1835:
1833:
1831:
1826:
1811:
1806:
1802:
1796:
1792:
1786:
1782:
1778:
1774:
1766:
1762:
1758:
1753:
1749:
1739:
1733:
1714:
1711:
1708:
1703:
1700:
1697:
1693:
1688:
1680:
1676:
1672:
1662:
1656:
1649:
1643:
1638:
1635:
1632:
1627:
1624:
1621:
1617:
1611:
1608:
1602:
1587:
1583:
1578:
1568:
1561:
1527:
1523:
1519:
1515:
1508:
1502:
1498:
1483:
1479:
1474:
1464:
1457:
1436:
1423:
1420:
1416:
1412:
1409:
1405:
1399:
1396:
1391:
1387:
1382:
1375:
1369:
1365:
1354:
1351:
1348:
1343:
1340:
1337:
1333:
1328:
1322:
1319:
1314:
1310:
1303:
1297:
1292:
1289:
1286:
1281:
1278:
1275:
1271:
1265:
1262:
1256:
1247:
1243:
1238:
1228:
1221:
1195:
1192:
1188:
1181:
1175:
1171:
1162:
1158:
1153:
1143:
1136:
1122:
1106:
1098:
1091:
1088:which is the
1072:
1066:
1061:
1058:
1055:
1050:
1047:
1044:
1040:
1034:
1031:
1026:
1021:
1011:
1000:
983:
977:
974:
969:
959:
948:
947:
946:
932:
923:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
873:
867:
864:
861:
858:
852:
846:
837:
823:
798:
794:
790:
787:
769:
751:
747:
739:
735:
716:
710:
702:
686:
678:
675:
660:
637:
631:
623:
615:
610:
606:
605:Wiener filter
603:
600:
599:Kalman filter
597:
594:
591:
589:
586:
583:
579:
576:
573:
571:
568:
565:
562:
559:
556:
553:
550:
548:
547:Least squares
545:
543:
540:
537:
534:
532:
529:
526:
523:
522:
521:
517:
509:
507:
505:
484:
469:
456:
451:
407:
393:
386:
382:
365:
341:
333:
329:
313:
308:
300:
296:
288:
279:
275:
265:
261:
254:
249:
236:
220:
215:
206:
203:
200:
194:
187:
177:
171:
161:
155:
149:
144:
131:
128:. Put into a
127:
124:(RV) of size
123:
122:random vector
119:
111:
109:
107:
104:
99:
96:
91:
84:
79:
75:
72:
68:
67:
66:
64:
63:
58:
54:
50:
44:
37:
33:
19:
5169:Quantization
5164:Oversampling
5155:Nyquist rate
5150:Downsampling
4999:
4907:
4888:
4869:
4851:
4833:
4815:
4808:the original
4802:
4793:the original
4777:
4760:. Springer.
4757:
4738:
4729:0-534-209343
4719:
4700:
4681:
4659:the original
4653:
4642:
4626:
4622:
4604:
4598:
4574:
4565:
4429:
4363:
4360:Applications
4351:
4168:
4165:
4159:
4155:World War II
4142:
4134:
4018:
4006:
3938:
3936:
3803:
3562:
3234:To find the
3233:
3118:
2746:
2253:independence
1845:
1829:
1827:
1666:independence
1437:
1094:
924:
838:
767:
619:
519:
452:
234:
125:
115:
100:
92:
88:
60:
48:
47:
5052:Z-transform
4800:Dan Simon.
4444:information
4436:probability
4370:experiments
4147:sample size
3993:estimator.
3867:results in
2514:of the pdf
2510:Taking the
1090:sample mean
674:independent
237:parameters
98:estimated.
5199:Categories
5184:Upsampling
5045:Techniques
5014:Sub-fields
4691:0387985026
4585:References
2749:derivative
736:and known
732:with zero
538:estimators
527:estimators
510:Estimators
57:parameters
53:statistics
5159:frequency
4781:. Wiley.
