Approximations to Distributions

Abstract

The exact probability distribution of estimators for finite samples is only available in convenient form for simple functions of the data and when the likelihood function is completely speci.ed. Frequently, these conditions are not satis.ed and the inference is based on approximations to the sample distribution. Typically, large sample methods based on the central limit the- orem are generally used. For example, if Tn is an estimator of the parameter based on a sample of size n, it is sometimes possible to obtain functionsbased on a sample of size n, it is sometimes possible to obtain functions () such that the distribution of the random variable pn(Tn -theta)=() coverges to the standard normal distribution as n tends to in.nity. In such a case, it is very common to approximate the distribution of Tn by a normal distribution with mean and variance 2()=n. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. distribution with mean and variance 2()=n. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. case, it is very common to approximate the distribution of Tn by a normal distribution with mean and variance 2()=n. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. distribution with mean and variance 2()=n. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. These asymptotic approximations can be good even for very small samples. The mean of independent draws from a rectangular distribution has a bell- shaped density for n as small as three. But it is easy to construct examples where the asymptotic approximation is bad even when the sample has hun- dreds of observations. It is therefore desirable to know the conditions under which the asymptotic approximations are reasonable and to have alternative methods available when these approximations do not work properly. Most of the material discussed here is closely related with the topic Asymptotic, higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. higher order which is presented as well in this Encyclopedia. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced prob- ability and numerical analysis textbooks is needed. For those with enough time and patience, chapters 15 and 16 of Feller (1971) are well worth reading. There is a good literature treating the theory and practice of approxima- tions to distributions, but introductory texts are relatively few. A very brief summary can be seen in Bickel and Doksum (1977), while some discussion is given in Johnson and Kotz (1970). The extension to asymptotic expansions can be seen in the excellent paper byWallace (1958), although it is outdated. For a good treatment of the subject, an incursion upon the advanced