4590:Citations
4275:≪
4240:≈
4196:−
4118:−
4093:−
4042:…
3912:σ
3906:≥
3896:^
3843:≥
3833:^
3782:σ
3745:
3726:∂
3716:∂
3700:−
3672:σ
3664:−
3649:−
3635:σ
3603:
3584:∂
3574:∂
3538:−
3518:−
3500:∑
3483:σ
3451:
3439:∂
3435:∂
3388:
3369:∂
3359:∂
3343:−
3302:
3290:∂
3286:∂
3171:−
3153:∑
3133:^
3099:−
3079:−
3061:∑
3043:−
3023:−
3005:∑
2988:σ
2945:−
2925:−
2907:∑
2890:σ
2867:−
2844:−
2826:∑
2809:σ
2777:
2765:∂
2761:∂
2713:
2704:
2692:^
2653:−
2630:−
2612:∑
2599:σ
2586:−
2576:π
2568:σ
2560:
2551:−
2525:
2480:−
2457:−
2439:∑
2426:σ
2413:−
2405:
2384:π
2376:σ
2327:−
2309:∏
2221:−
2194:σ
2181:−
2173:
2162:π
2154:σ
2089:σ
2035:becomes (
1961:σ
1948:−
1940:
1929:π
1921:σ
1803:σ
1783:σ
1712:−
1694:∑
1636:−
1618:∑
1572:^
1524:σ
1468:^
1352:−
1334:∑
1290:−
1272:∑
1232:^
1147:^
1059:−
1041:∑
1015:^
963:^
908:−
899:…
795:σ
748:σ
516:Estimator
489:θ
485:−
479:^
476:θ
435:^
432:θ
401:θ
394:π
359:θ
297:θ
289:⋮
276:θ
262:θ
246:θ
204:−
188:⋮
62:estimator
5140:Aliasing
5132:Sampling
4450:See also
4019:Given a
3939:equal to
2277:becomes
738:variance
611:Examples
85:Examples
4673:Sources
4440:optimal
4145:is the
4137:is the
3242:number
703:(AWGN)
677:samples
5093:(DTFT)
4983:Theory
4914:
4895:
4876:
4858:
4840:
4822:
4785:
4764:
4745:
4726:
4707:
4688:
4465:(BLUE)
4133:where
595:(MCMC)
577:(BLUE)
566:(MVUE)
130:vector
112:Basics
106:signal
5087:(DFT)
5081:(DCT)
4662:(PDF)
4557:Notes
4438:that
4432:noise
945:are:
699:with
653:, of
560:(MAP)
103:noisy
95:radar
4912:ISBN
4893:ISBN
4874:ISBN
4856:ISBN
4838:ISBN
4820:ISBN
4783:ISBN
4762:ISBN
4743:ISBN
4724:ISBN
4705:ISBN
4686:ISBN
4141:and
4062:UMVU
3961:and
1542:and
1208:and
1097:mean
768:i.e.
734:mean
76:The
4631:doi
2707:max
2701:arg
2402:exp
2251:By
2170:exp
1937:exp
1883:is
1099:of
34:or
5201::
5157:/
4741:.
4684:.
4651:,
4627:16
4625:,
4613:^
4349:.
4157:.
4016:.
3742:ln
3600:ln
3448:ln
3385:ln
3299:ln
2774:ln
2710:ln
2557:ln
2522:ln
2110:)
836:.
770:,
624:,
132:,
108:.
4968:e
4961:t
4954:v
4920:.
4901:.
4882:.
4864:.
4846:.
4828:.
4770:.
4751:.
4732:.
4713:.
4694:.
4633::
4416:)
4331:k
4328:m
4306:k
4302:/
4298:N
4278:N
4272:k
4260:2
4256:k
4250:2
4246:N
4234:)
4231:2
4228:+
4225:k
4222:(
4217:)
4214:1
4211:+
4208:N
4205:(
4202:)
4199:k
4193:N
4190:(
4182:k
4179:1
4143:k
4135:m
4121:1
4113:k
4110:m
4105:+
4102:m
4099:=
4096:1
4090:m
4085:k
4081:1
4078:+
4075:k
4048:N
4045:,
4039:,
4036:2
4033:,
4030:1
3969:A
3949:N
3921:N
3916:2
3902:)
3893:A
3887:(
3882:r
3879:a
3876:v
3852:I
3848:1
3839:)
3830:A
3824:(
3819:r
3816:a
3813:v
3786:2
3778:N
3773:=
3769:]
3765:)
3762:A
3759:;
3755:x
3751:(
3748:p
3734:2
3730:A
3720:2
3709:[
3704:E
3676:2
3667:N
3658:=
3655:)
3652:N
3646:(
3639:2
3631:1
3626:=
3623:)
3620:A
3617:;
3613:x
3609:(
3606:p
3592:2
3588:A
3578:2
3548:]
3544:A
3541:N
3535:]
3532:n
3529:[
3526:x
3521:1
3515:N
3510:0
3507:=
3504:n
3495:[
3487:2
3479:1
3474:=
3471:)
3468:A
3465:;
3461:x
3457:(
3454:p
3442:A
3412:]
3408:)
3405:A
3402:;
3398:x
3394:(
3391:p
3377:2
3373:A
3363:2
3352:[
3347:E
3340:=
3336:)
3331:2
3326:]
3322:)
3319:A
3316:;
3312:x
3308:(
3305:p
3293:A
3280:[
3275:(
3270:E
3266:=
3263:)
3260:A
3257:(
3252:I
3208:N
3188:]
3185:n
3182:[
3179:x
3174:1
3168:N
3163:0
3160:=
3157:n
3147:N
3144:1
3139:=
3130:A
3105:A
3102:N
3096:]
3093:n
3090:[
3087:x
3082:1
3076:N
3071:0
3068:=
3065:n
3057:=
3053:]
3049:A
3046:N
3040:]
3037:n
3034:[
3031:x
3026:1
3020:N
3015:0
3012:=
3009:n
3000:[
2992:2
2984:1
2979:=
2976:0
2955:]
2951:A
2948:N
2942:]
2939:n
2936:[
2933:x
2928:1
2922:N
2917:0
2914:=
2911:n
2902:[
2894:2
2886:1
2881:=
2877:]
2873:)
2870:A
2864:]
2861:n
2858:[
2855:x
2852:(
2847:1
2841:N
2836:0
2833:=
2830:n
2821:[
2813:2
2805:1
2800:=
2797:)
2794:A
2791:;
2787:x
2783:(
2780:p
2768:A
2733:)
2730:A
2727:;
2723:x
2719:(
2716:p
2698:=
2689:A
2664:2
2660:)
2656:A
2650:]
2647:n
2644:[
2641:x
2638:(
2633:1
2627:N
2622:0
2619:=
2616:n
2603:2
2595:2
2591:1
2582:)
2573:2
2564:(
2554:N
2548:=
2545:)
2542:A
2539:;
2535:x
2531:(
2528:p
2497:)
2491:2
2487:)
2483:A
2477:]
2474:n
2471:[
2468:x
2465:(
2460:1
2454:N
2449:0
2446:=
2443:n
2430:2
2422:2
2418:1
2409:(
2395:N
2390:)
2381:2
2372:(
2367:1
2362:=
2359:)
2356:A
2353:;
2350:]
2347:n
2344:[
2341:x
2338:(
2335:p
2330:1
2324:N
2319:0
2316:=
2313:n
2305:=
2302:)
2299:A
2296:;
2292:x
2288:(
2285:p
2264:x
2238:)
2232:2
2228:)
2224:A
2218:]
2215:n
2212:[
2209:x
2206:(
2198:2
2190:2
2186:1
2177:(
2159:2
2150:1
2145:=
2142:)
2139:A
2136:;
2133:]
2130:n
2127:[
2124:x
2121:(
2118:p
2098:)
2093:2
2085:,
2082:A
2079:(
2074:N
2052:]
2049:n
2046:[
2043:x
2023:]
2020:n
2017:[
2014:x
1993:)
1987:2
1983:]
1979:n
1976:[
1973:w
1965:2
1957:2
1953:1
1944:(
1926:2
1917:1
1912:=
1909:)
1906:]
1903:n
1900:[
1897:w
1894:(
1891:p
1871:]
1868:n
1865:[
1862:w
1830:N
1812:N
1807:2
1797:=
1793:]
1787:2
1779:N
1775:[
1767:2
1763:N
1759:1
1754:=
1750:]
1746:)
1743:]
1740:n
1737:[
1734:x
1731:(
1727:r
1724:a
1721:v
1715:1
1709:N
1704:0
1701:=
1698:n
1689:[
1681:2
1677:N
1673:1
1663:=
1657:)
1653:]
1650:n
1647:[
1644:x
1639:1
1633:N
1628:0
1625:=
1622:n
1612:N
1609:1
1603:(
1598:r
1595:a
1592:v
1588:=
1584:)
1579:2
1569:A
1562:(
1557:r
1554:a
1551:v
1528:2
1520:=
1516:)
1512:]
1509:0
1506:[
1503:x
1499:(
1494:r
1491:a
1488:v
1484:=
1480:)
1475:1
1465:A
1458:(
1453:r
1450:a
1447:v
1424:A
1421:=
1417:]
1413:A
1410:N
1406:[
1400:N
1397:1
1392:=
1388:]
1383:]
1379:]
1376:n
1373:[
1370:x
1366:[
1361:E
1355:1
1349:N
1344:0
1341:=
1338:n
1329:[
1323:N
1320:1
1315:=
1311:]
1307:]
1304:n
1301:[
1298:x
1293:1
1287:N
1282:0
1279:=
1276:n
1266:N
1263:1
1257:[
1252:E
1248:=
1244:]
1239:2
1229:A
1222:[
1217:E
1196:A
1193:=
1189:]
1185:]
1182:0
1179:[
1176:x
1172:[
1167:E
1163:=
1159:]
1154:1
1144:A
1137:[
1132:E
1107:A
1076:]
1073:n
1070:[
1067:x
1062:1
1056:N
1051:0
1048:=
1045:n
1035:N
1032:1
1027:=
1022:2
1012:A
987:]
984:0
981:[
978:x
975:=
970:1
960:A
933:A
911:1
905:N
902:,
896:,
893:1
890:,
887:0
884:=
881:n
877:]
874:n
871:[
868:w
865:+
862:A
859:=
856:]
853:n
850:[
847:x
824:A
804:)
799:2
791:,
788:0
785:(
780:N
766:(
752:2
720:]
717:n
714:[
711:w
687:A
661:N
641:]
638:n
635:[
632:x
584:.
470:=
466:e
408:.
405:)
397:(
366:.
363:)
354:|
349:x
345:(
342:p
314:,
309:]
301:M
280:2
266:1
255:[
250:=
235:M
221:.
216:]
210:]
207:1
201:N
198:[
195:x
181:]
178:1
175:[
172:x
165:]
162:0
159:[
156:x
150:[
145:=
141:x
126:N
45:.
38:.
20:)
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